chapt13 - Mechanical Vibrations Introduction • Mechanical...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 • Velocity-time and acceleration-time curves can be represented by sine curves of the same period as the displacement-time 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 spring-mass 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 50-kg block moves between vertical guides as shown. The block is pulled...
View Full Document

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.

Page1 / 28

chapt13 - Mechanical Vibrations Introduction • Mechanical...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online