Introduction
This book presents a unified approach to modeling electromechanical systems using Hamilton's principle It begins with an overview of Lagrangian dynamics in mechanical systems, followed by a detailed exploration of Lagrangian dynamics in electrical networks Subsequent chapters delve into a diverse range of electromechanical systems, including piezoelectric structures, providing comprehensive insights into their dynamics and interactions.
Lagrangian dynamics transforms classical vector dynamics by replacing vector quantities such as force and momentum with scalar quantities like energy and work This approach employs generalized coordinates instead of physical coordinates, enabling a formulation that is independent of the reference frame By analyzing systems globally rather than focusing on individual components, Lagrangian dynamics effectively eliminates interaction forces arising from constraints among the system's elementary parts It's important to note that the selection of generalized coordinates is not unique.
The variational form of dynamic equations is derived from Newton's laws using the principle of virtual work, which is adapted for dynamics through d'Alembert's principle This process ultimately leads to Hamilton's principle and the Lagrange equations applicable to discrete systems.
Hamilton's principle serves as an alternative to Newton's laws of motion and is considered a fundamental law of physics that cannot be derived However, its complex form may be challenging for inexperienced readers By deriving Hamilton's principle for a system of particles, we can enhance its understanding and acceptance as a viable alternative formulation in physics.
Lagrangian dynamics provides a framework for understanding mechanical systems in dynamic equilibrium, with Hamilton's principle offering a more general approach than Newton's laws This principle can be applied to distributed systems described by partial differential equations and extends to electromechanical systems Additionally, Hamilton's principle serves as the foundation for developing various numerical methods in dynamics, including the finite element method.
Kinetic state functions
Consider a particle travelling in the directionx with a linear momentum p According to Newton’s law, the force acting on the particle equals the rate of change of the momentum: f = dp dt (1.1)
The work done on a particle can be expressed as f dx = dp dt dx = dp dt v dt = v dp, where v represents the particle's velocity (v = dx/dt) The kinetic energy function T(p) is defined as the total work required to increase the momentum of the particle from 0 to p.
According to this definition,T is a function of the instantaneous momen- tump, with derivative equal to the instantaneous velocity dT dp =v (1.4)
Up to now, no explicit relation between p and v has been assumed; the constitutive equation of Newtonian mechanics is p=mv (1.5)
Substituting in Equ.(1.3), one gets
A complementary kinetic state function can be defined as the kinetic coenergy function (Fig.1.1)
0 p dv (1.7) which, as (1.3), is independent of the velocity-momentum relation Note, from Fig.1.1, thatT(p) and T ∗ (v) are related by v dv dp
Fig 1.1 Velocity-momentum relation for (a) Newtonian mechanics (b) special rela- tivity.
The total differential of the kinetic coenergy reads dT ∗ =p dv+v dp−dT dpdp=p dv (1.9) if (1.4) is used It follows that p= dT ∗ dv (1.10)
The kinetic coenergy is directly related to the instantaneous velocity (v) and its derivative corresponds to the instantaneous momentum The Legendre transformation outlined in Equation (1.8) facilitates the transition between independent variables, specifically from momentum (p in T(p)) to velocity (v in T*(v)), preserving the essential information about the material's behavior For a Newtonian particle, the expression for kinetic coenergy can be derived by integrating Equations (1.5) and (1.7).
4 1 Lagrangian dynamics of mechanical systems
In engineering mechanics, the term kinetic energy is commonly used, but it's important to note that T(p) and T ∗ (v) represent different variables despite having the same values for Newtonian particles Traditionally, T and T ∗ are treated as identical in Newtonian mechanics, largely due to the prevalence of displacement-based variational methods However, as we expand Hamilton’s principle to electromechanical systems in subsequent chapters, distinguishing between electrical and magnetic energy and coenergy functions will become essential Therefore, we will adopt the notation of kinetic coenergy T ∗ (v) in place of the classical kinetic energy T(v).
In transitioning from Newtonian mechanics to special relativity, it is crucial to note that the constitutive equation must be modified from \( p = mv \) to \( p = mv \sqrt{1 - v^2/c^2} \), where \( m \) represents the rest mass and \( c \) is the speed of light While these equations appear similar at low speeds, they significantly diverge at high speeds, resulting in different values for \( T \) and \( T^* \).
Generalized coordinates, kinematic constraints
Virtual displacements
A virtual displacement refers to an infinitesimal change in coordinates that occurs at constant time and adheres to the kinematic constraints of a system, while remaining arbitrary In this context, the notation δ is employed to denote these virtual changes, which follow similar rules as derivatives, excluding time involvement For systems with generalized coordinates \( q_i \) governed by holonomic constraints, the permissible variations must fulfill the condition \( \delta f = X_i \).
The same principles apply regardless of whether the variable t is explicitly included in the constraints, as virtual displacements are considered at constant time In the case of non-holonomic constraints, such as those represented in equations (1.15) and (1.16), it is essential that the virtual displacements adhere to specific conditions.
When comparing Equations (1.15) and (1.18), it becomes evident that if time is explicitly included in the constraints, the virtual displacements cannot be considered as actual displacements The differential displacements \( dq_i \) follow a specific trajectory that evolves over time, whereas the virtual displacements \( \delta q_i \) represent the difference between two distinct trajectories at a specific moment.
Consider a single particle constrained to move on a smooth surface f(x, y, z) = 0 The virtual displacements must satisfy the constraint equation
∂zδz = 0 which is in fact the dot product of the gradient to the surface, gradf =∇f = (∂f
∂z) T and the vector of virtual displacementδx= (δx, δy, δz) T : gradf.δx= (∇f) T δx= 0
8 1 Lagrangian dynamics of mechanical systems
The gradient ∇f is aligned with the normal vector n of the surface, indicating that virtual displacements occur within the tangent plane of the surface Additionally, when examining the reaction force F that restricts the particle's movement along the surface, it is important to note that in a smooth and frictionless system, this reaction force is also perpendicular to the surface.
F.δx=F T δx= 0 (1.19) the virtual work of the constraint forces on any virtual displacements is zero We will accept this as a general statement for a reversible system(without friction); note that it remains true if the surface equation de- pends explicitly ont, because the virtual displacements are taken at con- stant time.
The principle of virtual work
The principle of virtual work provides a variational formulation for the static equilibrium of a frictionless mechanical system In a system comprising N particles, each with position vectors x_i (where i ranges from 1 to N), the static equilibrium condition indicates that the resultant force R_i acting on each particle is zero Consequently, the dot product R_i · δx_i equals zero, highlighting the relationship between forces and displacements in this context.
Ri.δxi= 0 for all virtual displacements δxi compatible with the kinematic con- straints R i can be decomposed into the contribution of external forces appliedF i and the constraint (reaction) forcesF i ′
R i =F i +F ′ i and the previous equation becomes
For a reversible system (without friction), Equ.(1.19) states that the vir- tual work of the constraint forces is zero, so that the second term vanishes, it follows that
1.4 The principle of virtual work 9
In virtual work analysis, the external applied forces do no work on virtual displacements that align with the kinematic constraints, resulting in a zero value This principle is significant because it eliminates reaction forces from the equilibrium equations, transforms static equilibrium challenges into kinematic problems, and allows expression in generalized coordinates.
XQ k δq k = 0 (1.21) whereQ k is the generalized force associated with the generalized coordi- nateq k f ò w y x a
As an example of application, consider the one d.o.f motion amplification mechanism of Fig.1.4 Its kinematics is governed by x= 5asinθ y = 2acosθ
It follows that δx= 5acosθ δθ δy=−2asinθ δθ The principle of virtual work reads f δx+w δy= (f.5acosθ−w.2asinθ)δθ= 0 for arbitraryδθ, which implies that the static equilibrium forcesf andw satisfy f =w2
10 1 Lagrangian dynamics of mechanical systems
D’Alembert’s principle
D’Alembert’s principle extends the principle of virtual work to dynamics.
