Appendices Appendix a the Method of Curvilinear

APPENDICES

APPENDIX A

THE METHOD OF CURVILINEAR COORDINATES

In Appendix A the kinematics of point in curvilinear coordinates is considered. The formulas obtained are extended to the motion of any of mechanical systems. The theory, given in the Appendix, is widely used in studying the base material of the monograph.

§ 1. The curvilinear coordinates of point. Reciprocal bases

Suppose, the position of the point M in three-dimensional space is deﬁned by the radius-vector r = r(q1,q2,q3), i. e. the Cartesian coordinates of point 1 2 3 x1,x2,x3 are uniquely represented via the quantities q ,q ,q :

1 2 3 xk = Fk(q ,q ,q ) ,k=1, 2, 3 . (A.1)

If ∂x1 ∂x1 ∂x1 ∂q1 ∂q2 ∂q3 D(x ,x ,x ) ∂x ∂x ∂x 1 2 3 = 2 2 2 =0 , 1 2 3 1 2 3 D(q ,q ,q ) ∂q ∂q ∂q ∂x ∂x ∂x 3 3 3 ∂q1 ∂q2 ∂q3 then system of equations (A.1) is solvable for q1,q2,q3:

σ σ q = f (x1,x2,x3) ,σ=1, 2, 3 , (A.2) and the quantities q1,q2,q3 are called the curvilinear coordinates of point in space. From relations (A.2) it follows directly that, equating any curvilinear co- σ ordinate q to the constant quantity Cσ, we obtain the equation of coordinate surface σ f (x1,x2,x3)=Cσ ,σ=1, 2, 3 . The crossing of two coordinate surfaces gives a coordinate line, along which one coordinate is varied only. For example, the crossing of the coordinate 1 1 2 2 surfaces q = f (x1,x2,x3)=C1 and q = f (x1,x2,x3)=C2 gives a coordinate line, which the coordinate q3 varies along (Fig. A. 1).

213 214 Appendix A

Fig. A. 1

The crossing of the coordinate lines q1,q2,q3 is the point M. If through this point we construct the tangents to the coordinate lines in ascending order of the quantities q1,q2,q3,thenweobtainthe axes of curvilinear coordinates, which can make up as orthogonal (for example, the axes of spherical or cylindrical coordinates) as nonorthogonal systems. For the motion to be given in curvilinear coordinates it is necessary that the quantities q1,q2,q3 are given as time functions: qσ = qσ(t) ,σ= 1, 3 . (A.3) These functions are called equations of motion of point. Taking into account that the radius-vector r = r(q1,q2,q3) of the point M is a diﬀerentiable function, we obtain

3 ∂r dr = dqσ . ∂qσ σ=1 Denoting ∂r e = ,σ= 1, 3 , (A.4) σ ∂qσ we have 3 σ dr = eσdq . (A.5) σ=1 σ Note that |∂r/∂q | = |eσ| = Hσ,whereHσ are scale factors Lam´e.Using formulas (A.4), we obtain ∂x 2 ∂x 2 ∂x 2 H = 1 + 2 + 3 ,σ= 1, 3 . (A.6) σ ∂qσ ∂qσ ∂qσ

Formula (A.5) gives a decomposition of the vector dr using the axes of the σ curvilinear system of coordinates {q } with the basis {eσ}. The quantities Appendix A 215 dqσ from relations (A.5) are called contravariant components of the vector dr. A set of the vectors {eσ} is called a natural or fundamental basis of the curvilinear system of coordinates {qσ} at the point M. The tangential planes to the coordinate surfaces at the point M are called coordinate planes. They pass through the corresponding vectors of basis. For 3 example, the tangential plane to the surface q = C3 passes through the vectors e1 and e2. Denote by eτ a certain vector collinear to a vector of normal to the coor- τ dinate surface q = Cτ at the point M. Obviously, the system of all vectors {eτ } also makes up a certain basis. For the deﬁnition of that the basis {eτ } isuniqueweneed

τ τ 1 ,σ= τ, e · eσ = δ = (A.7) σ 0 ,σ= τ.

τ Here δσ are the Kronecker symbols. The basis {eτ } is called a reciprocal or dual basis relative to the fundamental one. The reciprocal basis can also be introduced by using the gradient operation (see the next section). Note that for any fundamental basis there exists a unique reciprocal basis and if the fundamental basis is orthonormal, then the reciprocal basis coincides with the fundamental one.

§ 2. The relation between a reciprocal basis and gradients of scalar functions

We assume that we have the certain function f(x1,x2,x3) in Cartesian coordinates of a point and this function can be represented in the curvilinear coordinates: f(q1,q2,q3). The diﬀerential of this function in Cartesian coordinates is as follows

3 ∂f df = dxk (A.8) ∂xk k=1 and in curvilinear coordinates

3 ∂f df = dqσ . (A.9) ∂qσ σ=1 The gradient of the function f is the vector

3 ∂f grad f = ik . ∂xk k=1 216 Appendix A

If we introduce Hamiltonian operator nabla 3 ∂ ∇ = ik , (A.10) ∂xk k=1 then the gradient of the function ftakes the form grad f = ∇f. 3 Taking into account that dr = k=1 dxkik, relation (A.8) can be represented as the scalar product df = ∇f · dr . (A.11) The question arises how in place of the formulae (A.10), which is valid for Cartesian coordinates, to ﬁnd a relation for the vector ∇ in curvilinear coordinates in such a way that the derivative df can be represented in the form (A.11) with dr in the form (A.5)? Substituting relation (A.5) into (A.11) and comparing with (A.9), we obtain the relation ∂f ∇f · e = . (A.12) σ ∂qσ It is easily checked that relation (A.12) is valid if 3 ∂f ∇f = eτ . (A.13) ∂qτ τ=1 Representation (A.13) is convenient to obtain the vectors of reciprocal basis. Really, using the concrete coordinate surface of the form (A.2) and taking into account relation (A.13), we have 3 ∂fσ ∇f σ =gradf σ = eτ = eσ . ∂qτ τ=1

§ 3. Covariant and contravariant components of vector

To the curvilinear system of coordinates qσ,σ=1, 2, 3, correspond both σ the fundamental basis eσ = ∂r/∂q ,σ=1, 2, 3, and the reciprocal basis eτ = ∇f τ ,τ =1, 2, 3. Any of the vectors a can be decomposed into as fundamental as reciprocal bases, i. e. can be represented in the form 3 3 σ τ a = a eσ , a = aτ e . (A.14) σ=1 τ=1 σ Here a are contravariant components of the vector a and aτ are covariant components of the vector a in the basis {eτ }. Further, we make use of the rule of dummy index, summation of repeated indices in the corresponding limits is implied. Then from formulas (A.14) and (A.7) we obtain · σ τ · σ τ σ σ · τ · τ a e = a eτ e = a δτ = a , a eσ = aτ e eσ = aτ δσ = aσ . Appendix A 217

Thus, we ﬁnd the simple formulas to obtain the components of the arbitrary vector a decomposed into considered bases, respectively:

σ σ a = a · e ,aσ = a · eσ . (A.15)

Relations (A.14) and (A.15) yield the rules of raising an index and missing an index:

σ σ τ σ τσ τ τ a = a · e = aτ e · e = g aτ ,aσ = a · eσ = a eτ · eσ = gτσa . (A.16)

Here gστ = gτσ = eσ · eτ , σ, τ =1, 2, 3, are elements of basic metric form or basicmetrictensorand gστ = gτσ = eσ · eτ , σ, τ =1, 2, 3, are components of complementary metric form or complementary metric tensor. Applying the metric tensor, it is not diﬃcult to obtain the transition formulas from the fundamental basis to the reciprocal one and vice versa:

τ τ σ σ τ στ eσ =(eσ · eτ )e = gστ e , e =(e · e )eτ = g eτ .

Note that if two vectors, represented in the same bases, are multiplied scalarly by each other, then the obtained relation turns out rather lengthy (nine addends): σ τ στ a · b = gστ a b = g aσbτ . If the vectors are decomposed into the diﬀerent bases, then the scalar product involves only three addends: · σ · τ σ τ σ σ a b = a bτ eσ e = a bτ δσ = a bσ = aσb . From the chain of relations · | || | | | aσ = a eσ = a eσ cos ϕ = eσ preσ a , τ · τ | || τ | | τ | a = a e = a e cos ψ = e preτ a we obtain the formulas for computing the projections of the vector a on the vectors of fundamental and reciprocal bases:

τ aσ a pr a = , pr τ a = . (A.17) eσ e τ |eσ| |e |

§ 4. Covariant and contravariant components of velocity vector

Let now the motion of the point M be considered in the curvilinear system of coordinates, in which case equations of motion (A.3) are known. By deﬁnition, the velocity is given by the vector v = dr/dt and therefore we have dr ∂r v = = q˙σ =˙qσe . dt ∂qσ σ 218 Appendix A

At the same time the velocity vector in fundamental basis can be represented σ as v = v eσ. Then, comparing with the previous formula, we obtain the following representations for the contravariant components of velocity vector:

vσ =˙qσ ,σ=1, 2, 3 . (A.18)

For the orthogonal fundamental basis {eσ} the relation for the modulus of velocity vector has the form σ 2 1 2 2 2 3 2 v = |v| = (v eσ) = (H1q˙ ) +(H2q˙ ) +(H3q˙ ) , (A.19) where, as we have shown earlier, Hσ are scale factors. According to formulas (A.16) and (A.18) the covariant components of velocity vector are the following

τ τ vσ = gστ v = gστ q˙ ,σ=1, 2, 3 . (A.20)

Consider another possible representation of the components vσ. For this purpose it is convenient to introduce the function v2 T = , (A.21) 1 2 which can be regarded as the kinetic energy of point with unit mass, what is marked by index "1". Function (A.21) can be rewritten as 1 1 1 T = v · v = q˙σe · q˙τ e = g q˙σq˙τ . (A.22) 1 2 2 σ τ 2 στ By relation (A.22), formula (A.20) can be represented now in the form ∂T v = 1 . (A.23) σ ∂q˙σ

As will be shown below, the function T1, given by formula (A.21), plays an important role in computing the covariant components of acceleration vector of point.

§5.Christoﬀelsymbols

In § 1 we introduce the vectors of fundamental basis ∂r e = ,σ=1, 2, 3 , σ ∂qσ which show the changes of radius-vector of point versus the changes of generalized coordinates. Let us study now the eﬀect of coordinates qτ ,τ=1, 2, 3, on the vector eσ,σ=1, 2, 3. For this purpose we consider the derivatives ∂e σ ,σ,τ=1, 2, 3 . (A.24) ∂qτ Appendix A 219

Since the vector can be represented in one of the forms of (A.14), where the covariant and contravariant components are computed by formulas (A.15), for the sought vectors we obtain ∂e ∂e ∂e ∂e σ = σ · e eρ , σ = σ · eρ e , ∂qτ ∂qτ ρ ∂qτ ∂qτ ρ ρ, σ, τ =1, 2, 3 .

The covariant and contravariant components of vectors (A.24) in the above relation are called the Christoﬀel symbols of the ﬁrst and second kinds and ρ are denoted by Γρ,στ and Γστ , respectively. Thus, we have ∂e ∂e Γ = σ · e , Γρ = σ · eρ . ρ,στ ∂qτ ρ στ ∂qτ Finally, the previous relations take the form: ∂e ∂e σ =Γ eρ , σ =Γρ e , ∂qτ ρ,στ ∂qτ στ ρ ρ, σ, τ =1, 2, 3 .

Obviously, by formulas (A.16) the Christoﬀel symbols are related as

ρ ρπ π Γ = g Γπ,στ , Γρ,στ = gρπΓ , στ στ (A.25) π,ρ,σ,τ =1, 2, 3 .

Represent the Christoﬀel symbols of the ﬁrst kind via the elements of basic metric tensor. Assuming that the mixed second derivatives are continuous in the coordinates of radius-vector of point, we obtain the chain of relations

∂e ∂2r ∂2r ∂e σ = = = τ . ∂qτ ∂qσ∂qτ ∂qτ ∂qσ ∂qσ Applying this formula twice, we can perform the following transformations: ∂e 1 ∂e ∂e Γ = σ · e = σ · e + τ · e = ρ,στ ∂qτ ρ 2 ∂qτ ρ ∂qσ ρ 1 ∂(e · e ) ∂(e · e ) ∂e ∂e = σ ρ + τ ρ − ρ · e − ρ · e = 2 ∂qτ ∂qσ ∂qτ σ ∂qσ τ 1 ∂(e · e ) ∂(e · e ) ∂e ∂e = σ ρ + τ ρ − τ · e − σ · e . 2 ∂qτ ∂qσ ∂qρ σ ∂qρ τ

This implies the formula for computing the Christoffel coefficients of the first kind: 1 ∂g ∂g ∂g Γ = ρσ + ρτ − στ . (A.26) ρ,στ 2 ∂qτ ∂qσ ∂qρ 220 Appendix A

The Christoﬀel coeﬃcients of the second kind can be determined, in turn, by formulas (A.25).

§ 6. Covariant and contravariant components of acceleration vector. TheLagrangeoperator

Represent the acceleration vector in the following way: dv d de w = = (vρe )=v ˙ρe + vρ ρ . dt dt ρ ρ dt We have de (q) ∂e ρ = ρ q˙σ , dt ∂qσ and therefore this formula can be represented as ∂e w =˙vρe + vρvσ ρ . (A.27) ρ ∂qσ Multiplying scalarly the above relation by the vectors eπ,weobtainthe contravariant components of acceleration vector:

π π π ρ σ w =¨q +Γρσq˙ q˙ .

Multiplying scalarly (A.27) by the vectors eπ, we obtain the covariant components of acceleration vector:

ρ ρ σ wπ = gπρq¨ +Γπ,ρσq˙ q˙ . (A.28)

Now we proceed to the obtaining of the second representation of covariant components of acceleration. We write the acceleration vector as: dv d deτ w = = (v eτ )=v ˙ eτ + v . dt dt τ τ τ dt However, since deτ (q) ∂eτ = q˙σ , dt ∂qσ this formula takes the form ∂eτ w =˙v eτ + v vσ . τ τ ∂qσ

Multiplying scalarly this relation on the vectors eρ, we obtain ∂eτ w =˙v + v vσ · e . (A.29) ρ ρ τ ∂qσ ρ Appendix A 221

Consider the last scalar product in formula (A.29). By the property of the vectors of reciprocal bases (A.7), we have

τ · τ e eρ = δρ = const .

Then ∂(eτ · e ) ρ =0. ∂qσ Henceweobtain ∂eτ ∂e · e = − ρ · eτ = −Γτ . ∂qσ ρ ∂qσ ρσ It follows that formula (A.29) can be rewritten as

− τ σ wρ =˙vρ Γρσvτ v . (A.30)

However, by the rule of raising an index (A.16) we have

π τ vτ = gτπv , Γπ,ρσ = gπτ Γρσ .

Therefore formula (A.30) becomes

π σ wρ =˙vρ − Γπ,ρσv v . (A.31)

Relations (A.26) give 1 ∂g ∂g ∂g Γ vπvσ = πρ + πσ − ρσ vπvσ . π,ρσ 2 ∂qσ ∂qρ ∂qπ

If in the right-hand side of this relation in the ﬁrst double sum we interchange the summation indices π and σ, then this sum coincides with the last double sum, given with the minus sign. In this case, collecting terms, we obtain 1 ∂g ∂T Γ vπvσ = πσ vπvσ = 1 , (A.32) π,ρσ 2 ∂qρ ∂qρ where T1 is a function, introduced by formula (A.21). Recall that according to (A.23) the covariant components of velocity vector are also represented by this function. Therefore we have d ∂T v˙ = 1 . (A.33) ρ dt ∂q˙ρ

Using relations (A.32) and (A.33), from formula (A.31) we ﬁnd the ﬁ- nal second representation of a covariant component of acceleration vector of point, namely d ∂T ∂T w = 1 − 1 . (A.34) ρ dt ∂q˙ρ ∂qρ 222 Appendix A

Fig. A. 2

Introducing the Lagrange operator d ∂ ∂ L = − , ρ dt ∂q˙ρ ∂qρ we rewrite representation (A.34) as

wρ = Lρ(T1) .

