Balkan Journal of Mechanical Transmissions RO manian Association of (BJMT) Balkan Association of ME chanical Power Transmissions Volume 2 (2012), Issue 1, pp. 61-72 Transmissions (BAPT) ISSN 2069–5497 (ROAMET)
AN EXTENSION OF THE ELECTROMECHANICAL ANALOGY IN THE DOMAIN OF HYDROSTATIC TRANSMISSIONS Part I. THE ELECTROMAGNETIC AND ELECTROMECHANICAL ANALOGIES Mircea RADULESCU ABSTRACT. The paper aims to expand the electromechanical analogy in other domains of technology: hydraulic, pneumatic, acoustic, sonic, and even in thermodynamics. In addition to the similarity of the equations and mathematical models, in the domain of fluidic systems we have highlighted the analogy of the circuit elements and some basic structures, for which the equivalent schemes are given. Analogy tables are presented, including the important sizes, units, symbols and generalized mathematical models applicable in all domains above and the advantages of the analogy and its limits of application are highlighted . KEYWORDS. Electromechanic / hydraulic analogy, generalized parameters, electric / mechanical / hydraulic resistance / inductance / capacity / impedance, analogy of sizes / equations, limits of the analogy. 3 NOMENCLATURE V, ∆V [ m ] volume; finite fluid volume variation Symbol Description ω; ω rad / s angular/natural frequency r r r 0 A ; B= ∇ × A electrodynamic vector potential z= x + jy complex number (j = − 1) −1 Be [ ] electric susceptance (R ;θ ; λ ) spatial polar coordinates 2 τ apparent power mass density BL [ N / m ] bulk modulus of elasticity of liquid C= S−1 [ F ] electric capacity / elastance ABBREVIATIONS e e P.C. particular case f Hz circular frequency B.C.; I.C. boundary conditions/initial conditions S.C.P.; S.D.P. lumped/distributed parameters system IH = H/ L [ − ] hydraulic slope E.S.F.; M.S.F. electric/magnetic quasi stationary field diffusion coefficient in KD L.F.; I.F. local (differential)form / integral form electrochemistry O.D.E.; P.D.E. ordinary/partial differential equations 3 KH [ m ] flow module in hydrodynamics E.Q.S.;M.Q.S. electric/magnetic quasistationary Le [ H ] electric inductance 1. INTRODUCTION 2− 5 MH [ s m ] hydraulic module in hydrodynamics The analogy is one of the basic methods used to nondimensional exponent of RH research various areas of physics specific n − hydraulic resistance to motion phenomena, being based on the formal identity of 3 mathematical models that describe the original Q; Qm [ /] s volumetric flow/leakage flow phenomenon and the similar phenomenon. Therefore, pp; = pi − p e pressure; drop/jump pressure the analogy method allows solving problems in a particular domain, using the results from another field, R= G −1 [ ] electric resistance / conductance e e where the phenomena of the two domains have the Re =vd ν −1[ − ] Reynolds number in hydraulic pipes same mathematical model (are analogue). linear/linearised hydraulic resistance R;'[ R Ns / m 5 ] The scale of analogy is the constant ratio between the L to fluid motion values of the two similar sizes, which belong to the n two domains and must verify the (formally identical) Ns non linear hydraulic resistance to R mathematical models which express the conduct of N 3n+ 2 motion of fluid m the two phenomena. The analogy is partial, if only s [ m] curvilinear coordinate; string deviation some of the quantities involved in the description of T [ K] thermodynamic temperature phenomena are analogue. During the dynamic U [Nm] scalar potential of the force modeling of complex systems, difficulties arise, Basic geometrical volume of rotary regarding their direct analysis, so that sometimes the V[ m3 / rot ] g hydraulic volumic machines application of methods of study and indirect analysis
