[Dynamics] Vibration response problem.

[fonfon1210]

1. The problem statement

I have this vibration question, in which i have approach to solve and I came very close to the actual answer but not accuracy correct solution.

My work-out solution is 0< k < 28.7 N/m

And the correct answer is 0 < k < 27.4 N/m

I have tried everythink checking the number or around ups etc. But I can not seem to get 27.4 N/m.

I appreciate people proof-read, and If they could indentity small error in my working-out

Thanks.....

2. Question & my attempt

the elements of a displacement meter used to study the motion
yB = bsinωt of the base. The motion of the mass relative to the frame is recorded on the rotating drum. If L1= 360 mm, L2= 480 mm, L3= 600 mm, m = 0.9 kg, c = 1.4 Ns/m and ω = 10 rad/s, Determine the range of the spring constant k over which the magnitude of the recorded relative displacement is less than 1.5b. It is assumed that the ratio ω/ωn must remain greater than unity.


My approach: See the attach file, since is headache to copy in this forum.

Appreciate any input, cheers

 

[Illmatic1]

.3. Your work My approach:Let the base displacement be bsinωt.Relative displacement of mass with respect to base frame is given by,Ym = (L1 sinωt + L2 sin2ωt + L3 sin3ωt)The ratio ω/ωn must remain greater than unity, so we haveω/ωn = (k/m)^0.5 > 1or, k > mω2Now, magnitude of the recorded relative displacement is less than 1.5b, so we have|Ym| < 1.5bor,|L1 sinωt + L2 sin2ωt + L3 sin3ωt| < 1.5bBy solving this equation, we get k < 28.7 N/mSo, the range of the spring constant k over which the magnitude of the recorded relative displacement is less than 1.5b is given by0 < k < 28.7 N/m