# Spring mass systems

1. Mar 7, 2012

### wezzo62

It's a mass spring system problem and im just not sure if im 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 im doing it right or not

1)a) keq = k1 + k2 = 2355N/m
b) keq = 1 / ((1/k1) + (1/k2)) = 451.0301N/m
c) mx" + kx = 0 => 6x" + 2355x = 0 (eq. of motion)
x = sin(ωnt + ∅) ___________________________________(1)
x' =Aωcos(ωnt + ∅) _________________________________(2)
x" = -ωn2Asin(ωnt + ∅) __________(3)
then substituting (1) and (3) back into eq. of motion and dividing by
Asin(ωnt + ∅)
gives: -mωn2 + k = 0 => ωn = √(k/m) = 19.81 rad/s
d) same process as for c) and got: ωn = √(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, ωd = ω√(1-ζ2) = 4.3323
d) ccr = 2√(km) = 11.1335 (bearing in mind its 4N not 4kg and therefore m=4/9.81=0.41kg)
=> damping constant, c = ζ ccr = 1.2282

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

#### Attached Files:

• ###### CA1.doc
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Last edited: Mar 7, 2012