The projections of acceleration on the vectors of fundamental basis can be found by formulas (A.17):

Lρ(T1) preρ w = . Hρ

§ 7. The case of cylindrical system of coordinates

As an example of application of the found formulas we consider the cylindrical system of coordinates q1 = ρ, q2 = ψ, q3 = z (Fig. A. 2). The Carte- sian coordinates of point x, y, z are represented via the cylindrical coordinates in the following way:

x = ρ cos ψ, y= ρ sin ψ, z= z. (A.35)

To the constant values of the generalized coordinates

ρ = C1 ,ψ= C2 ,z= C3 correspond the coordinate surfaces, passing through the point M(C1,C2,C3) (Fig. A. 2): a vertical cylinder of radius ρ; a vertical plane, which makes the Appendix A 223 angle ψ with the plane Oxz; a horizontal plane, raised oﬀ Oxy by z.The crossing of these coordinate surfaces at the point M gives the coordinate lines: the horizontal straight line O1M, the vertical straight line NM,and the circle of radius ρ centered at the point O1. The vectors of fundamental basis, which in accordance with formulas (A.4) and (A.35) have the form

∂r e = e = =cosψi +sinψj , 1 ρ ∂ρ ∂r e = e = = −ρ sin ψi + ρ cos ψj , (A.36) 2 ψ ∂ψ ∂r e = e = = k , 3 z ∂z are directed along the tangents to the coordinate curves in ascending order of the corresponding curvilinear coordinates. These vectors, as is shown in Fig. A. 2, make up an orthogonal but unnor- malized system since by formulas (A.6) and (A.35) we have

Hρ =1,Hψ = ρ, , Hz =1. (A.37)

The orthogonality of fundamental basis can also be established analytically. Really, formulas (A.36) imply that

e1 · e2 = −ρ cos ψ sin ψ + ρ sin ψ cos ψ =0,

e1 · e3 =0, e2 · e3 =0.

The reciprocal basis eρ,eψ,ez coincides in directions with the fundamental one and has the lengths 1 |eρ| =1, |eψ| = , |ez| =1. ρ The found bases permit us to construct the matrix of basic metric tensor ⎛ ⎞ 100 ⎝ 2 ⎠ (gστ )= 0 ρ 0 , (A.38) 001 and the matrix of complementary metric tensor ⎛ ⎞ 100 (gστ )=⎝01/ρ2 0⎠ . (A.39) 001

It is easily seen that the product of matrix (A.38) by matrix (A.39) is the unit matrix. For the computation of the Christoﬀel symbols of the ﬁrst kind 224 Appendix A we make use of formula (A.26). Since in matrix (A.38) the variable element 2 is g22 = gρρ = ρ only, then the only nonzero symbols are the following

Γ2,21 =Γ2,12 = −Γ1,22 = ρ. (A.40)

The Christoﬀel symbols of the second kind can be computed now by formulas (A.25), using the elements of matrix (A.39). Formulas (A.18) and (A.19) give

v1 = vρ =˙ρ, v2 = vψ = ψ,˙ v3 = vz =˙z, v = |v| = ρ˙2 +(ρψ˙)2 +˙z2 , (A.41) therefore 1 T = (˙ρ2 +(ρψ˙)2 +˙z2) . (A.42) 1 2 If contravariant components of velocity (A.41) are known, the covariant components can be obtained by formulas (A.20):

τ 1 v1 = g1τ v = v =˙ρ, τ 2 2 2 ˙ v2 = g2τ v = ρ v = ρ ψ, (A.43) τ 3 v3 = g3τ v = v =˙z.

The covariant components of accelerations can be found by means of representations (A.28). Since for the cylindrical system of coordinates the nonzero Christoﬀel symbols of the ﬁrst kind are the symbols, given by formulas (A.40) only, we have

σ σ τ 1 2 2 ˙ 2 w1 = g1σq¨ +Γ1,στ q˙ q˙ =¨q +Γ1,22q˙ q˙ =¨ρ − ρψ , σ σ τ 2 2 1 2 2 1 2 ¨ ˙ w2 = g2σq¨ +Γ2,στ q˙ q˙ = ρ q¨ +Γ2,12q˙ q˙ +Γ2,21q˙ q˙ = ρ ψ +2ρψρ,˙ σ σ τ 3 w3 = g3σq¨ +Γ3,στ q˙ q˙ =¨q =¨z. (A.44) Note that it is rather convenient to determine the covariant components of velocity and acceleration, applying the functions T1. Really, using formulas (A.23) with provision for (A.42) we can compute at once the covariant components of velocity (A.43) obtained above:

∂T v = v = 1 =˙ρ, 1 ρ ∂ρ˙ ∂T1 2 ˙ v2 = vψ = = ρ ψ, ∂ψ˙ ∂T v = v = 1 =˙z, 3 z ∂z˙ Appendix A 225 and by formulas (A.34) the covariant components of acceleration, which co- incide with relations (A.44):

d ∂T ∂T d w = w = 1 − 1 = ρ˙ − ρψ˙ 2 =¨ρ − ρψ˙ 2 , 1 ρ dt ∂ρ˙ ∂ρ dt d ∂T1 ∂T1 d 2 ˙ 2 ¨ ˙ w2 = wψ = − = (ρ ψ)=ρ ψ +2ρψρ,˙ dt ∂ψ˙ ∂ψ dt d ∂T ∂T w = w = 1 − 1 =¨z. 3 z dt ∂z˙ ∂z The projections of velocity and acceleration are obtained by formulas (A.17), taking into account lengths (A.37) of the vectors of fundamental basis:

v v ˙ v preρ =˙ρ, preψ = ρψ, prez =˙z, w − ˙ 2 w ¨ ˙ w preρ =¨ρ ρψ , preψ = ρψ +2ψρ,˙ prez =¨z.

§ 8. Covariant components of acceleration vector for nonstationary basis

Consider now a more general case when the radius-vector r depends not only on q =(q1,q2,q3) but on time t, i. e. it is the function of the form r = r(t, q). In particular, this is possible in the case when the curvilinear coordinates qσ give the position of points relative to the system of coordinates Ox1x2x3, which has a given motion relative to the stationary (absolute) sys- σ tem of coordinates O1ξ1ξ2ξ3. In this case even for the ﬁxed values of q the radius-vector r varies in time in virtue of the translational motion of system Ox1x2x3. The absolute velocity v is computed by formula ∂r ∂r v = r˙ = + q˙σ . (A.45) ∂t ∂qσ

Introducing, for short, the notation q0 = t (therefore q˙0 =1), we can express velocity (A.45) in the following way:

∂r v = q˙α ,α= 0, 3 . (A.46) ∂qα We emphasize that such a representation is introduced only for short and therefore we do not need to consider the problem in four-dimensional space. The coordinate vectors are, as before, only the vectors ∂r e (t, q)= ,σ= 1, 3 . σ ∂qσ 226 Appendix A

Thus, the nonstationary basis varies not only with the change from point to point but at each point in a time. Compute a covariant component of the acceleration w: dv ∂r d ∂r d ∂r w = w · e = · = v · − v · . (A.47) π π dt ∂qπ dt ∂qπ dt ∂qπ

Diﬀerentiating ﬁrst relation (A.45) with respect to q˙π and then with respect to qπ (π =1, 2, 3),wehave

∂v ∂r ∂v ∂2r ∂2r d ∂r = , = + q˙σ = . ∂q˙π ∂qπ ∂qπ ∂t∂qπ ∂qσ∂qπ dt ∂qπ

It follows that the addends, entering into relation (A.47), take the form

∂r ∂v 1 ∂v2 ∂T v · = v · = = 1 , ∂qπ ∂q˙π 2 ∂q˙π ∂q˙π d ∂r ∂v 1 ∂v2 ∂T v · = v · = = 1 . dt ∂qπ ∂qπ 2 ∂qπ ∂qπ

Finally, for wπ we obtain

d ∂T ∂T v2 w = 1 − 1 ,T= ,π= 1, 3 . (A.48) π dt ∂q˙π ∂qπ 1 2

Thus, Lagrange’s form of representation of the covariant component wπ does not change also in the case of nonstationary basis. According to representation (A.46) the kinetic energy T1 of a point with unitmassisasfollows v2 1 ∂r ∂r 1 T = = · q˙αq˙β = g q˙αq˙β ,α,β= 0, 3 . (A.49) 1 2 2 ∂qα ∂qβ 2 αβ

If in relation (A.49) we discriminate the addends, involving explicitly ∂r/∂q0 = ∂r/∂t,thenwehave

(2) (1) (0) T1 = T1 + T1 + T1 , 1 ∂r ∂r 1 T (2) = · q˙ρq˙σ = g q˙ρq˙σ , 1 2 ∂qρ ∂qσ 2 ρσ (A.50) (1) ∂r · ∂r σ σ T1 = σ q˙ = g0σq˙ , ∂t ∂q 1 ∂r 1 T (0) = = g . 1 2 ∂t 2 00

Note that in formulas (A.50) the metric coeﬃcients are the quantities gρσ, (2) ρ, σ = 1, 3, entering into the relation T1 only. Appendix A 227

By formulas (A.48), (A.49) the covariant components of acceleration vector in expanded form are the following

ρ α β wπ = gπρq¨ +Γπ,αβq˙ q˙ ,π,ρ= 1, 3 ,α,β= 0, 3 . (A.51) This formula is the extension of the ﬁrst representation of covariant component of acceleration (A.28) to the case of nonstationary basis. We emphasize that like the previous remark, in formula (A.51) the Christoﬀel symbols ourselves are only the following ∂e Γ = ρ · e , π,ρ,σ = 1, 3 , π,ρσ ∂qσ π and by the use of the vector e0 = ∂r/∂t, the quantities Γπ,ρ0, Γπ,00 denote only the functions ∂e ∂2r Γ = ρ · e = · e , π,ρ0 ∂t π ∂qρ∂t π ∂e ∂2r Γ = 0 · e = · e , π,00 ∂t π ∂t2 π π,ρ = 1, 3 . They are introduced here for brevity of notation and allow us to obtain in the case of nonstationary basis the formulas similar to those in the stationary case.

§ 9. Covariant components of a derivative of vector

In Chapter IV the relations for the covariant components of derivatives of vector are used. We obtain here the corresponding formulas for the vector a of arbitrary physical structure. Recall that in § 6 of this Appendix they already have been obtained as a result of the diﬀerentiation of velocity vector. Consider the representation of the vector a in reciprocal basis:

τ a = aτ e . Find the vector b, which is a derivative of the vector a: deτ b = a˙ =˙a eτ + a . τ τ dt Since we have deτ (t, q) ∂eτ = q˙α , dt ∂qα the previous formula takes the form ∂eτ b =˙a eτ + a q˙α . τ τ ∂qα 228 Appendix A

Multiplying this relation scalarly by the vectors eρ,weget ∂eτ b =˙a + a q˙α · e . ρ ρ τ ∂qα ρ Arguing as in § 6, we ﬁnd ∂eτ · e = −Γτ , ∂qα ρ ρα and therefore ﬁnally we have

− τ α bρ =˙aρ Γραaτ q˙ . (A.52)

The particular case of this formula is relation (A.30). Formula (A.52) is often used in Chapter IV. Note that in Chapter IV it is also obtained more general formulas. The formulas, found above, can be used to describe motion of representation point in the curvilinear coordinates q =(q1, ...,qs).Inthiscasethe indices π,ρ,σ,τ are varied from 1 to s =3N and for nonstationary system, α and β from 0 to s =3N. In Chapter IV, using a tangent space, the formulas of this Appendix are extended to mechanical systems, consisting of not only the mass points but the rigid and elastic bodies. In this case the covariant and contravariant components of the velocity vectors v and the acceleration vectors w of mechanical system, as well as for one point, are represented by the following function T 1 T = = g q˙αq˙β ,α,β= 0,s, 1 M 2 αβ where M is a mass of total system and T is its kinetic energy. APPENDIX B

STABILITY AND BIFURCATION OF STEADY MOTIONS OF NONHOLONOMIC SYSTEMS

Appendix B contain a brief survey of the works, devoted to questions of the existence, stability, and branching of a steady motion of conservative nonholonomic systems. This Appendix is the plenary report of A. V. Karapetyan with the same title, which was spoken in the International science conference on mechanics "The third Polyakhov readings" (St.Petersburg, February 4-6, 2003).

In studying the questions of existence, stability, and branching of steady motions of conservative nonholonomic systems two approaches [97, 98, 101, 333, 334] are usually applied. In the general case when steady motions of conservative nonholonomic systems correspond to the symmetries, to which the linear first integrals do not correspond (unlike the conservative holonomic systems), the methods of Lyapunov–Malkin and Andronov–Hopf (see [91, 94, 97, 99. 1985, 101, 333]) are used. These methods are based on the analysis of equations of perturbed motion and on the characteristic equation of the linearized equations of perturbed motion. The latter has always zero roots, the number of which is not so less as the dimension of a family of steady motions, which unperturbed steady motion belongs to. If the number of zero roots is equal to the above-mentioned dimension and the rest of roots have negative real parts, then the unperturbed motion is stable, in which case any perturbed motion sufficiently close to the unperturbed one tends asymptot- ically to a steady motion of the considered family but, generally speaking, not unperturbed motion (according to the Lyapunov–Malkin theory). On the boundary of domain of stability (in the space of parameters of problem) the characteristic equation has either zero root, either a pair of pure imaginary roots. In the first case another families of steady motions are branched off unperturbed steady motion and in the second case the families of periodic motions (the Andronov–Hopf bifurcation occurs). The described approach to the study of steady motions of conservative nonholonomic systems is also applied in the case when the nonholonomic constraints have a so-called "dissipative" effect [94, 99. 1981, 1985]. The second approach to the study of questions of existence, stability, and branching of steady motions of nonholonomic systems is based on the modified theory of Routh–Salvadori, Poincaré–Chetaev, and Smale (see [97, 98, 99. 1983, 100. 1994, 2000, 101, 333, 334]). It can be applied to the cases when to the symmetries of system correspond not only steady motions but also the linear first integrals. Consider this case in more detail. At first we consider the case when the linear integrals, corresponding to the symmetries of system, are given in explicit form.

229 230 Appendix B

Let 1 H = H (v, r)= (A (r) v · v)+(a (r) · v)+a (r)=h (B.1) 2 be a total mechanical energy of system and

K = K (v; r)=BT (r) v + b (r)=k =const (B.2) be a k-dimensional vector of linear integrals (the sign "T" means a transposition). Here v is an n-dimensional vector of quasivelocities (in particular, of impulses or generalized velocities), r ∈ M is an m-dimensional vector of determining coordinates such that the n × n-matrix A (r) of positive definite quadratic form, the n-dimensional vector a (r), and the scalar function a (r), entering together into a total mechanical energy, and also the n × k-matrix B (r) and the k-dimensional vector b (r) of the coefficients of the first integrals depend on these coordinates. Denote by M a configuration space of the system dim M n. According to the Routh theory, on the fixed levels of the first integrals K = k to the critical points of the functions H correspond steady motions, in which case to the minimum points correspond stable steady motions. Taking into account a structure of function (B.1) and the first integrals (B.2), the problem of obtaining the critical points of this function on the fixed levels of these integrals can be solved in two stages. At the first stage we determine a single minimum of the function H on the fixed levels k of the first integrals K =constwith respect to the variables v (in this case the variables r are regarded as parameters): min H = H vk(r); r , v K=k 1 H v (r); r = a(r)+ C(r)c · c − k 2 k k − −1 · A (r)a(r) a(r) = Wk(r) , (B.3)

T ck = ck(r)=k − b(r)+B (r)A(r)a(r) , −1 C(r)= BT(r)A−1(r)B(r) ,

−1 −1 vk(r)=A (r)B(r)C(r)ck − A (r)a(r) .

Here and below we assume that rankB (r)=k, ∀ r ∈ M, i. e. the integrals are independent of a whole configuration space. The function Wk (r) is called an effective potential, which depends, obviously, on the variables r ∈ M and the parameters k ∈ Rk. Then the problem of study of steady motions of system is reduced to the problem of the analysis of effective potential. Appendix B 231

Theorem 1. If the eﬀective potential takes a nondegenerate stationary value at the point r0 ∈ M, then the relation

r = r0, v = v0 = vk (r0) describes a steady motion. The point r0, at which the effective potential has a stationary value, depends on the constants k of the first integrals. This means that the points r0 (k), which are stationary in configuration space, make up k-parametric k families in the space k ∈ R , r ∈ M . The same families in the space k n k ∈ R , r ∈ M , v ∈ R make up the points r = r0 (k), v = v0 = vk (r0), which are stationary in the phase space, i. e. make up steady motions. Even for the fixed values of the constants k, the effective potential Wk (r) can take stationary values not only at the point r0 but, generally speaking, at the certain another points r1, r2, .... These points also depend on the ∗ constants k. In the general case for certain values of k the families r0 (k), ∗ r1 (k), r2 (k), ...canhavecommonpoints.Suchvaluesofk are called bifurcational by Poincaré. Obviously, the corresponding steady motions r = r0 (k), v = v0 = vk (r0) have common points if and only if the families r0 (k), r1 (k), r2 (k), ... have common points (see (B.3)). In addition, by construction of effective potential, for indices we have the following relation

2 2 ind δ H (v0, r0) |(2) = ind δ Wk (r0) .

The latter permits us to simplify substantially the construction of the bifurcational diagrams of Poincar´e–Chetaev and to restrict ourself by constructing the families r0 (k) ∪ r1 (k) ∪ r2 (k) ∪ ... in the space {k, r} only. Consider the set k Σh,k = h ∈ R, k ∈ R : h = hs (k) ,s=0, 1, 2, ... (B.4) of the space {h, k},where

h = hs (k)=H (vk (r);r); r = rs (k) ,s=0, 1, 2, ... .