It states that a problem of dynamic equilibrium can be transformed into a problem of static equilibrium by adding the inertia forces -mx¨ i to the externally applied forcesF i and constraints forces F ′ i
Indeed, Newton’s law implies that, for every particle,
Building on the previous section's analysis, we can derive that the total virtual work done by the constraint forces is zero when we sum over all particles involved.
The effective force is defined as the sum of applied external forces and inertia forces In this context, the virtual work of effective forces on virtual displacements that comply with constraints equals zero While this principle is fundamental, its application is challenging due to its reliance on vector quantities in an inertial frame, making it less straightforward than the principle of virtual work Hamilton's principle, discussed in the following section, provides a means to translate this concept into generalized coordinates.
If the time does not appear explicitly in the constraints, the virtual displacements are possible, and Equ.(1.22) is also applicable for the actual displacementsdx i = ˙x i dt
In a conservative force field, external forces can be represented as the gradient of a potential V that is independent of time, leading to the relationship P Fi.dxi = −dV If V does depend on time, the total differential will include a partial derivative with respect to time Additionally, the second term in this equation represents the differential of kinetic coenergy.
Hamilton’s principle
Lateral vibration of a beam
This article examines the transverse vibration of a beam, as illustrated in Fig 1.7, which is subjected to a distributed load p(x, t) It is assumed that the principal axes of the beam's cross-section align with the vibration plane, where v(x, t) represents the transverse displacements and the associated virtual displacements are also considered.
1.6 Hamilton’s principle 15 δv(x, t) satisfy the geometric (kinematic) boundary conditions and are such that δv(x, t 1 ) =δv(x, t 2 ) = 0 (1.29)
(the configuration is fixed at the limit timest 1 and t 2 ).
The Euler-Bernoulli beam theory simplifies analysis by ignoring shear deformations and presuming that the cross section remains perpendicular to the neutral axis This leads to the assumption that the uniaxial strain distribution, S11, varies linearly with the distance from the neutral axis, expressed as S11 = −zv′′, where v′′ represents the beam's curvature Consequently, the potential energy in this framework is identified as strain energy.
The bending stiffness of a beam is defined by the equation EI(v ′′ )² dx, where v ′′ represents the beam's curvature, E is the Young's modulus, and I is the geometric moment of inertia of the cross-section When considering only translational inertia, the kinetic coenergy is also taken into account.
0 ̺A( ˙v) 2 dx (1.31) where ˙v is the transverse velocity,̺is the density andAthe cross section area The virtual work of the non-conservative forces is associated with the distributed load: δW nc Z L
As in the previous section, δv˙ can be eliminated by integrating by part overt, and similarly, δv ′′ can be eliminated by integrating twice by part overx; one gets δV Z L
16 1 Lagrangian dynamics of mechanical systems and similarly
The expression in the bracket vanishes because of (1.29) Substituting the above expressions in Hamilton’s principle, one gets
The variational indicator must equal zero for all permissible variations δv that adhere to the kinematic constraints outlined in equation (1.29) This condition indicates that the dynamic equilibrium is described by a specific partial differential equation.
Besides, cancelling the terms within brackets, we find that the following conditions must be fulfilled inx= 0 and x=L,
The first equation indicates that at both ends of a beam, either the rotation is fixed (δv ′ = 0) or the bending moment is zero (EIv ′′ = 0) Similarly, the second equation states that at both ends, either the displacement is fixed (δv = 0) or the shear force is zero ((EIv ′′ ) ′ = 0) The conditions δv = 0 and δv ′ = 0 are referred to as kinematic boundary conditions, while EIv ′′ = 0 and (EIv ′′ ) ′ = 0 are known as natural boundary conditions, arising from the variational principle A free end allows for arbitrary values of δv and δv ′, leading to the conditions EIv ′′ = 0 and (EIv ′′ ) ′ = 0.
A clamped end condition results in both δv and δv' being zero, while a pinned end condition allows δv to be zero but permits δv' to vary, leading to the conclusion that EIv'' equals zero It is important to note that the kinematic and natural boundary conditions are energetically conjugate, specifically relating displacement to shear force and rotation to bending moment.
In Chapter 5, we will revisit the Euler-Bernoulli beam theory by incorporating a piezoelectric layer Advanced beam theories that consider shear deformations and rotary inertia of the cross-section are also available, utilizing different kinematic assumptions for the displacement field These theories yield new formulations for strain energy (V) and kinetic coenergy (T*), with Hamilton's principle applied in a manner consistent with previous discussions.
Lagrange’s equations
Vibration of a linear, non-gyroscopic, discrete system 19
The general form of the kinetic coenergy of a linear non-gyroscopic, dis- crete mechanical system is
The kinetic co-energy, represented by the equation 2x˙ T Mx˙ (1.47), involves a set of generalized coordinates, x, and a mass matrix, M This mass matrix is symmetric and semi-positive definite, ensuring that any velocity distribution yields a non-negative kinetic co-energy value When M is strictly positive definite, it indicates that all coordinates possess associated inertia, making it impossible to achieve a velocity distribution where T ∗ = 0 Additionally, the general form of the strain energy is also considered in this context.
2x T Kx (1.48) whereKis the stiffness matrix, also symmetric and semi-positive definite.
A rigid body mode is a set of generalized coordinates with no strain energy in the system.K is strictly positive definite if the system does not have rigid body modes.
The Lagrangian of the system reads,
Assuming that the virtual work done by non-conservative external forces can be expressed as δW nc = f T δx, the application of Lagrange's equations leads to the derivation of the equation of motion.
Dissipation function
In the literature, it is customary to define thedissipation function Dsuch that the dissipative forces are given by
20 1 Lagrangian dynamics of mechanical systems
If this definition is used, Equ.(1.46) becomes d dt à∂L
∂q i =Q i (1.52) where Q i includes all the non-conservative forces which are not already included in the dissipation function Viscous damping can be represented by a quadratic dissipation function If one assumes
2x˙ T Cx˙ (1.53) in previous section, one gets the equation of motion
The equation Mx¨ + Cx˙ + Kx = f (1.54) represents a dynamic system where C denotes the viscous damping matrix, which is symmetric and semi-positive definite In this article, we will explore several mechanical system examples to highlight key features of the method.
Disk of mass and moment of inertia m I q1 q 2 g o
Fig 1.8 Pendulum with a sliding mass attached with a spring (a) and (b): Point mass (c) Disk.
Example 1: Pendulum with a sliding mass
In the described system, a mass \( m \) slides frictionlessly along a massless rod within a constant gravitational field \( g \), while a linear spring with stiffness \( k \) connects the mass to the pivot point \( O \) of the pendulum This setup features two degrees of freedom, with \( q_1 \) representing the position of the mass along the rod.
1.7 Lagrange’s equations 21 q 2 (angle of the pendulum) as the generalized coordinates It is assumed that, whenq1= 0, the spring force vanishes.
The kinetic coenergy is associated with the point massm; its velocity can be expressed in two orthogonal directions as in Fig.1.8(b); it follows that
The initial contribution to the system's dynamics is due to gravity, with the reference altitude set at the pivot point O, while the second contribution arises from the strain energy in the spring, which is considered unstretched when q1 equals zero The Lagrange equations governing the motion are expressed as d/dt(m q1² q̇²) + m g q1 sin(q2) = 0 and mq̈1 - m q1 q̇² - m g cos(q2) + k q1 = 0.
When considering a mass m as a disk with a moment of inertia I sliding along a massless rod, it introduces an additional term to the kinetic coenergy This term accounts for the rotational kinetic coenergy of the disk The total kinetic coenergy of a rigid body is the combination of the translational kinetic coenergy of the mass concentrated at the center of mass and the rotational kinetic coenergy around that center of mass.
The disk shares equivalent potential energy with the point mass Additionally, for a uniform rod with a total mass M and length l, its moment of inertia about the pivot can be calculated.
