Dampled harmonic motion
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Dashpots
Equations are omitted for technical reasons - download the original pdf
The purpose of a car suspension system is to bring about the damping of oscillations in a tyre as it runs over a bump. We will be initially considering the case where the tyre runs over a single bump which effectively gives it a sudden sharp displacement from its equilibrium position. In order to bring about the damping the suspension system is fitted with a mechanical device called a dashpot. The dashpot is represented thus [Diagram goes here - download the original to see it.] Its function is to provide a linear resistive force [Equation goes here - download the original to see it.] where r is the dashpot constant. The whole car suspension system can be represented by [Diagram goes here - download the original to see it.] In this system the mass is actually represented by the chasis, and we are imagining that the tyre, dashpot and spring have no mass. However, we will deal with the solution to questions where the mass is represented by the object here labelled the tyre. The symmetry of the situation means that what applies to a tyre of mass m where the chasis is fixed and has no mass would apply to a chasis of mass m where the tyre is fixed and has no mass. In other mechanical systems where there is a linear resistive force, this force can also be represented by dashpot symbol. Example A mass, m = 2kg, is suspended by a spring of natural length [Equation goes here - download the original to see it.] and stiffness [Equation goes here - download the original to see it.] and by a dashpot with dashpot constant [Equation goes here - download the original to see it.]. It is subject to a sudden sharp displacement of 0.5m. Find its equation of motion and its phase lag. Sketch a graph showing the subsequent motion. Find its frequency and period. Solution The system can be represented by [Diagram goes here - download the original to see it.] Let [Equation goes here - download the original to see it.] represent its displacement at time t after it receives the shock, so [Equation goes here - download the original to see it.]. The general equation of motion for this system is [Equation goes here - download the original to see it.] [Example goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] So the equation of motion can also be written [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] So the phase lag of 0.463 rad corresponds to a time of 0.463s. At this time, the oscillation achieves its maximum amplitude of [Equation goes here - download the original to see it.] The angular frequency of the damped oscillation is 1.5 . The frequency is [Equation goes here - download the original to see it.] The period is [Equation goes here - download the original to see it.]
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Contents of
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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