Set (B.4) is called bifurcational by Smale: in this set we have the crossplot- tings of topological types of domains, of motions in configuration space, which are defined by the relation Wk (r) h, r ∈ M. Theorem 2. If the effective potential takes locally a strictly minimal sta- 0 0 tionary value for the fixed values k of the constants k at the point r0 k , 0 0 then r = r0 k , v = v0 k is a stable steady motion. Theorem 3. If the index of the second variation of effective potential is 0 0 0 odd at the point r0 k , then r = r0 k , v = v0 k is an unstable steady motion. Theorems 1 – 3 follow from the Routh–Salvadori theory [97, 98, 334] and correspond to the special form of the first integrals (B.1), (B.2). 232 Appendix B

Remark. If for the certain r0 ∈ M we have rankB (r)

Suppose, γα = γα (r, p) is an m-dimensional vector, composed of the elements of the α-th row of the matrix Γ,whereα = 1,k.If Dγ T Dγ α = α ,α= 1,k, Dr Dr then the system of km equations in partial derivatives ∂p = Γ (r, p) ∂r is completely integrable and has the family of solutions p = Φ (r) k,which depends on the k arbitrary constants k, and the determinant of the k × k- matrix Φ (r) is not equal to zero. The latter means that system (B.5) except for generalized integral of energy (B.6) allows the k linear integrals

K = Φ−1 (r) p = k =const. (B.7) Appendix B 233

Though the explicit form of these integrals is unknown the general theory of Routh–Salvadori permits us to assert that the stationary values of integral (B.6) on ﬁxed levels of integrals (B.7) correspond to the steady motions

r = r0, r˙ = 0, p = p0 (B.8) of system (B.5), in which case the locally strictly minimal values correspond to the stable steady motions. Obviously, steady motions (B.8) make up a k-parametric family since the k + m constants r0 and p0 in (B.8) satisfy the system of m equations DW = 0 . (B.9) Dr The function H has a minimum on steady motion (B.8) under conditions (B.7) if the function W under these conditions has a minimum at the point (r0, p0). The latter occurs if all the eigenvalues of the matrix

D2W (B.10) Dr2 are positive at the point (r0, p0). If the determinant of matrix (B.10) is negative at the point (r0, p0), then steady motion (B.8) is unstable. Obviously, to generate equations (B.9) and matrix (B.10) it is not required to know an explicit form of the ﬁrst integrals (B.7). The existence and structure of these integrals permit us to make use of the Routh–Salvadori theory and to aﬃrm reasonably that steady motion (B.8) is stable for all positive eigenvalues of matrix (B.10), which is symmetric under the condition that the matrices

Dγ α ,α= 1,k, Dr are symmetric. However for the bifurcational diagrams of Poincaré–Chetaev and Smale to be constructed it is necessary to know the solution of system (B.7)–(B.9) in the form of r = r0 (k) and the quantities h = h (k)= W (r0 (k) , p0 (k)), respectively, i. e. it is necessary to obtain the first integrals (B.7) in explicit form (though in terms of special functions, not necessarily in terms of the elementary ones as in the problem on the motion of dynamically symmetric ball on absolutely roughened plane). In the problem on the motion of circular disk on absolutely roughened plane these first integrals are known in the form of hypergeometric Gaussian series. This makes it possible [127. 1999, 2001] to study completely the problem on a steady rolling of disk on horizontal plane. APPENDIX C

THE CONSTRUCTION OF APPROXIMATE SOLUTIONS FOR EQUATIONS OF NONLINEAR OSCILLATIONS WITH THE USAGE OF THE GAUSS PRINCIPLE

The Gauss principle is applied to the construction of approximate solutions of equations of nonlinear oscillations, in particular, of the solution by the Bubnov–Galerkin method.

If the motion of mechanical system is incompletely defined, then it is rational to construct the equations, which permit us to determine completely this motion, using Gauss’ principle represented in the form δZ =0, (C.1) where the function Z is given by formula (3.8) of Chapter IV. Two accents of the symbol δ are used to emphasize that the second time derivatives of generalized coordinates are varied only. We make use of this principle to find the approximate solutions of the nonlinear equation mx¨ = F (t, x, x˙) , (C.2) where m is a mass of mass point, x is its coordinate in the case of linear motion, F is a projection of the force acting on the point. Suppose, we seek the motion of mass point in the interval [0,τ] in the form n ν x(t)= aν f (t) , (C.3) ν=1 ν where f (t) are linearly independent functions, aν are the sought parameters. The function x(t), given in the form (C.3), does not satisfy, generally speaking, differential equation (C.2) and therefore, substituting it into this equation, we obtain mx¨ − F (t, x, x˙)=R, (C.4) where R is a residual. From the mechanical point of view this residual is regarded as a force, under which the motion of point exactly satisfies the law (C.3). We shall assume that the motion in the form (C.3) is incompletely given in the sense that the parameters aν are not known. In order to find these parameters, it is necessary that the average value of the square force R in the interval [0,τ] is minimal by virtue of the varying of the accelerations only (as in the Gauss principle) i. e. the following relation τ 2 δ mx¨ − F (t, x, x˙) dt =0

235 236 Appendix C

is satisﬁed. In other words, we shall seek the coeﬃcients aν under the as- sumption that the error of mean square on the interval [0,τ] is minimum. Taking into account that in the Gauss principle the accelerations are varied only, we have τ mx¨ − F (t, x, x˙) δxdt¨ =0.

0 Substituting relation (C.3) into this equation, we obtain

τ n n n n ν ν ν ν δaν m aν f¨ − F t, aν f , aν f˙ f¨ dt =0. (C.5) ν=1 0 ν=1 ν=1 ν=1

The quantities δaν are arbitrary and independent. Therefore from equation (C.5) it follows that

τ n n n ν ν ν ν m aν f¨ − F t, aν f , aν f˙ f¨ dt =0,ν= 1,n. (C.6) 0 ν=1 ν=1 ν=1 The conditions, under which this system of algebraic equations has solutions different from zero, depend on the form of as the function F (t, x, x˙),as the functions f ν (t), ν = 1,n. The quantity R, introduced by formula (C.4), was regarded above as a force. Now we regard it as an error, which occurs for the function x(t),given in the form (C.3), to be satisfied equation (C.2). According to this approach the system of algebraic equations (C.6) with respect to the parameters aν becomes a system, which under certain assumptions permit us to find a partial approximate solution of equations (C.2) in the form (C.3). We apply now this method to determine the approximate periodic solutions of equation (C.2). For the sake of simplicity, we seek the periodic solutions in the form

x(t)=a1 cos ωt + a2 sin ωt . (C.7) Then system (C.6), in which the time τ is assumed to be equal to the period 2π/ω, can be represented in the following way:

2π/ω 2 − mω (a1 cos ωt + a2 sin ωt)− 0 −F (t, a1 cos ωt + a2 sin ωt , −a1ω sin ωt + a2ω cos ωt) cos ωtdt =0, (C.8) 2π/ω 2 − mω (a1 cos ωt + a2 sin ωt)− 0 −F (t, a1 cos ωt + a2 sin ωt , −a1ω sin ωt + a2ω cos ωt) sin ωtdt =0. Appendix C 237

Equations (C.8) are used to construct approximately the solution of equation (C.2) in the form (C.7) by the Bubnov–Galerkin method. They are usually deduced from the fundamental equation of dynamics. Recall that, as is shown for Example VI. 3 considered in § 4 of Chapter VI, the clariﬁcation of the methods of Ritz and Bubnov–Galerkin by means of the integral variational principles, can be found in the work of G. Yu. Dzhanelidze and A. I. Lur’e [56]. The above method for obtaining the approximate solutions of equations (C.2) can easily be extended to the case of arbitrary mechanical system with s degrees of freedom. In this case the Gauss principle (C.1) is used in integral form, i. e. we assume that

τ (MW − Y)δWdt =0. (C.9)

0 Recall that we have d ∂T ∂T MW − Y = − − Q eσ ,δW = δq¨σe . dt ∂q˙σ ∂qσ σ σ

Here T is a kinetic energy of system, Qσ is a generalized force, corresponding σ σ to the generalized coordinate q , eσ and e are the vectors of fundamental and reciprocal bases, respectively. Therefore equation (C.9) can be rewritten as τ d ∂T ∂T − − Q δq¨σdt =0. dt ∂q˙σ ∂qσ σ 0 It follows that the functions qσ(t),givenas

n σ σ ν q (t)= aν f (t) ,σ= 1,s, (C.10) ν=1 can be regarded as an approximate solution of Lagrange’s equations if the σ parameters aν satisfy the following equations

τ d ∂T ∂T − − Q f¨ν dt =0,σ= 1,s, ν = 1,n. (C.11) dt ∂q˙σ ∂qσ σ 0

Here the functions qσ(t) are assumed to be given in the form (C.10). The applying of formulas (C.11) to the solution of nonlinear system of diﬀerential equations, which describes the steady-state oscillations of a certain electromechanical system, using the Bubnov–Galerkin method, can be found in the work [262]. APPENDIX D

THE MOTION OF NONHOLONOMIC SYSTEM WITH OUT REACTIONS OF NONHOLONOMIC CONSTRAINTS

In Appendix D the motion of nonholonomic systems in the case when the reactions of constraints are lacking is considered. By the Mei Fengxiang terminology such a motion is called a free motion of nonholonomic system. The free motion of the Chaplygin sledge is studied. A realizing of the free motion of nonholonomic systems acted by external forces is discussed.

§ 1. Existence conditions for "free motion" of nonholonomic system

The motion of nonholonomic system is deﬁned by forces, constraints, and the initial data. In the work of Mei Fengxiang [362. 1994] the notion of free motion of nonholonomic system, which is regarded as a motion under zero values of reactions of nonholonomic constraints, is introduced. In this work, in particular, the free motion of the Chaplygin sledge is considered. In the work [362] existence conditions for a free motion of nonholonomic system are given. Below we obtain these conditions. The motion of mechanical system with the ideal nonholonomic constraints

ϕκ(t, q, q˙)=0,q=(q1, ... ,qs) , κ = 1,k, (D.1) is described by Lagrange’s equations of the ﬁrst kind in curvilinear coordinates d ∂T ∂T − = Q + R ,σ= 1,s. (D.2) dt ∂q˙σ ∂qσ σ σ Here the generalized reactions of nonholonomic constraints take the form ∂ϕκ R =Λκ ,σ= 1,s, κ = 1,k. σ ∂q˙σ

Equations (D.2) can be written by using the Christoﬀel symbols of the ﬁrst kind κ τ α β ∂ϕ M(g q¨ +Γ q˙ q˙ )=Q +Λκ , στ σ,αβ σ ∂q˙σ σ, τ = 1,s, α,β = 0,s, q0 = t, q˙0 =1. This system can be solved as the algebraic system with respect to q¨τ , τ = 1,s: κ τ Δστ ∂ϕ α β q¨ = Q +Λκ − MΓ q˙ q˙ . (D.3) Δ σ ∂q˙σ σ,αβ

239 240 Appendix D

Here Δ is a determinant of the matrix (Mgστ ), Δστ is an algebraic comple- ment with (σ, τ ) number. We diﬀerentiate equations of constraints (D.1) with respect to time

dϕκ ∂ϕκ ∂ϕκ ∂ϕκ ≡ + q˙τ + q¨τ =0, κ = 1,k, τ = 1,s, (D.4) dt ∂t ∂qτ ∂q˙τ and substitute solutions (D.3) into formulas (D.4). Then we obtain κ κ κ κ ∂ϕ ∂ϕ τ Δστ ∂ϕ ∂ϕ α β + q˙ + Qσ +Λκ − MΓσ,αβ q˙ q˙ =0, ∂t ∂qτ Δ ∂q˙τ ∂q˙σ (D.5) κ = 1,k, σ,τ = 1,s, α,β = 0,s.

The Lagrange multipliers Λκ, κ = 1,k, can be determined from this system if the corresponding determinant is not equal to zero. Assuming Λκ =0, κ = 1,k, from relations (D.5) we obtain necessary and suﬃcient conditions for the existence of free motion of nonholonomic system. In the case when the constraints are stationary and the kinetic energy is independent of time they have the form ∂ϕκ Δ ∂ϕκ q˙τ + στ Q − MΓ q˙ρq˙τ =0, ∂qτ Δ ∂q˙τ σ σ,ρτ κ = 1,k, ρ,σ,τ = 1,s.

Just these conditions under (7) number are given in the work [362]. If in place of Lagrange’s equations of the ﬁrst kind we take Maggi’s equations σ d ∂T − ∂T − ∂q˙ σ σ Qσ λ =0,λ= 1,l, σ = 1,s, (D.6) dt ∂q˙ ∂q ∂v∗ then in the second group of equations for a free motion of nonholonomic system we have zeros: σ d ∂T − ∂T − ∂q˙ κ σ σ Qσ l+κ =0, = 1,k, σ = 1,s. (D.7) dt ∂q˙ ∂q ∂v∗

§ 2. Free motion of the Chaplygin sledge

Consider the case when the center of mass of the Chaplygin sledge is situated above a runner. Let x, y be coordinates of the center of mass C of sledge in a horizontal plane and θ be an angle of its rotation. Then the kinetic energy of system is as follows M J T = (˙x2 +˙y2)+ θ˙2 , 2 2 Appendix D 241 where M is a mass of sledge, J is a moment of inertia of sledge about the vertical axis, passing through the center of mass. On the motion of sledge it is imposed the nonholonomic constraint

ϕ ≡ x˙ sin θ − y˙ cos θ =0. (D.8)

Denote q1 = x, q2 = θ, q3 = y and introduce the quasivelocities

1 2 3 v∗ =˙x, v∗ = θ,˙ v∗ =˙x sin θ − y˙ cos θ.

This implies that

v3 x˙ = v1 , θ˙ = v2 , y˙ = v1 tg θ − ∗ . ∗ ∗ ∗ cos θ We generate the Maggi’s equations:

Mx¨ − Qx +(My¨ − Qy)tgθ =0,

Jθ¨ − Qθ =0, (D.9) 1 (My¨ − Q ) − =Λ, y cos θ where Qx,Qy,Qθ are generalized exterior forces. Condition (D.7) takes the form 1 (My¨ − Q ) − =0. (D.10) y cos θ Diﬀerentiating equation of constraint (D.8) in time, we obtain

x¨ − y¨ctg θ = −θ˙(˙x ctg θ +˙y) . (D.11)

Consider the motion of the Chaplygin sledge under the conditions

Qx = Qy = Qθ =0. (D.12)

Then the ﬁrst equation of system (D.9) is as follows

x¨ = −y¨tg θ. (D.13)

Substituting (D.13) into relation (D.11), we have

y¨ = θ˙(˙x +˙y tg θ)cos2 θ.

Now we represent condition (D.10) (assuming Qy =0)as

θ˙(˙x +˙y tg θ)cosθ =0. (D.14) 242 Appendix D

Taking into account the equation of constraints y˙ =˙x tg θ and assuming that cos θ =0 , equation (D.14) becomes

θ˙x˙ =0. (D.15)

We notice that the obtained condition (D.15) of a free motion of the Chaplygin sledge imposes a restriction on the choice of initial data. Really, if ˙ ˙ x˙ t=0 =˙x0 , y˙ t=0 =˙y0 , θ t=0 = θ0 , then, according to formula (D.15), the following relation ˙ θ0x˙ 0 =0 (D.16) is satisﬁed. Restriction (D.16) allows the following choice of initial data: ˙ x˙ 0 =0, θ0 =0 . (D.17)

Since nonholonomic constraint (D.8) are satisﬁed, this implies the initial condition for y˙: y˙0 =0. (D.18) To the initial data (D.17), (D.18) corresponds a motion such that the center of mass of sledge rests and the sledge uniformly rotates round it. In this case the force, preventing the displacement of sledge in transverse direction relative to a runner, is lacking and we say that with initial data (D.17), (D.18) the sledge moves (rotates) to be free. Condition (D.16) allows another choice of the initial data: ˙ θ0 =0, x˙ 0 =0 , y˙0 =0 , x˙ 0 sin θ0 − y˙0 cos θ0 =0.

In this case the center of mass of sledge has a linear and uniform motion along the initial orientation of runner and the sledge does not rotate. Condition (D.16) also allows the following obvious choice of the initial data: ˙ x˙ 0 =˙y0 = θ0 =0. This corresponds to the rest of sledge if the exterior forces are lacking. If the equation of constraint is obtained under condition (D.14), then in place of relations (D.15) we have

θ˙y˙ =0. (D.19)

The investigation of possible motions under condition (D.19) leads to the same three free motions of the Chaplygin sledge. Thus, if conditions (D.12) and (D.16) are satisﬁed, then the Chaplygin sledge moves free in the above-mentioned sense. If not the sledge has a stan- dard motion proper for nonholonomic system. In this case we need to ﬁnd a Appendix D 243 reaction of constraint in order to check wether this nonholonomic constraint is nonretaining.