61 are required, allowing the complete knowledge of the 6. Any analogy must respect the principle of studied phenomenon, based on observations and conservation of energy, so that the condition of experimental research performed on similar models. compatibility from an analogy to the other derived from it, is to be a biunique correspondence of The biunivocal correspondence between the original powers for the two domains of the similar phenomenon and its model allows that, on the basis phenomena; for this reason, the equations from of rules and assumptions established a priori, the the mathematical model of the analog electrical variables that can not be assessed on the primary circuit must be isomorphic with the mathematical (original) system can be determined on the analog model equations of the studied (original) system, model; based on information obtained from the model, as defined in the biunique correspondence. some conclusions can be drawn on the original behavior. Frequently, the original system (a The analogy of non electrical elements and circuits mechanical, acoustic, hydraulic, pneumatic or thermal (mechanic, acoustic, thermic, hydraulic, pneumatic, system) is studied on an analog electric model, this sonic etc.) with the electrical ones allows the modeling providing the research of the phenomenon on an of the original system, based on some active circuit equivalent electric or electronic circuit, which allows elements (sources) connected to passive elements the processing of results and the implementation of (resistances, inductances, capacities, perditances) solving methods from the electric field in the area of through specific junction elements. Since the analogy interest. Similar models in the electric field are of sizes and mathematical models also aims an preferred, as their structure and operation have been analogy of the physical laws specific to analogue improved based on results obtained in the theory of circuits (systems), some theorems and laws of the electrical circuits. In these circumstances, the analogy electric disciplines can be properly translated into laws can be a process of synthesis of complex non electric and theorems expressing the phenomena of the circuits, based on synthesis algorithms specific to domain of interest (the original domain); example: electric or electronic circuits. Kirchhoff’s theorems, Ohm’s law, the law of electromagnetic induction, the transfiguration A detailed approach of the studies regarding the theorem, superposition theorem, Y↔ transfiguration analogy is shown by Olson (1958), Kinsler (1962), etc. (as shown by Mocanu, 1979, ora, 1982, Levi (1966), Hackenschmidt (1972), Stanomir (1989), Stanomir, 1989). and Fransua (1999). 3. GENERALIZED PARAMETERS 2. REQUIREMENTS OF THE ELECTROMECHANICAL ANALOGY The need to extend the electromechanical analogy to other non electric domains, which derive from The electromechanic analogy has the advantage that mechanics, becomes obvious with the approach of it can easily adapt to the following requirements interdisciplinary researches with applicability in specific to the study of physical phenomena by science vanguard fields like space flight engineering, modeling (as shown by Stanomir, 1982). mecatronics etc. There is also a need to establish a 1. The model should ensure a broad representation common language based on the essential role of the of the original, that is to allow highlighting its extended analogy between the domains unknown properties; they must be better known corresponding to physical systems that aim the than the ones of the original, or must be easily interdisciplinary science objectives. experimentally modeled; The degree of generality of the analogy can be 2. The bonds of the non electric system must be significantly increased if the quantities involved in the holonome, scleronome and linear and the system description of the phenomena are brought to must have a linear equivalent graph and only dimensionless forms and if a common language, with elementary one port structures. general validity is adopted. In order to highlight the domain of application for some extended analogies, 3. The specific dynamic process of the original generalized power variables are introduced: f (flow) system to be studied on the analog model will and e (effort), and on this basis we can define the strictly collapse (limit) to the domain of interest; generalized energy variables (as shown by Buculei, 4. Based on the scheme of the original model, we 1993, Rădulescu, 2005). can establish the equivalent electrical scheme; for We can generalize some non electric sizes, similar to example, the series / parallel connection of those associated with passive circuit elements elements that goes into the original, must be (resistances, inductances, capacities, perditances replaced with similar elements of circuit, connected etc.); these quantities are introduced taking into properly, in series / parallel; account the analogy of some basic laws from the 5. The physical character of a quantity of the original disciplines with non electric profile, with laws and system must be maintained to the study on the equations of circuit and electrical machines theory. analogue system; for example, the hydraulic Table 1 gives a generalization of the basic parameters potential must have as analogous the electric that characterize some systems/devices and the potential, etc. elements of machines and circuits.
