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cpburris
Gold Member
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Not a homework problem, just a question. What is a periodic driving force, specifically what is periodic about it? Is it the magnitude of the force that is periodic?
BvU said:In isolation this doesn't make much sense. The ##\omega << \omega_0## helps, though.
You wrote the eqn of motion for a harmonic oscillator without damping.
So any initial conditions that cause a considerable amplitude can give rise to large acceleration.
If there is some damping, this will be for a certain time only (transient solution), and ultimately only the driving force determines the motion of the oscillator.
Without actually solving the thing and dealing with the initial conditions, it is a little difficult to say very much about it.
Usually the mass is in ##\omega_0## and the equation is written as
$$\ddot x+\omega_0^2 x = F_0\, \cos(\omega t)$$Any solution to the homogeneous equation
$$\ddot x+\omega_0^2 x = 0$$ can be added to a solution of the complete equation. And such a solution has two integration constants, e.g. ##x(0)## and ##\dot x(0)##.
If we follow your account, and let the oscillator move with the driver,
we could try a solution for the inhomogeneous equation
that looks like ##x(t)=A \cos(\omega\, t+\phi)##, fill it in and get
$$A\left (-\omega^2 + \omega_0^2 \right) \cos(\omega\, t+\phi) = F_0 \, \cos(\omega t)$$ Which must be satisfied at all t. From which ##\phi = 0## (more than just in agreement with "pretty much in phase with the driver" -- which gives me the impression some damping is considered to be present...)
and $$ A = {F_0 \over \omega_0^2-\omega^2}$$
[edit]saw your edit but was too far along already...
A periodic driving force is a type of external force that is applied to a system at regular intervals, causing the system to oscillate or vibrate with a specific frequency.
A periodic driving force can change the behavior of a system, causing it to exhibit resonance, damping, or other dynamic responses depending on the frequency and amplitude of the force.
Some examples of periodic driving forces include sound waves, electromagnetic fields, and mechanical vibrations.
A periodic driving force is typically quantified using parameters such as frequency, amplitude, and phase, which describe the intensity and timing of the force.
Understanding periodic driving forces is crucial in fields such as engineering, acoustics, and signal processing, as it allows for the analysis and design of systems that can efficiently and accurately respond to external forces.