The total kinetic coenergy of the system is represented by the equation M = ̺l, which incorporates both translational and rotational energy, with the moment of inertia I 0 calculated at the pivot point Additionally, the potential energy includes a term of -m g l cos(q2)/2, considering that the center of mass of the rod is located at its midpoint.
22 1 Lagrangian dynamics of mechanical systems
Example 2: Rotating pendulum
The rotating pendulum, depicted in Figure 1.9(a), consists of a point mass m attached to a massless rod that pivots around a vertical axis at a constant angular velocity Ω, all while operating within a vertical gravitational field g.
In a system with a constant Ω and a single degree of freedom represented by the coordinate θ, the kinetic coenergy can be effectively expressed by projecting the velocity of a point mass onto an orthogonal frame This frame includes one axis that is tangent to the circular trajectory of the pendulum's rotation about the vertical axis with θ fixed, and another axis that is tangent to the mass's trajectory in the pendulum's plane when it does not rotate vertically The projected components of velocity are thus lΩsinθ and lθ, maintaining orthogonality in their relationship.
The first term in the equation is quadratic in ˙θ, specifically represented as T2 ∗ in (1.39), while the second term, T0 ∗ in (1.39), is independent of ˙θ and represents the potential of centrifugal forces When considering the reference altitude at the pivot, the gravitational potential is expressed as V = -gmlcosθ.
2 h(lθ)˙ 2 + (lΩsinθ) 2 i +gmlcosθThe corresponding Lagrange equation reads
1.7 Lagrange’s equations 23 ml 2 θ¨−ml 2 Ω 2 sinθcosθ+mglsinθ= 0
For small oscillations nearθ= 0, the equation can be simplified using the approximations sinθ≃θand cosθ= 1; this leads to θ¨+g lθ−Ω 2 θ= 0
One sees that the centrifugal force introduces a negative stiffness Figure1.9(c) shows the evolution of the frequency of the small oscillations of the pendulum withΩ; the system is unstable beyondΩ=ω 0
Example 3: Rotating spring mass system
A spring mass system rotates in the horizontal plane at a constant velocity Ω, characterized by a single degree of freedom represented by the spring's extension The absolute velocity of the point mass m can be effectively projected in the moving frame (x, y), with its components expressed as ( ˙u, uΩ).
Once again, there is a quadratic contribution,T2 ∗, and a contribution in- dependent of the generalized velocity,T 0 ∗ (potential of centrifugal force).
The kinetic coenergy of the rotating mechanism is constant and can be omitted when formulating the Lagrange equation In the absence of gravitational forces, the potential energy is linked to the spring's extension, with the spring force considered zero when the displacement is zero.
Lagrangian dynamics of mechanical systems leads to the Lagrange equation, represented as \( m\ddot{u} + (k - m\Omega^2)u = 0 \) or \( \ddot{u} + (\omega_n^2 - \Omega^2)u = 0 \), where \( \omega_n^2 = k/m \) This equation mirrors the linearized form of the previous example, indicating that the system becomes unstable when \( \Omega > \omega_n \).
Fig 1.10 Rotating spring-mass systems (a) Single axis (b) Two-axis.
Example 4: Gyroscopic effects
In the system depicted in Fig 1.10(b), the constraint along y = 0 has been replaced by an orthogonal spring, resulting in a configuration with two degrees of freedom (d.o.f.) This system is characterized by the generalized coordinates x and y, which represent displacements along moving axes that rotate at a constant speed, Ω Assuming small displacements, the stiffness values k1 and k2 denote the global stiffness along the x and y axes, respectively Additionally, there is viscous damping along the x-axis, characterized by a damping coefficient c1 The absolute velocity in the rotating frame is given by (˙x - Ωy, y˙ + Ωx), which contributes to the kinetic coenergy of the point mass.
As in the previous example, we disregard the constant term associated with the rotation at constant speed of the supporting mechanism Upon expandingT ∗ , one gets
In the analysis of gyroscopic forces, the term T1* representing the first order in generalized velocities emerges for the first time, particularly illustrated in the simplest system shown in Fig 1.10(b) Additionally, the potential energy V is linked to the extension of the springs, with the assumption that small displacements occur.
The damping force can be handled either by the virtual work, δWnc=−c1xδx˙ or with dissipation function (1.53) In this case,
2c1x˙ 2 The Lagrange equations read m¨x−2mΩy˙+c 1 x˙ +k 1 x−mΩ 2 x= 0 my¨+ 2mΩx˙+k 2 x−mΩ 2 y = 0 or, in matrix form, withq= (x, y) T ,
# are respectively the mass, damping and stiffness matrices, and
(1.56) is the anti-symmetric matrix of gyroscopic forces, which couples the mo- tion in the two directions; its magnitude is proportional to the inertia (m)
26 1 Lagrangian dynamics of mechanical systems and to the rotating speedΩ The contribution−Ω 2 M is, once again, the centrifugal force Note that, with the previous definitions of the matrices
M, G, K and C, the various energy terms appearing in the Lagrangian can be written
2q˙ T Cq˙ Note that the modified potential
2q T ³ K−Ω 2 M ´ q (1.58) is no longer positive definite ifΩ 2 > k 1 /mork 2 /m.
Let us examine this system a little further, in the particular case where k1 =k2=k and c1 = 0 Ifω 2 n =k/m, the equations of motion become ¨ x−2Ωy˙+ ³ ω n 2 −Ω 2 ´ x= 0 ¨ y+ 2Ωx˙+ ³ ω n 2 −Ω 2 ´ y= 0
To analyze the stability of the system, let us assume a solution of the formx=Xe pt ,y=Y e pt ; the corresponding eigenvalue problem is
Nontrivial solutions of this homogenous system of equations require that the determinant be zero, leading to the characteristic equation p 4 + 2p 2 (ω n 2 +Ω 2 ) + (ω n 2 −Ω 2 ) 2 = 0The roots of this equation are
Lagrange’s equations with constraints
The eigenvalues are entirely imaginary for all values of Ω, as illustrated in Figure 1.11, which depicts the evolution of natural frequencies with Ω, commonly referred to as a Campbell diagram Unlike the previous example, the system remains stable beyond Ω = ω n, thanks to the stabilizing effect of gyroscopic forces.
Fig 1.11 Campbell diagram of the system of Fig.1.10(b), in the particular case k 1 = k 2 and c 1 = 0.
In scenarios where the n generalized coordinates are dependent, the virtual changes in configuration, denoted as δq k, must adhere to a specific set of m constraint equations, as outlined in Equation (1.18).
The system's degrees of freedom are given by \( n - m \) In Hamilton's principle, the variations \( \delta q_i \) are constrained by Equation (1.59), making the transition from Equation (1.45) to (1.46) unfeasible This challenge can be addressed using Lagrange multipliers, which involves incorporating a linear combination of the constraint equations into the variational indicator.
28 1 Lagrangian dynamics of mechanical systems where the Lagrange multipliers λ l are unknown at this stage Equation (1.60) is true for any set ofλ l Adding to Equ.(1.45), one gets
In this equation, the variations δq k can be chosen freely as independent variables, necessitating that the corresponding expressions within brackets equal zero The remaining terms in the sum lack independent variations δq k, allowing us to choose Lagrange multipliers λ l to eliminate them as well Consequently, this leads to the result d dt(∂L).
The second term on the right side signifies the generalized constraint forces, represented as linear functions of the Lagrange multipliers This results in a system of equations containing n + m unknowns, which include the generalized coordinates q_k and the Lagrange multipliers λ_l By integrating these with the constraint equations, we derive a comprehensive set of n + m equations For non-holonomic constraints described by the form (1.15), the equations can be expressed accordingly.
X l=1 λ l a lk k= 1, , n (1.62) with the unknownq k , k= 1, , nandλ l , l= 1, , m If the system is holo- nomic, with constraints of the form (1.13), the equations become g l (q 1 , q 2 , q n ;t) = 0 l= 1, , m (1.63) d dt(∂L
∂q k k= 1, , n (1.64)This is a system ofalgebro-differential equations This formulation is fre- quently met in multi-body dynamics.