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Table 1 . Generalized parameters Parameters t Displacement: qt()()()=∫ fτ d τ , q 0 = 0 0 t Pulse: ℑ=()()()t∫ eτ d τ , ℑ= 0 0 e t
General General Generalized power: P= ef = Π W ; Generalized energy: Et()()()=∫ PdEτ τ , 0 = 0 J 0 −1 Action: At( ) = qt( ) ℑ ( t) J ; Impedance: ZefRjXY=/ =+= ; j =− 1 Admittance: Y= G − jB S = −1 x m , strokelength− for linear hydraulic volumic m achines( L . M .) Displacement: qm = ϕ rad , angle− for rotary hydraulic volumic machines( R . M .) & v= q m/, s linearvelocity of the mobile equipment of the machine− L. M . Velocity: w = & ω= ϕ rad/, s angular velocity of the rotative parts of t he machine− R. M . F N , force reduced to the stroke of the machine− L. M . Effort: φ = M Nm , torque reduced to the rotor axis of the machi ne− R. M . m kg , mass reduced to the stroke axis− LM . . Mass: M = J Nm2 s , momentum of gyration reduced to the rotor a xis− R. M . Mechanical Mechanical
kv Ns/ m − viscous friction constant − L . M . Damping constant : kw = kω Nms − viscous friction constant − R. M . (Rm ) dφ kx N/, m elasticconstant , forlineardeformations− LM .. Mechanical stiffness: k = = q dq kϕ Nm/, rad elastic constant , forangulardeformati ons− R .. M 1 2 2 Elementary work: δL= φ ⋅ δ q J ; Overall energy: WWW=+=M wKq + J c p2 ( m ) V/2/,π m3 rad V− basic geometricvolume − R .. M g g Geometric capacity: K = A orA m3 /, m thepistonareaoftheactivechamber− LM. . 1 2 Hydraulic resistance to motion (friction): 5 d() p RL [ Ns / m ] , linear resistance () Re< Re cr R = = H Q n3 n + 2 RN [ Ns / m ]; n∈( 1;2 , nonlinearresistance () Re > Re cr Hydraulic leakage conductance: G= Qpm/5 / N H Hydraulic resistance to acceleration (hydraulic inertance/inductance): 2 5 −1 5 2 Hydraulic L= pQ/ = HNs / m ; Hydraulic mobility: M= L m/ Ns H H H Hydraulic capacity: C= Qpm/5 / N ; H Hydraulic resistance to deformation / hydraulic capacity: D= C−1 Nm/ 5 H H ∂φ Hydraulic stiffness: R= Nm ⋅ rad −2 h∂q Q =0 2− 1 − 11 − Hydraulic compliance: Λh =C H/ K = R h [ Nmrad ]
63 Fransua, 1999). 4. THE ELECTRIC / MAGNETIC ANALOGY On the basis of this rationing, the ideal passive Table 2 shows the analogy between the equations of elements of the mechanical lumped parameter the electric steady field and the ones of the magnetic system, noted R , L , C , as well as the sources of field. Most of the sizes and definition relationships m m m generalized force ( Φ ) or velocity ( w ) are represen have the same shape in both domains (volumic m m ted using symbols presented in the second part of this density of the energy and forces, stored energy, flux / paper (as mentioned by Fransua, 1999 and circulation of the fields, generalized forces etc.). Most Radulescu, 2005). laws and theorems are also similar (Kirchhoff’s theorems, the constitutive relationships, the When adopting the mechanical scheme one must uniqueness and superposition theorems etc.), as have in mind the compliance between d’Alembert’s shown by Levi (1966) and ora (1982). principle and Kirchhoff’s first law, as well as the correspondence between the passive elements of the Table 3 presents the summarized analogy by two analogue circuits. When drafting the analogue comparing the definitions and properties of the scheme for the circuit corresponding to a mechanic electrostatic, electrokinetic and magnetic fields. For system one can distinguish the following stages: particular cases (P.C.) we have assumed the three mediums are linear, isotropic and homogenous. In this 1. The compounding bodies are reduces to material case there is also a similarity between some points which correspond to nodes of mechanic definitions and laws of the three domains (as network with the velocity in ratio to a standard mentioned by Stan, 