How Do You Solve the Driven Harmonic Oscillator with an Arbitrary Driving Force?

In summary, for a driven harmonic oscillator with an arbitrary driving force, the expression for x(t) can be found by solving the characteristic equation a^2 + 2ba + w = 0. This solution, denoted as a1, can then be used to find the ordinary differential equation satisfied by G. To solve for x(t), you can differentiate x(t) and plug it back into the original differential equation.
  • #1
mhyuen
3
0
1. Driven Harmonic Oscillator with an arbitrary driving force:
f(t)=x"+2bx'+w^2 x
Let x(t) be expressed by x(t)= g(t)*exp(a1*t), where a1 is a solution to the characteristic equation a^2 + 2ba+w=0 for the above second order differential equation. Find the ordinary differential equation that is satisfied by G. There's more parts, but I am just really stuck at the first part.
3.I'm not even really sure where to start. I am thinking i have to differentiate x(t) to get x'(t) and x"(t). then i plug that back into the original differential. is that correct?
 
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  • #2
Try it and see what you get.
 

FAQ: How Do You Solve the Driven Harmonic Oscillator with an Arbitrary Driving Force?

What is a Driven Harmonic Oscillator?

A Driven Harmonic Oscillator is a physical system that exhibits oscillatory behavior when subjected to an external force or driving force. It is a type of simple harmonic oscillator that is constantly being driven by an external source.

How is a Driven Harmonic Oscillator different from a Simple Harmonic Oscillator?

A Driven Harmonic Oscillator is different from a Simple Harmonic Oscillator in that it has an external force acting on it, whereas a Simple Harmonic Oscillator does not. This external force causes the Driven Harmonic Oscillator to have a different amplitude and frequency compared to a Simple Harmonic Oscillator.

What factors affect the behavior of a Driven Harmonic Oscillator?

The behavior of a Driven Harmonic Oscillator is affected by several factors, including the frequency and amplitude of the external force, the mass and stiffness of the system, and any damping present. These factors can change the amplitude, frequency, and phase of the oscillations.

What is resonance in a Driven Harmonic Oscillator?

Resonance in a Driven Harmonic Oscillator occurs when the frequency of the external force matches the natural frequency of the system. This causes the amplitude of the oscillations to increase significantly, leading to large and potentially damaging vibrations in the system.

How is a Driven Harmonic Oscillator used in real-world applications?

Driven Harmonic Oscillators have many real-world applications, including in musical instruments, electrical circuits, and mechanical systems such as bridges and buildings. By understanding the behavior of Driven Harmonic Oscillators, scientists and engineers can design systems that are more efficient and less prone to failure.

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