Conservation laws
Jacobi integral
When generalized coordinates are independent, the Lagrange equations form a set of n second-order differential equations, necessitating 2n initial conditions to define the system's configuration and velocity at t=0 In certain situations, these systems may possess first integrals of motion, which include derivatives of the variables that are one order lower than the differential equations The most renowned of these first integrals is the conservation of energy, which is a specific instance of the broader Jacobi integral relationship.
If the system is conservative (Q k = 0) and if the Lagrangian does not depend explicitly on time,
The total derivative ofL with respect to time reads dL dt n
On the other hand, from the Lagrange’s equations (taking into account thatQ k = 0)
Substituting into the previous equation, one gets dL dt n
30 1 Lagrangian dynamics of mechanical systems
Recall that the Lagrangian reads
L=T ∗ −V =T 2 ∗ +T 1 ∗ +T 0 ∗ −V (1.68) where T 2 ∗ is a homogenous quadratic function of ˙q k , T 1 ∗ is homogenous linear in ˙q k , and T 0 ∗ and V do not depend on ˙q k
According to Euler’s theorem on homogenous functions, if T n ∗ is an homogeneous function of order n in some variables qi, it satisfies the identity
It follows from this theorem that
∂q˙ k ả ˙ q k = 2T 2 ∗ +T 1 ∗ and (1.67) can be rewritten h=T 2 ∗ −T 0 ∗ +V =C t (1.70) This result is known as a Jacobi integral, or also a Painlev´e integral If the kinetic coenergy is a homogeneous quadratic function of the velocity,
The equation T ∗ + V = C t (1.71) represents the integral of energy conservation, applicable to conservative systems This is relevant for systems where the Lagrangian is not explicitly time-dependent, as indicated in Equation (1.65), and where the kinetic coenergy is defined as a homogeneous quadratic function of the generalized velocities (T ∗ = T 2 ∗).
The previously discussed equation [Equ.(1.23)] highlights a significant relationship with the current conditions Notably, equation (1.65) indicates that the potential is independent of time (t), while the condition T ∗ = T 2 ∗ suggests that the kinematic constraints also do not explicitly depend on time, as referenced in equations (1.38) and (1.39).
Ignorable coordinate
In a conservative system, if a generalized coordinate (denoted as q_s) is not explicitly present in the Lagrangian, meaning it appears only in its derivative form (˙q_s) and satisfies the condition ∂L/∂q_s = 0, this coordinate is referred to as ignorable This concept is derived from Lagrange's equations, highlighting the significance of ignorable coordinates in the analysis of dynamical systems.
∂q˙ s =C t and, since V does not depend explicitly on the velocities, this can be rewritten p s = ∂L
∂q˙ s =C t (1.72) p s is thegeneralized momentum conjugate to q s [by analogy with (1.10)]. Thus,the generalized momentum associated with an ignorable coordinate is conserved.
The existence of the first integral is highly dependent on the choice of coordinates, and it may remain concealed when inappropriate coordinates are utilized These ignorable coordinates, often referred to as cyclic coordinates, frequently correspond to rotational dimensions.
32 1 Lagrangian dynamics of mechanical systems
Example: The spherical pendulum
The spherical pendulum, as shown in Fig 1.12, is defined by two generalized coordinates, θ and φ Its kinetic coenergy and potential energy are essential components of its overall energy dynamics.
V =−mglcosθ and the Lagrangian reads
The Lagrangian is independent of time and the coordinate φ, making the system suitable for the two first integrals previously mentioned Additionally, because the kinetic energy is a homogeneous quadratic function of ˙θ and ˙φ, the principle of conservation of energy is applicable.
As for the ignorable coordinate φ, the conjugate generalized momen- tum is p φ =∂T ∗ /∂φ˙=ml 2 φ˙sin 2 θ=C t
The equation illustrates the conservation of angular momentum around the vertical axis Oz, indicating that the moments created by external forces, such as the pendulum's cable and gravity, effectively cancel each other out.
More on continuous systems
Rayleigh-Ritz method
The Rayleigh-Ritz method, known as the Assumed Modes method, is an approximation technique that converts partial differential equations into ordinary differential equations This method enables the representation of continuous systems through discrete approximations, facilitating easier analysis and computation.
1.10 More on continuous systems 33 expected to approximate the low frequency behavior of the continuous system To achieve this, it is assumed that the displacement field (as- sumed one-dimensional here for simplicity, but the approximation applies in three dimensions as well) can be written v(x, t) n
The equation X i=1 ψ i (x)q i (t) (1.73) represents a sum of assumed modes, ψ i (x), that are continuous and fulfill geometric boundary conditions, although they do not meet natural boundary conditions The time-dependent functions, q i (t), serve as the generalized coordinates for the approximate discrete system When the set of assumed modes is complete, such as in Fourier series or power series, the approximation approaches the exact solution as the number of modes, n, increases.
To demonstrate this method, we revisit the lateral vibration of the Euler-Bernoulli beam By approximating the transverse displacement as outlined in equation (1.73), we can easily convert the strain energy described in equation (1.30).
2q T Kq (1.74) whereK is the stiffness matrix, defined by
EIψ ′′ i (x)ψ j ′′ (x)dx (1.75) Similarly, the kinetic coenergy is approximated by
2q˙ T Mq˙ (1.76) where the mass matrix is defined as
0 ̺Aψi(x)ψj(x)dx (1.77) The reader familiar with the finite element method will recognize the form of the mass and stiffness matrices, except that the shape functionsψi(x)
Lagrangian dynamics of mechanical systems are established across the entire structure, adhering to geometric boundary conditions The matrices K and M are symmetric, ensuring that the potential energy V and kinetic energy T* align with the forms outlined in section 1.7.1, resulting in the differential equation (1.50) Additionally, when the trial functions ψi(x) correspond to the system's vibration modes φi(x), the matrices K and M, as defined by equations (1.75) and (1.77), become diagonal due to the orthogonality of the mode shapes, leading to a set of decoupled equations.
General continuous system
In preparation for the analysis of piezoelectric structures discussed in Chapter 4, we utilize the standard notations S_ij for the strain tensor and T_ij for the stress tensor These notations are essential for understanding the constitutive equations governing linear elastic materials.
T ij =c ijkl S kl (1.78) where c ijkl is the tensor of elastic constants The strain energy density reads
0 TijdSij (1.79) from which the constitutive equation may be rewritten
Green strain tensor
In mechanical engineering, particularly in beam theory, the infinitesimal definition of strain in linear elasticity is often adequate for analysis However, for problems involving large displacements and prestresses, a strain measure invariant to global system rotation is necessary, ensuring that rigid body motion results in S ij = 0 The Green strain tensor provides this representation, allowing for accurate assessment of strain in continuous bodies.
1.10 More on continuous systems 35 points before deformation, andA ′ B ′ be the same segment after deforma- tion; the coordinates are respectively:A :xi, B :xi+dxi, A ′ : xi +ui,
B ′ :x i +u i +d(x i +u i ) Ifdl 0 is the initial length ofABanddl the length ofA ′ B ′ , it is readily established that dl 2 −dl 0 2 = (∂u i
)dx i dx j (1.82) The Green strain tensor is defined as
The Green strain tensor is symmetric and incorporates both a classical linear component, which measures strain in linear elasticity, and an additional quadratic component that addresses large rotations The equation dl² - dl²₀ = 2Sᵢⱼ dxⁱ dxⱼ indicates that when Sᵢⱼ equals zero, the segment length remains unchanged, even with significant displacements This tensor effectively captures the complexities of large rotations in material deformation.
Sij =S ij (1) +S ij (2) (1.85) whereS ij (1) is linear in the displacements, andS ij (2) is quadratic.
Geometric strain energy due to prestress
The lateral stiffness of strings and cables is influenced by their axial tension force, while long rods experience modified lateral stiffness under large axial forces; compressive forces decrease the natural frequency, whereas tensile forces increase it When the axial compressive load surpasses a certain threshold, the rod buckles, resulting in a buckling load that reduces the natural frequency to zero Geometric stiffness plays a crucial role in structures subjected to significant dead loads, as these loads contribute substantially to the system's strain energy.
