Forced and dampled harmonic motion
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Forced oscillations
We now consider the situation where a car suspension system travels over a bumpy road. The road surface is modelled by a function [Equation goes here - download the original to see it.], where t is the time at which the wheel hits the surface whose height is [Equation goes here - download the original to see it.]. For example, suppose [Equation goes here - download the original to see it.] then the surface would be represented by a sine curve. [Diagram goes here - download the original to see it.] We want to be able to predict what will happen to the car suspension system as it travels over this surface. The function describing the 'surface' is generally called a forcing function. [Diagram goes here - download the original to see it.] The diagram shows one wheel as it travels over the bumps. We want to know what the passengers will experience, which is how the chasis of mass m will move. As before, we will answer the related question of what would happen to an object of mass m subjected to a forcing function [Equation goes here - download the original to see it.]. [Diagram goes here - download the original to see it.] The mass m will be subjected to a forcing function [Equation goes here - download the original to see it.]. In order to determine the response of the system, we need a 'fixed' point or reference level from which we can measure the displacement. The displacement of the point P is determined by the forcing function. Let us take the fixed level to be the position of the length of the spring when hanging at static equilibrium not subject to a forcing function. [Diagram goes here - download the original to see it.] In this situation the spring will have length [Equation goes here - download the original to see it.]where d is the extension and the mass will be subject to the tension and the weight which are in eqilibrium, so as usual [Equation goes here - download the original to see it.] So we will determine the response of the system by finding how this point will move when the point P is subjected to the forcing function [Equation goes here - download the original to see it.]. The forcing function [Equation goes here - download the original to see it.] gives the displacement of about, say, its static equilibrium level. [Diagram goes here - download the original to see it.] As the point P is forced upwards by [Equation goes here - download the original to see it.]the length of the spring will shorten by [Equation goes here - download the original to see it.]. If [Equation goes here - download the original to see it.]is the increase in the length of the spring due to the displacement of the system about the equilibrium level then the length of the spring at time will be [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] In case this looks odd you should realise that is the reference level that is fixed in our description of the oscillating system; the upper end is not fixed - it is also oscillating, so the diagram could be redrawn thus [Diagram goes here - download the original to see it.] The extension of the spring is [Equation goes here - download the original to see it.] So the tension is [Equation goes here - download the original to see it.] It is negative because it acts upwards that is in the direction opposite to that of the displacement. The particle is subject to the forces [Equation goes here - download the original to see it.] And [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] The resultant is [Equation goes here - download the original to see it.]and obeys Newton's second law. So [Equation goes here - download the original to see it.] Note we are measuring the forcing function in the opposite direction to the sense in which we measure the displacement. This eliminates the untidy negative sign that would appear in front of [Equation goes here - download the original to see it.]. This is a inhomogeneous second order constant coefficient differential equation and its method of solution should be familiar to you. [Example 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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