§ 3. The possibility of free motion of nonholonomic system under active forces

By nonholonomic constraints (D.1) the law of system motion in L–space and in K–space can be represented, respectively, as

K K K MWL = YL + RL ,MW = Y + R . (D.20)

In the case of ideal constraints RL =0and in studying a free motion of nonholonomic system we have, in addition, RK =0. Therefore equations (D.20) take the form

K K MWL = YL ,MW = Y . (D.21)

Equations (D.21) imply that in the case of free motion the component WK of the vector of acceleration of system W is the function of variables t, q, q˙, given in the following way

YK (t, q, q˙) WK = . (D.22) M On the other hand, as is shown in § 1 of Chapter IV, the vector-function WK (t, q, q˙) is uniquely deﬁned by equations of constraint (D.1). Then the force YK (t, q, q˙), entering into relation (D.22), can be called the control force K Ycontrol, under which the incomplete program of motion, given in the form (D.1), is realized. Thus, by formulas (D.21) a free motion of nonholonomic system can be regarded as a motion such that the active force Y has a component, belonging to L–space only, and the control force Ycontrol belonging to K–space only. Applying this approach to a free motion of nonholonomic system we obtain that in accordance with the theory of constrained motion K K control∇ κ the control force Ycontrol = Ycontrol has the form Ycontrol =Λκ ϕ . control Here Λκ is a generalized control force, under which the constraint with κ number is realized. Note that the same approach is usedused in Chapter III to solve the problem of ﬂight dynamics on the directing of a mass point on a target by the curve of pursuit. The concept of study of free motion of nonholonomic systems, developed in the work of Mei Fengxiang [362. 1994], can also be of another practical importance. For example, in the treatise [226] it is considered a controllable motion of nonholonomic systems. The control is chosen from the condition that the nonholonomic system has a given program motion. In this case the control forces also provide generation of forces equal to the corresponding reactions of nonholonomic constraints. Under these control forces the reactions 244 Appendix D of nonholonomic constraints are equal to zero and therefore by the terminology of the work [362. 1994] such a controllable motion is a free motion of nonholonomic system. In the book [226] the possibility of small deviations of the obtained generalized coordinates and velocities from the required ones is taken into account. This is gone along with the occurrence of small reactions of nonholonomic constraints, which are regarded in the considered problem as disturbances. Finally, the original problem is reduced to the conditional problem of adaptive control with unknown disturbances. The algorithms of control and also the estimation of so far as a program motion of system is realized for a given accuracy of stabilization are given. APPENDIX E

THE TURNING MOVEMENT OF A CAR AS A NONHOLONOMIC PROBLEM WITH NONRETAINING CONSTRAINTS

The turning movement of a car with its possible sideslip is considered as a nonholonomic problem with nonretaining constraints. The four possible types of the car motion are studied.

§ 1. General remarks

The complete theory of the motion of a car with deformable wheels is developed by N. A. Fufaev and detailed in his book [130]. The treatise by V. F. Zhuravlev and N. A. Fufaev [72] is devoted to mechanics of systems with nonretaining constraints. In this treatise the Boltzmann–Hamel equations are used for studying the motion of nonholonomic systems, and the possibility of restoring nonholonomic constraints is investigated on the basis of behaviour of solution curves in the common space of generalized coordinates and quasivelocities. In this Appendix E the Maggi equations, which make it possible to easily determine the generalized reaction forces of nonholonomic constraints, are applied; the beginning and stop of wheels sideslip being determined by these constraint forces. We now return to examples II. 4 and II. 5 considered in Chapter II. Pay attention that the Boltzmann-Hamel and Maggi equations are formed there for the realized constraints (4.16). In this case the turning of a car is studied, saying ﬁguratively, under the "dynamic control when the turning moment ˙ L1(t), resisting moment L2(θ), and restoring moment L3(θ) are applied to the rotating front axle (see Fig. II. 4). This scheme required introducing four generalized coordinates ϕ, θ, ξC ,ηC , that was reasonable from the methodical point of view, for in this case we get an example, in which with two constraints (4.16) we have to obtain two Boltzmann-Hamel’s equations or two Maggi’s equations. This mathematical model can be of interest in studying the motion of wheeled robot vehicles, development of which is given much attention at present (see, for ex., works by V. N. Belotelov, V. I. Kalyonova, A. V. Karapetyan, A. I. Kobrin, A. V. Lenskii, Yu. G. Martynenko, V. M. Morozov, D. E. Okhotsimskii, M. A. Salmina [146-148, 423]). Let us go to the "kinematic control under which the turn of the front axle is determined by a driver as a certain time function θ = θ(t).Insuch a scheme a turning car has three degrees of freedom. In this case, we shall consider nonholonomic constraints

1 ˙ ϕ ≡−ξC sin ϕ +˙ηC cos ϕ − l2ϕ˙ =0, (E.1) 2 ˙ ϕ ≡−ξC sin(ϕ + θ)+η ˙C cos(ϕ + θ)+l1ϕ˙ cos θ =0, (E.2)

245 246 Appendix E which should be satisﬁed by the car motion, as nonretaining. The active forces F1(t) and F2(t) have the same meaning as in examples II. 4 and II. 5, of Chapter II.

§ 2. The turning movement of a car with retaining (bilateral) constraints

We shall study the car motion in the horizontal plane with respect to the ﬁxed system of coordinates Oξηζ (see Fig. E. 1). We shall set the position of a car by generalized coordinates q1 = ϕ (the angle between the longitudinal 2 3 Cx-axis of the car and the Oξ-axis ), q = ξC ,q = ηC (the coordinates of point C). The angle θ is equal to the angle between the front axle and a perpendicular to the Cx-axis . It is a given time function:

θ = θ(t) .

Two nonholonmic constraints (E.1) and (E.2), expressing the absence of side slipping of the front and rear axles of the car are imposed on the car motion. The kinetic energy of the system consists of the kinetic energies of the car body and front axle and is calculated according to the formula:

∗ ˙ 2 2 ∗ 2 ˙2 ˙ ˙ 2T =M (ξC +˙ηC )+J ϕ˙ +J2θ +2J2ϕ˙θ +2M2l1ϕ˙(−ξC sin ϕ +˙ηC cos ϕ) , ∗ ∗ 2 M = M1 + M2,J= J1 + J2 + M2l1 . (E.3) Using the expression for virtual elementary work

δA = Qϕδϕ + QξC δξC + QηC δηC , we shall ﬁnd the generalized forces acting on the car, as was done in examples II. 4, II. 5, Chapter II. For the rear drive car we obtain the following

Fig. E. 1 Appendix E 247 expressions: Q1 ≡ Qϕ =0, ≡ − ˙ Q2 QξC = F1(t)cosϕ F2(vC )ξC /vC , Q ≡ Q = F (t)sinϕ − F (v )˙η /v , (E.4) 3 ηC 1 2 C C C 2 2 vC = ξ˙C +˙ηC . In order to form the Maggi equations describing the vehicle motion we introduce new nonholonomic variables by to the formulas:

1 2 ˙ v∗ =˙ϕ, v∗ = −l2ϕ˙ − ξC sin ϕ +˙ηC cos ϕ, 3 ˙ v∗ = l1ϕ˙ cos θ − ξC sin(ϕ + θ)+η ˙C cos(ϕ + θ) , and write the reverse transformation

1 1 2 2 1 2 2 2 3 q˙ ≡ ϕ˙ = v∗ , q˙ ≡ ξ˙C = β v∗ + β v∗ + β v∗ , 1 2 3 (E.5) 3 ≡ 3 1 3 2 3 3 q˙ η˙C = β1 v∗ + β2 v∗ + β3 v∗, where l cos ϕ cos θ + l cos(ϕ + θ) β2 = 1 2 , 1 sin θ 2 cos(ϕ + θ) 2 −cos ϕ β2 = ,β3 = , sin θ sin θ (E.6) l sin ϕ cos θ + l sin(ϕ + θ) β3 = 1 2 , 1 sin θ sin(ϕ + θ) sin ϕ β3 = ,β3 = − . 2 sin θ 3 sin θ The ﬁrst group of the Maggi equations in this case consists of a single equation

1 2 3 − ∂q˙ − ∂q˙ − ∂q˙ (MW1 Q1) 1 +(MW2 Q2) 1 +(MW3 Q3) 1 =0. (E.7) ∂v∗ ∂v∗ ∂v∗

The expressions MWσ may be calculated using kinetic energy by the formulas d ∂T ∂T MW = − ,σ= 1, 3 . σ dt ∂q˙σ ∂qσ Finally, using expressions (E.3), (E.4), (E.5), (E.6), we represent the equation of motion (E.7) in the following expanded form: ∗ ¨ ¨ J ϕ¨ + J2θ + M2l1(−ξC sin ϕ +¨ηC cos ϕ)+ 2 ∗ ¨ − 2 − ˙ +β1 (M ξC M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) F1(t)cosϕ + F2(vC )ξC /vC )+ 3 ∗ − 2 − +β1 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) F1(t)sinϕ + F2(vC )˙ηC /vC )=0. (E.8) The equations of constraints (E.1) and (E.2) should be added to this equation. 248 Appendix E

If the initial conditions and analytic representation of the functions F1(t), F2(vC ) are given, then after numerical integrating the nonlinear system of diﬀerential equations (E.1), (E.2), (E.8) we shall ﬁnd the law of the car motion: ϕ = ϕ(t),ξC = ξC (t),ηC = ηC (t). (E.9) Now we can determine the generalized reaction forces. The second group of Maggi’s equations will be written as follows:

1 2 3 − ∂q˙ − ∂q˙ − ∂q˙ Λ1 =(MW1 Q1) 2 +(MW2 Q2) 2 +(MW3 Q3) 2 , ∂v∗ ∂v∗ ∂v∗

1 2 3 − ∂q˙ − ∂q˙ − ∂q˙ Λ2 =(MW1 Q1) 3 +(MW2 Q2) 3 +(MW3 Q3) 3 , ∂v∗ ∂v∗ ∂v∗ or in the extended form for the rear drive vehicle: 2 ∗ ¨ − 2 − ˙ Λ1 = β2 (M ξC M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) F1(t)cosϕ + F2(vC )ξC /vC )+ 3 ∗ − 2 − +β2 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) F1(t)sinϕ+(E.10)

+F2(vC )˙ηC /vC ) , 2 ∗ ¨ − 2 − ˙ Λ2 = β3 (M ξC M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) F1(t)cosϕ + F2(vC )ξC /vC )+ 3 ∗ − 2 − +β3 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) F1(t)sinϕ+(E.11)

+F2(vC )˙ηC /vC ) . After inserting expressions (E.9) into these formulas we ﬁnd the law of varying the generalized reaction forces

Λi =Λi(t),i=1, 2 . These functions allow us to investigate the possibility of realizing the nonholonomic constraints (E.1), (E.2). If the reaction forces appear to exceed the forces provided by Coulomb’s frictional forces, then these constraints will not be realized and the vehicle will begin to slip along the axles to which the wheels are fastened. In order to write the conditions of the beginning of side slipping of the wheels in the analytical form, it is necessary to establish the relation between the determined generalized reactions Λ1, Λ2 and reaction forces RB, RA applied to the wheels from the road (see Fig. E. 1). This is a question of principal importance, so let us consider the relation between the generalized reaction force Λ of the nonholonomic constraint and the reaction force R for the following quite general case. Assume that the equation of the nonholonomic constraint sets the condition of the fact that for the plane motion the velocity v of a point of mechanical system along the direction of the unit vector n is equal to zero, i. e. assume that constraint equation written in vectorial form is as follows: ϕn = v · n =0. Appendix E 249

This equation in a scalar form appears as

n ϕ =˙xnx +˙yny =0.

If the constraint is ideal, then the reaction force R can be represented as

R = Rxi + Ryj = ∂ϕn ∂ϕn =Λ i + j =Λn , ∂x˙ ∂y˙ where i and j are unit vectors in x− and y− directions. Hence, the generalized reaction force Λ is equal to the projection of the constraint reaction force R onto the direction of vector n. It is clear that this representation of the vector R in the form Λn can be extended also to the constraints (E.1), (E.2). Writing these constraints in the vector form 1 ϕ = vB · j =0, (E.12) 2 ϕ = vA · j1 = vA · (−i sin θ + j cos θ)=0, (E.13) where j1 is the unit vector of the ordinate axis of the movable frame Ax1y1 of the car front axle, we obtain

RB =Λ1j , (E.14)

RA =Λ2(−i sin θ + j cos θ) . (E.15) Remark that if the constraints (E.12) and (E.13) are violated, then nonzero values of ϕ1 and ϕ2 are equal to projections of velocities of the points B and A onto the vectors j and j1, correspondingly. In this case the resulting friction forces applied to the wheels may be represented as

fr − fr 1 RB = Λ1 sign(ϕ )j , fr − fr 2 − RA = Λ2 sign(ϕ )( i sin θ + j cos θ) .

fr fr Finding the positive quantities Λ1 and Λ2 will be reported below.

§ 3. The turning movement of a rear-drive car with nonretaining constraints

General remarks. Let us return to the question considered in the pre- vioius paragraphs. Note that the Maggi equation (E.8) was obtained for the satisﬁed constraints (E.1), (E.2), i. e. when these nonholonomic constraints were retaining (bilateral). Let us study the vehicle motion in the case when the constraints (E.1), (E.2) may be nonretaining, i. e. when side slipping of the front or rear wheels 250 Appendix E

(or both front and rear wheels simultaneously) begins. The dynamic conditions of realizing the kinematic constraints (E.1), (E.2) is the requirement that the forces of interaction between the wheels and the road should not exceed the corresponding Coulomb’s friction forces. For the driven front wheels in accordance with formula (E.15) this is expressed by inequality:

| | fr Λ2

fr where F2 ,k2 are the frictional force and the coeﬃcient of friction between front wheels and the road, respectively, N2 is the normal pressure of the front axle. When considering the rear driving wheels it is necessary to take into account that the value of this wheels-road interaction force FB is determined by the vector sum of the driving force F1 and side reaction force RB given by formula (E.14) (see Fig. E. 2). To provide the absence of side slipping of the rear axle, the following condition should be satisﬁed (the introduced notation is analogous to the notation used for the front axle): 2 2 fr FB = (F1) +(Λ1)

According to Fig. E. 2 this means that the end of the force vector FB should fr not go beyond the circle of radius F1 . Otherwise the road will not be able to develop such reaction value |Λ1| that is required for realization of the nonholonomic constraint (E.1). Thus, this constraint becomes nonretaining, the side velocity component of driven wheels appears, and Coulomb’s friction fr force F1 starts acting to them from the road. This Coulomb’s friction force fr F1 arises from simultaneous action of the driving force F1 and side friction fr force Λ1 ,sothat 2 ≡ fr 2 ≡ 2 2 fr 2 (FB) (F1 ) (k1N1) =(F1) +(Λ1 ) . (E.18)

Fig. E. 2 Appendix E 251

Note that at the beginning of side slipping the driver sets

F1 =0.

