Spring Mass System Problem Solutions: Check Here

[wezzo62]

It's a mass spring system problem and I am just not sure if I am doing it right or not. If you could have a look at the problem attached and my solutions below and just let me know if I am doing it right or not

1)a) k_eq = k₁ + k₂ = 2355N/m
b) k_eq = 1 / ((1/k₁) + (1/k₂)) = 451.0301N/m
c) mx" + kx = 0 => 6x" + 2355x = 0 (eq. of motion)
x = sin(ω_nt + ∅) ___________________________________(1)
x' =Aωcos(ω_nt + ∅) _________________________________(2)
x" = -ω_n²Asin(ω_nt + ∅) __________(3)
then substituting (1) and (3) back into eq. of motion and dividing by
Asin(ω_nt + ∅)
gives: -mω_n² + k = 0 => ω_n = √(k/m) = 19.81 rad/s
d) same process as for c) and got: ω_n = √(451.0301/6) = 8.67rad/s

Im now a little stuck/confused on e) and f) and its making me think what I've done so far is wrong! this is really annoying me so please could someone tell me if any of this is right.

2)a) log decrement, δ = ln(10/5) = 0.6931
b) damping factor, ζ = δ/2pi = 1.0888
c) natural frequency, ω = √(76/4) = √19
=> damped natural frequency, ω_d = ω√(1-ζ²) = 4.3323
d) c_cr = 2√(km) = 11.1335 (bearing in mind its 4N not 4kg and therefore m=4/9.81=0.41kg)
=> damping constant, c = ζ c_cr = 1.2282

I think most of Q2 should be right . . ? any help much appreciated, thanks

 

[KataKoniK]

it is important to always double check your work and make sure you are following the correct procedures. From what I can see, your approach and calculations for the first part (a-d) seem to be correct. However, for part e and f, there are a few things that need to be clarified.

In part e, you are asked to calculate the critical damping coefficient (ccr). This is typically defined as the damping coefficient at which the system will return to equilibrium without any oscillations. In your calculation, you have used the equation ccr = 2√(km), which is correct. However, you have used the mass of 4N instead of 4kg. The mass should be in kilograms, so the correct calculation would be ccr = 2√(4*9.81) = 8.8588 Ns/m.

In part f, you are asked to calculate the damping constant (c). In your calculation, you have used the damping factor (ζ) instead of the damping coefficient (c). The damping constant is defined as c = ζ * ccr, so the correct calculation would be c = 1.0888 * 8.8588 = 9.6348 Ns/m.

Overall, it seems like you have a good understanding of the concepts and equations involved in this mass spring system problem. Just be sure to double check your units and calculations to avoid any mistakes. I hope this helps clarify things for you. Good luck with your problem!