Damped Vibration

This video shows the free vibration of a single degree of freedom mass-spring-damper system with 10% damping.

One could use the real (or imaginary) part of equation x = x0*exp[ (-zeta+/-1j*sqrt(1-zeta^2) )*wn*t] to describe this movement; which is a circular motion with decreasing amplitude in the complex plane. wn = sqrt(k/m) and zeta = c/(2*sqrt(k*m)).

x = x0*exp[ (-zeta+/-1j*sqrt(1-zeta^2) )*wn*t] = x0*exp(-zeta*wn*t)*exp(1j*sqrt(1-zeta^2)*wn*t)
x0: amplitude, depends on initial conditions
exp(-zeta*wn*t): decay term, it is equal to 1 at time =0 and approaches zero as times approaches infinity.

exp(1j*sqrt(1-zeta^2)*wn*t): a circular motion in the complex plane with angular speed of wd = sqrt(1-zeta^2)*wn. this creates real and imaginary parts in the form of cos(wd*t) and sin(wd*t).

Category:

Science & Technology

Tags:

• Damping Ration
• Decay
• Free Vibration

[technohunger]

nice