Inverse Laplace of a Mass-Spring System

  • #1

Homework Statement



Given a transfer function in the Laplace Domain

Detemine an expression for x(t), given f(t) is a sinusodial input with frequency omega = root(k2/m2) and amplitude of 1 N (initial conditions equal 0)


Homework Equations


[URL]http://latex.codecogs.com/gif.latex?X_1/F=(m_2&space;s^2+k_2)/(m_1&space;m_2&space;s^4+k_2&space;(m_1+m_2)s^2&space;)[/URL]

Inverse laplace 1/s^2 = t.u(t)
Inverse laplace (omega/s^2+omega^2) = sin(omega.t) . u(t)

The Attempt at a Solution



I divided the transfer function by m2 to obtain omega^2. I then brought the F over to the LHS as a sin function in the laplace domain (omega/s^2+omega^2). I have obtained the following equation

[URL]http://latex.codecogs.com/gif.latex?X_1=(1/s^2)&space;.w/((s^2&space;m_1+w^2&space;((m_1+m_2)/m_2&space;))[/URL]


What is the next step? I am given inverse laplace transforms for 1/s^2 and omega/s^2+omega^2
 
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Answers and Replies

  • #2
gabbagabbahey
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[URL]http://latex.codecogs.com/gif.latex?X_1/F=(m_2&space;s^2+k_2)/(m_1&space;m_2&space;s^4+k_2&space;(m_1+m_2)s^2&space;)[/URL]

Inverse laplace 1/s^2 = t.u(t)
Inverse laplace (omega/s^2+omega^2) = sin(omega.t) . u(t)

The Attempt at a Solution



I divided the transfer function by m2 to obtain omega^2. I then brought the F over to the LHS as a sin function in the laplace domain (omega/s^2+omega^2). I have obtained the following equation

[URL]http://latex.codecogs.com/gif.latex?X_1=(1/s^2)&space;.w/((s^2&space;m_1+w^2&space;((m_1+m_2)/m_2&space;))[/URL]


What is the next step? I am given inverse laplace transforms for 1/s^2 and omega/s^2+omega^2
First, I think you have a small error in your equation. I get

[tex]X_1(s)=\frac{\omega}{s^2[m_1s^2+(m_1+m_2)\omega^2]}[/tex]

since [itex]m_1m_2s^4+(m_1+m_2)k_2s^2=m_2s^2[m_1s^2+(m_1+m_2)\omega^2][/itex]

Second, use partial fraction decomposition. Say


[tex]\frac{\omega}{s^2[m_1s^2+(m_1+m_2)\omega^2]}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{m_1s^2+(m_1+m_2)\omega^2}[/tex]

and find the constants [itex]A[/itex], [itex]B[/itex], [itex]C[/itex] and [itex]D[/itex].
 
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