Principle of Virtual Work Degrees of Freedom Associated with the concept of the lumped-mass approximation is the idea of the NUMBER OF DEGREES OF FREEDOM. This can be defined as “the number of independent co-ordinates required to specify the configuration of the system”. The word “independent” here implies that there is no fixed relationship between the co- ordinates, arising from geometric constraints. Modelling of Automotive Systems 1 Degrees of Freedom of Special Systems A particle in free motion in space has 3 degrees of freedom z particle in free motion in space r has 3 degrees of freedom y x 3 If we introduce one constraint – e.g. r is fixed then the number of degrees of freedom reduces to 2. note generally: no. of degrees of freedom = no. of co-ordinates –no. of equations of constraint Modelling of Automotive Systems 2 Rigid Body This has 6 degrees of freedom y 3 translation P2 P1 3 rotation P3 . x 3 e.g. for partials P1, P2 and P3 we have 3 x 3 = 9 co-ordinates but the distances between these particles are fixed – for a rigid body – thus there are 3 equations of constraint. The no. of degrees of freedom = no. of co-ordinates (9) - no. of equations of constraint (3) = 6. Modelling of Automotive Systems 3 Formulation of the Equations of Motion Two basic approaches: 1. application of Newton’s laws of motion to free-body diagrams Disadvantages of Newton’s law approach are that we need to deal with vector quantities – force and displacement. thus we need to resolve in two or three dimensions – choice of method of resolution needs to be made. Also need to introduce all internal forces on free-body diagrams – these usually disappear when the final equation of motion is found. 2. use of work with work based approach we deal with scalar quantities – e.g. work – we can develop a routine method – no need to take arbitrary decisions. Modelling of Automotive Systems 4 Free body diagram T θ mg Modelling of Automotive Systems 5 Principle of Virtual Work The work done by all the forces acting on a system, during a small virtual displacement is ZERO. Definition A virtual displacement is a small displacement of the system which is compatible with the geometric constraints. P P1 2 b a b P2 bδθ P1 aδθ a e.g. This is a one-degree of freedom system, only possible movement is a rotation. work done by P1 = P1(- aδθ) work done by P2 = P2(bδθ) Modelling of Automotive Systems 6 Total work done = P1(- aδθ) + P2(bδθ) = δW By principle of Virtual Work δW= 0 therefore: P1 (- aδθ) + P2(bδθ) = 0 - a P1 + b P2 = 0 P1a = P2b Modelling of Automotive Systems 7 D’Alembert’s Principle Consider a rigid mass, M, with force FA applied A = &x& FA (acceleration) M From Newton’s 2nd law of motion A F = Ma = M&x& or A F − M&x& = 0 Modelling of Automotive Systems 8 I Now, the term ( − M&x& ) can be regarded as a force – we call it an inertial force, and denote it F –thus we can then write: I F = −M&x& FA + FI = 0 In words – the sum of all forces acting on a body (including the inertial force) is zero – this is a statics principle. In fact all statics principles apply if we include inertial forces, including the Principle of Virtual Work. Modelling of Automotive Systems 9 Virtual Work and Displacements Using the concept of virtual displacements, and virtual work, we can derive the equations of motion of lumped parameter systems. Example 1 Mass/Spring System x k m Here number of degrees of freedom =1 Co-ordinate to describe the motion is x Now consider free-body diagram, at some time t Modelling of Automotive Systems 10 Total _ force : −m&x&− kx δW = ()− m&x&− kx δx = 0 Hence − m&x&− kx = 0 or m&x&+ kx = 0 R (reactive force) m inertial force − m&x& m restoring force kx mg (gravity force) General one degree of freedom system If q1 is the co-ordinate used to describe the movement then the general form of δW is as follows: δW = Q1δ q1 we call δq1 – generalised displacement Q1 – generalised force. From principle of virtual work δW = Q1δ q1 = 0 ∴Q1 = 0 Modelling of Automotive Systems 11 Example Referring to the mass/spring system again x= q1 k m x δW = (− mq&&1 − kq1 )δq1 = 0 ∴ Q1 = −mq&&1 − kq1 = 0 mq&&1 + kq1 = 0 Modelling of Automotive Systems 12 Example 2 Simple pendulum − mlθ&2 (inertial force –radial) 0 θ − mlθ&& (inertial force – tangential) P mg This is another one degree of freedom system. During a virtual displacement, δθ, the virtual work done is δW = (−ml&θ& − mgsin θ) lδθ = 0 δW = 0 (PVW) ∴ml&θ& + mg sin θ = 0 or g &θ& + sin θ = 0 l Modelling of Automotive Systems 13 Example 3 – two degrees of freedom system x1 x2 k k m m free-body diagrams. kx1 -mx 1 -mx2 m m k(x2-x1) Modelling of Automotive Systems 14 For LH mass: []− kx1 − m&x&1 + k()x2 − x1 δx1 = δW1 []− k()x2 − x1 − m&x&2 δx2 = δW2 δW = δW1 + δW2 = ()Q1 δx1 + ()Q2 δx2 For δW = 0 for all δx1,δx2 the Qi quantities must be zero. Hence: m&x&1 + kx1 − k(x2 − x1 ) = 0 m&x&2 + k(x2 − x1 ) = 0 or m 0 &x&1 k − kx1 0 + = 0 m&x&2 − k k x2 0 Modelling of Automotive Systems 15 n degree of freedom systems Having discussed single and two degree of freedom systems, and introduced the concept of generalised forces we can now consider the general case of an n degree of freedom system. A virtual displacement must be consistent with the constraints on the system. The motion can be described by n independent, generalised co-ordinates, q1,q2 ,....,qn . Hence a virtual displacement can be represented by small changes in these co-ordinates:- δ q1,δ q2 ,...,δ qn Suppose only one co-ordinate, qi (1 ≤ i ≤ n) is given a small, imaginary displacement, δ qi. As a result every particle in the system will be, in general, displaced a certain amount. The virtual work done will be of the form δW = Qiδ qi where Qi is an expression relating directly to the forces acting on the system. Qi is the generalised force associated with qi . Modelling of Automotive Systems 16 From the principle of virtual work δW = Qiδ qi = 0 Since, δ qi is finite, we get Qi = 0 This must be true for i = 1,2,...,n. Q1 = 0 Q = 0 I.e. 2 these are the equations M of motion of the system Qn = 0 Modelling of Automotive Systems 17 The generalised forces have component parts 1) inertial forces (mass x acceleration) 2) elastic or restraining forces 3) damping forces (energy dissipation) 4) external forces 5) constraint forces I E D A δW=Qiδqi =δW +δW +δW +δW inertial elastic damping external I E D A = ()Qi + Qi + Qi + Qi δ qi (Noting, as before, that the constraint forces do no virtual work) Then the equations of motion are: I E D A Qi + Qi + Qi + Qi = 0 i =1,2,...,n These are the n equations of motion. We will examine each of these components now, in more detail. The aim is to relate these component forces to the generalised co-ordinates q1,q2 ,....,qn. Modelling of Automotive Systems 18 Inertial Forces (See also Handout) The position of the ith particle of mass, in the system, is, in general, related to the n generalised co-ordinates, and time (if the constraints are independent of time) then the position of the ith particle depends only on the n generalised co- ordinates. Thus r r ri =ri ()q1,q2,...,qn,t (1) Now we suppose that the system is in motion and that we represent the inertial force on the ith particle (using D’Alembert’s Principle) as r − mi&r&i (2) We now give the system an arbitrary virtual displacement – this can be represented in terms of generalised co-ordinates by δq1,δq2 ,...,δqn . The virtual displacement of the Ithparticle can be represented by r δr (3) and the virtual work done by the inertia force on the Ith particle is simply r r (−mi &r&i ). δri (4) (note that this is a scalar product). From this result we get the total virtual work as I r r δW = ∑(−mi&r&i ). δri (5) i Modelling of Automotive Systems 19 Using equation (1) we have n r r ∂ri δri = ⋅ δq j ∑ (6) i=1 ∂q j Hence n ∂rr δW I = (−m &r&) ⋅ i ⋅ δq ∑ i i ∑ j (7) i j =1 ∂q j and re-arranging n ∂r I &r& i δW = ∑δq j ∑()− miri ⋅ (8) j =1 i ∂rj However, the generalised inertial forces, Qj, are effectively defined by n I r I δW = ∑(−mi&r&i ) ⋅ ∑Q jδq j (9) i j =1 Comparing (8) and (9) we have r I r ∂ri Q j = ∑(−mi&r&i ) ⋅ (10) i ∂q j It is shown in the handout notes that I ∂ ∂T ∂T Q j = − ( )+ (11) ∂t ∂q& j ∂q j Modelling of Automotive Systems 20 where T is the total KE 1 r r T = ∑ m i r&i ⋅ r&i (12) i 2 Elastic Forces Consider a simple spring: ←FS • FE x For static equilibrium FE = FS (external force) = (internal spring force) Modelling of Automotive Systems 21 Suppose we define the POTENTIAL ENERGY, V, as the work done by the external force to extend the spring a distance χ FE κχ extension of χ spring 1 2 external work done = kx = V 2 ∴ V= f(x)=V(x)− a function of x 1 2 Here V = kx 2 The work done by the internal spring force, W, is equal and opposite to V 1 W = −V = − kx2 2 Modelling of Automotive Systems 22 Now consider a small, virtual displacement, δx .