Amplitude problem — Car shock absorbers going over a bump

[robax25]

According to the question we can write also e^bt/2m=1/4

 

[haruspex]

F=force acts on the spring

No, it can't be that. To fit in dimensionally with the rest of your equation it has to be an acceleration, not a force. Also, the rest of that equation is time-independent, so it cannot be anything that varies over time.
I believe it should be the driving amplitude (resulting from the bumps in the road) multiplied by the spring constant.
See http://farside.ph.utexas.edu/teaching/315/Waves/node13.html for a suitable model for this problem.

ω'= damped angular velocity

Your equation has ω, not ω'. They are different.

If the height of each bump is 5cm, what amplitude of driving oscillation is that?
What is the value of the spring constant?
What is the undamped frequency (ω) of the spring?
All of those can be deduced from the given information.

 

[robax25]

the amplitude of driving oscillation is 2.5cm
spring constant k=73575N
ωo(natural frequency)=7 rad/s
m=1500kg
According to your link,
X0=ωo²Xo/√((ωo²-ω²)²+v²ω²) v= damping constant
v=ωom/(2π*m/ΔE/E)
v=7rad/s*1500kg/(2π*1500kg/0.5) As energy loss is 50%:

v=0.557 kg/s

ωo=ω as driven frequency X0=ωo²Xo/vωo
= 31.4cm

 

[haruspex]

Car's shock observer works fine,damping the deflection to half each oscillation

It has at last dawned on me what this is saying.
It means that if the car were standing still and you were to push it down by 10cm and let go then it would bounce up and down, with the next down being only 5cm below equilibrium. From this you can calculate the damping constant (but it is not trivial).

I think it is clear that the answer to the question should be significantly less than the height of the bumps, so no more than 1mm. (I calculate it to be a lot less.)

 

[robax25]

I did not consider speed of the car, it has to consider. I will ask him again.

 

[haruspex]

I did not consider speed of the car, it has to consider. I will ask him again.

No need to bother your prof again, I understand the question statement now.
First, solve the differential equation just for the car bouncing up and down on its springs. So this is damped (with unknown constant) but no forcing. (Or maybe you have a standard solution for that in your notes.)
Using that solution, figure out how much the amplitude reduces each period. You are told that it halves, so this allows you to find the damping constant.

Use the given velocity to find the forcing frequency.

 

[robax25]

the differential equation would be like that,
d²x/dt² +2Qωo dx/dt + ω²x=0 here Q=daming factor, Q= b/2√mk
b=daming constant kg/s

 

[haruspex]

the differential equation would be like that,
d²x/dt² +2Qωo dx/dt + ω²x=0 here Q=daming factor, Q= b/2√mk
b=daming constant kg/s

Ok. Can you use the information in the question about the halving and the equation at https://en.m.wikipedia.org/wiki/Damping_ratio#Logarithmic_decrement to find the damping ratio?