Forced and dampled harmonic motion
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Critical and non-critical damping
We have seen that the general equation of motion for a system subject to linear resistive force and consequently subject to damping is [Equation goes here - download the original to see it.] The theory of homogenous second order constant coefficient differential equations tells us that this has solutions depending on the nature of the roots to the auxiliary equation [Equation goes here - download the original to see it.] Suppose [Equation goes here - download the original to see it.] are the two roots of this equation. Then, there are three cases: 1. If [Equation goes here - download the original to see it.] are real and distinct [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] In the case of a linear resistive force the roots [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] will both be negative. This indicates a situation where there is 'overdamping'. The system does not oscillate but experiences an immediate exponential decrement returning it to the equilibrium position. [Diagram goes here - download the original to see it.] 2. If [Equation goes here - download the original to see it.] is a real repeated root, the solution takes the form [Equation goes here - download the original to see it.]. The system does not travel beyond the equilibrium position, and is brought back to the equilibrium position. The motion is not oscillatory. [Diagram goes here - download the original to see it.] 3. If [Equation goes here - download the original to see it.] are complex conjugate numbers so that [Equation goes here - download the original to see it.] then the solution is [Equation goes here - download the original to see it.] Again the requirement that the linear resistive force acts in the opposite direction to the velocity entails that is a negative quantity. Hence, the system oscillates but the amplitude of the oscillations is damped. [Diagram goes here - download the original to see it.] For oscillations to take place there must be complex roots. Examining the equation [Equation goes here - download the original to see it.] with the auxiliary equation [Equation goes here - download the original to see it. We require that the discriminant. [Equation goes here - download the original to see it.] e define Equation goes here - download the original to see it.] o be the damping factor. The requirement for oscillations to take place is, therefore, Equation goes here - download the original to see it.] hen [Equation goes here - download the original to see it.] we have the case of the single repeated root Equation goes here - download the original to see it.] with solution [Equation goes here - download the original to see it.] This situation, where [Equation goes here - download the original to see it.], is called critical damping. It returns the system to the equilibrium position at the fastest possible rate without overshooting the equilibrium position. When such things as gun recoil mechanisms are designed, this value is the value that is selected. [Diagram goes here - download the original to see it.]
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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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