2005). branch point; The analogy between the stationary electric and 2. The elements of circuit are inserted between the magnetic fields can help solving some theoretical and nodes; experimental problems regarding the study of electric 3. The equations of equilibrium of forces ( φ ) are circuits using already known results in electrostatics / mk magnetostatics or vice versa. A good example would written for every node; be determining some electrostatic characteristics 4. The mathematical model is solved (the integral using experimental models in electro kinetics using differential equations) taking into consideration the electrolytic tanks or electroconductive paper. When initial conditions and the velocity of the node w is choosing the physical model one must take into j obtained. consideration the condition that the two analogue models have similar configurations. The electromechanical analogy allows solving issues related to vibrations of complex mechanical systems The previous observations mentioned prove their by replacing them with equivalent electrical network, utility mainly in the didactic field and mostly in which can be easily studied on the basis of “standard” interdisciplinary practical courses (electromechanics, results obtained in the theory of electrical circuits. magnetohydrodynamics, mecatronics, robotics etc. Using this analogy is only possible if the mechanical 5. THE ELECTROMECHANICAL ANALOGY system studied is linear, if its vibration amplitudes are The electromechanical analogy (E.M.A.) is used in the small enough. It is essential to obtain the correct study of simple oscillatory electric systems, on the configuration of the electrical scheme of the vibrating basis of some models of elementary mechanic mechanical system, given the following systems and in some cases the more complex recommendations (as shown by Fransua, 1999; discrete mechanic systems are analyzed using Stanomir, 1989): analogue electrical networks, considering the formal equivalence between Lagrange’s equations and • for the analogy of impedances, to a mechanical Kirchhoff’s laws, as well as the practical possibilities of series assembly corresponds a parallel electrical measurement of the state parameters of electrical circuit and vice versa. circuits. In table 4 is presented the E.M.A. of the basic • for the analogy in admittances, to a mechanical sizes for the I and II type analogies. parallel assembly corresponds a parallel electric In the E.M.A., the ideal passive elements that form a circuit and to a mechanic series assembly mechanical lumped parameter (discrete) system are corresponds a series electric circuit. represented using analogue symbols of the elements Usually, the equivalent electromechanical scheme is R, L, C specific to electrical lumped parameter circuits determined, which corresponds to the mathematical as mentioned by Nedelcu (1978), Iacob (1980) and model, based on which one can determine the Stanomir (1989). mechanical impedance or mechanical mobility. The ideal active elements (the mechanical sources) For example, Fig. 1 presents an elementary are adopted in analogy to the sources specific to mechanical system (autovehicle and trailer) in electric circuits and are associated a positive sense translation movement (a) and its analogue mechanical and polarities which correspond to the positive sign of scheme (b). the mechanical power in the circuit (as shown by
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Table 2 . The electric magnetic analogy of parameters Electrics Denomination Magnetism r r r r Ert( ;) [/] V m Strength of the vectorial Hrt( ;) [/] Am field r r r r Drt(;)[ C / m 2 ] Flux density Br(;)[ t T ] r r Moment pe [ C⋅ m ] pm [ Nm/ T ] r r r r Polarization / Pp= dp e / d V M= dpm / d V magnetization r r r r r Torque Ce= p e × E Cm= p m × B rr r rr r F= ∇( p ⋅ E ) Force F= ∇( p ⋅ B ) rer e rmr r m 2 Volumetric force 2 fe=ρ v E −()1/ 2 E ∇ ε ( ε′( τ ) = 0 ) fm =×− JB()1/ 2 H ( ′( τ ) = 0 ) Theorem of generalized ∂We ( q0; x ) ∂Wm (ϕ; x ) Xk = − forces Xk = − ∂xk ∂xk r rrr1 rr r Maxwellian tensions r rrr1 rr r T= EDn ⋅− DEn ⋅ T= HBn ⋅− HBn ⋅ n () 2 () n () 2 () r r ⇒ r r ⇒ Tn = n ⋅ T e Tn = n ⋅ T m ε F/ m Vacuum permittivity/ H/ m 0 [ ] 0 [ ] permeability χe Susceptivity χm εr=1 + χ e Relative permittivity/ r=1 + χ m permeability