Possible types of the car motion. We shall explain possible different types of motion of the mechanical model of a car. In Fig. E. 3 in the phase space of variables qσ, q˙σ,σ = 1, 3, we see the representation of two hypersurfaces. The first one corresponds to the constraint given by equation (E.12), and the second one corresponds to the constraint given by equation (E.13). In an explicit form these constraints are presented by formulas (E.1), (E.2). For simultaneous realization of nonholonomic constraints (E.1) and (E.2) the point of the phase space should be located in the line of intersection of these hypersurfaces. This corresponds to the I-st type of the car motion fr (bold curve I in Fig. E. 3). If the first constraint is violated (FB = F1 ) and the second constraint is satisfied, then the representation point is located at the hypersurface ϕ2 =0(II-nd type of motion). If the second constraint is violated, but the first constraint is fulfilled ϕ1 =0, then the representation point belongs to hypersurface ϕ1 =0(III-rd type of motion). In the case if both constraints are violated, the representation point does not belong to hypersurfaces, as this takes place the vehicle moves in the presence of side friction forces acting on the front and rear axles (IV-th type of motion). From any type of motion the representation point can change to any other type of motion. For example, in the I-st type of motion, if inequality (E.17) is not satisfied , the vehicle becomes released of the constraint (E.1). If in this case inequality (E.16) is still satisfied, then constraint (E.2) keeps on working, thus, the representation point can move only over the hypersurface ϕ2 =0(the car changes to the II-nd type of motion). Here two cases of possible restoring the I-st type of motion should be distinguished. In some area G1 the solution curves go through the curve I, without stop-

Fig. E. 3 252 Appendix E ping there (see Fig. E. 3). This instantaneous realization of the constraint (E.1) corresponds to the stop of side motion of the rear axle in one direction and change of the same axle to the side motion in the reverse direction. In contrast to this the behaviour of solution curves within the area G2 charac- terizes restoration of the constraint ϕ1 =0and change from the II-nd type of motion to the I-st one. Without preliminary studies of behaviour of solution curves in the common space of generalized coordinates and quasivelocities [72], it is possible 1 to find out in which area G1 or G2 the equation ϕ =0turned out to be satisfied, in the following manner. By the values of phase variables, such that the constraint (E.1) is fulfilled, let us calculate the reaction Λ1 by the formula (E.10). If for the obtained value of Λ1 the inequality (E.17) is satisfied, then the constraint ϕ1 =0becomes retaining (bilateral) (the solution curve is within the area G2), otherwise this constraint is not restored (the solution curve is within the area G1). In investigating the II-nd type of motion it is necessary also to ensure that inequality (E.16) is satisfied, for if it is violated the vehicle will change to the IV-th type of motion. If constraint (E.1) is restored and constraint (E.2) is violated at the same time, then the III-rd type of motion will occur. Note that for the sake of simplicity, it was assumed in the foregoing that the static and dynamic coefficicients of Coulomb’s friction force are equal to each other. The difference of these quantities could be taken into account in a similar way as it has been done in § 4 of Chapter I, when studying accelerated motion of a car with the possible slipping of its driving wheels. Let us write out the equations of motion for the turning car four types of motion cosidered. I-st type of motion. For this motion both constraints (E.1) и (E.2) are fulfilled : ϕ1 =0,ϕ2 =0. Maggi’s equation for a rear-wheel drive vehicle takes the form (E.8), which should be integrated together with the equations of constraints (E.1) and (E.2). Having obtained the law of motion

ϕ = ϕ(t) ,ξC = ξC (t) ,ηC = ηC (t) , the generalized reactions can be found from (E.6), (E.10), (E.11)

Λ1 =Λ1(t) , Λ2 =Λ2(t) . By these values, fulﬁllment of inequelities (E.16) and (E.17) is being checked. When one of them is violated, the vehicle changes to the II-nd or III-rd type of motion, and when both of them are violated simultaneously it changes to the IV-th type. II-nd type of motion. For this type of motion only the second constraint is fulﬁlled: ϕ1 =0 ,ϕ2 =0. Appendix E 253

The rear axle of the vehicle executes lateral motion, therefore the lateral fr frictional force Λ1 , calculated by formula (E.18) is applied to it. As this takes place, if ϕ1 > 0, then according to formula (E.12) the rear wheels sideslip in the positive direction of the y-axis. Therefore, the lateral frictional force is opposed to the y-axis, and if ϕ1 < 0, it is aligned with the y-axis (see Fig. E. 1). Let us obtain Maggi’s equations in the presence of one constraint (E.2). Letusgotoquasivelocitiesbyformulas: 1 2 v∗ =˙ϕ, v∗ = ξ˙C , 3 ˙ v∗ = −ξC sin(ϕ + θ)+η ˙C cos(ϕ + θ)+l1ϕ˙ cos θ. Let us ﬁnd the inverse transformation: 1 ˙ 2 3 1 3 2 3 3 ϕ˙ = v∗ , ξC = v∗ , η˙C = β1 v∗ + β2 v∗ + β3 v∗ , where 3 − 3 3 β1 = l1 cos θ/ cos(ϕ + θ) ,β2 =tg(ϕ + θ) ,β3 =1/ cos(ϕ + θ) . (E.19) Now we may get two Maggi’s equations for the rear-wheel drive vehicle ∗ ¨ −¨ − fr 1 J ϕ¨ + J2θ + M2l1( ξC sin ϕ +¨ηC cos ϕ) Λ1 sign(ϕ )l2+ 3 ∗ − 2 − +β1 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) − fr 1 F1(t)sinϕ + F2(vC )˙ηC /vC +Λ1 sign(ϕ )cosϕ)=0, ∗ ¨ 2 ˙ M ξC − M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) − F1(t)cosϕ + F2(vC )ξC /vC − − fr 1 3 ∗ − 2 − Λ1 sign(ϕ )sinϕ + β2 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) F1(t)sinϕ+ fr 1 +F2(vC )˙ηC /vC +Λ1 sign(ϕ )cosϕ)=0. (E.20) From the second group of Maggi’s equations there remains one equation for determination of the generalized reaction Λ2. For the vehicle with driving rear wheels it is as follows: 3 ∗ 2 Λ2 = β (M η¨C + M2l1(¨ϕ cos ϕ − ϕ˙ sin ϕ) − F1(t)sinϕ+ 3 (E.21) fr 1 +F2(vC )˙ηC /vC +Λ1 sign(ϕ )cosϕ) . The equations of motion (E.20) are integrated together with the constraint equation (E.2). If the dynamic condition (E.16) for the constraint (E.2) to be realized holds for the obtained value of Λ2, then the II-nd type of motion continues. If the condition (E.16) is violated, then the vehicle will change to IV-th type of motion. In the course of checking inequality (E.16) it is necessary to keep watching if the constraint ϕ1 =0begins to hold. If this constraint is realized under certain obtained values of t, qσ, q˙σ,σ = 1, 3, and if inequality (E.17) 254 Appendix E

holds for the value Λ1 calculated by formula (E.10), then the constraint ϕ1 =0is restored, the rear axle ceases to execute lateral motion and the car changes to the I-st type of motion. If inequality (E.17) is not fulﬁlled for the value Λ1 calculated by formula (E.10), then the car keeps the II-nd type of motion (rear axle begins lateral motion in the opposite direction). Theoretically the car may change from the II-nd type of motion to the III-rd one: for this purpose, at a certain time instant inequality (E.16) must cease to hold and simultaneously the constraint ϕ1 =0must be restored. III-rd type of motion. This motion is studied in a similar way to the II-nd type. Now the following should be fulﬁlled: ϕ1 =0,ϕ2 =0 . Due to side slipping of the front axle of the car this front axle is acted upon by the side friction force fr Λ2 = k2N2 . (E.22) In order to form Maggi’s equations for this nonholonomic problem with one constraint (E.1) let us change to quasivelocities by using the formulas: 1 2 v∗ =˙ϕ, v∗ = ξ˙C , 3 ˙ v∗ = −ξC sin ϕ +˙ηC cos ϕ − l2ϕ.˙ This corresponds to the reverse transformation: 1 ˙ 2 3 1 3 2 3 3 ϕ˙ = v∗ , ξC = v∗ , η˙C = β1 v∗ + β2 v∗ + β3 v∗ , where 3 3 3 β1 = l2/ cos ϕ, β2 =tgϕ, β3 =1/ cos ϕ. (E.23) Two Maggi equations for the car with driving rear wheels have the form: ∗ ¨ −¨ fr 2 J ϕ¨ + J2θ + M2l1( ξC sin ϕ +¨ηC cos ϕ)+Λ2 sign(ϕ )l1 cos θ+ 3 ∗ − 2 − +β1 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) − fr 2 F1(t)sinϕ + F2(vC )˙ηC /vC +Λ2 sign(ϕ )cos(ϕ + θ)) = 0 , (E.24) ∗ ¨ 2 ˙ M ξC − M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) − F1(t)cosϕ + F2(vC )ξC /vC − − fr 2 3 ∗ − 2 − Λ2 sign(ϕ )sin(ϕ + θ)+β2 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) − fr 2 F1(t)sinϕ + F2(vC )˙ηC /vC +Λ2 sign(ϕ )cos(ϕ + θ)) = 0 .

The generalized reaction Λ1 is expressed as 3 ∗ − 2 − Λ1 = β3 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) F1(t)sinϕ+(E.25) fr 2 +F2(vC )˙ηC /vC +Λ2 sign(ϕ )cos(ϕ + θ)) . Appendix E 255

The equations of motion (E.24) are integrated together with the constraint equation (E.1). If the dynamic condition (E.17) for realizing the constraint (E.1) is satisfied for the value of Λ1 obtained by formula (E.25), then the III-rd type of motion continues. If the condition (E.17) is violated, then the car changes to the IV-th type of motion. In the course of checking inequality (E.17) it is necessary to keep watching if the constraint ϕ2 =0begins to be realized. If this constraint is realized under certain calculated values t, qσ, q˙σ,σ = 1, 3, then these values of the variables should be substituted in formula (E.11). If for the obtained Λ2 the inequality (E.16) is satisfied, then the constraint ϕ2 =0is restored, the front axles ceases to execute lateral motion, and the car changes to the I-st type of motion. If for the calculated value of Λ2 inequality (E.16) is not satisfied, then the car continues the III-rd type of motion (the front axle begins lateral motion in the opposite direction). Theoretically the III-rd type of motion can change to the II-nd one: for this purpose, at a certain instant inequality (E.17) must cease to hold, and at the same time the constraint ϕ2 =0must be restored. IV-th type of motion. For such motion the following must take place:

ϕ1 =0 ,ϕ2 =0 .

This means that the car moves as a holonomic system when its wheels are fr fr acted upon by side frictional forces Λ1 and Λ2 set by formulas (E.18) and (E.22). The motion of the rear wheel-drive car is determined by the following Lagrange equations of the second kind: ∗ ¨ ¨ J ϕ¨ + J2θ + M2l1(−ξC sin ϕ +¨ηC cos ϕ)− − fr 1 fr 2 Λ1 sign(ϕ )l2 +Λ2 sign(ϕ )l1 cos θ =0, ∗ ¨ 2 ˙ M ξC − M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) − F1(t)cosϕ + F2(vC )ξC /vC − − fr 1 − fr 2 Λ1 sign(ϕ )sinϕ Λ2 sign(ϕ )sin(ϕ + θ)=0, (E.26) ∗ 2 M η¨C + M2l1(¨ϕ cos ϕ − ϕ˙ sin ϕ) − F1(t)sinϕ + F2(vC )˙ηC /vC + fr 1 fr 2 +Λ1 sign(ϕ )cosϕ +Λ2 sign(ϕ )cos(ϕ + θ)=0. In course of calculation of motion by equations (E.26) it is necessary to keep watching if either function ϕ1 or ϕ2 vanishes, or both functions ϕ1 and ϕ2 do so for the current values of

t, qσ , q˙σ ,σ= 1, 3 . (E.27)

1 If ϕ =0holds for the values (E.27), then Λ1 should be calculated for these values of variables by formula (E.25). If for this value of Λ1 inequality (E.17) is satisﬁed, then the car changes to the III-rd type of motion, otherwise it keeps the motion of the IV-th type. 256 Appendix E

If it turns out that ϕ2 =0for the values (E.27), then for these values of variables Λ2 should be calculated by formula (E.21). If the inequality (E.16) is satisfied for this value of Λ2, then the car changes to the II-nd type of motion, otherwise it keeps the motion of the IV-th type. If it turns out that for the values (E.27) the both functions ϕ1 and ϕ2 vanish simultaneously, then Λ1 and Λ2 should be found from formulas (E.10), (E.11). If both inequalities (E.16) and (E.17) are fulfilled for these values, then the car changes to the I-st type of motion. If only inequality (E.16) is satisfied, then the II-nd type of motion begins. If only inequality (E.17) is fulfilled, then from this point on the car will execute the III-rd type of motion.

§ 4. Equations of motion of a turning front-drive car with non-retaining constraints

Consider a motion of a front-drive car. All necessary changes in the equations of motion are caused by the fact that the application point of the force F1(t) changes. Now the force is applied to the point A and aligned with the axis Ax1 (see Fig. E. 1). So, for a front-drive car the expressions of generalized forces appear as

Q1 ≡ Qϕ = l1F1(t)sinθ, ≡ − ˙ Q2 QξC = F1(t)cos(ϕ + θ) F2(vC )ξC /vC , Q ≡ Q = F (t)sin(ϕ + θ) − F (v )˙η /v , 3 ηC 1 2 C C C 2 2 vC = ξ˙C +˙ηC .

Below we present the equations of motion for the four types of motion. I-st type of motion. The car motion without slipping. In this case the Maggi equations have the form ∗ ¨ ¨ J ϕ¨ + J2θ + M2l1(−ξC sin ϕ +¨ηC cos ϕ) − l1F1(t)sinθ 2 ∗ ¨ − 2 − +β1 (M ξC M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) F1(t)cos(ϕ + θ) ˙ 3 ∗ − 2 − +F2(vC )ξC /vC )+β1 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) F1(t)sin(ϕ + θ)

+F2(vC )˙ηC /vC )=0. (E.28) As this takes place, the generalized constraint reaction forces are expressed as

2 ∗ ¨ − 2 − Λ1 = β2 (M ξC M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) F1(t)cos(ϕ + θ) ˙ 3 ∗ − 2 +F2(vC )ξC /vC )+β2 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) (E.29)

−F1(t)sin(ϕ + θ)+F2(vC )˙ηC /vC ) , Appendix E 257

2 ∗ ¨ − 2 − Λ2 = β3 (M ξC M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) F1(t)cos(ϕ + θ) ˙ 3 ∗ − 2 +F2(vC )ξC /vC )+β3 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) (E.30)

−F1(t)sin(ϕ + θ)+F2(vC )˙ηC /vC ) . 2 3 2 3 In equation (E.28) and relations (E.29), (E.30) the quantities β1 ,β1 ,β2 ,β2 , 2 3 β3 ,β3 should be calculated according to formulae (E.6). In the case of a front-drive car, inequalities, the fulﬁllment of which means the realization of constraints (E.1) and (E.2), appear as

| | fr Λ1

3 3 3 where coeﬃcients β1 ,β2 ,β3 are determined from (E.19), and the quantity fr Λ1 is deﬁned as fr Λ1 = k1N1 . (E.33)

The relation for determining the generalized reaction Λ2 is 3 ∗ − 2 − Λ2 = β3 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) F1(t)sin(ϕ + θ)+ fr 1 +F2(vC )˙ηC /vC +Λ1 sign(ϕ )cosϕ) .

III-rd type of motion. The car front axle executes side (lateral) motion. In this case the, for a front-drive car the two Maggi equations take the form ∗ ¨ ¨ J ϕ¨ + J2θ + M2l1(−ξC sin ϕ +¨ηC cos ϕ) − l1F1(t)sinθ+ fr 2 3 ∗ − 2 − +Λ2 sign(ϕ )l1 cos θ + β1 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) 258 Appendix E

Fig. E. 4

− fr 2 F1(t)sin(ϕ + θ)+F2(vC )˙ηC /vC +Λ2 sign(ϕ )cos(ϕ + θ)) = 0 , ∗ ¨ 2 ˙ M ξC − M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) − F1(t)cos(ϕ + θ)+F2(vC )ξC /vC − − fr 2 3 ∗ − 2 − Λ2 sign(ϕ )sin(ϕ + θ)+β2 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) − fr 2 F1(t)sin(ϕ + θ)+F2(vC )˙ηC /vC +Λ2 sign(ϕ )cos(ϕ + θ)) = 0 . 3 3 3 Here the quantities β1 ,β2 ,β3 are deﬁned from (E.23), and the friction force fr Λ2 is expressed as 2 2 fr 2 (k2N2) =(F1) +(Λ2 ) . (E.34)

In this case the generalized constraint reaction force Λ1 is 3 ∗ − 2 − Λ1 = β3 (M η¨C + M2l1(¨ϕ cos ϕ ϕ˙ sin ϕ) F1(t)sin(ϕ + θ)+

fr 2 +F2(vC )˙ηC /vC +Λ2 sign(ϕ )cos(ϕ + θ)) .

IV-th type of motion. Both car axles execute side motion. Equations of motion of a front-drive car slipping on the horizontal surface (level) are ∗ ¨ ¨ J ϕ¨ + J2θ + M2l1(−ξC sin ϕ +¨ηC cos ϕ) − l1F1(t)sinθ+

fr 1 fr 2 +Λ1 sign(ϕ )l2 +Λ2 sign(ϕ )l1 cos θ =0, ∗ ¨ 2 ˙ M ξC − M2l1(¨ϕ sin ϕ +˙ϕ cos ϕ) − F1(t)cos(ϕ + θ)+F2(vC )ξC /vC − − fr 1 − fr 2 Λ1 sign(ϕ )sinϕ Λ2 sign(ϕ )sin(ϕ + θ)=0, ∗ 2 M η¨C + M2l1(¨ϕ cos ϕ − ϕ˙ sin ϕ) − F1(t)sin(ϕ + θ)+F2(vC )˙ηC /vC + fr 1 fr 2 +Λ1 sign(ϕ )cosϕ +Λ2 sign(ϕ )cos(ϕ + θ)=0. fr fr The values of side friction forces Λ1 and Λ2 are deﬁned from (E.33) and (E.34). Taking into account conditions (E.31), (E.32) of constraints realization and formulae for deﬁning the values of friction forces (E.33), (E.34), the logic Appendix E 259 of change from one type of motion to another is the same as in the case of a rear-drive car (see § 3).