Consider a continuous system in a prestressed state (T ij 0 , S ij 0 ) indepen- dent of time, and then subjected to a dynamic motion (T ij ∗ , S ij ∗ ) The total stress and strain state is (Fig.1.13)
36 1 Lagrangian dynamics of mechanical systems
Fig 1.13 Continuous system in a prestressed state.
It is impossible to account for the strain energy associated with the pre- stress if the linear strain tensor is used If the Green tensor is used,
S ij ∗ =S ij ∗ (1) +S ij ∗ (2) (1.87) it can be shown (Geradin & Rixen, 1994) that the strain energy can be written
The additional strain energy, represented by the equation Ω ∗ c ijkl S ij ∗ (1) S kl ∗ (1) dΩ (1.89), accounts for the linear deformation that occurs beyond the prestress level This term is significant in scenarios where no prestress is applied.
The geometric strain energy due to prestress is represented by the equation Ω ∗ T ij 0 S ij ∗ (2) dΩ (1.90), where T ij 0 denotes the prestressed state and the strain tensor's quadratic component Unlike the always-positive V ∗, the value of V g can vary, being either positive or negative based on the prestress's sign A positive V g contributes to the system's rigidity, while a negative value results in a softening effect In discrete systems, V g is expressed in a general form.
2x T K g x (1.91) whereK g is thegeometric stiffness matrix, no longer positive definite since
The geometric stiffness plays a crucial role in the overall stiffness of rotating helicopter blades, while the reduction of natural frequencies in civil engineering structures is commonly attributed to dead loads.
Lateral vibration of a beam with axial loads
Fig 1.14 Euler-Bernoulli beam with axial prestress.
Consider again the in-plane vibration of a beam, but subjected to an axial loadN0(x) (positive in traction) The displacement field is u=u 0 (x)−z∂w
∂x v= 0 w=w(x) The axial preload atx is
The Green tensor is in this case
∂x) 2 ] (1.93) and, assuming large rotations but small deformations,
∂x and (∂u/∂x) 2 can be neglected It follows that the linear part of the Green tensor is
(as in section 1.6.1), and the quadratic part
Accordingly, the additional strain energy due to the linear part
38 1 Lagrangian dynamics of mechanical systems is identical to (1.30), and the geometric strain energy due to prestress is
N 0 (x)(w ′ ) 2 dx (1.97) where the axial preloadN 0 (x) is positive in traction.
Example: Simply supported beam in compression
A simply supported beam under a constant axial compression load P can have its first natural frequency estimated using the Rayleigh-Ritz method with a single mode approximation, represented by the equation w=qsin(πx).
With this assumption, the Lagrangian reads
Note that V g contributes negatively to the potential energy because the loadP is compressive The potential energy can be rearranged
P cr ] (1.100) where P cr = π 2 EI/L 2 is the well known Euler’s critical buckling load. The Lagrangian (1.99) is that of a single d.o.f oscillator; the correspond- ing natural frequency is ω 2 1 = π 4 EI ̺AL 4 [1− P
The first term represents the exact natural frequency for a simply supported beam without prestress, while the second term accounts for the correction due to axial loading It is observed that a compressive load decreases the natural frequency (ω1), reaching zero when the load (P) equals the critical load (P cr) In contrast, a tensile load increases ω1 The accurate determination of the natural frequency is achieved through an approximation technique, as the assumed mode shape is precisely the exact mode of the problem Any alternative assumption that meets the geometric boundary conditions would result in a higher ω1, since the Rayleigh-Ritz method typically overestimates natural frequencies.
Introduction
Electrical networks consist of passive components like resistors, capacitors, and inductors, alongside active elements such as voltage and current sources The interconnection of these components is governed by Kirchhoff’s rules, which outline the fundamental constraints within electrical circuits.
Kirchhoff’s current rule (KCR) stipulates that no electrical charge can accumulate at a node in the network : the algebraic sum of the currents entering any node must be zero.
Kirchhoff’s voltage rule (KVR) states that the voltage between any two points in a network remains constant regardless of the path taken This principle asserts that the algebraic sum of voltage drops around any closed loop in a circuit must equal zero.
This chapter explores the variational approach for analyzing electrical networks, offering an alternative to the direct method based on Kirchhoff’s rules Notably, this approach can be integrated with the variational analysis of mechanical systems to study the dynamics of electromechanical systems The discussion is grounded in quasi-static electromagnetic field theory, which assumes that the electromagnetic field's time variation is slow enough to disregard interactions between the electric and magnetic fields This condition necessitates that the device size, l, is significantly smaller than the wavelength, satisfying the criterion l/λ ≪ 1.
Fig 2.1 Capacitor (a) network schematic (b) constitutive relation.
Constitutive equations for circuit elements
The Capacitor
A capacitor consists of two conductive surfaces separated by a dielectric material When a charge of q Coulombs is applied to one surface and removed from the other, the current flowing through the capacitor is defined as the rate of change of charge, expressed by the equation i = dq/dt.
During the charging of a capacitor, a voltage difference (e) develops between its conductive surfaces The static relationship between this voltage and the charge (q) can be quantified, defining the capacitor's constitutive relation, e(q).
The electrical energy W e (q) stored in a capacitor is the work done in charging the capacitor from no charge toq Since the power input isP ei(Fig 2.1.a),
This integral is the area below the curve in Fig.2.1.b It follows that e= dW e dq (2.3)
Usual capacitors are nearly linear, and their constitutive equations can be written
2.2 Constitutive equations for circuit elements 43 q (2.4)
As for the kinetic energy in section 1.2, a complementary state function can be defined by the Legendre transformation
W e ∗ (e) is calledelectrical coenergy function of the capacitor; it represents the area above the curve in Fig.2.1.b The total differential of the electrical coenergy is dW e ∗ =q de+e dq− ∂We
∂q dq=q de where (2.3) has been used It follows that q = dW e ∗ de (2.7) and
For a linear capacitor with the constitutive equation (2.4),
The Inductor
When current flows through a conductor, it generates a magnetic field whose strength is proportional to the current Conversely, a changing magnetic field within a conductor induces a voltage This induction is significant in conductors with tightly packed coils, where the magnetic behavior is typically linear in air However, adding a ferromagnetic core to the coil greatly enhances the magnetic flux density, resulting in nonlinear and hysteretic behavior According to Faraday’s law, the voltage across an inductor is directly related to the rate of change of flux linkage.
Fig 2.2 Inductor (a) network schematic (b) constitutive relation. e= dλ dt (2.10)
The unit of flux linkage is the Weber or volt-second, and in an ideal inductor, flux linkage (λ) solely depends on the instantaneous current A constant current results in a constant λ, leading to an electromotive force (e) of zero, causing the ideal inductor to act like a perfect conductor or short-circuit For a linear inductor, λ is expressed as λ = Li, where L represents inductance, measured in Henrys or Webers per Ampere The magnetic energy (Wm) stored in an ideal inductor is defined as the work done when changing its magnetic state from no flux linkage to a certain flux linkage λ, calculated by integrating the power delivered to the circuit.
0 i dλ (2.12) where (2.10) has been used; it represents the area below the curve of Fig 2.2.b It follows that i= dW m dλ (2.13)
If the coil exhibits a linear behavior as in (2.11),
A magnetic coenergy function W m ∗ is defined by the Legendre transfor- mation
2.2 Constitutive equations for circuit elements 45
W m ∗ (i) =λi−W m (λ) (2.15) The total differential of the magnetic coenergy is dW ∗ =λ di+i dλ−dWm dλ dλ=λ di using (2.13) It follows that λ= dW m ∗ (i) di (2.16) and
It represents the area above the curve of Fig.2.2.b For a linear inductor (2.11),
Voltage and current sources
Fig 2.3 (a) Real source and its voltage-current characteristic (b) Ideal voltage source model (c) Ideal current source model.
