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MIT16_07F09_Lec24

MIT16_07F09_Lec24 - J Peraire S Widnall 16.07 Dynamics Fall...

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J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 1.0 Lecture L24 - Pendulums A pendulum is a rigid body suspended from a fixed point (hinge) which is offset with respect to the body’s center of mass. If all the mass is assumed to be concentrated at a point, we obtain the idealized simple pendulum . Pendulums have played an important role in the history of dynamics. Galileo identified the pendulum as the first example of synchronous motion, which led to the first successful clock developed by Huygens. This clock incorporated a feedback mechanism that injected energy into the oscillations (the escapement, a mechanism used in timepieces to control movement and to provide periodic energy impulses to a pendulum or balance) to compensate for friction loses. In addition to horology (the science of measuring time), pendulums have important applications in gravimetry (the measurement of the specific gravity) and inertial navigation. Simple Pendulum Consider a simple pendulum of mass m and length L . The equation of motion can be derived from the conservation of angular momentum about the hinge point, O , ¨ I O θ = mgL sin θ . Since the moment of inertia is simply I O = mL 2 , we obtain the following non-linear equation of motion, θ ¨ + g sin θ = 0 . (1) L 1
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Multiplying this equation by θ ˙ , we can write, d ( 1 θ ˙ 2 g cos θ ) = 0 , dt 2 L which implies that θ ˙ 2 (2 g/L ) cos θ = constant. Setting θ = θ max , when θ ˙ = 0 we have, θ ˙ = ± 2 g (cos θ cos θ max ) . L This equation cannot be integrated further in an explicit manner. Its solution must be expressed in terms of, so called, elliptic functions. The period of the oscillation, T , is obtained by multiplying by four the time it takes for the pendulum to go from θ = 0 to θ = θ max . Thus, 4 θ max T = . (2 g/L ) 0 cos θ cos θ max Again, this is an integral which cannot be evaluated explicitly, but can be approximated, assuming that θ max is not very large, as (the algebra is omitted here), L θ 2 max T 2 π g 1 + 16 . (2) Small amplitude approximation If we assume that the amplitude of pendulum’s oscillation is small, then sin θ θ , and the equation of motion, given by 1, becomes linear, θ ¨ + g θ = 0 . (3) L This expression is much simpler than equation 1, and has solutions of the form, θ = A sin ω n t + B cos ω n t , where ω n = g/L is the natural frequency of oscillation. It is clear that these solutions are periodic, and the period is given by 2 π L T = = 2 π . (4) ω n g Setting
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