Page 1 © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Contents 19 - 2 Introduction Free Vibrations of Particles. Simple Harmonic Motion Simple Pendulum (Approximate Solution) Simple Pendulum (Exact Solution) Sample Problem 19.1 Free Vibrations of Rigid Bodies Sample Problem 19.2 Sample Problem 19.3 Principle of Conservation of Energy Sample Problem 19.4 Forced Vibrations Sample Problem 19.5 Damped Free Vibrations Damped Forced Vibrations Electrical Analogues Page 2 © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Contents 19 - 2 Introduction Free Vibrations of Particles. Simple Harmonic Motion Simple Pendulum (Approximate Solution) Simple Pendulum (Exact Solution) Sample Problem 19.1 Free Vibrations of Rigid Bodies Sample Problem 19.2 Sample Problem 19.3 Principle of Conservation of Energy Sample Problem 19.4 Forced Vibrations Sample Problem 19.5 Damped Free Vibrations Damped Forced Vibrations Electrical Analogues © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Introduction 19 - 3 • 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. Page 3 © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Contents 19 - 2 Introduction Free Vibrations of Particles. Simple Harmonic Motion Simple Pendulum (Approximate Solution) Simple Pendulum (Exact Solution) Sample Problem 19.1 Free Vibrations of Rigid Bodies Sample Problem 19.2 Sample Problem 19.3 Principle of Conservation of Energy Sample Problem 19.4 Forced Vibrations Sample Problem 19.5 Damped Free Vibrations Damped Forced Vibrations Electrical Analogues © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Introduction 19 - 3 • 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. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Free Vibrations of Particles. Simple Harmonic Motion 19 - 4 • 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, ( ) 0 = + - = + - = = kx x m kx x k W F ma st ? ? d • General solution is the sum of two particular solutions, ( ) ( ) t C t C t m k C t m k C x n n ? ? cos sin cos sin 2 1 2 1 + = ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = • x is a periodic function and ? 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 ? ? cos sin 2 1 + = 0 2 x C = n v C ? 0 1 = ( ) ( ) t C t C x v n n n n ? ? ? ? sin cos 2 1 - = = ? Page 4 © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Contents 19 - 2 Introduction Free Vibrations of Particles. Simple Harmonic Motion Simple Pendulum (Approximate Solution) Simple Pendulum (Exact Solution) Sample Problem 19.1 Free Vibrations of Rigid Bodies Sample Problem 19.2 Sample Problem 19.3 Principle of Conservation of Energy Sample Problem 19.4 Forced Vibrations Sample Problem 19.5 Damped Free Vibrations Damped Forced Vibrations Electrical Analogues © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Introduction 19 - 3 • 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. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Free Vibrations of Particles. Simple Harmonic Motion 19 - 4 • 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, ( ) 0 = + - = + - = = kx x m kx x k W F ma st ? ? d • General solution is the sum of two particular solutions, ( ) ( ) t C t C t m k C t m k C x n n ? ? cos sin cos sin 2 1 2 1 + = ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = • x is a periodic function and ? 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 ? ? cos sin 2 1 + = 0 2 x C = n v C ? 0 1 = ( ) ( ) t C t C x v n n n n ? ? ? ? sin cos 2 1 - = = ? © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Free Vibrations of Particles. Simple Harmonic Motion 19 - 5 ( ) ? ? + = t x x n m sin = = n n ? p t 2 period = = = p ? t 2 1 n n n f natural frequency ( ) = + = 2 0 2 0 x v x n m ? amplitude ( ) = = - n x v ? ? 0 0 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 ? ? + . n ? 0 2 0 1 x C v C n = = ? Page 5 © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Contents 19 - 2 Introduction Free Vibrations of Particles. Simple Harmonic Motion Simple Pendulum (Approximate Solution) Simple Pendulum (Exact Solution) Sample Problem 19.1 Free Vibrations of Rigid Bodies Sample Problem 19.2 Sample Problem 19.3 Principle of Conservation of Energy Sample Problem 19.4 Forced Vibrations Sample Problem 19.5 Damped Free Vibrations Damped Forced Vibrations Electrical Analogues © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Introduction 19 - 3 • 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. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Free Vibrations of Particles. Simple Harmonic Motion 19 - 4 • 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, ( ) 0 = + - = + - = = kx x m kx x k W F ma st ? ? d • General solution is the sum of two particular solutions, ( ) ( ) t C t C t m k C t m k C x n n ? ? cos sin cos sin 2 1 2 1 + = ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = • x is a periodic function and ? 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 ? ? cos sin 2 1 + = 0 2 x C = n v C ? 0 1 = ( ) ( ) t C t C x v n n n n ? ? ? ? sin cos 2 1 - = = ? © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Free Vibrations of Particles. Simple Harmonic Motion 19 - 5 ( ) ? ? + = t x x n m sin = = n n ? p t 2 period = = = p ? t 2 1 n n n f natural frequency ( ) = + = 2 0 2 0 x v x n m ? amplitude ( ) = = - n x v ? ? 0 0 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 ? ? + . n ? 0 2 0 1 x C v C n = = ? © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics Ninth Edition Free Vibrations of Particles. Simple Harmonic Motion 19 - 6 ( ) ? ? + = 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 p ? ? ? ? ? ? + + = + = = t x t x x v n n m n n m ? ( ) ( ) p ? ? ? ? ? ? + + = + - = = t x t x x a n n m n n m sin sin 2 2 ? ?Read More

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