EG260-Harmonic excitation 2

# EG260-Harmonic excitation 2 - A.K Slone EG-260 Dynamics(1...

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2010 1 of 1 EG-260 DYNAMICS I – Harmonic Excitation 2 EG-260 DYNAMICS I – Harmonic Excitation 2 .......................................................... 1 1. Damped SDOF harmonically excited systems ................................................ 2 1. The particular solution .................................................................................... 3 2. The complete solution ..................................................................................... 7 2.1. Example 2.1 ..................................................................................................... 8 3. The transient and steady state responses. ...................................................... 11 3.1. Example 3.1 ................................................................................................... 16 4. Resonance in damped systems .................................................................. 18 5. Alternative solution methods ........................................................................ 21 5.1. The geometrical method. ............................................................................... 21 5.2. The frequency response method. ................................................................... 22 5.3. The transform method. .................................................................................. 26 6. Harmonic excitation of the base .................................................................... 28 6.1. Force transmitted to the mass ........................................................................ 35 6.2. Example 6.2 ................................................................................................... 36 7. Rotating imbalance ........................................................................................ 37 7.1. Force transmitted to the base ......................................................................... 41 7.2. Example 7.2 ................................................................................................... 42 7.3. Example 7.3 ................................................................................................... 45 8. Measurement. ................................................................................................ 49 8.1. Measurement devices .................................................................................... 49

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2010 2 of 2 1. Damped SDOF harmonically excited systems In this document the case of a viscously damped SDOF system subject to harmonic excitation is considered. Using Newton’s Second Law of motion the equation of motion for the system shown in From Figure 1 is: ( ) t F kx x c x m = + + & & & (1) For harmonic excitation the forcing term is assumed to be ( ) ( ) t cos F t F ω 0 = thus the equation of motion is: ( ) t cos F kx x c x m ω 0 = + + & & & (2) Equation (2) may be divided by m to give: ( ) t cos f x m k x m c x ω 0 = + + & & & (3) f c F (t) Friction-free surface k c x(t) mg N f k F Free body diagram Figure 1 Viscously damped SDOF spring-mass system m