Page 105 - The Mechatronics Handbook

0066_Frame_C07 Page 15 Wednesday, January 9, 2002 3:39 PM

When there are no mechanical elements in the system, the dynamical equations take the form of
conservation of charge and the Faraday–Henry law of flux change.

dQ
------- = I (Conservation of charge) (7.38)
dt
df V (Law of flux change) (7.39)
------ =
dt

where φ = NΦ is called the number of flux linkages, and N is an integer. In electromagnetic circuits the
analog of mechanical constitutive properties is inductance L and capacitance C. The magnetic flux in an
inductor, for example, often depends on the current I.
f = fI() (7.40)

For a linear inductor we have a definition of inductance L, i.e., φ = LI. If the system has a mechanical
state variable such as displacement x, as in a magnetic solenoid actuator, then L may be a function of x.
In charge storage circuit elements, the capacitance C is defined as

Q = CV (7.41)

In MEMS devices and in microphones, the capacitance may also be a function of some generalized
mechanical displacement variable.
The voltages across the different circuit elements can be active or passive. A pure voltage source can
maintain a given voltage, but the current depends on the passive voltages across the different circuit
elements as summarized in the Kirchhoff circuit law:

d
Q
----- Lx()I + ----------- + RI = Vt() (7.42)
dt Cx()
Lagrange’s Equations of Motion for Electromechanical Systems

It is well known that the Newton–Euler equations of motion for mechanical systems can be derived using
an energy principle called Lagrange’s equation. In this method one identifies generalized coordinates
{q k }, not to be confused with electric charges, and writes the kinetic energy of the system T in terms of
generalized velocities and coordinates, T(q ˙ k , q k ). Next the mechanical forces are split into so-called
conservative forces, which can be derived from a potential energy function W(q k ) and the rest of the
forces, which are represented by a generalized force Q k corresponding to the work done by the kth
generalized coordinate. Lagrange’s equations for mechanical systems then take the form:

()
d ∂Tq ˙ ,q k ) ∂T ∂Wq k
(
k
----------------------------- – -------- + ------------------ = Q k (7.43)
dt ∂q ˙ k ∂q k ∂q k
For example, in a linear spring–mass–damper system, with mass m, spring constant k, viscous damping
constant c, and one generalized coordinate q 1 = x, the equation of motion can be derived using, T =
1 2 1 2
-- mx ˙ , W = kx , Q 1 = −c , in Lagrange’s equation above. What is remarkable about this formulationx ˙
--
2 2
is that it can be extended to treat both electromagnetic circuits and coupled electromechanical problems.
As an example of the application of Lagrange’s equations to a coupled electromechanical problem,
consider the one-dimensional mechanical device, shown in Fig. 7.14, with a magnetic actuator and a
capacitance actuator driven by a circuit with applied voltage V(t). We can extend Lagrange’s equation to

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