§ 5. Calculation of motion of a certain car

As an example, let us consider the motion of a hypothetical compact 2 2 motor car with M1 = 1000 kg ; M2 = 110 kg ; J1 = 1500 kg·m ; J2 =30kg·m ; l1 =0.75 m ; l2 =1.65 m ; k1fr =0.4 ; k2fr =0.4 for the power characteristics: −1 F2(vC )=k2vC N ; k2 = 100 N·s·m . The following car motion is studied. In the beginning the vehicle moves rectilinearly (the planes of the front and rear wheels are parallel) during eight seconds, in this case ϕ = π/6. During this time the function F1(t) changes by the law F1(t) = 200t (F1 is measured in Newtons, t is measured in seconds), i. e. at the initial time F1(0) = 0, and at the end of rectilinear motion F1(8) = 1600. Graphs of dependences of coordinates on time are presented in Fig. E. 4. After eight seconds of rectilinear motion the driver starts to turn the steering wheel at a smooth manner at the angle θ = π(t − 8)/8 ,thatis,in two seconds the angle θ is equal to π/4 . For this motion F1(t) = 1600 .By the computed values of constraint reactions Λ1 and Λ2 we get graphs shown in Fig. E. 5 . It follows from the graphs that inequality (E.17) is satisﬁed, but condition (E.16) is violated when t1 =9.5147 , θ(t1)=0.5948 .Thus,when 8

Fig. E. 5 260 Appendix E the same time we check if the condition ϕ2 =0 is fulﬁlled. As follows from Fig. E. 6 , it starts to be satisﬁed at the moment t2 =9.8415;inthiscaseas we can see from calculations, the constraint reaction force |Λ2| becomes close in value to the friction force between wheels and the road, and the front axle stops moving in side directions.

Fig. E. 6

Fig. E. 7

Fig. E. 8 Appendix E 261

Thus, for t1

§ 6. Reasonable choice of quasivelocities

Previously, when studying possible types of the car motion, we had to use different forms of the equations of motion (E.8), (E.20), (E.24), (E.26). This makes certain difficulties, especially when numerically integrating the given systems of differential equations with the help of computer. For similar problems with nonholonomic nonretaining constraints N. A. Fufayev [72] suggests to use a single form of Boltzmann-Hamel’s equations. Let us see, how this idea may be applied in the case of using Maggi’s equations in analogous problems. (We notice that for solving similar problems the equations of motion of nonholonomic systems with variable kinematic structure can be effective [221]). Quite different forms of the equations of motion (E.8), (E.20), (E.24), (E.26) were obtained due to the fact that for different types of the car motion new transition formulas for quasivelocities were chosen every time, or the generalized coordinates were used directly to get the Lagrange equations of the second kind. Now we shall use the form of Maggi’s equations for all the four types of motion, the generalized velocities being always expressed in terms of quasivelocities by the same formulas (E.5). In these formulas the 1 quasivelocities have a certain physical meaning: v∗ is the angular velocity 2 3 of rotation of the car body, v∗ and v∗ are, according to formulas (E.12) and (E.13), the side velocities of the rear and front axles, correspondingly, aligned with the vectors j and j1. If the nonholonomic constraints (E.1), (E.2) are 2 3 realized, quasivelocities v∗ and v∗ vanish, and if these constraints turn out to be nonretaining, then these quasivelocities have real nonzero values (except for the instant stops of axles in their side motion). For the motion of the I-st type we still use the equation of motion (E.8) (or (E.21)) and the formulas for determination of the generalized reactions (E.10), (E.11). For the II-nd type of motion, if the constraint ϕ2 =0holds, then the generalized reaction Λ2 calculated by the formula (E.11) arises. In this case the equation of motion (E.8) should be completed with the differential equation (E.10), where Λ1 is changed for the projection of the side friction force − fr 1 ( Λ1 sign(ϕ )) acting on the rear axle during its side slipping. It is necessary to add the constraint equation (E.2) to these differential equations. 262 Appendix E

For the III-rd type of motion Λ1 is calculated in the same way by formula (E.10), and equation (E.11), in which the reaction Λ2 is replaced with the − fr 2 projection of the side friction force ( Λ2 sign(ϕ )), is added to equation (E.8). The constraint equation (E.1) is added to these differential equations. Maggi’s equations are linear combinations of the Lagrange equations of the second kind, therefore, in order to keep the uniformity of differential equations and for the IV-th type of motion corresponding to the holonomic problem, it is convenient to use the form of Maggi’s equations. Eventually the equations of motion will take the form (E.8), (E.10), (E.11), where Λ1 − fr 1 − fr 2 and Λ2 are replaced with ( Λ1 sign(ϕ )) and ( Λ2 sign(ϕ )). The logic of change from one type of motion to another is the same as in § 3. Pay attention that the obtained equations of motion have a singularity at θ =0. Therefore, the difficulties may occur in calculations, when turning begins with rectilinear motion. In this case, instead of some possible modifica- tions of the system of differential equations, which we used in the calculations given above, we can advise to change initially to the special system of curve- linear coordinates suggested in works [423]. APPENDIX F

CONSIDERATION OF REACTION FORCES OF HOLONOMIC CONSTRAINTS AS GENERALIZED COORDINATES IN APPROXIMATE DETERMINATION OF LOWER FREQUENCIES OF ELASTIC SYSTEMS

A new method for determination of lower frequencies of mechanical systems consisting of elastic bodies connected to each other is oﬀered. The conditions of connection of bodies are written as holonomic constraints, the reactions of which are considered as generalized coordinates. Therefore the number of degrees of freedom proves to be equal to the number of constraints.

On the possibility of introducing generalized reaction forces as Lagrangean coordinates. This Appendix presents a development of the method suggested in the Chapter VI. The equation of frequencies (6.12) of this chapter makes it possible, if necessary, to determine any number of the system’s natural frequencies for a reasonably great number N of dynamically considered oscillation modes of the system elements. However, as a rule, it is necessary to know only several ﬁrst frequencies and modes. When calculating them one can use the following approximate approach to this problem. The potential energy of the system consisting of elastic bodies connected to each other can be represented as a positively deﬁned quadratic form of the generalized constraint reactions introduced

1 n Π= c Λ Λ , (F.1) 2 ij i j i,j=1 when considering all the natural vibration modes of the system’s elements quasi-statically. Recall that the coeﬃcients of this form are calculated by formulas (5.15), (5.13) of Chapter VI. In quasistatics the deformed state of all system elements is uniquely determined by setting the quantities Λi, i = 1,n. The given elastic system comes to this state as a result of the fact that its points have obtained displacements, which can be found as linear functions of the reactions Λi, i = 1,n. Hence the position of all points of the system at the time t is uniquely determined by setting the quantities Λi, i = 1,n. Therefore, they can be considered as the generalized Lagrange coordinates; and the kinetic energy of the system can be represented in the form

1 n T = a Λ˙ Λ˙ . (F.2) 2 ij i j i,j=1

263 264 Appendix F

Here aij, i, j = 1,n, are some constants, the calculation procedure for which will be shown below through a number of examples. Lagrange’s equations of the second kind corresponding to expressions (F.1) and (F.2) are

n (aijΛ¨ j + cijΛj)=0,i= 1,n. j=1

By assuming as in § 5 of Chapter VI

Λi = Λ˜ i cos(pt+ α) ,i= 1,n, we come to the following equation of frequencies:

2 det[cij − p aij ]=0. (F.3)

When calculating the factors aij and cij of this determinant, one need not know the natural frequencies and natural modes of oscillation of the system’s elements. It is essential that these factors can be determined rather simply for the bars of variable section too. Let us start analyzing this approach with solving the problem of approximate determining the first natural frequency and mode of bending oscillations of the cantilever of variable cross-section. Bending oscillations of the cantilever of variable cross-section. Let us assume that at the end x = l the bar is rigidly clamped and that the area of cross-section and the moment of inertia of this section are defined correspondingly as follows: x S(x)=A(ξ)S(l) ,J(x)=B(ξ)J(l) ,ξ= , 0 ξ 1 . (F.4) l Here A(ξ) and B(ξ) are some prescribed functions. Note that they may be step functions too. Let us introduce into consideration the deflection of neutral layer of the cantilever y(x, t). As the bar is rigidly clamped at the end x = l,then ∂y y(l, t)=0, =0. (F.5) ∂x x=l We shall consider these two conditions as holonomic constraints imposed on the motion of a free bar. The constraint reaction forces are the bending moment M =Λ1 and the lateral force Q =Λ2 applied to the end x = l of the free bar (see Fig. F. 1). The motion of the free bar under the action of these forces can be represented as, first, translational motion (motion of the center of mass C), secondly, rotation about the center of mass and, third, bending. This bending deformation in quasistatics can be found in the following manner. Appendix F 265

Fig. F. 1

The acceleration of the center of mass Wc and the angular acceleration ϕ¨ at the time t are

Λ2(t) Λ1(t)+(l − xc)Λ2(t) Wc = , ϕ¨ = . (F.6) l l − 2 ρ 0 S(x)dx ρ 0 S(x)(xc x) dx

Here ρ is the density, and xc is the coordinate of the center of mass. The intensity of inertia forces caused by translational and rotation motion of the bar appears as

q(x, t)=−ρ(Wc +¨ϕ(x − xc))S(x) . (F.7)

The bending moment in section x, corresponding to the load q(x, t),isequal to x M(x, t)= q(x1,t)(x − x1)dx1 . (F.8) 0 The deﬂection caused by the action of the bending moment M(x, t) satisﬁes the equation ∂2y EJ(x) = M(x, t) . ∂x2 This equation in dimensionless variables

y x M(x, t)l y¯ = ,ξ= ,L(ξ,t)= (F.9) l l EJ(l) takes the form: ∂2y¯ B(ξ) = L(ξ,t) . (F.10) ∂ξ2 Formulas (F.4), (F.6)–(F.9) imply that the dimensionless moment L(ξ,t) is equal to L(ξ,t)=Λ¯ 1(t)f1(ξ)+Λ¯ 2(t)f2(ξ) . (F.11) Here Λ (t)l Λ (t)l2 Λ¯ (t)= 1 , Λ¯ (t)= 2 , 1 EJ(l) 2 EJ(l) ξ A(η)(c − η) f1(ξ)= (ξ − η) dη , 0 a 266 Appendix F ξ (c − η)A(η)(1 − c) A(η) f2(ξ)= − (ξ − η) dη , (F.12) 0 a b 1 1 1 1 a = A(ξ)(c − ξ)2 dξ; b = A(ξ) dξ , c = ξA(ξ) dξ . 0 0 b 0 Integrating (F.10) and taking into account the constraint equation (F.5) produce 2 1 f (η)(η − ξ) y¯(ξ,t)= Λ¯ (t)h (ξ) ,h= k dη . (F.13) k k k B(η) k=1 ξ The potential energy of the bar 1 l M 2(x, t) Π= dx 2 0 EJ(x) can be represented by using the formulas (F.4), (F.9), (F.11) as

EJ(l) 2 Π= c¯ Λ¯ Λ¯ , (F.14) 2l ij i j i,j=1 where 1 fi(ξ)fj(ξ) c¯ij = dξ . 0 B(ξ) The kinetic energy of the system ρ l ∂y 2 T = S(x) dx , 2 0 ∂t as follows from formulas (F.4), (F.9), (F.13), is 1 2 1 T = ρS(l) l3 a¯ Λ¯˙ Λ¯˙ , a¯ = A(ξ)h (ξ)h (ξ) dξ . (F.15) 2 ij i j ij i j i,j=1 0 Equation (F.3) and expressions (F.14), (F.15) imply that the dimensionless frequencies p∗ related to the required frequencies p as 1 EJ(l) p = p∗ (F.16) l2 ρS(l) are the roots of the equation

2 det [¯cij − p∗a¯ij]=0,i,j=1, 2 . (F.17)

For oscillations with the frequencies pk,k=1, 2, in accordance with expression (F.13) we obtain: ˜ ˜ y¯k(ξ,t)=(Λ¯ k1h1(ξ)+Λ¯ k2h2(ξ)) cos(pkt + α) ,k=1, 2 . Appendix F 267

˜ ˜ The quantities Λ¯ k1, Λ¯ k2,k=1, 2, satisfy the equations

− 2 ¯˜ − 2 ¯˜ (¯c21 p∗ka¯21)Λk1 +(¯c22 p∗ka¯22)Λk2 =0,k=1, 2 . This yields that the ﬁst two modes of oscillation of the cantilever can be approximately represented as

X (ξ) c¯ − p2 a¯ Y (ξ)= k ,X(ξ)=h (ξ) − 12 ∗k 12 h (ξ) ,k=1, 2 . k k 1 − 2 2 Xk(0.5) c¯22 p∗ka¯22 The exact solutions for the cantilever of wedge and cone shape were obtained by Kirchhoﬀ in 1879. These solutions are given in many books, in particular, in the reference book by E. Kamke [421] (Chapter IV, paragraphs 4.22, 4.24). For the wedge, where

A(ξ)=ξ, B(ξ)=ξ3 , natural frequencies p∗ are the roots of the equation √ J1(κ)I0(κ)=I1(κ)J0(κ) ,κ=2 p∗ .

Here J0(κ) and J1(κ) are Bessel’s functions of the first kind, and I0(κ) and I1(κ) are modified Bessel’s functions of the first kind. Natural modes corresponding to the natural frequencies p∗ are as follows: √ √ X(ξ) J (κ)I (κ ξ) − I (κ)J (κ ξ) Y (ξ)= ,X(ξ)= 0 1 √ 0 1 . X(0.5) ξ In the case of a cone, where

A(ξ)=ξ2 ,B(ξ)=ξ4 , the equation of frequencies, and functions X(ξ) take the form:

κ(J0(κ)I1(κ)+I0(κ)J1(κ)) = 4J1(κ)I1(κ) , √ √ √ √ √ √ I (κ)[J (κ ξ) − κ ξ J (κ ξ)] J (κ)[I (κ ξ) − κ ξ I (κ ξ)] X(ξ)= 1 1 √ 2 0 + 1 1 √ 2 0 . ξ ξ ξ ξ By using the suggested approximate approach we obtain — for the wedge: p∗1 =5.3187 ,p∗2 =17.3006 , 5ξ ξ3 1 ξ ξ2 ξ3 h (ξ)=1− +2ξ2 − ,h(ξ)= − + − , 1 2 2 2 6 2 2 6 — for the cone: p∗1 =8.73521 ,p∗2 =25.1813 , 7 5ξ2 2ξ3 1 ξ ξ2 ξ3 h (ξ)= − 3ξ + − ,h(ξ)= − + − . 1 6 2 3 2 6 2 2 6 268 Appendix F

The exact values of the first two frequencies are as follows: — for the wedge: p∗1 =5.3151 ,p∗2 =15.2072 , — for the cone: p∗1 =8.71926 ,p∗2 =21.1457 . The second frequency error for the wedge as well as for the cone is great enough. Therefore this approximate method can be used only for determination of the first frequency and the first mode for the cantilever of variable cross-section. ThefirstnaturalmodesforthewedgeandtheconeareshowninFig.F.2. Solid curves correspond to the approximate solution, and dashed curves over them correspond to the exact solution. For visualization of differences between the depicted curves, the deflection at ξ =1/2 is taken as a unit of measurement for each of them. The cone is a more flexible bar than the wedge and thus the first natural mode of the cone for ξ<1/2 is located higher than the corresponding curve for the wedge. For the cone the mass per unit of length decreases while approaching to the end by the quadratic law, and for the wedge the linear law is applied. For the bar of constant cross-section the mass per unit of length is constant. Pay attention to the following fact. The first frequency error for the cone is equal to 0.2%, for the wedge it is equal to 0.07%. For the bar of constant cross-section we have: — approximately: p∗1 =3.516035 ,p∗2 =22.7125 , —exactly: p∗1 =3.516015 ,p∗2 =22.0345 . Thus, the first frequency error makes up only 5.7 · 10−4% .Uponcompar- ison of given above errors for the cone, the wedge and the bar of constant cross-section we can expect that for the bar of constant cross-section with the mass localized at the end the approximate solution will become practically

Y1 6

0.2 0.4 0.6 0.8 1 ξ Fig. F. 2 Appendix F 269 the exact one. Really, in this case we obtain: m A(ξ)=1+γδ(ξ) ,B(ξ)=1,γ= 2 . m1

Here δ(ξ) is the Dirac delta-function, m2 is the load mass, m1 is the bar mass. For γ =1we have: — approximately: p∗1 =1.5572990 ,p∗2 =16.6203 , —exactly: p∗1 =1.5572976 ,p∗2 =16.2501 . We see that the first frequency error decreased six times relative to the case when γ =0. For the cantilever with the disk at its end we obtain quite accurate solution if we consider the presence of disk at the end as the third and the forth holonomic constraints. This system with four degrees of freedom makes it possible to determine to a rather high accuracy not only the first frequency but the second and the third ones. So let’s analyze the following problem. Determination of the lower natural frequencies of bending oscillations of the cantilever of variable cross-section with a disk at its end. In the rotor dynamics, the urgent problem is accurate determination of first two critical critical speeds of the cantilever shaft with a disk at its end. We remind that the values of these critical speeds are proportional to natural frequencies of the cantilever with a disk. Actually, as for instance in the case of marine screw (water propeller) or airscrew, there is not a disk at the shaft end but a body of rather complicated shape. There are methods allowing us to determine the moment of inertia of this body relative to the axis that is perpendicular to the shaft axis. Let us assume that this moment is set in the form 2 I = m2R , where m2 = γρlS(l) is mass of the body, and R = rl is its radius of inertia. Notice that with given functions A(ξ) and B(ξ) the required natural frequencies p∗ will depend on two parameters γ and r. In the case of the bar of constant cross-section the exact values of the frequencies p∗ are found from the equation V (x)+γxU(x) S(x)+γxV (x) √ det =0,x= p∗ . (F.18) S(x) − γr2x3T (x) T (x) − γr2x3U(x)