ε= ε −1D′() E Differential permittivity/ = −1B′() H diff 0 permeability diff 0 r⇒ r Law of polarization/ r⇒ r P=ε Xe ⋅ E magnetization M= Xm ⋅ H rt 0 rt Permanent polarization/ Pp Mp magnetization r r r r Electro / magnetomotive uem = Edr ⋅ umm = Hdr ⋅ ∫ force ∫ ()C ()C r r r r ψ=∫∫ D ⋅ d σ Flux φ=∫∫ B ⋅ d σ ()Σ ()Σ r r r r ψ* =∫∫ E ⋅ d σ Flux of the field ( ε = ct .) φ* =∫∫ H ⋅ d σ ()Σ ()Σ r r V= V( rt; ) Scalar potential V= V( rt; ) er e r r mr m r r Irotational field ∇ ×E = 0 ; E= −∇ V e ∇ ×H = 0 ; H= −∇ V m u= V − V Voltage / magnetic u= V − V e e1 e 2 m m1 m 2 r tension r Solenoid field ∇⋅=D0( q e = 0) ∇ ⋅B = 0 1 r r Volumetric density of the 1 r r w= E ⋅ D w= H ⋅ B e 2 energy m 2 1 Energy in the capacitor/ 1 We= q e V e inductor Wm = φ i 2 r2 r V +ρ ε −1 = 0 ; Poisson’s equation A + J = 0 ; e v r 0 r r scalar: E= −∇ V vector: B= ∇ × A e Potential rr rr ρv=0 ⇒ V e = 0 P.C.: Laplace’s equation J=0 ⇒ A = 0 The two schemes include specific notations for the Fig. 2 shows an example of elementary mechanical two types of motion, but they can also be written in a system in rotary movement (gear box): the cinematic generalized form, as shown in Table 1. scheme (a) and the analogue calculus scheme (b).
65 Table 3 . The electrostatic electrokinetic magnetic analogy Electrostatic field Electrokinetic field Magnetic field Differential equation of the field lines r r r r r r r r r dr× E = 0 dr× J = 0 dr× H = 0 Characteristics of the material Electric permittivity Electric conductivity/resistivity Magnetic permeability/reluctivity ε= ε ε [F / m ] (not ) −1 0 r −1 = 0 r = ν [H / m ] ; σρρ=e; e == ρρρ0 r [ m ] ν magnetic reluctivity Global energetic relations Electrostatic energy Law of degradation of electric Magnetic energy energy rr rr 1 t r r 1 We=ρ ve VdV + ρ se Vd σ Wm = AJdV ⋅ +∇⋅×() AHd σ 2 ∫∫∫ ∫∫ W= wdV ; w= EJd ⋅ τ 2 ∫∫∫ ∫∫ ()()D Σ J∫∫∫ J J ∫ ()D ()Σ ()D 0 (Joule Lenz) Particular forms of the Maxwell’s equations Constitutive relations rThe relectric r conduction law Constitutive relations r⇒ r r r⇒ r r J=σ ( E + E i ) (Ohm’s law) D=ε E + P p r rrrr B= ⋅ H + 0 M p ⇒ r r r r PC: Ei =⇒=0 JEEJσ ⇔ = ρ ⇒ r r r r PC: ε→= εct; Pp =⇒= 0 D ε E PC: →= ct; Mp =⇒= 0 B H Electric fluxr law (Gauss’s law) Law of electricityr conservation Magnetic fluxr law (Gauss’s law) & (L.F.) ∇ ⋅D =ρv ⇔ (L.F.) ∇ ⋅J = −ρv ⇔ (L.F.) ∇ ⋅B =0 ⇔ r r r r r r & ⇔B ⋅ d σ = 0 (I.F.) ⇔∫∫ D ⋅ dσ = q e (I.F.) ∫∫ Jd⋅σ = − qte ( ) (I.F.) ∫∫ ()Σ ()Σ ()Σ r r & P.C.: ρv =0 ⇒ ∇ ⋅D = 0( L.F. ) ⇔ P.C.: ρv =0 ⇒ ∇ ⋅J = 0( L.F. ) ⇔ r r r r ⇔∫∫ D ⋅ d σ = 0 (I.F.) ⇔∫∫ J ⋅ d σ = 0 (I.F.) ()Σ ()Σ P.C.: the flux of the field vector through one field tube (stationary field ) r r r r r r ∫∫ D⋅ d σ = Ψ ∫∫ J⋅ dσ = i ∫∫ B⋅ d σ = Φ ()Σ ()Σ ()Σ Law of electrostatic potential E.Q.S. field: r M.Q.S. field: law of magnetic circuits (Ve ); irrotational field: Faraday’s law (v = 0) (Ampere’sr r law ) r r r r r ∇ ×H = J (D.F. ) ⇔ ∇×E =0 ⇒ E= −∇ V e (L.F. ) ⇔ ∂B r r r r ∇ ×E = −()L.F. ⇔ r r ⇔Edr ⋅ = V − V (I.F.) ∂t ⇔ Hdr ⋅= Jd ⋅ σ (I.F.) ∫ e1 e 2 ∫ ∫∫ r r ∂Φ ()()C Σ ⇔E ⋅ dr =− (I.F.) r r ∫ ∂t PC: J=0 ⇒ H =−∇ V ; r r r m & V scalar magnetic potential PC: B=0 ⇒ E = −∇ V e ; m Ve scalar electric potential Usual circuit elements (stationary field ) Electric capacitor: Electric conductance: Magnetic permeance: q i r r Φ r r C=e ; uVV = − G== R−1; u =⋅ E dr Λ==m R−1; u = H ⋅ dr eu Cee1 2 eu e R ∫ mu m m ∫ C R ()C m ()C Electric elastance: Electric resistance: Magnetic reluctance: l l l ds ds ds A S= = C−1 F − 1 R= = G −1 [] R = = Λ −1 e∫ εS e e∫ σS e m∫ A m Wb 0 0 0
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v t v t k 1( ) 2 ( ) MA M1 v2
z1 k J1; k v F( t ) e 1 ω1 m1 m2 kv ωA MT z2 J2 k a k a ω v1 v2 2 ωT M2 ke ke N 1 ke N1 N2 1 2 N2
+ + kv ω1 ω2 kv k F 1 m1 m2 v2 MA z1 z2 J ω J1 kv 2 kv A 1 2
N′ N′ N1′ N2′ 1 b 2 b Fig. 1. Example of equivalent scheme for a Fig. 2. Example of equivalent scheme for a mechanical system in rotation movement mechanical system in translation movement (φ ↔ u ; w↔ i ), the two analogue circuits would When drafting these schemes, we have used the II m e m have had different configurations, because type analogy ( φ ↔ i ; w↔ w ) which assures the m m e d’Alembert’s principle would have been incorrectly compatibility of d’Alembert’s principle and Kirchhoff’s associated to Kirchhoff’s second law. first law. If the I type analogy would have been used Table 4 . The electromechanic analogy of parameters Generalised parameters in mechanics Electric analogy I type (Z) II type (Y) t t t Displacement: qtm()()= ∫ w m τ d τ Charge: qte ()()= − ∫ iτ d τ Flux: Φe = − ∫u e ()τ d τ 0 0 0 o Current: Voltage: ue ( t) Velocity: wm = qm () t o o o o Velocity of i: i ( t) Acceleration: wm = q m Velocity of u e: ue () t Mass: M Inductance: Le Capacitance: Ce −1 −1 Stiffness: kq Inverse of capacitance: Ce Inverse of inductance: Le
