Automotive Suspension: harmonic oscillator system, differential equations

[clarineterr]

Homework Statement

You are working at a company that designs suspension systems. Some guy from the marketing department asks you to design a shock absorber that "bounces twice", meaning that after the initial bump, the spring should expand, compress, expand again and then gradually settle back to rest.

Describe the position graph of this situation.

Explain briefly why such behavior is impossible, if you are using a harmonic oscillator to model the suspension system

Suggest a choice for a harmonic oscillator that would approximate this behavior.

Homework Equations

The Attempt at a Solution

The graph is sinusoidal, goes through two positive peaks and then levels out. I need help explaining why it is impossible...but it guess its basically that the spring will never actually stop oscillating? I have no idea how to approximate this behavior.

 

[mighty2000]

The position graph of this situation would show a sinusoidal wave with two complete oscillations, starting from the rest position and ending at the same rest position. The initial bump would cause the spring to compress and then expand, resulting in the first peak. The second peak would occur as the spring expands again after the initial compression. The spring would then gradually settle back to its rest position.

Such behavior is impossible if we are using a harmonic oscillator to model the suspension system because a harmonic oscillator follows the principle of simple harmonic motion, which states that the motion of the system is periodic and the period of oscillation is constant. In simpler terms, the motion of a harmonic oscillator is repetitive and does not change over time. Therefore, it is not possible for the spring to expand, compress, and then expand again in a single oscillation as it would violate the principle of simple harmonic motion.

To approximate this behavior, a damped harmonic oscillator could be used. A damped harmonic oscillator experiences a gradual decrease in amplitude over time, which would result in the spring eventually settling back to its rest position. This would closely approximate the desired behavior of the shock absorber "bouncing twice". However, it is important to note that even with a damped harmonic oscillator, the spring would not exhibit exactly two bounces, as the amplitude would continue to decrease over time.