Driven Damped Harmonic Oscillator, f = ma?

This is because f(t) is a time-varying force while F=ma assumes a constant acceleration. Therefore, substituting in a for \ddot{x} is not a valid solution method for the problem.
  • #1
dimensionless
462
1
Driven Damped Harmonic Oscillator, f != ma??

Let's say I've got a driven damped harmonic oscillator described by the following equation:
[tex]A \ddot{x} + B \dot{x} + C x = D f(t)[/tex]

given that [tex] f = ma[/tex] why can't I write

[tex]A \ddot{x} + B \dot{x} + C x = D ma[/tex]

substitute [tex]\ddot{x} = a[/tex] to get

[tex]A \ddot{x} + B \dot{x} + C x = D m \ddot{x}[/tex]

and then rearrange to get

[tex](A - D m) \ddot{x} + B \dot{x} + C x = 0[/tex]

I know that's not how the problem is solved, but what is to stop me from solving it that way?
 
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  • #2
Are you assuming constant acceleration?
 
  • #3
You're forgetting that f(t) is a time-varying force. You also have forces that depend on velocity and position, which is why what you wrote isn't correct. You only got the time-varying force.
 
  • #4
dimensionless said:
Let's say I've got a driven damped harmonic oscillator described by the following equation:
[tex]A \ddot{x} + B \dot{x} + C x = D f(t)[/tex]

given that [tex] f = ma[/tex] why can't I write
The F in Newton 's F=ma is the net force, which does not equal the function f(t) in your problem.
 

FAQ: Driven Damped Harmonic Oscillator, f = ma?

1. What is a driven damped harmonic oscillator?

A driven damped harmonic oscillator is a physical system that exhibits periodic motion, or oscillations, about an equilibrium position. It is driven by an external force and also experiences damping, which causes the amplitude of the oscillations to decrease over time.

2. How is the motion of a driven damped harmonic oscillator described?

The motion of a driven damped harmonic oscillator is described by the equation f = ma, where f is the force acting on the system, m is the mass of the oscillator, and a is the acceleration. This equation takes into account both the driving force and the damping force.

3. What factors affect the behaviour of a driven damped harmonic oscillator?

The behaviour of a driven damped harmonic oscillator is affected by several factors, including the amplitude and frequency of the driving force, the mass and stiffness of the oscillator, and the amount of damping present in the system. These factors can cause changes in the amplitude, frequency, and phase of the oscillations.

4. How does the damping force affect the motion of a driven damped harmonic oscillator?

The damping force in a driven damped harmonic oscillator causes the amplitude of the oscillations to decrease over time. This is because the damping force acts in the opposite direction of the motion, dissipating the energy of the system and reducing the amplitude of the oscillations.

5. What is the resonance frequency of a driven damped harmonic oscillator?

The resonance frequency of a driven damped harmonic oscillator is the frequency at which the amplitude of the oscillations is at its maximum. This frequency is determined by the mass and stiffness of the oscillator, as well as the amount of damping present. It is important to avoid driving the oscillator at its resonance frequency, as this can cause the amplitude of the oscillations to become very large and potentially damage the system.

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