Damping coefficient: kw Resistance: Re Conductance: Ge o o o Inertial: φi ()t= M w Inductive: uL= Lit e () Capacitive: ic= Cu e e () t −1 −1 Elastic: φq= k q q m Capacitive: u= Cqt() Inductive: i= L Φ Effort Effort C e e L e e current Voltage Voltage Damping: φw= k w w m Resistive: uR= Rit e ( ) Conductive: iG= G e u e r Irrotationalr field r φm= −∇ U m E= −∇ V e H= −∇ V m Generalized force r r ∂ri ∂We ∂Wm Xk= F i ⋅ Xk = + Xk = − ∂qm ∂xk ∂xk k qe Φe Stationary characteristic φm= φ m(w m ) ue= u e ( i ) um= u m( Φ e ) Impulse of the body: r r Charge of the capacitor: Flux of the inductor: Φe = Li ℑ(t) = M w qe= C e u e
Mechanic power: Pm= φ m w m Electric power: Pe= u e i
67 Table 4 . Continuation Elementary mechanical work: Elementary electric work δLm= φ m δ q m δLe= u e δ q e δLe= i e δφ e Mechanical reactance: Electric reactance: X Susceptance: B = e X= ω ω −1 k −1 − 1 e 2 mM - q Xe= ω L e- ω C e Ze
φm Ue I Mechanical impedance: Zm = Electric impedance: Ze = Admittance: Ye = wm I Ue Zm= k w + jX m Ze= R e + jX e Ye= G e − jB e 2 1 2 1 2 1 qc Kinetic: Wk= M w m Solenoid: WL= L e i Capacitor: WC = 2 2 2 Ce 2 1 2 1 qe 1 2 Potential: Wp= k q q m Capacitor: WC = WL= L e i 2 2 Ce 2 Resistance: Conductance: t 2 t t Electric energy:
Mechanicenergy Dissipative: Wd= kwd w τ 2 2 ∫ WJ= Ri e ()τ d τ WJ= Gu e R ()τ d τ 0 ∫ ∫ 0 0 Total energy 2 1 1 2 q WW= + w2 + kq 2 WW= + Li + e t d(M mqm ) t J e 2 2 Ce r r Action Am= ℑ ( tqt) m ( ) Ae= Φ ( tqt) e ( ) model in which they occur, we can obtain the dynamic Table 5 and Table 6 give two commonly used characteristics in the frequency domain, as mentioned mathematical models with generalized (unified) by Olson (1958), Iacob (1980) and Stanomir (1989). notations corresponding to the previously defined generalized parameters. Table 5 shows the basic The mechanic phenomena specific to continuous features of the linear oscillator with damping and material spectrums can be modeled on the basis of harmonic excitation, seen as a model for the study of the equations of the electromagnetic field, using the systems with concentrated parameters. calculus of variations and having in view the Lagrange density function and Hamilton’s equations. Therefore, The mechanical system with concentrated parameters Table 6 summarizes the model with distributed shows a basic structure which materializes the basic parameters, which has a general character, allowing effects (inertial, elastic, dissipative) and is the study of analog phenomena in several fields characterized by the constitutive equation that (electrical, mechanical, hydraulic, acoustic, sonic, indicates the dependence between a kinematic size etc.). (displacement, velocity or acceleration) and an effort size (force or torque); based on this dependency we Table 7 presents the study of a mechanical circuit (Rm, can identify the mechanical impedance. Lm, Cm) compared to the analogue electrical circuit; in both assemblies (series / parallel) are shown two The mechanical system with distributed parameters types of analogies (in impedance analogy type I) and can be one dimensional or two dimensional, where (in admittance analogy type II), based on the waves arise (in particular small oscillations) for which mathematical model in Table 5. the equations of the mathematical model are linear. The modeling of non electric systems can be based From the above it is confirmed that the base criterion on electric systems with distributed parameters whose for choosing the right type of analogy is the mathematical model contains the telegraph, the correspondence between Kirchhoff’s theorems and second order differential equations with partial deri the two base equations which interfere in the vates which relate the parameters u (t, x ) and i (t, x). description of the analogue (non