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Unformatted text preview: Mechanical Vibrations Introduction • Mechanical vibration is the motion of a particle or body which oscillates about a position of equilibrium. Most vibrations in machines and structures are undesirable due to increased stresses and energy losses. • Time interval required for a system to complete a full cycle of the motion is the period of the vibration. • Number of cycles per unit time defines the frequency of the vibrations. • Maximum displacement of the system from the equilibrium position is the amplitude of the vibration. • When the motion is maintained by the restoring forces only, the vibration is described as free vibration . When a periodic force is applied to the system, the motion is described as forced vibration . • When the frictional dissipation of energy is neglected, the motion is said to be undamped . Actually, all vibrations are damped to some degree. Free Vibrations of Particles. Simple Harmonic Motion • If a particle is displaced through a distance x m from its equilibrium position and released with no velocity, the particle will undergo simple harmonic motion , ( ) = + − = + − = = kx x m kx x k W F ma st ¡ ¡ δ • General solution is the sum of two particular solutions , ( ) ( ) t C t C t m k C t m k C x n n ω ω cos sin cos sin 2 1 2 1 + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = • x is a periodic function and ω n is the natural circular frequency of the motion. • C 1 and C 2 are determined by the initial conditions: ( ) ( ) t C t C x n n ω ω cos sin 2 1 + = 2 x C = n v C ω 1 = ( ) ( ) t C t C x v n n n n ω ω ω ω sin cos 2 1 − = = ¡ Free Vibrations of Particles. Simple Harmonic Motion ( ) φ ω + = t x x n m sin = = n n ω π τ 2 period = = = π ω τ 2 1 n n n f natural frequency ( ) = + = 2 2 x v x n m ω amplitude ( ) = = − n x v ω φ 1 tan phase angle • Displacement is equivalent to the x component of the sum of two vectors which rotate with constant angular velocity 2 1 C C G G + . n ω 2 1 x C v C n = = ω Free Vibrations of Particles. Simple Harmonic Motion ( ) φ ω + = t x x n m sin • Velocitytime and accelerationtime curves can be represented by sine curves of the same period as the displacementtime curve but different phase angles. ( ) ( ) 2 sin cos π φ ω ω φ ω ω + + = + = = t x t x x v n n m n n m ¡ ( ) ( ) π φ ω ω φ ω ω + + = + − = = t x t x x a n n m n n m sin sin 2 2 ¡ ¡ Simple Pendulum (Approximate Solution) • Results obtained for the springmass system can be applied whenever the resultant force on a particle is proportional to the displacement and directed towards the equilibrium position. for small angles, ( ) g l t l g n n n m π ω π τ φ ω θ θ θ θ 2 2 sin = = + = = + ¡ ¡ : t t ma F = ∑ • Consider tangential components of acceleration and force for a simple pendulum, sin sin = + = − θ θ θ θ l g ml W ¡ ¡ ¡ ¡ Sample Problem 19.1 A 50kg block moves between vertical guides as shown. The block is pulled...
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This note was uploaded on 11/09/2009 for the course ENG 91301 taught by Professor Lui during the Spring '08 term at Hong Kong Institute of Vocational Education.
 Spring '08
 LUI
 Stress

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