Here 1 1 S(x)= (ch x +cosx) ,T(x)= (sh x +sinx) , 2 2 1 1 U(x)= (ch x − cos x) ,V(x)= (sh x − sin x) 2 2 are the Krylov functions. In approximate determination of the frequencies p∗ we shall consider the conditions of rigid ﬁxing (F.5) as two holonomic constraints as before. We 270 Appendix F shall denote now their reaction forces: the bending moment M(t) and lateral force Q(t) by Λ1(t) and Λ2(t) correspondingly. The condition that the deﬂection y(0,t) is equal to the displacement of mass m2, and the angle of rotation of the bar’s end ∂y ϕ = ∂x x=0 is equal to the angle of the body rotation will be considered as two holonomic constraints imposed on motion of the free bar. The reaction forces of these constraints are the lateral force Λ3(t) and bending moment Λ4(t).Theyare applied to the bar at the cross-section x =0. Positive directions of reactions, applied to the bar are shown in Fig. F. 3. Formulas (F.6) in this case will take the form:

Λ2(t)+Λ3(t) Λ1(t) − Λ4(t)+(l − xc)Λ2(t) − xcΛ3(t) Wc = , ϕ¨ = . l l − 2 ρ 0 S(x)dx ρ 0 S(x)(xc x) dx The intensity of inertial forces q(x, t) will be calculated by formula (F.7) as before; formula (F.8) will take the form: x M(x, t)=Λ4(t)+xΛ3(t)+ q(x1,t)(x − x1) dx1 . 0 When going to dimensionless variables we obtain:

4 L(ξ,t)= Λ¯ k(t)fk(ξ) . k=1 Here Λ (t)l Λ (t)l2 Λ¯ (t)= 1 , Λ¯ (t)= 2 , 1 EJ(l) 2 EJ(l) Λ (t)l2 Λ (t)l Λ¯ (t)= 3 , Λ¯ (t)= 4 . 3 EJ(l) 4 EJ(l)

The functions f1(ξ) and f2(ξ) are set by formulas (F.12), and the functions f3(ξ) and f4(ξ) are as follows: ξ (η − c)cA(η) A(η) f3(ξ)=ξ + − (ξ − η) dη , 0 a b

Fig. F. 3 Appendix F 271 ξ A(η)(η − c) f4(ξ)=1+ (ξ − η) dη . 0 a Formulas (F.13), (F.14), (F.17) remain valid, but their indices i, j and k run nowfrom1to4. When calculating the kinetic energy it is necessary to take into account the kinetic energy of the disk, therefore the factors a¯ij of determinant (F.17) in this case are as follows: 1 2 a¯ij = A(ξ)hi(ξ)hj (ξ) dξ + γhi(0)hj(0) + γr ϕi(0)ϕj (0) ,i,j= 1, 4 . 0 (F.19) Here dh (ξ) ϕ (ξ)= i ,i= 1, 4 . i dξ In the case of the bar of constant cross-section the equation (F.18) allows us to calculate the natural frequencies exactly and so to estimate an error of this approximate method. The radius of inertia for the thin disk R is equal to R1/2,whereR1 is the radius of the disk and therefore R1 =2lr. If the shaft of radius r1 and the disk of thickness h are made of the same material then for r1 = l/20 and h = R1/20 we obtain

γ = 160 r3 . (F.20)

Assuming that γ and r are related to each other with this expression and r varies within the range from 0 to 1/2, let us follow the change of error for the ﬁrst, second and third frequencies. Upon calculations we obtain the following values for the error in percentage terms (%):

r =0.000 1.5 · 10−4 0.56 2.67

r =0.125 3.7 · 10−5 9/5 · 10−2 0.85 r =0.250 1.4 · 10−6 3.7 · 10−4 0.40 r =0.500 − 9.0 · 10−6 3.9 · 10−5 0.35 The first column corresponds to the first frequency, the second column corresponds to the second frequency and the third one corresponds to the third frequency. We see that the higher frequency, the greater error. For r 0.125 the error for the first frequency is close to the limits of accuracy which is provided by the software package "Mathematica 5.2". In this regard one can say that this method permits to determine the first frequency exactly. Therefore it may be used both for the rotor dynamics and for testing the programs for analysis of complicated mechanical systems. In rotor engineering it is important to have the analytical dependence of the first natural frequency on the system’s parameters. This method based on consideration of four holonomic constraints does not allow us to do that 272 Appendix F as it leads to the solution of algebraic equation of the fourth order. But if we limit ourselves to consideration of only two constraints at the end where the disk is located, then the required first frequency will be determined in analytical form as a root of biquadratic equation. Let us prove, that this simple solution also makes it possible to find the first frequency accurately enough. When getting this solution it is reasonable to measure the coordinate of the bar cross-section not from the free end but from the end that is rigidly clamped. Formulas (F.4) and (F.17) remain valid, but now S(l) and J(l) will correspond not to the rigidly clamped end, but to the place of disk fixation. The bending moment Λ1(t) and lateral force Λ2(t), applied to the end x = l, are constraint reactions and considered in this problem as the generalized coordinates. Their positive directions, as well as the positive direction of the moment M(x, t) applied to the cross-section x, are shown in Fig. F. 4. The dimensionless bending moment L(ξ,t) introduced by formula (F.9) is equal in this case to

L(ξ,t)=Λ¯ 1(t)f1(ξ)+Λ¯ 2(t)f2(ξ) , Λ (t)l Λ (t)l2 (F.21) f (ξ)=1,f(ξ)=1− ξ, Λ¯ (t)= 1 , Λ¯ (t)= 2 . 1 2 1 EJ(l) 2 EJ(l)

Expression (F.11), as seen, survives and therefore the potential energy will be written in the form (F.14). Integrating equation (F.10) and taking into account that ∂y¯ y¯(0,t)= =0, ∂ξ ξ=0 imply expression (F.13), where now ξ fk(η)(η − ξ) hk(ξ)= dη , k =1, 2 . (F.22) 0 B(η) As the deﬂection is represented in the same form (F.13), the kinetic energy will be written in the same form (F.15) too. The factors a¯ij in this case should

Fig. F. 4 Appendix F 273

be calculated by formulas (F.19), but now hi(0) should be replaced with hi(1), and ϕi(0) should be replaced with ϕi(1). When calculating for the bar of constant cross-section the error of the ﬁrst and second frequencies in percentage terms (%) for the same relation (F.20) between γ and r, we obtain:

r =0.000 0.47 58 r =0.125 8.4 · 10−2 15.6 r =0.250 2.6 · 10−3 0.21 r =0.500 1.1 · 10−5 1.1 · 10−3

For r 0.25 we can say that for the first frequency we obtain the exact value. Notice, however that for r =0.25 the disk diameter is equal to the shaft length, and for r =0.5 it is two times greater. For such relation between these quantities for the assumed values r1 = l/20 and h = R1/20 this disk can not be regarded as a perfectly rigid body. It is necessary to take into account the influence of its compliance on the natural frequencies of the system. It is feasible but it will require additional calculations, the basic framework of which will be shown through the example of the cantilever with a flexible bar at its end. This example will require no new mathematical apparatus. It is reduced to the same calculations as above. Determination of the first three frequencies of the cantilever with a flexible bar at its end. Let us analyze the problem, when the bar executing longitudinal oscillations in the mechanical system depicted in Fig. VI. 2 is absent (see Fig. F. 5). Within the frames of such problem we have three constraints and three reaction forces correspondingly. The bending moment Λ1(t) and the lateral force Λ2(t) are applied to the cantilever as is shown in Fig. F. 4. The third reaction force is the lateral force Λ3(t) applied to the bar which is perpendicular to the cantilever. Both kinetic and potential energy of the cantilever are determined by the formulas given above. Therefore it is necessary to take into account only the second bar. When released from the constraints it becomes free and similar to the bar shown in Fig. F. 1, but now the bending moment M(t)=Λ1(t) and lateral force Q(t)=Λ3(t) are applied not to the end of the bar but to the cross-section x∗ = zl. Therefore, the constraint equations will be written in the form ∂y y(x∗,t)=0, =0. (F.23) ∂x x=x∗ 274 Appendix F

We shall not provide the parameters of the second bar with indices when considering the question how the deflection curve will change depending on the place of application of the reactions. We shall do that upon obtaining expressions for the potential energy of its deflection and for the deflection curve. Formulas (F.6) in this case will appear as

Λ3(t) Λ1(t)+(x∗ − xc)Λ3(t) Wc = , ϕ¨ = , l l − 2 ρ 0 S(x) dx ρ 0 S(x)(xc x) dx and formula (F.7) remains valid. The bending moment M(x, t) applied to the left of the cross-section x = x∗ is set by expression (F.8), and the bending moment applied to the right of cross-section takes the form l M(x, t)= q(x1,t)(x1 − x) dx1 ,x∗

¯ (2) ¯ (2) Ln(ξ,t)=Λ1 (t)f1n(ξ)+Λ3 (t)f3n(ξ) ,n=1, 2 .

Fig. F. 5 Appendix F 275

Here Λ (t)l Λ¯ (t)l2 Λ¯ (2)(t)= 1 , Λ¯ (2)(t)= 3 , 1 EJ(l) 3 EJ(l) ξ (c − η)A(η) f (ξ)= (ξ − η) dη , 0 ξ z, 11 a 0 ξ (c − η)(z − c)A(η) A(η) f (ξ)= − (ξ − η) dη , 0 ξ z, 31 a b 0 1 A(η)(η − c) f (ξ)= (ξ − η) dη , z ξ 1 , 12 a ξ 1 (η − c)(z − c)A(η) A(η) f32(ξ)= + (ξ − η) dη , z ξ 1 . ξ a b (F.24) We remind that the values a, b, c, included in these expressions are calculated ¯ (2) ¯ (2) by formulas (F.12). The index "2"of the quantities Λ1 (t) and Λ3 (t) means that transition to the dimensionless variables corresponds to the parameters l, E and J(l) of the second bar (see Fig. 5). The functions A(ξ), B(ξ) and the values a, b, c, should be also provided with index "2"hereinafter, but for the sake of simplicity they are omitted. Integrating equation (F.10) for L(ξ,t)=L1(ξ,t),andthenforL(ξ,t)= L2(ξ,t), and taking into account the constraint equations (F.23) imply

(2) (2) y¯(ξ,t)=Λ¯ (t)h11(ξ)+Λ¯ (t)h31(ξ) , 0 ξ z, 1 3 (F.25) ¯ (2) ¯ (2) y¯(ξ,t)=Λ1 (t)h12(ξ)+Λ3 (t)h32(ξ) ,zξ 1 , where z f (η)(η − ξ) f (ξ)= k1 dη , 0 ξ z, k1 B(η) ξ ξ fk2(η)(ξ − η) fk2(ξ)= dη , z ξ 1 ,k=1, 3 . z B(η) By using the unit function 1 ,x 0 , U(x)= 0 ,x<0 , we represent expressions (F.25) as

¯ (2) ¯ (2) y¯(ξ,t)=Λ1 (t)h1(ξ)+Λ3 (t)h3(ξ) , 0 ξ 1 . (F.26) Here hk(ξ)=hk1(ξ)U(z − ξ)+hk2(ξ)U(ξ − z) . (F.27) 276 Appendix F

The potential energy of deformation of the second bar has to be calculated independently for its right and left sections. Calculating and summing these energies produce EJ(l) Π= (¯c(2)(Λ¯ (2)(t))2 +2¯c(2)Λ¯ (2)(t)Λ¯ (2)(t)+¯c(2)(Λ¯ (2)(t))2) , (F.28) 2l 11 1 13 1 3 33 3 where z f 2 (ξ) 1 f 2 (ξ) c¯(2) = k1 dξ + k2 dξ , k =1, 3 , kk B(ξ) B(ξ) 0 z z 1 (2) f11(ξ)f31(ξ) f12(ξ)f32(ξ) c¯13 = dξ + dξ . 0 B(ξ) z B(ξ) Adding the potential energy of bending of the ﬁrst bar to potential energy (F.28), we represent their sum in the form

E J (l ) 3 Π= 1 1 1 c¯ Λ¯ (1)Λ¯ (1) . (F.29) 2l ij i j 1 i,j=1 Here index "1" means that this quantity corresponds to the ﬁrst bar. The ¯ (1) dimensionless variables Λi ,i= 1, 3, are introduced by the formulas:

2 ¯ (1) Λ1(t)l1 ¯ (1) Λk(t)l1 Λ1 (t)= , Λk (t)= ,k=2, 3 . E1J1(l1) E1J1(l1)

Note that in these formulas J1(l1) corresponds not to the place of rigid ﬁxing, as it was in the beginning of this Appendix, but to the point where the ﬁrst bar is connected to the second one (see Fig. F 5). In formulas (F.24) and (F.28) all quantities refer to the second bar. In- troduction of the parameters

3 E1J1(l1)l2 l2 α = 3 ,β= E2J2(l2)l1 l1 allows us to represent the potential energy (F.28) of the second bar as

E1J1(l) (2) ¯ (1) 2 −2 (2) ¯ (1) ¯ (1) −1 (2) ¯ (1) 2 Π2 = α (¯c11 (Λ1 (t)) β +2¯c13 Λ1 (t)Λ3 (t)β +¯c33 (Λ3 (t)) ) . 2l1

This implies that the factors c¯ij in expression (F.29) are as follows:

(1) −2 (2) (1) c¯11 =¯c11 + αβ c¯11 , c¯12 =¯c12 , −1 (2) (1) (2) c¯13 = αβ c¯13 , c¯22 =¯c22 , c¯23 =0, c¯33 = αc¯33 . Here in accordance with formulas (F.14), (F.21) 1 1 1 2 (1) dξ (1) (1 − ξ) dξ (1) (1 − ξ) dξ c¯11 = , c¯12 = , c¯22 = . 0 B1(ξ) 0 B1(ξ) 0 B1(ξ) Appendix F 277

The kinetic energy of the ﬁrst bar will be represented by using expressions (F.15), (F.21), (F.22) in the form

1 2 T = ρ S (l ) l3 a¯(1)Λ¯˙ (1)Λ¯˙ (1) , 1 2 1 1 1 1 ij i j i,j=1 1 (1) (1) (1) a¯ij = A1(ξ)hi (ξ)hj (ξ) dξ , 0 ξ ξ (1) (ξ − η) dη (1) (1 − η)(ξ − η) dη h1 (ξ)= ,h2 (ξ)= . 0 B1(η) 0 B1(η) Let us calculate the kinetic energy of the second bar now. The assump- tion that amplitude of oscillations of the bars under consideration is small allows us, as was noted in § 2 of Chapter VI, to calculate the kinetic energy of translational motion of the second bar along the axis independently from the kinetic energy of its motion in the direction that is perpendicular to its axis. The kinetic energy of translational motion of the second bar is m l2 T = 2 1 (h(1)(1)Λ¯˙ (1) + h(1)(1)Λ¯˙ (1))2 , 21 2 1 1 2 2 1 m2 = ρ2S2(l2) l2 A2(ξ) dξ . 0 Displacements of the cross-sections of the second bar in the direction perpendicular to the bar axis are caused, first, by rotation of the bar about the cross-section x∗ = zl2, and, secondly, by the deflection defined by expression (F.26). Therefore we have:

− ¯ (2) (2) ¯ (2) (2) y2(ξ,t)=l2(ψ(t)(z ξ)+Λ1 (t)h1 (ξ)+Λ3 (t)h3 (ξ)) . Here (1) ¯ (1) ¯ (1) dhk ψ(t)=ϕ1(1)Λ1 (t)+ϕ2(1)Λ2 (t) ,ϕk(1) = ,k=1, 2 . dξ ξ=1

(2) (2) Index "2" of the functions h1 (ξ) and h3 (ξ) means that these functions deﬁned by expressions (F.27) are calculated for the parameters of the second bar. Taking into account that

¯ (2) −2 ¯ (1) ¯ (2) −1 ¯ (1) Λ1 = αβ Λ1 , Λ3 = αβ Λ3 , the kinetic energy l2 2 1 ∂y2 T22 = ρ2 S2(x) dx , 2 0 ∂t 278 Appendix F will be represented as 1 1 3 ¯˙ (1) ¯˙ (1) − T22 = ρ2S2(l2)l2 A2(ξ)((ϕ1(1)Λ1 + ϕ2(1)Λ2 )(z ξ)+ 2 0 −2 ¯˙ (1) (2) −1 ¯˙ (1) (2) 2 +αβ Λ1 h1 (ξ)+αβ Λ3 h3 (ξ)) dξ . Introducing into consideration the third parameter ρ S (l )l γ = 2 2 2 2 , ρ1S1(l1)l1 the total kinetic energy of the second bar appears as follows:

