EG260-Harmonic excitation 1

# EG260-Harmonic excitation 1 - A.K Slone EG-260 Dynamics(1...

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2008 1 of 17 EG-260 DYNAMICS I – Harmonic Excitation 1 EG-260 DYNAMICS I – Harmonic Excitation 1. ........... 1 Introduction to harmonic excitation. ............................. 2 1. Equation of motion . ................................................. 2 2. Harmonic excitation of an undamped system. ......... 4 2.1. Example. ................................................................. 8 2.2. The phenomenon of beats. ................................... 12 2.3. Resonance. ........................................................... 14

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2008 2 of 17 Introduction to harmonic excitation A system is said to be subject to a forced vibration whenever external energy is supplied to the system during vibration. The harmonic excitation refers to the excitation due to a sinusoidal external force of a single frequency. In what follows the harmonic excitation as applied to a spring-mass-damper SDOF system is considered. Harmonic excitation may be solved mathematically fairly straightforwardly and are easily simulated under laboratory conditions. In addition, harmonic excitations are frequently experienced in a number of real life applications, for instance in many rotating machines such as fans, electric motors, where there may be some element of imbalance. For more complex situations where there may be a complex forcing term Fourier analysis may be used to express the complex forcing term as a series of harmonic terms. 1. Equation of motion A force F (t) acts on a viscously damped spring-mass system as in Figure 1.
A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2008 3 of 17 From Newton’s Second Law of motion the equation of motion is: ( ) t F kx x c x m = + + & & & (1) As with any second order differential equation the solution to the non-homogenous equation (1) is formed by obtaining a solution to the homogenous equation and the particular solution and adding the two solutions together. The homogenous equation: 0 = + + kx x c x m & & & (2) represents the free vibration of the system. f c F (t) Friction-free surface k c x(t) mg N f k F m Free body diagram Figure 1 Viscously damped SDOF spring-mass system

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2008 4 of 17 As the system is damped the free vibration dies out due to the effect of damping. There are three possible damping conditions: Underdamped Overdamped Critically damped Thus the general solution to equation (1) eventually reduces over time to particular solution, x p (t), i.e. to the steady state vibration. Steady state vibration will be present as long as the forcing function is present. 2. Harmonic excitation of an undamped system.
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EG260-Harmonic excitation 1 - A.K Slone EG-260 Dynamics(1...

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