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Chapter Five: Linear Oscillations and Normal Modes
By R. Douglas Gregory
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From Classical Mechanics: An Undergraduate Text
OVERVIEWKEY FEATURES The key features of this chapter are the properties of free undamped oscillations, free damped oscillations, driven oscillations, and coupled oscillations. Oscillations are a particularly important part of mechanics and indeed of physics as a whole. This is because of their widespread occurrence and the practical importance of oscillation problems. In this chapter we study the classical linear theory of oscillations, which is important for two reasons: (i) the linear theory usually gives a good approximation to the motion when the amplitude of the oscillations is small, and (ii) in the linear theory, most problems can be solved explicitly in closed form. The importance of this last fact should not be underestimated! We develop the theory in the context of the oscillations of a body attached to a spring, but the same equations apply to many different problems in mechanics and throughout physics. In the course of this chapter we will need to solve linear second order ODEs with constant coefficients. For a description of the standard method of solution see Boyce & DiPrima [8]. 5.1 BODY ON A SPRINGSuppose a body of mass m is attached to one end of a light spring. The other end of the spring is attached to a fixed point A on a smooth horizontal table, and the body slides on the table in a straight line through A. Let x be the displacement and ? the...
Copyright Cambridge University Press 2006 under license agreement with Books24x7
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