What Are the Two Possible Values of C in a Damped Harmonic Oscillator Equation?

[clipperdude21]

1. The equation of motion is Ma(t) +rv(t) + Kx(t)=0
a) Look for a solution of this equation with x(t) proportional exp(-Ct) and find two possible values of C.

Homework Equations

3. No clue... Please help if you can!

 

[Kurdt]

How are x(t), v(t) and a(t) related?

 

[Moetasim]

I am familiar with the concept of a damped harmonic oscillator. This equation of motion represents the behavior of a system that experiences both a restoring force (Kx(t)) and a damping force (rv(t)) that opposes its motion.

To find a solution to this equation, we can assume that the displacement of the system, x(t), is proportional to exp(-Ct), where C is a constant. This assumption is based on the fact that exponential functions are often used to model damped systems.

Plugging this assumption into the equation of motion, we get Ma(t) +rv(t) + Kx(t) = M(C^2-K)x(t) = 0. This means that C^2-K = 0 in order for the equation to hold true. Therefore, there are two possible values of C: C = √K and C = -√K.

These two values represent the two possible solutions for the damped harmonic oscillator. The first solution, C = √K, represents an underdamped system where the displacement decreases exponentially over time. The second solution, C = -√K, represents an overdamped system where the displacement decreases more rapidly than in the underdamped case.

In summary, the equation of motion for a damped harmonic oscillator suggests that the displacement of the system is proportional to an exponential function with a coefficient of √K or -√K. This understanding can help us analyze and predict the behavior of damped systems in various scientific fields.