An ideal voltage source delivers a consistent voltage regardless of the current, while an ideal current source provides a stable current irrespective of the voltage In contrast, real sources exhibit linear characteristics, as illustrated in Fig 2.3.a, operating between a maximum open-circuit voltage (E0) and a maximum short-circuit current (I0).
A real source can be modelled by ideal sources combined with resistors; in Fig 2.3.b, an ideal voltage source is connected in series with a resistance
R, leading to the characteristic es=E0−Ris
Alternatively, combining a current sourceI 0 in parallel with a resistance
R, one gets the characteristics e s =RI 0 −Ri s
Modern power electronics enable the development of voltage amplifiers that function almost like ideal voltage sources within specific frequency ranges, as well as current amplifiers that operate similarly to ideal current sources in designated frequency ranges.
Kirchhoff’s laws
Electrical networks are formed by connecting passive components and power sources, which create relationships among the variables that characterize each element These relationships are governed by Kirchhoff's laws, which outline the fundamental principles of electrical interconnections.
Kirchhoff’s Current Rule (KCR) asserts that the total current entering a node in an electrical circuit must equal the total current leaving that node, ensuring that the sum of currents at the node is zero This principle embodies the conservation of electric charge, highlighting that electric charge cannot build up at any point within the network.
Kirchhoff’s Voltage Rule (KVR) asserts that the total voltage drops across all elements in a closed loop must equal zero, highlighting that the electric potential at any point remains constant regardless of the path taken to reach it.
In the analysis of a passive network, the following requirements must be satisfied;
Hamilton’s principle for electrical networks
Hamilton’s principle, charge formulation
In this formulation, the generalized variables are the charges and currents. Admissible variationsδq i must satisfy Kirchhoff’s current rules, and the current and charges variables must satisfyi k = ˙q k =dq k /dt.
By analogy with Equ.(1.22), we can write the virtual work expression
(e i −dλi dt )δq i = 0 (2.19) whereM is the number of circuit elements The first contribution to this sum can be separated into its conservative and non-conservative parts;
The electrical energy function of the system is denoted as W, while E k represents the generalized voltage associated with nonconservative elements, corresponding to the independent generalized charge variable q k The number of independent generalized charge coordinates is indicated by n e The negative sign in δWe reflects that when a conservative circuit element does work, the electrical energy within that element decreases Additionally, the second contribution to the sum can be reformulated for clarity.
2.4 Hamilton’s principle for electrical networks 49
The first term on the right side represents a total time derivative that can be eliminated by integrating over the interval [t1, t2], provided that the system configuration is known at both t1 and t2, leading to δqi(t1) = δqi(t2) = 0 The second term on the right side of equation (2.21) is then reformulated accordingly.
X i=1 λiδii =δW m ∗ (2.23) after permutingd/dt and δ, and using (2.17) Finally, integrating (2.19) between two fixed configurations att1 and t2 and combining Equ.(2.20)- (2.23), one gets
The actual path is that which cancels the variational indicator (2.25) with respect to all admissible charge variations δq i of the path between two instantst 1 and t 2 , and such that δq i (t 1 ) = δq i (t 2 ) = 0.
The magnetic coenergy function, W m ∗, represents the total magnetic coenergies of individual inductors in a network, expressed through the currents i j Conversely, the electrical energy function, W e, is the aggregate of electrical energies from individual capacitors, represented in terms of qj For the system to be valid, the currents and charges must adhere to Kirchhoff’s current law, with the relationship i j = dq j / dt The difference W m ∗ - W e defines the network's Lagrangian There exists a complete analogy between these equations, where P E k δq k corresponds to the virtual work done by nonconservative elements, indicated as δW nc.
Hamilton’s principle, flux linkage formulation
In this section, we explore the dual formulation involving the generalized variables of flux linkages (λ k) and voltages (e k) It is essential that the admissible variations adhere to Kirchhoff's voltage rule, ensuring that the relationship between flux linkage and voltage is maintained, specifically that e k equals the derivative of λ k with respect to time (dλ k /dt).
By analogy with (2.19), the virtual work expression reads
In the equation (i k − dq k dt )δλ k = 0, where N represents the number of circuit elements, the first contribution can be divided into conservative and non-conservative components Notably, as indicated in (2.12), the conservative part is associated with the magnetic energy W m of all conservative elements within the network.
In the context of network analysis, I_k represents the generalized currents associated with nonconservative elements, while λ_k denotes the conjugate flux linkage coordinates The variable n_e indicates the number of independent flux linkage coordinates Following the methodology outlined in the previous section, the second contribution to equation (2.26) can be reformulated accordingly.
The first term on the right side, representing a total time derivative, will become zero after integrating between the specified system configurations at times t1 and t2, leading to δλk(t1) = δλk(t2) = 0 Consequently, the second term on the right side of equation (2.28) is reformulated.
X k=1 q k δe k =δW e ∗ (2.30) where we have used (2.8) and the commutability ofδand (˙) Finally, upon integrating Equ.(2.26) betweent 1 andt 2 , assuming that the configuration is fixed att 1 and t 2 , and combining with (2.27)-(2.30), one gets
2.4 Hamilton’s principle for electrical networks 51
The actual path is that which cancels the variational indicator (2.32) with respect to all admissible flux linkage variations δλ k of the path between two instantst 1 and t 2 , and such that δλ k (t 1 ) =δλ k (t 2 ) = 0.
The electrical coenergy function, denoted as W ∗, represents the total electrical coenergies of individual capacitors in a network and is expressed in terms of the voltage e k Meanwhile, the magnetic energy function, W m, accounts for the sum of magnetic energies of individual inductors, defined by the independent flux linkage variables λ k For these parameters to be valid, they must adhere to Kirchhoff’s voltage rule, ensuring that e k equals the derivative of λ k with respect to time The difference between W ∗ and W m constitutes the Lagrangian of the network within this framework.
Discussion
Hamilton's principle reveals striking similarities between mechanical systems and electrical networks The Lagrangian is defined as the coenergy, which relies on the time derivatives of the generalized coordinates, subtracted by the energy that is dependent solely on the generalized coordinates, excluding their time derivatives.
L=W m ∗ (i k )−W e (q k ) q k = electric charge, i k = ˙q k (b) flux linkage formulation
L=W e ∗ (e k )−W m (λ k ) λ k = flux linkage, e k = ˙λ k The virtual work of the nonconservative elements reads, respectively δW nc =Q i δq i Q i = generalized force δW nc =E k δq k E k = generalized voltage δW nc =I k δλ k I k = generalized current
In electrical networks, work expressions are classified as positive when an element supplies energy and negative when it absorbs energy For instance, in the case of a resistor (R), the relationship is defined by the equation e = -Ri In the charge formulation, the virtual work expression can be represented as e δq = -Ri δq = -Rq δq˙, while in the flux linkage formulation, it is expressed as i δλ = -e.
An ideal voltage source E(t) contributes e δq = E(t)δq in the charge formulation, but does not affect the flux linkage formulation due to the fixed nature of the voltage over time While voltage sources do not appear in the virtual work expression, they are integral to the flux linkage formulation of Hamilton's principle, serving as part of the admissibility criteria for voltages and flux linkages.
Nonconservative element e=dõ=dt i =dq=dt
Fig 2.4 The virtual work increment delivered to the network by the element is e δq or i δλ.
An ideal current source I(t) contributes to the virtual work expression in the flux linkage formulation by adding I(t)δλ, but it does not affect the charge formulation due to the restriction on current variation In this context, current sources are included in the charge formulation of Hamilton’s principle as part of the admissibility requirements.