mechanic) system. One can observe that the I type analogy regards Table 6 shows the basic features of systems with Kirchhoff’s first theorem and the mechanics equation distributed parameters, taking into account the two resulting from Newton’s first law (the equilibrium of generalized basic sizes involved in relationships (1) sizes e). The II type analogy regards Kirchhoff’s and (2), intensive and complementary sizes: e (effort) second theorem and the mechanics equation which and f (flow); these sizes can be identified in each of results from Newton’s second law (the balance of the the areas subject to observation in this work (electric, sizes f). A particular case is the electroacoustic namely, non electric), so based on the mathematical analogy (E.A.A.), which allows modeling the acoustic
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systems using elementary electric systems. For The E.A.A. is frequently used in electroacoustics (as example, Fig. 3 shows the acoustic (a) and the shown by Stanomir, 1989) where various analogue electric scheme (b) for a classic microphone inhomogeneous converting subsystems interfere, (as shown by Iacob, 1980). The schemes have been specific to unconventional conversion microdrives. adopted considering the II type analogy ( u ↔ v; i e Table 8 presents the study of a section of acoustic line ↔p ) and the notations correspond to those in Table a (pipe) with distributed parameters based on the 8. The mathematical model of the scheme mechanical acoustic analogy (Type I direct, corresponds to an oscillatory system without losses, respectively, type II reverse), obtained from the whose scheme is reduced to a series combination reference mathematical model from Table 6. (C+ L ) connected in parallel to C . eeq e e3 For acoustic systems with distributed parameters the L geometric dimensions of the system are comparable Ce e N k 0 1 e1 to the wavelength λu , for which there is a phase shift, ±F ± due to propagation. When designing technical ke 2 C Ce F m e1 3 systems a mechanic mechanic analogy is sometimes N C ke k e2 m needed, being regarded in some cases also as a 3 e0 similitude. Next, we present an example of mechanic N a b 2 mechanic analogy. Fig. 3. The mechanical scheme and the analogue electric scheme of the microphone Table 5 . The basic features of the linear oscillator (for S.C.P.) Parameter Definition and calculus equations Other formulae, notations Resonance values −1/2 1/2 2 2 2 Pulsation Natural: ω0 =(MCmm ) = () KM mm / ; pseudo pulsation: ωp = ωδω0 −= 0 1 − ζ Mechanical Rm δ resistance R Damping factor: ζ = = Rm= 2ω0 M m m R ω cr (damping factor) mcr 0 Damping δ=R/ 2 M = ζω δrez=R m/ 2 M m ≅ ω0 ς coefficient m m 0 cr Driving ω γ= ω/ ω 2 frequency 0 γrez =1 − 2 ζ ω0 Mm 1 1 Quality factor Q= =ω0 = Mm K m Q0 = 2δ Rm R m 2ζ Static q=φ/ Kp = /; ω2 p = φ / M φ amplitude of perturbation q= 2ζ ( q ) displacement st0 m 0 0 m 0 st 1 max && & 2 Generalized mathematical model: q+2δ q + ω0 qp = sin ω t ; −δt Transitory component: qt= q0 e sin (ω p + ϕ 0 ) ; Permanent component: qp = q1 sin (ω t − ϕ ) Characteristic && & 2 2 2 equation x+2δ x + ω 0 x = 0 ; solution: x1,2=−±δ δ − ω 0 q p qst q =st ≅ q1 = q1 = ()1 max 2 2 2 Amplitude 2 2 2 2 2 2 2ζ 1 − ζ ()ω0 − ω + ()2 δω ()1−γ + 4 ζ γ ≅qst /2ζ = p /2 δω 0 2δω γ ϕ = arctg ϕ = arctg ϕ=arctg ζ −2 − 2 Phase 2 2 2 rez ω0 − ω (1− γ ) Q Dynamic −1 1 1 2 ξ = ξrez = ≅ Q0 amplification ξ=q1 / q st = 1 − γ 22 22 2 factor (1−γ ) + 4 γ ζ 2ξ 1 − ζ Logarithmic 2 decrement δ1= ln(qk / q k + 1 ) ; δ1 =2/1 πζ − ζ ≅ 2 πζ() ζ 1 ; π/ δ1 = 1/2 ζ over amplitude coefficient Dynamic elastic K=φ/ ZKM =− ωωω2 + jK = jZ ; q= zz; =+ x jyZ ; =+ R jX constant D m m m m 1 m m m 2π2p 2 ω 2 W γ 2 2 W = = 2π Q0 Energy 4 22 22− 2 2 2 Wrez = ω(1− γ ) + γ Q Wrez Q (1−γ ) + γ 2 0 ω0
69 Table 6. The basic features of mechanical systems with distributed parameters (for S.D.P.)
Quantities; Definitions, formulas, mathematical models and their solutions characteristic Without losses Without General case (R’ ; L’ ; C’ ; G’ ≠ 0) Without perditance (G’ = 0) constants (R’ = G’ = 0) distortion Wave −1 L′ 1 T===ω; ω = 2 π f period 00 00 R′ 2δ1
Damping R′ G′ R′ δ2 Constants Constants δ = ; δ1 = ; δ2 = = 2 δ δ1 = 0 δ= δ 1 = 0 δ= δ = coeff. 2H′ 2C′ H′ 1 2 x L 1 1 a Time: δ3 = [s ] ; x= L ⇒ T p = ; wave propagation speed: a = ; a0 = ; wavelength: λu =aT 0 = a a H′ C ′ H′ C 0 ′ f0
