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# set13 - \$ Set 13 Vibrations Part 1 Kyle A Gallivan...

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a39 a38 a36 a37 Set 13: Vibrations Part 1 Kyle A. Gallivan Department of Mathematics Florida State University Ordinary Differential Equations Fall 2009 1

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a39 a38 a36 a37 Mechanical and Electrical Vibration au ′′ + bu + cu = F ( t ) a > 0 , c > 0 , b 0 simple model of mass hanging on a vertical spring mass sliding on a surface connected to a horizontal spring current in a simple RLC circuit generalizations to linear systems for structures and circuits Mu ′′ + Du + Ku = F ( t ) M, D, K R n × n generalizations to systems with nonlinear elements, e.g., transistors 2
a39 a38 a36 a37 Mechanical Vibration mass hanging on a vertical spring F = ma Newton and F s = - kL Hooke’s Law for a spring unweighted length equilibrium position yields length of spring + L > 0 mg = kL, m > 0 , k > 0 , g > 0 u ( t ) represents displacement from equilibrium where spring is stretched/mass moved down u > 0 spring is compressed/mass moved up u < 0 3

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a39 a38 a36 a37 Mechanical Vibration mu ′′ = forces mg > 0 force of gravity stretching the spring • - k ( L + u ) force of spring resisting stretching and compression (note the sign change is as L + u changes sign) • - γu , γ > 0 force of damping slowing velocity (note the sign change as u changes sign) damping from friction of surface, guides, enclosing fluid, etc. damping device: dashpot 4
a39 a38 a36 a37 Mechanical Vibration mu ′′ = forces mu ′′ = mg - γu - k ( L + u ) mu ′′ = mg - kL - γu - ku mu ′′ = - γu - ku mu ′′ + γu + ku = 0 u (0) = initial displacement u (0) = initial velocity If there are other forces varying with time then mu ′′ + γu + ku = F ( t ) 5

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a39 a38 a36 a37 Mechanical Vibration Undamped ( γ = 0 ) and free ( F ( t ) = 0 ) vibrations satisfy mu ′′ + ku = 0 mr 2 + k = 0 r ± = - γ ± radicalbig γ 2 - 4 mk 2 m = ± i radicalbigg k m u ( t ) = A cos ω 0 t + B sin ω 0 t steady sinusoidal oscillation ω 0 is called the natural frequency where ω 2 0 = k m period of oscillation T = 2 π ω 0 6
a39 a38 a36 a37

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set13 - \$ Set 13 Vibrations Part 1 Kyle A Gallivan...

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