Lagrange’s equations
Lagrange’s equations, charge formulation
In the charge formulation of Lagrange’s equations, n generalized charge coordinates,q k , are selected such that their time derivatives, ˙q k , constitute independent loop currents in the network The Lagrangian is
The equation L( ˙q k , q k ) = W m ∗ ( ˙q k ) − W e (q k ) describes the relationship between magnetic coenergy and electrical energy in a network Here, W m ∗ ( ˙q k ) represents the magnetic coenergy, calculated as the total of the individual inductors' coenergies, while W e (q k ) denotes the electrical energy, derived from the sum of the individual capacitors' energies Additionally, the virtual work done by nonconservative elements is articulated through independent generalized coordinates, adhering to the principle of work equality.
The equation E k δq k = Σ i ǫ i δq i represents the relationship between independent generalized coordinates and nonconservative elements This formulation leads to Lagrange's equations, expressed as d dt(∂L), which are essential for analyzing dynamic systems.
∂q k =E k k= 1, , n (2.34) whereE k is the generalized voltage associated with the generalized charge coordinateq k
Lagrange’s equations, flux linkage formulation
The flux linkage formulation is dual of the charge formulation;nindepen- dent flux linkage coordinatesλ k are selected, which automatically satisfy Kirchhoff’s voltage rule The Lagrangian is
The equation L( ˙λ k , λ k ) = W e ∗ ( ˙λ k ) − W m (λ k ) describes the relationship between electrical coenergy and magnetic energy in a network Here, W e ∗ ( ˙λ k ) represents the total electrical coenergy derived from the independent voltages ˙λ k, which is the cumulative electrical coenergy of all capacitors Conversely, W m (λ k ) denotes the magnetic energy of the network, calculated from the independent flux linkage variables and representing the total magnetic energy of all inductors Additionally, the virtual work done by nonconservative elements is articulated through generalized flux linkage coordinates, adhering to the principle of work equality.
The equation I i δλ i represents a sum over independent flux-linkage generalized coordinates on the left side, while the right side encompasses all nonconservative elements This formulation leads to the derivation of Lagrange’s equations, expressed as d dt(∂L).
∂λ k =I k k= 1, , n (2.36) where I k is the generalized current associated with the generalized flux linkageλ k
Example 1
Consider the electrical network of Fig.2.5 We shall write the dynamic equations of the network using successively the flux linkage formulation and the charge formulation.
We select flux linkage coordinates λi at all ungrounded nodes; this leads to 3 independent coordinatesλ 1 ,λ 2 ,λ 3 Since the voltage drops across the capacitorsC 1 andC 2 are respectively ˙λ 2 −λ˙ 1 and ˙λ 3 −λ˙ 2 The electrical coenergy reads
Fig 2.5 Electrical network, flux linkage formulation.
The flux linkage across the inductor isλ 1 (the grounded node is taken as reference,λ= 0) and the magnetic energy is
On the other hand, the virtual work of the nonconservative elements is δWnc =I δλ2−λ˙3
The primary contribution involves the ideal current generator, while the secondary contribution pertains to the resistor R, where the current flowing through the resistor is expressed as ˙λ 3 /R It is important to note that the virtual work associated with the resistor is negative due to its nature as a dissipative element In the context of Lagrange's equations, the generalized currents are defined as I 1 = 0, I 2 = I, and I 3 = −λ˙ 3 /R.
This system of three ordinary differential equations, in the three variables λ 1 ,λ 2 ,λ 3 , governs the dynamics of the network.
Fig 2.6 Electrical network, charge formulation.
In this article, we revisit the charge formulation by defining a set of two loops, each associated with charge variables q1 and q2, which represent the loop currents The chosen positive direction for the currents is clockwise, as illustrated in Fig 2.6 However, the currents q1 and q2 are not independent due to the presence of a current generator, leading to the admissibility condition q2 = q1 + I(t), where I(t) is equivalent to q0 By integrating this relationship, we derive that q2 equals q1 plus q0.
In this context, q 0 represents a fixed input to the system, specifically the electric charge supplied by the current source Consequently, the variations in virtual charge are equal, expressed as δq 2 = δq 1, leading to the conclusion that there is only one generalized coordinate, q 1 The Lagrangian for the system is defined accordingly.
(q 1 +q 0 ) 2 and the virtual work of the nonconservative elements is δW nc =−Rq˙ 2 δq 2 =−R( ˙q 1 + ˙q 0 )δq 1
There is no contribution from the current source in this formulation. Lagrange’s equations relative toq 1 is
C 2 with ˙q 0 = I(t) Comparing with the flux linkage formulation, we note that, in this case, the charge formulation is more compact and involves only one generalized variable instead of three.
Fig 2.7 Electrical network, charge formulation.
Example 2
In this article, we will derive the dynamic equation of the electrical network illustrated in Fig 2.7, utilizing both the charge formulation and the flux linkage formulation This example incorporates ideal voltage and current sources, allowing for a comparative analysis of their contributions to the virtual work of nonconservative elements Our focus will be on formulating the Lagrangian and determining the virtual work associated with these nonconservative elements, as Lagrange’s equations can be straightforwardly obtained from these foundational components.
The current loops, as illustrated in Fig 2.7, reveal that the currents ˙q2 and ˙q3 are interdependent due to the current source, adhering to the admissibility condition ˙q3 = ˙q2 + I, where I(t) equals ˙q0 Consequently, this leads to the relationship q3 = q2 + q0 and δq3 = δq2, as q0 remains unaffected by virtual variations Additionally, the contributions from all conservative elements to both magnetic coenergy and electrical energy are outlined.
The virtual work of nonconservative elements, such as resistors and voltage sources, is represented by the equation δW nc =−R( ˙q 1 −q˙ 2 )(δq 1 −δq 2 ) +E δq 1 It is important to note that while the current source does not contribute to δW nc, it is still present in both W m ∗ and W e, associated with ˙q 0 and q 0, respectively.
Introduction
This chapter explores the dynamics of composite systems that integrate mechanical and electrical variables, focusing on the crucial role of electromechanical transducers in converting mechanical energy to electrical energy and vice versa Common examples of these transducers include microphones, loudspeakers, electrical motors, magnetic suspensions, capacitive accelerometers, and microelectromechanical systems (MEMS), with piezoelectric transducers discussed in a subsequent chapter The chapter begins with a review of the constitutive equations for the most common lossless lumped parameter electromechanical transducers.
This article presents Hamilton's principle for electromechanical systems, outlines the Lagrange equations, and provides several examples that establish the dynamic equations governing classical electromechanical systems.
Constitutive relations for transducers
Movable-plate capacitor
A movable plate capacitor serves as a conservative energy-storing transducer, facilitating the conversion of electrical energy into mechanical energy and vice versa In this system, the charge on the capacitor is denoted as q, while the voltage across the plates is represented by e The displacement of the movable plate is indicated by x, and the external force needed to maintain the plate's equilibrium against electrostatic attraction is labeled as f This electromechanical transducer is considered ideal, with the electrical side functioning as a pure capacitor.
3.2 Constitutive relations for transducers 63 itance, and the mechanical construction is massless and without stiffness or damping.
In a movable-plate capacitor, the constitutive relations can be expressed through equations that define voltage and force as functions of the independent variables: movable plate displacement (x) and electrical charge (q) Specifically, the relationships are represented as e = e(x, q) and f = f(x, q).
The equations governing the electrostatic field can be derived from theoretical principles or through experimental methods It is essential that these equations ensure that when the charge (q) is zero, the electric field is nonexistent, resulting in a force (f) of zero for all positions (x).
The total power supplied to the capacitor combines both electric power, denoted as ei, and mechanical power, represented by f v Consequently, the net work performed on the capacitor over a time interval dt can be expressed as dW = ei dt + f v dt, which simplifies to dW = e dq + f dx.
In a conservative system, the work is transformed into stored electrical energy, denoted as dW e The total electrical energy, W e (x, q), encompasses both electrical and mechanical work and is derived by integrating from the reference state to the coordinates (x, q) Once the electrical energy function W e (x, q) is established, the constitutive equations can be retrieved through differentiation.
