example_IV_2_02 - Example IV.2.2 Find the response of the...

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1 Example IV.2.2 Find the response of the single DOF system shown below. z y(z) y 0 cos( π z/2a) x k m a a L L L v c Equation of Motion T = 1 2 m ˙ x 2 R = 1 2 c ˙ x 2 U = 1 2 k x y ( ) 2 Applying Lagrange’s equations: m ˙ ˙ x + c ˙ x + k x = ky t ( ) = f t ( ) Fourier series for excitation The road roughness is periodic in the space variable z; that is, y(z) = y(z+L). Specifically we can write for -L/2 < z < L/2: f z ( ) = 0 ; L/ 2 < z < a = ky 0 cos π z / 2a ( ) a < z < a = a < z < Using z = vt, we can write the road roughness as a function of time t as:
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2 f t ( ) = 0 ; L /2v < t < a / v = ky 0 cos ω t a/ v < t < a / v = a / v < t < where ω = π v/2a . Also, y(t) is an even periodic function in time with a fundamental period and fundamental frequency of: T = L v Ω = 2 π T = 2 π v L From this we can write: f 0 = k T y t ( ) dt T/2 = k T y t ( ) a/v a/v y t ( ) 0 for a / v < t < = k T y 0 cos ω t dt a/v = k T y 0 sin ω t ω t = a/v t = = k T y 0 ω a / v ( ) −ω a / v ( ) ω = kv L y 0 2sin π / 2 ( ) π v / 2a = 4 ka y 0 π L
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3 f cj =
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example_IV_2_02 - Example IV.2.2 Find the response of the...

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