(Hard) work done by damped, driven oscillator as function of time

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SUMMARY

The discussion focuses on solving the differential equation for a damped, driven harmonic oscillator under a constant force. The equation mx'' + Tx' + kx = F is established, where the complementary solution is derived as x(t) = x0e^-(Tt)cos(wt) + ((v0 + T * x0) / w)e^(-Tt)sin(wt). The participant notes that the initial frequency w is not equal to the natural frequency w0, and seeks guidance on obtaining the particular solution and calculating work as a function of time. The participant contemplates integrating the forced solution to find work, emphasizing the need for clarity on the approach.

PREREQUISITES
  • Understanding of second-order linear differential equations
  • Familiarity with concepts of damped harmonic motion
  • Knowledge of Wronskian and variation of parameters
  • Basic principles of work-energy in physics
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  • Study the method of variation of parameters for non-homogeneous differential equations
  • Learn about calculating work done by a force in oscillatory systems
  • Explore the relationship between damping ratio and natural frequency in oscillators
  • Investigate numerical methods for solving differential equations when analytical solutions are complex
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Physics students, engineers, and anyone studying dynamics of mechanical systems, particularly those interested in damped and driven oscillators.

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Homework Statement



Force F = const is applied to H.O. initially at rest with mass m, freq w0, damping T. Find x(t). Find work as function of time.

Homework Equations


mx'' + Tx' + kx = F for F= Constant

The Attempt at a Solution



First obtain complimentary solution for free H.O. which I get after some work is x(t) = x0e^-(Tt)coswt + ((v0 + T *x0) / w )*e^(-Tt)sinwt. This agrees with textbook, but NOTE: w here is not equal to w0 for initial frequency and v0 can be taken to be zero. Now...if I try to apply variation of constant and use Wronskian I get a mess for the integrals. So where do I go from here to get my particular solution and then if I obtain it how to I obtain work as function of time?

Thanks[/B]
 
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For work could I just plug x(t) = x(particular) + x(complimentary) into my initial ode and integrate w.r.t. x?
 
However, x free should 0 as the oscillator is at rest so I just need the forced solution for F=const.
 

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