(a) Prove that, given a non-constant solution to an...

(a) Prove that, given a non-constant solution to an overdamped mass-spring system, there is at most one time where $u\left(t_{\star}\right)=0$. (b) Is this statement also valid in the critically damped case?

Key Concepts

Overdamped Systems

An overdamped system is one in which the damping is so large that the characteristic equation of the second?order differential equation has two distinct real roots. This means that the system does not oscillate; instead, the solution decays monotonically towards equilibrium. Because of this monotonic behavior, a non-constant solution can cross zero at most once, as any additional crossing would require a change in the direction of the motion, which is not possible given the strictly exponential decay governed by the two distinct exponential terms.

Critically Damped Systems

A critically damped system occurs when the damping is exactly at the threshold between oscillatory and non-oscillatory behavior, producing a repeated real eigenvalue in the characteristic equation. This leads to solutions that, while still non-oscillatory, include a polynomial factor multiplying the exponential decay term. Because of this added complexity, the zero-crossing behavior in this case can be different, and the uniqueness of a zero crossing is not guaranteed in the same way as for the overdamped case.

Zero Crossing Property

The zero crossing property refers to the moments in time when the solution of a differential equation equals zero. For systems like the overdamped mass-spring model, the nature of the solution—typically a linear combination of decaying exponentials—ensures that if there is a zero, it occurs uniquely. Analysis of the derivative and the monotonic nature of the solution confirms that more than one zero crossing would violate the underlying dynamics imposed by the damping terms.

Linear Second-Order Differential Equations

The problem at hand is framed within the context of linear second-order differential equations, which model many physical systems such as mass-spring-damper systems. In these equations, the behavior of solutions is deeply influenced by the roots of the characteristic polynomial, which dictates whether the system is underdamped, overdamped, or critically damped. Understanding how these roots affect the solution structure is key to analyzing properties like zero crossings and the overall motion of the system.

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