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EG260 Multiple Degrees of Freedom Systems

# EG260 Multiple Degrees of Freedom Systems - A.K Slone...

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Unformatted text preview: A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2010 1 of 41 EG-260 DYNAMICS I – Multiple Degrees of Freedom EG-260 DYNAMICS I – Multiple Degrees of Freedom ................................................. 1 1. Introduction ............................................................................................................ 2 2. Two degrees of freedom. ......................................................................... 2 2.1. Finding eigenvalues and eigenvectors. .................................................. 6 2.2. Physical interpretation – natural frequency and mode shape. ........... 9 2.3. Example 1. ............................................................................................. 11 2.3.1. Example 1 applying the initial conditions ........................................... 13 3. Forced response for more than one degree of freedom. .................... 20 3.1. Equations of motion of forced vibration. ............................................ 20 4. Example of a practical 2 dof system .................................................... 22 5. Torsional vibration. .............................................................................. 27 5.1. Example of torsional vibration analysis. ............................................. 28 5.1.1. Example of torsional mode shapes. ..................................................... 32 5.1.2. Example of torsional amplitudes. ........................................................ 35 6. The vibration absorber. ........................................................................ 37 A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2010 2 of 41 1. Introduction Although many systems may be usefully analysed assuming a SDOF system, there are a number of instances where this simplification is not applicable and the system must be modelled as a multiple degree of freedom system (MDOF). The number of degrees of freedom of a system is dependent on the number of moving parts and the directions in which they may move. Each degree of freedom will have an associated natural frequency thus increasing the opportunity for the occurrence of resonance. 2. Two degrees of freedom. Figure 1 shows three configurations that will result in two degrees of freedom. Figure 1a shows two masses, hence two degrees of freedom, connected in series by two springs. Figure 1b shows a single mass that is free to move in two directions x 1 and x 2 , hence two degrees of freedom. Figure 1c shows a single mass with both a translational and a rotational degree of freedom, hence the system has two degrees of freedom. All the cases shown are undamped. Figure 2 shows the force diagram for the two mass system displayed by Figure 1a. Gravitational forces are not considered as the static deflection balances the gravitational force. A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2010 3 of 41 x 2 k 1 k 2 x 1 a) m 1 m 2 b) m k 1 k 2 x 2 x 1 c) W θ k, k θ I m x Figure 1 Schematic for two degrees of freedom systems m 1 m 2 k 2 (x 2- k 2 (x 2- x 1 ) k 1 x 1 x 1 x 2 Figure 2 Force diagram for Figure 1a A.K. Slone A....
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EG260 Multiple Degrees of Freedom Systems - A.K Slone...

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