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# L48 - Fall 2003 Math 308/501502 4 Linear Second-Order...

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Fall 2003 Math 308/501–502 4 Linear Second-Order Equations 4.8 Free Mechanical Vibrations Fri, 03/Oct c 2003, Art Belmonte Summary Background material on spring-mass-dashpot systems may be found in Section 4.1 of your textbook. For electric circuits, we refer you to the component laws and Kirchhoff’s laws on pages 119–120. Familiarize yourselves with the following terms. spring equilibrium spring-mass equilibrium mass m [inertia] damping constant b [due to a dashpot, friction, etc.] spring constant k [stiffness] Motion of a vibrating spring The governing DE of motion is my 00 + by 0 + ky = F ( t ) , where the right-hand side is a function which represents an external force. In this section, F ( t ) = 0, so that the motion is unforced . Electric circuits Remarkably, the mathematical treatment of certain circuits directly parallels that of mass-spring systems. For example, LQ 00 + RQ 0 + 1 C Q = E ( t ) or LI 00 + RI 0 + 1 C I = E 0 ( t ) . Harmonic motion Putting the DEs into SLF, we make the following identifications. c = b 2 m , ω 0 = r k m , f ( t ) = F ( t ) m , x = y c = R 2 L , ω 0 = r 1 LC , f ( t ) = E ( t ) L , x = Q c = R 2 L , ω 0 = r 1 LC , f ( t ) = E 0 ( t ) L , x = I This gives the equation of harmonic motion, where c and ω 0 are nonnegative: x 00 + 2 cx 0 + ω 2 0 x = f ( t ) . Simple harmonic motion When c = 0 and f ( t ) = 0, we have x 00 + ω 2 0 x = 0, whence x = c 1 cos ω 0 t + c 2 sin ω 0 t via §4 . 3 techniques. The natural frequency of the motion is ω 0 = k / m and its period is T = 2 π/ω 0 = 2 π m / k . Damped harmonic motion Here c > 0. There are three cases according to the nature of the roots of the characteristic equation r 2 + 2 cr + ω 2 0 = 0, namely r = - c q c 2 - ω 2 0 , which we’ll label r 1 and r 2 , respectively. When c < ω 0 , the motion is underdamped . We have x ( t ) = e - ct ( c 1 cos ω t + c 2 sin ω t ) , with ω = q ω 2 0 - c 2 . When c > ω 0 , the motion is overdamped . We have x ( t ) = c 1 e r 1 t + c 2 e r 2 t ; here r 1 < r 2 < 0. When c = ω 0 , the motion is critically damped . We have x ( t ) = c 1 e - ct + c 2 te - ct . Note that in the overdamped and critially damped cases, there is no oscillation. Hand Examples Example A Plot y = - 3 cos 2 t + sin 2 t over the interval 0 t 2 π . Then place this function in the form y = A cos t - φ) and graph it.

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L48 - Fall 2003 Math 308/501502 4 Linear Second-Order...

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