Operatio s δ γ (s )= ( R′ + HsG ′ )( ′ + Cs ′ ) γ ()s= ( R′ + HsCs ′ ) ′ γ(s ) = s HC′ ′ = γ (s ) = s + 2 nal a a γ(j ω ) = factor factor jω Complex γω()(j=+ RHjGCj′ ′ ω )( ′′ + ωγδω ); =+ j γj ω =
Propagation Propagation () = −HC′′ω2 + jRC ω ′′ a
Operatio s + 2δ 2δ Zs( )= ( R′ + HsG ′ )( ′ += Cs ′ ) Z Z= Z 1 + Zc→ Z c nal c c 0 c c 0 0 s + 2δ1 s
R′+ jω H ′ H′ ρa0 Z( j ω ) = ; Z = = wave impedance Impedance Impedance Complex c c0 RG′′− CH ′′ω2 + j ω ( RC ′′ + HG ′′ ) C′ s
2 2 Load (terminal) 1 2 R′ 1 2 R′ impedance ZRjxZZc=+s s; c =⇒= s R s H′ + ( H ′ ) + ; Xs = −+ H′( H ′ ) + 2C′ ω 2C′ ω
Coupled ∂ ()e ∂ () f ∂ ()f ∂ () e =−R′ + f H ′ ; =−R′ e + H ′ eq. ∂x ∂ t ∂x ∂ t ∂ 2()e ∂ 2 () e ∂ () e =HC′′ ++( RCGH ′′′′ ) + RGe ′′ 2 2 Decoupled ∂x ∂ t ∂t eq. ∂ 2()f ∂ 2 () f ∂ () f
Mathematical model model Mathematical =HC′′ ++( RCGH ′′′′ ) + RGf ′′ ∂x2 ∂ t 2 ∂t d2( e ) −+(R′ HsG ′ )( ′ + Cs ′ ) = e 0 ; dx 2 Systemic model d2( f ) −+(R′ HsG ′ )( ′ + Cs ′ ) = f 0 dx 2 γx− γ x esx( , ) = Ce1 + Ce 2 General γx− γ x fsx( , ) = Ce1 + Ce 2 1 =esx(,) eshx2 ()γ + esh γ () Lx − Solution Solution sh(γ L ) {}1 Final 1 =fsx(,) fshx2 ()γ + fsh γ () Lx − sh(γ L ) {}1
ch()γ x Zshc () γ x e() j ω e1 () j ω = 1 ; fj()ω sh()()γ x ch γ x fj() ω Operational Operational Z 1 Matrix input c output complex relationship e∗() jω chjL((γω )) [ Zj ( ω )/ Z ( jshjL ωγω )](( )) e∗ ( j ω ) 2 c c 0 1 = ∗ ∗ Complex Complex f() jω [Zc ( jω )/ Zj c ( ωγω )](( sh jL )) ch (( γω jL )) f( j ω ) 2 0 1
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Table 7. Comparison between a mechanical system, an electric series circuit and an electric parallel circuit
Mechanical system / device Type of analogy Analogue electric circuit Type I (direct) in impedance
Zm↔ Z e
f ↔ u; v ↔ i; R m ↔Re; Physical −1 Mm ↔Le; Km= C m ↔ C e t t 1 1 MvtRvt& ()()+ + vd()τ τ = f Lit&()()+ Rit + i()τ d τ = u m mC ∫ k e eC ∫ k m 0 e 0 alogue electric circuit F U v() t=cos ()ω t + ϕ ; Type II (inverse) in admittance i() t=cos ()ω t + ϕ ; −1 Zm f ↔ i; v ↔ u; R m ↔ Ge= R ; Ze e Mm 1 M ↔ C ; K−1 = C ↔ L ; Le 1 Qm =ω0 = m e m m e Qm =ω0 = Rmω0 R m C m Reω0 R e C e Mathematical Zm↔ Y e 2 2 (/ω ω 0 )− 1 (/ω ω 0 )− 1 ϕ = arctg ϕ = arctg ωR C ωR C m m e e
Type I (direct) in impedance
Zm↔ Z e
Physical f ↔ u; v ↔i; R m ↔ Re; −1 Mm ↔ Le; Km= C m ↔ C e
• t • t 1 1 1 CftGft()()+ + fd()τ τ = v Z= R + jω L − (impedance) Cut()()+ Gut + u()τ d τ = v m mM ∫ k e eL ∫ i m 0 ωC e 0 ft()= ZV cos(ω t + ϕ ) 1 ut()= ZIe cos(ω t + ϕ ) m Y= G + jω C − (admittance)
Model of the mechanic system / device and of the an 2 ωL 2 (/ω ω 0 )− 1 (/ω ω 0 )− 1
Mathematical ϕ = arctg R −1/2 ϕ = arctg R ωM m ω = (LC ) natural frequency ωL e m 0 e Table 8 . The mathematical model of an acoustic line with D.P., established using the electroacoustic analogy
Domain Type of oscillating harmonic system Mechanical system with Acoustic system with constant Uniform one dimensional Type longitudinal motion section ( S=ct .) acoustic system ′ ′ ′ ′ ′ ′ + Rm; M m + Ra; M a + Rs; M s
v Cm′ x q Ca′ x v Cs′ x f p p model Gm′ Ga′ Gs′ Physical
∂f' ' ∂ v ∂pa ' ' ∂q ∂p' ' ∂ v +Rv+M = 0 +Rq+M = 0 +Rv+M = 0 ∂xm m ∂ t ∂xa a ∂ t ∂xs s ∂ t Equations
' ' −1 'ρ ' S ' ' −1 M = ρS;C =(ES) M= ;C= ; M = ρ;C =E ; m m aS a E s s ' ' R' + j ωM ' Rm + jωM m R' + j ωM ' s s Zu = a a Zu = ' ' Zu = ' ' Gm + j ωC m ' ' Gs + j ωC s Ga + j ωC a Type I analogy (in impedance)
Mathematical model 0 E 0 E Zu = ρcS; c= 0 ρc E Zu =ρc;c= ρ Zu = ;c= ρ Systemparameters S ρ
71 Table 8 . Continuation ′ ′ ′ ′ ′ ′ + Cm; G m + Ca; G a + Cs; G s
f Mm′ x p Ma′ x p M s′ x v q v model Rm′ Ra′ Rs′ Physical
∂v' ' ∂ f ∂q' ' ∂ p ∂v' ' ∂ p +Gf+C = 0 +Gp+C = 0 +Gp+C = 0 ∂xm m ∂ t ∂xa a ∂ t ∂xs s ∂ t Equations
' ' −1 'ρ ' S ' ' −1 M = ρS;C =( ES ) ; M= ;C= ; M = ρ;C =E ; m m aS a E s s ' ' G' + j ωC ' Gm + j ωC m G' + j ωC ' s s Zu = a a Zu = ' ' Zu = ' ' Type II analogy (in admittance) Rm + jωM m ' ' Rs + j ωM s Ra + j ωM a
Mathematical model 0 1 E 0 1 E Z=u ;c= 0 S E Zu = ;c= ρcS ρ Zu = ;c= ρc ρ Systemparameters ρc ρ ρ density; E Young’s modulus; p acoustic pressure; q acoustic flow; v velocity of the medium
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RADULESCU, M. (2005). The Electrohydromechanic Analogy, ISIRR8 Szeged.
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