3.8we - 3.8: MECHANICAL AND ELECTRICAL VIBRATIONS KIAM...

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3.8: MECHANICAL AND ELECTRICAL VIBRATIONS KIAM HEONG KWA 1. Mechanical Vibrations Consider a mass m attached to an elastic spring 1 of length l , which is suspended from a rigid horizontal support. The equilibrium position of the mass is the point where the mass will hang at rest if no external forces act upon it. In equilibrium, the mass causes an elongation Δ l of the spring in the downward (positive) direction. Thus, by Hooke’s law, the spring exerts a restoring force F s = - k Δ l , where k is a numerically positive constant which measures the stiffness of the spring. It is called the spring constant . By Newton’s second law, the restoring force F s is precisely balanced by the weight W = mg of the mass in this position, so that (1.1) k Δ l = mg. Here g represents the gravitational acceleration. We are interested in studying the motion of the mass when it is initially displaced from its equilibrium position or is acted on by an external force. In describing the motion of the mass, it is convenient to measure its displacement u ( t ), measured (positive) downward, from the equilibrium position at time t . We now analyze the forces acting on the mass. (i) The mass has a weight of W = mg at all time. Date : February 1, 2011. 1 An elastic spring has the property that if it is stretched or compressed a distance Δ l which is small compared to its natural length l , then it will exert a restoring force F s of magnitude k Δ l in consequence of Hooke’s law. Hooke’s law is only an approximation, but it works very well for most springs in real life, as long as the spring isn’t compressed or stretched so much that it is permanently bent or damaged. The constant k is called the spring constant. It measures the stiffness of the spring. 1
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2 KIAM HEONG KWA (ii) At time t , the elongation of the spring is Δ l + u ( t ). Hence the spring force F s which tends to restore the spring to its natural position is now a function of t given by (1.2) F s ( t ) = - k l + u ( t )] . (iii) In addition, the mass and spring may be immersed in a medium, such as oil, which impedes the motion of the mass. This induces a damping or resistive force F d which always acts in the direction opposite to the direction of motion of the mass. For simplicity, we assume that (1.3) F d ( t ) = - γu 0 ( t ) for a numerically positive damping constant γ . The negative sign is due to the fact that the damping force F d tends to impede the motion of the mass. (iv) Finally, the mass may be acted upon by an external force F ( t ). In view of the forces acting on the mass, we get that
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This note was uploaded on 11/11/2011 for the course MATH 415.01 taught by Professor Kwa during the Winter '11 term at Ohio State.

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3.8we - 3.8: MECHANICAL AND ELECTRICAL VIBRATIONS KIAM...

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