For each of the following statements, determine if it is true or false. Give som
ID: 3110088 • Letter: F
Question
For each of the following statements, determine if it is true or false. Give some explanations based on the real part of the root of the characteristic equation and the natural and external frequencies. (a) The mass position r(t) in an undamped, free mass-spring system (second-order ODE) is a periodic function in time. (b) The mass position r() in an overdamped, free mass-spring system (second-order ODE) will decay in time to zero. (c) The mass position r(t) in an underdamped, free mass-spring system (second-order ODE) will grow in time. (d) Solutions to the equation x" + x' + 4x = sin3t after a long time will be dominated by a sine wave of frequency omega = 3. (e) The mass-spring model described by the equation x" + 4x = sin 3t will have resonance. (f) The mass spring model described by the equation x" + x' + 4x = cos 2t will have resonance.Explanation / Answer
a) True
in undamped system the DE is mx''+kx=0
x''+k/m x =0
its solution is inth form of x=c1 cos ( sqrt(k/m)) t + c2 sin ( sqrt(k/m)) t
which is periodic
b) True
for over damped system roots are roots are negative and uneual
thus solution is in the form of x=c1 e-m1t + c2 e-m2t
thus as time progress x will dacays to zero
c) False
in underdamped system
solution is in th form of x= e-mt ( c1 cos ( m2t) + c2 sin ( m2 t) )
because of negative exponential x is decaying
d) True
its a underdamped system
thus its natural responce goes to zero as time progress
thus as time progress forced responce will dominant the system
e) False
here natural frequency w0=2
external frequency = 3
no resonance
f) False
here
w0=2
2*w0zeta = 1 thus zeta = 1/4
damped natural frequency = 2 sqrt( 1-0.252) = 1.9365
External frequency=2
No resonance
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