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

In summary, the problem involves finding the position and work of a H.O. under the influence of a constant force. The complimentary solution for the H.O. is x(t) = x0e^-(Tt)coswt + ((v0 + T *x0) / w )*e^(-Tt)sinwt, with w not equal to w0 and v0 taken to be zero. The next step is to apply the variation of constant method and use the Wronskian to obtain the particular solution. To find the work as a function of time, the forced solution can be plugged into the initial ODE and integrated with respect to x, with x free being 0 as the H.O. is initially
  • #1
NucEngMajor
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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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  • #2
For work could I just plug x(t) = x(particular) + x(complimentary) into my initial ode and integrate w.r.t. x?
 
  • #3
However, x free should 0 as the oscillator is at rest so I just need the forced solution for F=const.
 

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

1. What is a damped, driven oscillator?

A damped, driven oscillator is a system that consists of a mass attached to a spring and subjected to an external force or driving force. It also experiences damping, which is the dissipation of energy due to friction or resistance. This system is commonly used to model various physical phenomena such as pendulums, musical instruments, and electrical circuits.

2. How does damping affect the work done by a damped, driven oscillator?

Damping reduces the amplitude of the oscillations of a damped, driven oscillator over time, resulting in a decrease in the total work done by the system. This is because some of the energy is dissipated as heat due to friction, rather than being converted into work.

3. What is the equation for work done by a damped, driven oscillator as a function of time?

The equation for work done by a damped, driven oscillator as a function of time is given by W(t) = 1/2kA^2e^(-2βt)cos^2(ωt+φ) where k is the spring constant, A is the amplitude, β is the damping coefficient, ω is the angular frequency, and φ is the phase angle.

4. How does the driving force affect the work done by a damped, driven oscillator?

The driving force in a damped, driven oscillator adds energy to the system and can increase the amplitude of the oscillations. This results in an increase in the total work done by the system. The amount of work done by the driving force depends on its frequency and amplitude.

5. What is the relationship between work done and energy in a damped, driven oscillator?

The work done by a damped, driven oscillator is directly related to the energy in the system. As the system oscillates, energy is transferred between kinetic and potential energy. The total work done by the system over time is equal to the change in energy, which can be calculated using the equation W(t) = ΔE = 1/2kA^2e^(-2βt).

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