Solve Dynamical Systems Homework: 20kg Spring/Damper System

[Velcro7]

Homework Statement

A mass of 20kg is supported by a spring and damper as shown. The system is at rest for t<0. At t=0 a mass of 2kg is added to the 20kg mass as shown. The system vibrates as shown in the accompanying figure. Determine the spring constant k and the damping coefficient c for the system.

Homework Equations

mx"+cx'+kx=0
c=2ζ√(2mk)
δ=ln(xn/(xn+1) and δ=(2πζ)/(√(1-ζ²) where π=symbol for pi

The Attempt at a Solution

m=20+2=22kg

22x"+kx'+cx=0

xo=.02m
xf=.08m

δ=ln(xf/xo) = ln(.08/.02) = 1.386

δ=(2πζ)/(√(1-ζ²) which becomes

1.386√(1-ζ²)=2πζ which becomes 4πζ²=(1.386²)(1-ζ²) which becomes

ζ=√((1.386²)/((4π²)+(1.386²))) = 0.215 (ζ<1 so system is underdamped)

c=2ζ√(mk) = 2(0.215)√(22*k) = 2.017√(k) N-s/m

I am not sure if I found c correctly since I added the 2kg mass. Also I am unsure as how to find k.

 

[KataKoniK]

Thank you for your post. It seems like you have made a good attempt at solving the problem. However, there are a few things that need to be addressed in order to find the correct values for k and c.

Firstly, when solving for the damping coefficient, you correctly used the formula c=2ζ√(mk). However, the value of ζ that you used (0.215) is incorrect. In the formula δ=ln(xf/xo), xf and xo should be the amplitudes for the first and second peaks, respectively. In this case, xf=0.08m, but xo should be the amplitude of the first peak, which is not given in the problem statement. Without knowing this value, we cannot accurately solve for ζ and therefore cannot find the correct value for c.

Similarly, when trying to solve for k, we need to know the value of the natural frequency ω. This can be found using the formula ω=√(k/m), where m is the total mass of the system (22kg in this case). However, we also need the value of ω in order to solve for k. This can be found by using the formula ω=2πf, where f is the frequency of the oscillations (given in the figure as 0.5 Hz). Therefore, we can solve for k by setting these two equations equal to each other and solving for k.

In summary, in order to accurately solve for k and c, we need to know the amplitude of the first peak and the natural frequency of the system. Once we have this information, we can use the equations you have listed to find the correct values for k and c.

I hope this helps. Let me know if you have any further questions or need clarification on any of the concepts discussed.