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Unformatted text preview: Fall 2003 Math 308/501–502 4 Linear Second-Order Equations 4.9 Forced Mechanical Vibrations Fri, 03/Oct c 2003, Art Belmonte Summary When x 00 + cx + ω 2 x = A cos ω t , we have a sinusoidal forcing term. The constant A is the amplitude of the driving force, whereas ω is the driving frequency . If c = 0, we have forced undamped harmonic motion whereas when c > 0, the we have forced damped harmonic motion . Forced undamped harmonic motion When the driving frequency ω is close but not equal to the natural frequency ω of the system, a phenomenon called beats is exhibited. When ω = ω , a phenomenon called resonance is exhibited. Forced damped harmonic motion In this case the solution is consists of the sum of a transient response and a steady-state repsonse . NOTE There is more detail in this section than is contained in this summary. It is especially pertinent in electrical and electronic engineering. We shall, however, omit it here for brevity. Hand Examples Example C [beats] Plot the function cos 9 t- cos 10 t over 0 ≤ t ≤ 4 π . The phenomenon exhibited is called “beats.” Use a trig identity to write the function as a product of sines. This shows that thewrite the function as a product of sines....
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