A certain forced and underdamped mechanical oscillation with a tunable viscous d

Question

A certain forced and underdamped mechanical oscillation with a tunable viscous damper can be modeled by the second-order ordinary differential equation

4y'' + ay' + y = cos wt for w> 0 and 0 < a < 4

W=omega and a=alpha.

a) Use characteristic polynomials and the method of undetermined coefficients to solve the differential equation.

b) Identify the transient and steady-state system responses in your solution.

Amplitude of steady-state response as a function of damping coefficient and external forcing frequency

Let R be the amplitude of the mechanical system's steady-state response to the external force. c) Express R analytically as R(w, a), a function of w and a.

d) Express R(1, a), lim R(w, a and lim R(w, a) in terms of a.

e) Interpret your results from part d) in the context of the mechanical system.

f) Graph R as a function of w for various values of a.

g) Set a = 0.4, graph R(w, 0.4), and numerically evaluate !!

1) The maximum steady-state amplitude for this damping coefficient, and

2) The forcing frequency that maximizes the steady-state amplitude. h) Compute ∂R/∂w and ∂R/∂a, and analyze their signs for various values of wand a

i) Interpret your partial derivative analysis in terms of the mechanical system.

Let w1 be the forcing frequency that maximizes R for a given value of a.

j) Express w1 analytically as a function of a, and determine its domain.

k) Express Rmax = R(w1, a) as a function of a, valid on the domain of w1.

g) Compute w1(0.4) and Rmax(0.4), and compare to your numerical results from part g).

m) Summarize what you've learned. Tell me what you can about the amplitude of the steady-state system response to a periodic external force as it depends on frequency of the external force and the damping coefficient.

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