Forced and dampled harmonic motion
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Damped and forced vibrations
Damped vibrations occur when the amplitude of an oscillating system progressively decreases. [Diagram goes here - download the original to see it.] To study damped vibrations we must begin by revising the study of undamped vibrations. One example of undamped oscillations occur when an object oscillates under simple harmonic motion. Simple harmonic motion occurs when the acceleration of an object is proportional to its displacement and acts in the opposite direction. If the displacement is [Equation goes here - download the original to see it.] then the velocity is [Equation goes here - download the original to see it.] and the acceleration is [Equation goes here - download the original to see it.] then simple harmonic motion is represented by the second order differential equation [Equation goes here - download the original to see it.] Damped oscillations occur when an equation takes the forM. [Equation goes here - download the original to see it.] Expressions like [Equation goes here - download the original to see it.] arise in situation where a particle is subject to linear resistive forces. It is these linear resistive forces that lead to the damping of the oscillation. A linear resistive force is a force acting on a particle whose magnitude is proportion to the speed at which the object is moving and whose direction opposes the velocity of the object. For example, suppose an object is suspended by a spring and is immersed in a bucket of treacle. It is displaced from its equilibrium position and starts to oscillate. However, the treacle progressively slows down the mass, so the amplitude of the oscillations progressively decreases. Consequently, the mass undergoes damped oscillation. [Diagram goes here - download the original to see it.] Let us show how a system of this kind can lead to a differential equation of the form [Equation goes here - download the original to see it.] Let the mass of the particle be M, let the stiffness of the spring be k, let the natural length of the spring be [Equation goes here - download the original to see it.] and let the magnitude of the resistive force of the treacle be [Equation goes here - download the original to see it.] where r is a constant and [Equation goes here - download the original to see it.] is the velocity of the mass. The resistive force is understood to oppose the motion of the particle. We will suppose the spring is displaced by [Equation goes here - download the original to see it.] from its equilibrium position. As usual, we start to study such systems, by considering what happens at the equilibrium position first. When the mass hangs under its own weight in the equilibrium position the forces may be drawn thus [Diagram goes here - download the original to see it.] The mass is subject to two forces, the tension in the spring pulling it upwards and the weight pulling it downwards. The tension is given by kd where d is the extension of the spring up to the equilibrium point. The mass is given by [Equation goes here - download the original to see it.]. Since the system is in equilibrium [Equation goes here - download the original to see it.] Now suppose that the mass is displaced from the equilibrium position by a further [Equation goes here - download the original to see it.]. Now it is subject to three force. 1. The tension due to the extension of the spring.2. Its weight 3.The linear resistive force arising from the opposition to motion of the treacle. [Diagram goes here - download the original to see it.] In the diagram the mass is about to move upwards so R points downwards. The total extension is d + x and the tension is [Equation goes here - download the original to see it.] The negative sign indicates that the tension is in the opposite direction to the displacement. The weight is, as usual [Equation goes here - download the original to see it.] The linear resistive force is [Equation goes here - download the original to see it.] It acts as to oppose the motion. This is shown by the negative sign. In our diagrams, if the particle is moving down then R pulls it up; if the particle is moving up, R pulls it down. The resultant force acting on the particle is [Equation goes here - download the original to see it.] and by Newton's second law this is [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] Since [Equation goes here - download the original to see it.] we have [Equation goes here - download the original to see it.] which is the equation of damped oscillations. To find the equation governing the motion of damped oscillator we need to solve this differential equation. But this is a second order, linear, homogenous constant coefficient differential equation and you should be familiar with its solution.
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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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