Mass Spring Oscillator with External Force - Study Material

Case study: Mass Spring Oscillator with
External Force
Contents
Introduction.................................................................................................................................................2
Objective.................................................................................................................................................2
Methodology...............................................................................................................................................3
Problem.......................................................................................................................................................4
Solution to the problems.............................................................................................................................5
Conclusion...................................................................................................................................................9
Reference..................................................................................................................................................10

Introduction
Objective
1. To be able to derive and determine the function damped oscillation
2. To be able to derive and determine the function of steady-state solution
3. To be able to derive and determine the function of displacement from equilibrium of the mass
at time t
4. To be able to use the derived function of the steady – state to solve application question
The study of vibration of a spring that has been attached mass on its free end is a context of thought
that has been used to study the several concepts of vibration which included under-damping, over-
damping and critically damped, but this report will only emphasis on the concept of determination of
under-damped vibration.
Under-damping case will occurs when the parameters of the system of vibration are such that (0< ξ <1)
and in this approach the discriminate ωn√ξ−1 becomes negative.

Methodology
An equation of vibrations of a mass-spring under-damped system when an external force is applied is
shown below
m d2 y
d t2 +b dy
dt + ky=F0Cosγt
The function of a damped oscillation is given by:
Yh(t) = A
e− ( b
2m ) tsin( √ 4 m−b2
√ 2 m t+∅)
While, the function of a steady-state solution is given by:
Yp(t) = F 0
( k −m γ2 )
2
+b2 γ 2 [ ( k−m γ 2 ) cos ( γt ) +bγ sin ( γt ) ]
Finally, the function of the displacement from equilibrium of the mass at time t is given by:
Y(t) = Ae− ( b
2m )tsin( √ 4 m−b2
√ 2 m t+∅) + F 0
( k −m γ2 )
2
+b2 γ 2 sin(γt +θ ¿

Problem
Mass Spring Oscillator with External Force Consider the vibrations of a mass-spring underdamped
system when an external force is applied:
m d2 y
d t2 +b dy
dt + ky=F0Cosγt
1. Show that the damped oscillation (solution of the corresponding homogeneous equation) is
given by:
Yh(t) = A
e− ( b
2m ) tsin( √ 4 m−b2
√ 2 m t+∅)
2. Show that the steady-state solution (particular solution) is such that:
Yp(t) = F 0
( k −m γ2 )
2
+b2 γ 2 [ ( k−m γ 2 ) cos ( γt ) +bγ sin ( γt ) ]
3. Show that the displacement from equilibrium of the mass at time t is given by:
Y(t) = Ae− ( b
2m )tsin( √ 4 m−b2
√ 2 m t+∅) + F 0
( k −m γ2 )
2
+b2 γ 2 sin(γt +θ ¿
4. Use the above model to determine the steady state solution if a 64 lb weight is attached to a
vertical spring, causing it to stretch 3 in. upon coming to rest at equilibrium. The damping
constant for the system is 3 lb-sec/ft and an external force F(t) = 3 cos12t is applied to the
weight.