MATH 217: Case Study on Mass Spring Oscillator with External Force

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Added on 2023/06/11

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This case study delves into the vibrations of a mass-spring underdamped system subjected to an external force, addressing key aspects such as damped oscillation, steady-state solutions, and displacement from equilibrium. The study derives and determines the functions for damped oscillation and steady-state solutions, culminating in a comprehensive function for displacement at time t. The practical application of the model is demonstrated through a problem involving a weight attached to a spring, with calculations to determine the steady-state solution. The analysis encompasses the derivation of equations for damped oscillation, steady-state solution, and displacement from equilibrium, alongside a practical application to determine the steady-state solution for a specific scenario. Desklib provides a platform for students to access similar solved assignments and past papers for academic support.

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

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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) = Ae−( b
2 m ) 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 γ2 [ ( k−m γ2 ) 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
2 m )tsin( √ 4 m−b2
√ 2 m t +
∅ ) + F 0
( k−m γ2 )2
+b2 γ2 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) = Ae−( b
2 m ) 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 γ2 [ ( k−m γ2 ) 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
2 m )tsin( √ 4 m−b2
√ 2 m t +
∅ ) + F 0
( k−m γ2 )2
+b2 γ2 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.

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Solution to the problems
Question 1

Question 2

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Question 3
Question 4

Yp(t) = F 0
( k−m γ2 )2
+b2 γ2 [ ( k−m γ2 ) cos ( γt ) +bγ sin ( γt ) ]
Weight = 64 lb = 284.69N
Coefficient (b ) = 3 lb – sec/ft = 43.782 N*s/m
F(t) = 3 cos12t
Weight = mg
Mass = weight/ g
Mass = 284.69/9.81
Mass = 29.02 Kg
Time at rest = 0
Taking the spring constant =k = 200 N/m
γ =
√ k
m
=
√ 200
29.02
= 2.63
Substituting into the equation
F(t) = 3 cos12t
F = 3*co12*0
F = 3N = 0.6744 lb
Yp(t) = F 0
( k−m γ2 ) 2
+b2 γ2 [ ( k−m γ2 ) cos ( γt ) +bγ sin ( γt ) ]
Yp(t) =
3
( 200−29 .02∗2.632 )2
+43.7822 ¿ 2.632 [ ( 200−29.02∗2.63 ) cos ( 2.63∗0 ) +43.782∗2.63∗sin ( 2.63∗0 ) ]
= 0.02798

= 0.028
Conclusion
To conclude, the objective of this project was to be able to derive and determine the function
damped oscillation, to be able to derive and determine the function of steady-state solution, to be able
to derive and determine the function of displacement from equilibrium of the mass at time t an to be
able to use the derived function of the steady – state to solve application question, which all were
achieved.

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Reference
J.P. Den Hartog. Mechanical Vibrations. New York: Dover Publications ,1934.
W.T. Thomson. Theory of Vibrations with Applications. 4th ed. Prentice-Hall, 1993
R.K. Nagle, E.B. Saff and A.D. Snider. Fundamentals of Differential Equations and Boundary Value
Problems. 6th ed. Pearson, Addison Wesley,2012.
D.G. Zill and M.R. Cullen. Differential Equations with Boundary Value Problems. 6th ed. Brooks/Cole
Publishing Company, 2005.
William F. Trench. Elementary Differential Equations with Boundary Value Problems. Brooks/Cole, 2001.
William E. Boyce and Richard C. DiPrima. Elementary Differential Equations. 7th ed. John Wiley & Sons,
2001.
C. Henry Edwards & David E. Penney. Differential Equations, Computing and Modeling. 4th ed. Pearson,
Prentice Hall, 2008.