Forced and dampled harmonic motion
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Resonance and frequency response
Let us imagine a system without damping and not subject to a forcing function. Then it will have equation of motion [Equation goes here - download the original to see it.] Or [Equation goes here - download the original to see it.] With solution [Equation goes here - download the original to see it.] And equation [Equation goes here - download the original to see it.] Where A is the amplitude and [Equation goes here - download the original to see it.]is the phase angle. These will be constants determined by the initial conditions. The system will exhibit simple harmonic motion and will oscillate indefinitely. This is the consequence of the assumption that there are not resistive forces, so there is no way to dampen the oscillation. The system will oscillate with angular frequency [Equation goes here - download the original to see it.] This is called the natural frequency of the system. For simplicity we will write the equation of simple harmonic motion [Equation goes here - download the original to see it.] With solution: [Equation goes here - download the original to see it.] Now suppose the system is subject to a forcing function [Equation goes here - download the original to see it.] Then the equation of motion is [Equation goes here - download the original to see it.] To find the steady state solution to this system let us try [Equation goes here - download the original to see it.] Hence, on substituting into the equation of motion we obtain [Equation goes here - download the original to see it.] That is [Equation goes here - download the original to see it.] The response amplitude is C to the input amplitude of the forcing function B. So [Equation goes here - download the original to see it.] As [Equation goes here - download the original to see it.]tends to zero, hence [Equation goes here - download the original to see it.] In other words a system without any damping when subject to a forcing function that is at the natural frequency of the system will exhibit a response with infinite amplitude! In practice, of course, some energy is dissipated - but the output response still increases to a maximum as the frequency of the forcing function approaches the natural frequency of the system. This response is called resonance. [Diagram goes here - download the original to see it.] Even when there is damping there will be resonance at the natural frequency. The general equation for a system with damping is [Equation goes here - download the original to see it.] The natural frequency is [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] If [Equation goes here - download the original to see it.]is a periodic function, then as the frequency of [Equation goes here - download the original to see it.] approaches [Equation goes here - download the original to see it.] the amplitude of the steady state solution gets progressively bigger and bigger.
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Contents of
Forced and dampled harmonic motion
1 Damped and forced vibrations
2 Dashpots
3 Critical and non-critical damping
4 Amplitude of successive oscillations
5 Forced oscillations
6 Resonance and frequency response
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