γ 3 T = ρ S (l ) l3 a¯(2)Λ¯˙ (1)Λ¯˙ (1) . 2 2 1 1 1 1 ij i j i,j=1

(2) Analytic expressions for the factors aij , dependent on the functions A2(ξ) and parameters α and β, are rather intricate and thus not given here. Note that they are easily found with the software package "Mathematica 5.2". The kinetic energy of both bars is

1 3 T = ρ S (l ) l3 a¯ Λ¯˙ (1)Λ¯˙ (1) , a¯ =¯a(1) + γa¯(2) . 2 1 1 1 1 ij i j ij ij ij i,j=1

We’ll find the required natural frequencies p∗ by solving equation (F.17). Notice that in formula (F.16) of transition to dimensional frequencies all quantities correspond to the first bar at the point of its connection to the second bar. Comparison with the bars of constant cross-section. The problem of bars of constant cross-section has been solved exactly by the methods of mathematical physics. As this takes place the equation of frequencies is obtained by equating the determinant of sixth order to zero. Its elements are the Krylov functions, the arguments of which depend on the parameters α, γ and z. This intricate transcendental equation, the computational solution of which was a matter of some difficulty even for modern computers, was used for testing the method suggested in § 3 of Chapter VI. Note that calculation of first three frequencies by using this suggested method does not create any difficulties. As stated above, the problem under investigation is a particular case of the problem discussed in §§ 5 and 6 of Chapter VI. When the bar executing longitudinal vibration is absent, the frequency determinant is a determinant of third order. Comparing the roots of the transcendental equation with the roots of the frequency equation shows that the first frequency is determined with four valid significant digits in the second approximation, the second frequency is determined with the same accuracy in the fourth approximation, Appendix F 279 and the third one is obtained with the same accuracy in the sixth approximation. If the first and the second bars are made of the same material and have the same cross-sections, then at z =1/2 the solution depends on a single parame- 3 ter β = l2/l1, for in this case γ = β and α = β . The calculations show that for the first frequency the error decreases as β rises, and for β =1/8, 1/4, 1/2, 2 it is equal to 0.22, 0.12, 0.056, 0.0022 percent (%) correspondingly. Note that in this example in case β 0.25, it is reasonable to consider the second bar as a concentrated mass located at the end of the cantilever beam and to use the method presented in the beginning of this Appendix. Let us discuss briefly the errors of the method under consideration for the second and third frequencies. Let us examine this problem through the example of the bars, differing only in length. For α = β = γ =1and z =1/2 the exact and approximate values of first three dimensionless frequencies p∗ are as follows: 1.44851 , 6.20782 , 14.0641 , 1.44876 , 6.24235 , 14.1204 . The errors in percentage terms (%) are equal correspondingly to 0.017 , 0.56 , 0.40 . If the second bar is symmetrically positioned in relation to the first one, there exists a mode of oscillations such that the first bar does not oscillate, and both halves of the second bar oscillate like a cantilever of length l = βl1/2. The first frequency of the cantilever oscillation in the dimensionless variables is 4 p∗ =3.516 . (F.30) β2 This frequency in the series of frequencies of the system consisting of two bars has the number n. This number increases as β decreases. For example, for β =1/4 it will be the ninth frequency, and the third root of equation (F.17) will correspond to it. Let us find this root in the explicit form. When the second bar does not oscillate, then the bending moment Λ1 and the lateral force Λ2 applied to the end of the first bar vanish. Therefore in this oscillation mode only the lateral force Λ3 applied to the middle of the second bar is not equal to zero. Under the action of this force the second bar moves translationally and bends so that the application point of the force Λ3 is immovable. In quasistatics the intensity of inertial forces is constant in this case, therefore either the second or the third root of equation (F.17) at z =1/2 is equal to 1 ξ c 4 2 p∗ = ,c= f (ξ) dξ , f(ξ)= (ξ − η) dη , a β2 0 0 1 1 a = h2(ξ) dξ , h(ξ)= f(η)(ξ − η) dη . 0 ξ 280 Appendix F

When calculating we obtain 4 p∗ =3.530 . (F.31) β2

This frequency exceeds its exact value obtained by formula (F.30) by 0.40 %. For β =1/2 the frequency approximately defined by expression (F.31) corresponds to the exact value of the fourth frequency, for β =1and β =2 it corresponds to the third frequency, and for β =4it corresponds to the second frequency. Hence this approximate method makes it possible to determine the first frequency for any values of the system parameters with a rather high degree of accuracy, and for some values of the parameters it allows us to define the second and the third frequencies as well. APPENDIX G

THE DUFFING EQUATION AND STRANGE ATTRACTOR

The nonhomogeneous Duﬃng equation with a linear resistance is studied. In this case the possibility of arising of strange attractors and periodical solutions with a period multiple to the period of excitation, depending on the excitation level, is investigated by a numerical method. The Appendix presents the ﬁrst part of the paper by P. E. Tovstik and T. M. Tovstik [425]. The prob- ability properties of a strange attractor considered in the second part of this paper are not covered in the Appendix. A more completed table of solution properties is given. The relationship of strange attractors with the classical theory of motion stability is presented in the monograph by G. A. Leonov [426].

In § 4 of Chapter VI the nonhomogeneous Duﬃng equation (4.9) has been obtained for describing the lateral vibration of a beam with the supports ﬁxed in the logitudinal directiion. Taking into consideration a linear resistance and some changes in notation, in dimensionless variables it appears as

d2x dx + c + x + x3 = b cos ωt , (G.1) dt2 dt where π 2 EJ 2L4 ω = Ω,b= √ f0 . L ρS π4E JS

Here Ω and f0 are the frequency and the amlitude of the disturbed force, respectively, c is a coefficient characterizing damping, E and ρ are the coefficient of elasticity and density of beam material, respectively, L is the length of beam, S is the cross-section area of the beam, J is the moment of inertia of cross section of the beam with respect to zero line. Equation (G.1) includes three parameters — c, b,andω. Let us fix two of them (ω =1,c=0.25) and vary parameter b in a wide range 0 b 100. Equation (G.1) has been integrated numerically under the arbitrary given initial conditions x(0), x˙(0) belonging to the domain of ([−5.0, 5.0] × [−5.0, 5.0]). It has been determined, how the existance of strange attractors and limiting solutions with a period multiple to the period of excitation depends on the value of b and the initial conditions. To this end, the range 0 b 100 has been partitioned with a step 0.1, and for each bi =0.1i the qualitative character of a limiting trajectory has been defined. The neighbour values of bi with the same qualitative characteristics have been united in intervals. When subsequently varying i from 0 up to 1000, 55 intervals with different qualitative behaviour of solutions have been found in total.

281 282 Appendix G

Таблица G. 1.

bnk bnk

0.0 − 2.911 52.13A, 2, 2 3.0 − 9.621, 152.2 − 52.722, 2 9.7 − 11.911 52.8 − 53.124, 4 12.0 − 14.821, 153.22A, A 14.9 − 22.911 53.3 − 54.31A 23.0 − 35.521, 154.3 − 54.72A, 1 35.6 − 38.622, 254.8 − 54.93A, 5, 1 38.7 − 38.924, 455.08A, 15, 15, 15, 10, 10, 5, 1 39.044, 4, 3, 355.1 − 57.92A, 1 39.1 − 39.224, 458.011 39.3 − 39.39 2 8, 858.1 − 58.22A, 1 39.44A, A, 8, 858.33A, A, 1 39.52A, A 58.434, 4, 1 39.64A, A, 10, 10 58.5 − 58.732, 2, 1 39.7 − 40.62A, A 58.843, 3, 2, 1 40.7 − 41.21A 58.9 − 59.032, 2, 1 41.33A, 5, 559.1 − 60.831, 1, 1 41.4 − 41.61A 60.9 − 62.021, 1 41.7 − 44.32A, 362.13A, 1, 1 44.4 − 48.613 62.2 − 62.343, 3, 1, 1 48.7 − 49.123, 362.4 − 62.733, 1, 1 49.226, 662.8 − 67.421, 1 49.33A, 6, 667.5 − 77.311 49.42A, A 77.4 − 91.521, 1 49.5 − 50.61A 91.641, 1, 1, 1 50.724, 491.7 − 92.731, 1, 1 50.828, 892.8 − 100.021, 1 50.9 − 52.01A Table G. 1 contains the results. In it, for corresponding values of b there are: —thenumbern of different limiting solutions, which can be obtained when changing the initial conditions, — the multiplicity k of a period kT of the limiting solution, the number of values of k given through a comma being equal to n, — in the case, when a periodical solution is not kT-periodical, but a strange attractor, the number k is substituted in Table G. 1 for the letter A. Forexample,forb =54.9 there are three (n =3) different stable limiting solutions: a strange attractor, a 5T -periodical solution, and a T -periodical solution. If we compare this table with the results obtained in the paper [425], then we see that this table only complements them. A greater number Appendix G 283 of intervals b has been got as a result of decreasing a partition step. This demonstrates that a more detailed study of the variation interval of b can lead to arising of new intervals that are qualitatively different from the considered ones. It follows from the table, that strange attractors occur in the range 39.4 b 62.1. Note that for b>100 strange attractors also occur, but in this case the vibration amplitude is so great that this vibration can hardly be modelled by the Duffing equation. We illustrate the dependence of limiting solutions on the initial conditions through the example b =4.0. As follows from Table G. 1, for this value of b there are two stable T -periodical solutions. In Fig. G. 1 on the part of the plane −5 x(0), x˙(0) 5 the domains of initial conditions that lead to the first or second solutions are given. These solutions are presented in Fig. G. 2. The part of the plane −50 x(0), x˙(0) 50, which is a hundred times greater, has been also regarded, in this case it turned out that the structure of the domain is more complicated than in Fig. G. 1. It is impossible to describe here all the considered variation range of b. We limit ourselves to four consecutive intervals in the range 40.0 b 45.0 and regard four successive values of b. The results are given in the form of the Poincaré diagrams (see Fig. G. 3), on which points with the coordinates x(mT ), x˙(mT ) for integer m are dotted. The Poincaré diagrams (or cross- sections) are the powerful means, which make it possible to determine the qualitative character of solution behaviour and find out bifurcations, i. e. the transfers from one qualitative state to another.

Fig. G. 1 284 Appendix G

x 2 2

1 1

t π 2π

–1

–2

Fig. G. 2

Fig. G. 3 Appendix G 285

If a limiting solution is kT-periodical, then a diagram has k different dots. The number of dots for a strange attractor depends on duration of integration. In Fig. G. 3 each of the strange attractors contains 800 dots. Consider consecutively the diagrams shown in Fig. G. 3. On the first of them (for b =40.0) two strange attractors 1 and 2 are depicted, which can be obtained under certain initial conditions. As b increases, attractors approach to each other, and for b =41.0 there is only one attractor under any initial conditions. For b =43.0 we have two stable limiting solutions: a strange attractor 1 and a 3T -periodical solution 2 depicted by three dots on the Poincaré diagram. As b grows further, a strange attractor vanishes, and there remains only one 3T -periodical solution depicted by three dots in Fig. G. 3 for b =45.0. References

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Abstract constraints, 151, 152 Contravariant components, 215, 216 Acceleration vector for arbitrary of tangent vector, 105 mechanical system, 107 of velocity vector, 218 Amplitude-frequency Coordinate characteristics, 164 line, 213 Andronov–Hopf bifurcation, 229 plane, 215 Appell function, 200 surface, 213 Appell’s equations, 101, 123 Covariant components, 216 Appell’s form equations with of velocity vector, 218 third-order constraints, 123 Curves of static bend (deflection), 158 Approximate periodic Curvilinear coordinates, 213 solutions, 236 Cylindrical system of coordinates, 222 Approximate solution of Lagrange’s equations, 237 D’Alembert–Lagrange principle, 13, 207 Axes of curvilinear coordinates, 214 Dynamic compliances, 156 Dynamic control of the motion of a car, Basic metric form, 217 245 Basic metric tensor, 217 Dynamic Euler equations, 200 Basis of the Lie algebra, 200 Basis of s-dimensional Lie algebra Effective potential, 230 with the commutator, 199 Elastic constraints, 165, 167, 168 Bending oscillations of the cantilever Equations of motion, 214 of variable cross-section, 264 of nonholonomic systems in quasico- Bubnov–Galerkin method, 237 ordinates, 200 in quasicoordinates, 211 Chaplygin’s equations, 30, 39 Equations of noncomplete program in quasicoordinates, 43 of motion, 96 Chaplygin’s type equations, 32, Equations of nonholonomic systems in 42, 197 the Poincaré–Chetaev Chetaev’s postulate, 76, 92–93 variables, 200 Chetaev’s type constraints, 76 Equations, represented in Maggi’s Christoffel symbols form, for third-order of first kind, 219 constraints, 121 of second kind, 219 Euclidean structure of the tangent Coefficients of influence, 156 space, 106 Complementary metric form, 217 Complementary metric tensor, 217 Formula for computing the Christoffel Condition of a free motion of the coefficients of the first Chaplygin sledge, 242 kind, 219 Condition of the ideality of Free (unconstrained) motion of constraints, 110 nonholonomic system, 239 Conditions of N. G. Chetaev, 202, 203, 207 Gaussian function, 116 Configuration space of the system, Gaussian principle, 103, 109 230 generalized, 119, 183 Constraints completely defined by their General (fundamental) equation of analytic representations, 9 dynamics, 110–111

327 328 Index

Generalized control force, 127 Maggi’s equations, 30, 201 Generalized D’Alembert–Lagrange second group, 30, 101 principle, 202 Mangeron–Deleanu principle, 135 Generalized forces, 6 Manifold of positions of the mechanical corresponding to system, 105 quasivelocities, 94 Maximum principle of Generalized impulses, 7 Pontryagin, 185 Generalized operator Metric tensor, 106 Appell, 114 Mixed problem of dynamics, 128 Lagrange, 114 Motion of dynamically symmetric ball Generalized problem of on absolutely roughened plane, P.L. Chebyshev, 126 233 Generalized reactions, 82, 150 Gradient of the function, 215 Natural (fundamental) basis, 215 Necessary and sufficient conditions for Hamel–Boltzmann equations, 31, 44 the existence of free motion of Hamel–Novoselov equations, 32, nonholonomic 43, 197 systems, 240 Hamiltonian nabla operator, 216 New class of control problems, 128 Harmonic coefficients of Newton’s determinacy principle, 136 influence, 156 Noncomplete program of motion, 100 High-order program Nonholonomic bases, 28 constraints, 135 Nonlinear second-order nonholonomic constraints, 116 Ideal constraints Normal (natural) forms (modes) of holonomic, 3, 4, 81 oscillations, 153 nonholonomic, 29, 81 Normal (natural) frequency, 153 Ideal control, 134 Ideality condition of Objects of nonholonomicity, 31 control, 135 Introducing generalized reaction Parametrization of constraints, 195 forces as Lagrangean Permutable relations, 206, 207 coordinates, 263 Poincaré–Chetaev equations, 34 Poincaré–Chetaev parameters, 202 Kinematical characteristics, 194 Poincaré–Chetaev–Rumyantsev Kinematic control of the motion equations, 44, 201, 206 of a car, 245 Poincaré diagrams, 283 Kronecker symbols, 215 Poincaré equations, 44, 200 Poincaré parameters, 194 Lagrange multipliers, 82, 150 Possible types of the car motion, 251 Lagrange operator, 6, 222 Principle of virtual accelerations, 110 Lagrange’s equations Principle of virtual displacements, 111 of first kind, 4, 7 Principle of virtual velocities, 110 of first kind in generalized Program constraints, 116, 119, 135 coordinates, 11 for nonholonomic systems, 33 Quasicoordinates, 41 of second kind with multipliers, 11 Quasivelocities, 41 undetermined, 33 Lamé factors, 214 Reciprocal (dual) basis, 107, 215 Linear transformation of forces, 94 Representation point, 1, 2 Index 329

Residual, 235 Tangent space, 106 Rule of dummy index, 216 Rules of raising and missing an Udwadia–Kalaba equations, 34 index, 217 Variation of the generalized Series in resonance velocity, 67 frequencies, 186 Variations of coordinates, 12, 106 Set, bifurcational by Smale, Vector of generalized impulse, 113 231 Velocity vector of mechanical Steady motions, 231 system, 113 of conservative nonholonomic Virtual displacements, 12, systems, 229 106, 112 Steady rolling of disk on horizontal Virtual elementary work, 7, plane, 233 84, 106 Strange attractors, 281 Virtual velocity, 112 Structural constants of Lie Voronets equations, 40 algebra, 199 Voronets–Hamel coeﬃcients of ﬁrst Subspace kind, 198 of motions, 8 Voronets–Hamel equations, 44 of reactions, 8 Voronets–Hamel type Suslov–Jourdain principle, 68 equations, 198