As in the previous chapter, the complementary state function calledelec- trical coenergy function is defined by the Legendre transformation
W e ∗ (x, e) =eq−We(x, q) (3.5) The total differential of the coenergy is dW e ∗ =q de+e dq−∂W e
∂q dq and it follows from (3.4) that
Assuming that the capacitor is electrically linear, one can find the explicit form of the state functionsW e (x, q) andW e ∗ (x, e) In this case, e= q
C(x) (3.7) whereC(x) is the capacitance corresponding to the positionxof the mov-
A movable-plate capacitor consists of two flat plates separated by a constant distance, where the capacitance is defined as C(x) = εA/x, with ε representing the dielectric constant and A the plate area The electrical energy function W_e(q, x) is derived by integrating from a reference state to the point (q, x) In a conservative system, this integral is path-independent By considering a path made up of two straight segments, from (0,0) to (x,0) and then from (x,0) to (x,q), we can simplify the calculations, noting that the force f and electric field e are zero along the first segment, while the distance dx remains constant along the second.
2C(x) (3.8) and, from the Legendre transformation (3.5) and the constitutive equation (3.7),
The constitutive equations of the linear movable plate capacitor follow from (3.4) and (3.6),
The state functions outlined in equations (3.8) and (3.9) were derived under the assumption of a conservative system, independent of the specific form of f(x, q) presented in equation (3.1) Once the explicit expressions for W e and W e ∗ are determined, the mechanical force required to counterbalance the electrostatic force within the capacitor can be calculated using equations (3.10) and (3.11).
Movable-core inductor
An ideal movable-core inductor serves as a conservative energy-storing transducer, analogous to a movable-plate capacitor In this context, λ represents the coil's flux linkage, e denotes the input voltage, i signifies the input current, x indicates the iron core's displacement, and f is the external force necessary to maintain the core's equilibrium against magnetic attraction Assuming the electrical side functions as a perfect inductor without hysteresis and that the mechanical structure is massless and frictionless, we can express the constitutive equations for i and f in terms of λ and x, treating them as independent variables.
The constitutive equations can be derived from either electromagnetic field theory or experimental data, assuming single-valued functions without hysteresis It is essential that when λ equals 0, there is no magnetic field or attraction present Consequently, the force f that counteracts the magnetic attraction must equal 0 for all values of x, represented by the equation f(x, 0) = 0.
The total power supplied to the solenoid combines both electrical power (ei) and mechanical power (f v) Consequently, the net work done on the solenoid can be expressed as dW = ei dt + f v dt, which simplifies to i dλ + f dx (3.14).
In a conservative transducer, the work is transformed into stored magnetic energy within the solenoid, represented as dW m This stored magnetic energy can be articulated using independent variables, denoted as Wm(x, λ), leading to the expression dW m = ∂Wm.
∂x dx (3.15) and, by comparison with (3.14), one recovers the constitutive equations i= ∂W m
If the system is conservative, the total stored magnetic energy is obtained by integrating (3.14) along any path from the reference state to (x, λ).
As for the movable plate capacitor, the magnetic coenergy function is defined by the Legendre transformation
Upon taking the total differential of the coenergy and taking into account (3.16), one finds an alternative form of the constitutive equations λ= ∂W m ∗
When the constitutive relation between the flux linkage and the current is linear,
3.2 Constitutive relations for transducers 67 λ=L(x)i (3.19) whereL(x) is the inductance of the coil when the core is in the positionx.
To evaluate the magnetic stored energy \( W_m(x, \lambda) \), one can explicitly integrate from the reference state to the point \( (x, \lambda) \) In a conservative system, this integral is path-independent, allowing the use of a straightforward path consisting of two segments: from \( (0,0) \) to \( (x,0) \) and from \( (x,0) \) to \( (x, \lambda) \) Notably, when \( \lambda = 0 \), the condition \( f = 0 \) can be applied to simplify the calculations.
Fig 3.4 Movable-core inductor: integration path for the magnetic energy.
The magnetic coenergy is obtained from (3.17), using (3.19) and (3.20):
The constitutive relations can be recovered from (3.16) and (3.18): f = ∂W m
Moving-coil transducer
A moving-coil transducer functions as an energy transformer, converting electrical power to mechanical power and vice versa It comprises a permanent magnet that generates a uniform magnetic flux density perpendicular to the gap, along with a coil that can move axially within this gap The system operates with variables such as the coil's velocity (v), the external force (f) that maintains equilibrium against electromagnetic forces, the voltage difference (e) across the coil, and the current (i) flowing into the coil.
In an ideal transducer, we overlook the electrical resistance, self-inductance, mass, and damping of the coil, which can be addressed by incorporating resistance and inductance into the electrical circuit or adding mass and damping to its mechanical model The voice coil actuator is a widely used component in mechatronics, commonly found in applications such as electromagnetic loudspeakers, and it also serves as a sensor in devices like geophones.
The constitutive equations for moving-coil transducers are derived from Faraday’s law and the Lorentz force law According to Faraday’s law, the voltage increment (de) over an elementary length (dl) in the direction of current flow, induced by the coil's motion, is expressed as de = ~v × B · dl.
On the other hand, a charge particle moving in an electromagnetic field (electric fieldE~ and magnetic flux densityB) is subjected to the Lorentz~ force f~=q(E~ +~v×B~) (3.25)
In the macroscopic realm, the magnetic force predominates, allowing us to neglect the electrostatic contribution When examining a current comprised of numerous charged particles, specifically electrons, moving through a conductor, the total force exerted by the field on an infinitesimal segment of the conductor, denoted as \( df = i \, dl \times B \), can be expressed mathematically.
By applying equation (3.24) to an elementary length \( dl = r d\theta \) for one complete turn of the coil, it is determined that the voltage increment in the direction of the current flow can be expressed as \( de = \mathbf{v} \times \mathbf{B} \cdot dl = -v B r d\theta \) This relationship illustrates the orthogonal nature of \( \mathbf{B} \), \( \mathbf{v} \), and \( dl \) as shown in Fig 3.5(b).
Integrating over θ, assuming that B is uniform in the gap, the voltage drop in the coil, in the direction of the current, is e= 2πnrBv=T v (3.27) where
T = 2πnrB (3.28) is thetransducer constant, equal to the product of the length of the coil exposed to the magnetic flux, 2πnr, and the magnetic flux density B.
On the other hand, the Lorentz force of the magnetic field acting on the element dl=rdθ of one turn of the coil with a current i follows from (3.26). df =irdθB (3.29)
The external force necessary to balance the total magnetic field force on the conductor's turns is depicted in Fig 3.5(a) By integrating equation (3.29) over the conductor's length exposed to the magnetic flux density, we derive the force as f = −i2πnrB = −Ti (equation 3.30), where T represents the transducer constant (equation 3.28) Equations (3.27) and (3.30) define the constitutive equations for the movable-coil transducer, as illustrated in Fig 3.6.
70 i e=dõ=dt v =dx=dt f e= Tv àTi = f
Fig 3.6 Symbolic representation of a moving-coil transducer.
Notice that the transducer constantT appearing in Faraday’s law (3.27), expressed in volt.sec/m, is the same as that appearing in the Lorentz force (3.30), expressed inN/Amp.
The total power in a moving-coil transducer is the sum of electric power (ei) and mechanical power (fv), represented by the equation ei + fv = T vi - T iv = 0 This indicates a constant equilibrium between the electrical power absorbed and the mechanical power delivered, highlighting the transducer's role as a perfect electromechanical converter However, in practice, the transfer is imperfect due to factors such as eddy currents, flux leakage, and magnetic hysteresis, resulting in varying values of T.
In the case of the movable core inductor, the magnetic energy \( W_m \) is constant and can be referenced as \( W_m = 0 \) The constitutive equation can be expressed in terms of flux linkage \( \lambda = T(x - x_0) \), where \( x_0 \) represents an arbitrary reference state This formulation enables the expression of the coenergy function effectively.
W m ∗ (x, i) =λi−Wm(x, λ) =T i(x−x0) (3.33) Using (3.18), we recover the constitutive equations f =−∂W m ∗