Question

Suppose that a simple spring-mass system can be modeled by the 2nd-order Non- homogeneous ODE stated below. Answer the following questions concerning the properties of this spring-mass system. 4 + 4 = 2 cos(2t); }(0) = 4 (0) = 0 (i) is the spring-mass system an underdamped system, critcally-damped system, overdamped system, or a system with no damping? [Select] (ii) Can this system ever achieve resonance? Select] (iii) is the spring-mass system characterized by the ODE stable? Select] (iv) Does the equilibrium position of the spring in this system return to y(t) = 0 ast approaches infinity? (Select)

Answer 1

Consider the given spring-mass system given using equations:

y'' + 4y = 2 cos(2t) ; y(0) = 0 and y'(0) = 0;

The given system shows an underdamped response, i.e., a sinusoidal wave increasing its amplitude in a ramp.

The system can achieve resonance. This is because the natural frequency of oscillation of the system, in the undamped case is given as

wn = (coefficient of y)1/2 ; when coefficient of y'' is made 1.

Thus, natural frequency of the system is 2 radians/second. Also, the input frequency is 2 radians per second. Thus the system can achieve resonance.

The spring mass system characterized by homogeneous part of the ode is marginally stable. This is because the poles of the system without an input have only imaginary parts and no real part.

These can be calculated as follows.

let d/dt = D

thus, using the given equation, we get, for zero input conditions,

D2 y + 4y = 0

or, y (D2 + 4) = 0

or, D2 = -4

or D = 2i and D = -2i

Since the poles of the system are purely imaginary, the system is marginally stable.

However, given the input 2cos(2t), the response of the system is unbounded (i.e., it tends to infinity as time tends to infinity).