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Unformatted text preview: 12 February 2009 Summary page : Solutions to the ODE’s for the dynamics of springs . Spring with mass m and stiffness constant k . Solving , with and , should be pretty much automatic for us now. The roots of the corresponding characteristic polynomial are m k i . Hence, the general solution for the displacement from the relaxed position is t c t c t x sin cos ) ( 2 1 where the system frequency is m k . Differentiating yields the equation t c t c t v t x cos sin ) ( ) ( 2 1 for the velocity. Given initial conditions ) ( x x and ) ( v v , the equation for the displacement becomes t v t x t x sin cos ) ( (Note: It is interesting to work out the units for the equation of a spring. In the cgs system, the mass m is in grams, the displacement x is in centimeters, and time is in seconds. The spring constant k is in g/s 2 which gives k x units of force or dynes in the cgs system.) In the phase-amplitude form, the above solution is...
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This note was uploaded on 10/19/2009 for the course MATH 38 taught by Professor Any during the Spring '08 term at Tufts.
- Spring '08