EG260 General forced response

# EG260 General forced response - A.K. Slone EG-260 Dynamics...

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 1 of 22 EG-260 DYNAMICS I – General Forced Response 1. Introduction ...................................................... 2 1.1. Impulse Response. ............................................ 3 1.2. Under-damped system ..................................... 6 1.3. Undamped system ............................................ 7 1.4. Response to unit impulse ................................. 7 1.5. Example ............................................................. 7 2. Response to arbitrary input .......................... 10 3. Response to arbitrary periodic input ........... 11 3.1. Fourier theorem. ............................................. 12 3.2. Euler Equations for Fourier series. .............. 13 3.3. Fourier Summary. .......................................... 16 3.4. Fourier Example 1. ......................................... 17 3.5. Fourier Example 2. ......................................... 20

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 2 of 22 1. Introduction Previously study has been concerned with the topic of harmonic excitation as applied to the excitation of a SDOF system by a sinusoidal force operating at a single frequency. Here the response of a SDOF system to a generalised harmonic force is considered. The SDOF system is linear therefore the principle of superposition may be applied to calculate the response to a variety of forces based on the response to a specific force. Superposition If x 1 and x 2 are solution to the linear equation of motion 0 2 = + x x w then so is 2 2 1 1 x a x a x + = , where a 1 and a 2 are constants. The concept of superposition may be applied to linear equations with non-zero right hand sides. For example, if x 1 is a particular solution of 1 2 f x x = + w and x 2 is a particular solution of 2 2 f x x = + w then 2 1 x x x p + = is a particular solution of 2 1 2 f f x x + = + w Types of applied force . Periodic – forces that repeat over time. Aperiodic (non-periodic) – forces that do not repeat over time, e.g. a step function. Transient – one that reduces to zero after a finite time, e.g. impulse or shock loading. Types of applied loadings causing vibration.
A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 3 of 22 Earthquakes modelled as sum of decaying periodic or harmonic forces applied to buildings. High winds on structures modelled as either impulsive forces or step loadings. Fluid loadings from sea motion on ships and marine structures. Air providing both lift and drag forces on aircraft. Rough or uneven surfaces providing a variety of forces to cars aeroplanes etc. Manufacturing processes resulting in either random or periodic or aperiodic or transient applied forces causing vibration. 1.1.

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## This note was uploaded on 05/18/2010 for the course ENGINEERIN EG 260 taught by Professor Stone during the Spring '10 term at Swansea UK.

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EG260 General forced response - A.K. Slone EG-260 Dynamics...

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