a. Electrodynamic shakers are commonly used in the dynamic testing of products. One possible…

a. Electrodynamic shakers are commonly used
in the dynamic testing of products. One possible configuration of a
shaker/test-object system is shown in Figure P10.18(a). A simple, linear,
lumped-parameter model of the mechanical system is shown in Figure P10.18 (b).
Note that the driving motor is represented by a torque source Tm.
Also, the following parameters are indicated:

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Jm = equivalent moment of
inertia of motor rotor, shaft, coupling, gears, and shaker platform

r1 = pitch circle radius of the
gear wheel attached to the motor shaft

r2 = pitch circle radius of the
gear wheel rocking the shaker platform

 = lever arm from
the rocking gear center to the support location of the test object

mL = equivalent mass of the test
object and support fixture

kL = stiffness of the support
fixture

bL = equivalent viscous damping
constant of the support fixture

ks = stiffness of the suspension
system of the shaker table

bs = equivalent viscous damping
constant of the suspension system.

Note that, since the inertia effects are
lumped into equivalent elements, it can be assumed that the shafts, gearing,
platform, and support fixtures are light. The following variables are of interest:

ωm = angular speed of the
drive motor

vL = vertical speed of motion of
the test object

fL = equivalent dynamic force
(in spring kL) of the support fixture

fs = equivalent dynamic force
(in spring ks) of the suspension system.

a. Obtain an expression for the motion
ratio

b. Using x = [ωm, fs,
fL, vL]T as the state vector, u = [Tm] as the
input, and y = [vL, fL]T as the output vector,
obtain a complete state-space model for the system.

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