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By Reza N. Jazar

Advanced Vibrations: a latest Approach is gifted at a theoretical-practical point and explains mechanical vibrations thoughts intimately, focusing on their functional use. comparable theorems and formal proofs are supplied, as are real-life purposes. scholars, researchers and working towards engineers alike will get pleasure from the undemanding presentation of a wealth of themes together with yet now not constrained to useful optimization for designing vibration isolators, and temporary, harmonic and random excitations.

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Therefore, the Fourier series of f (t) is F cos T /3 2π j t dt T 2π j t dt T 2πj 1 4πj 3F 1 cos + cos −1 2 2 3 2 3 π j 2 ⎧ ⎨− 9F2 2 j = 1, 2, 4, 5, 7, 8, . . 216) 38 1 Vibration Kinematics Fig. 32 An even periodic function ∞ 9F 4 f (t) = F − 3 2π 2 j =1,2,4,5,7,8,... 223) −π This integral is zero if j = k. 225) This completes the proof of Eq. 218). The bounded integrals of sin and cos happens frequently in Fourier series calculations. Example 28 (Even and odd functions) A function f (t) is called even if f (−t) = f (t) and is called odd if f (−t) = −f (t).

To have a power density of 1 W/m2 , we need a sound with 120 dB loudness. 130) Decibel is also used for the ratio of a physical quantity, usually power or acceleration, relative to a specified or implied reference level. Decibel is dimensionless when we use it to express the ratio of a physical quantity in the same units. Example 17 (Libration of harmonic oscillators) Consider a point mass m and an attraction force towards a fixed origin O on the line of x-axis. 2 Kinematics of Vibrations 27 The equation of motion of the point is a linear equation with constant coefficients.

21. 22. 23. 24. 1 Vibration Kinematics (d) x(t1 ) = x1 , x(t ¨ 2 ) = x¨2 , (e) x(t ¨ 1 ) = x1 , x(t ¨ 2 ) = x¨2 , (f) x = x1 + x2 , x = 2 sin(ωt − π/6), x1 = sin(ωt − π/3). Wave combination. Determine the combined wave of x = x1 + x2 = X sin(ωt + ϕ) if (a) x1 = 3 sin(ωt + π3 ) and x2 = 4 sin(ωt + π4 ), (b) x1 = 3 cos(ωt + π3 ) and x2 = 4 sin(ωt + π4 ), (c) x1 = 3 cos(ωt + π3 ) and x2 = 4 cos(ωt + π4 ). Wave decomposition. The combination of the waves x1 and x2 is x = x1 + x2 = 4 sin(ωt + π4 ). What is x2 if (a) x1 = 3 sin(ωt + π3 ), (b) x1 = 3 cos(ωt + π3 ), (c) x1 = 3 sin(ωt − π3 ).

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