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Multi-degree of Freedom Systems Motivation: Many systems are too complex to be represented by a single degree of freedom model. Objective of this chapter: Understand vibration of systems with more than one degree of freedom.

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Free-vibration of undamped two- degree of freedom system We learn how to analyze free vibration by considering an example k1k1 k2k2 k3k3 m1m1 m2m2 x1x1 x2x2

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Deriving equations of motion m1m1 k1x1k1x1 k 2 (x 1 -x 2 ) m2m2 k 2 (x 2 -x 1 ) k3x2k3x2 Interpretation of coefficients First equation: k 1 +k 2 is the force on m 1 needed to move it slowly by one unit while m 2 is held stationary. –k 2 is the force on m 1 need to hold it steady if m 2 is displaced slowly by one unit. Second equation: k 1 +k 2 is the force on m 2 needed to move it slowly by one unit while m 1 is held stationary. –k 2 is the force on m 2 need to hold is steady if m 1 is displaced slowly by one unit. Maxwell reciprocity theorem: the force on m 1 need to hold it steady if m 2 is displaced slowly by one unit=force on m 2 need to hold is steady if m 1 is displaced slowly by one unit.

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Special case Let m1=m2=m and k1=k2=k3=k. Then:

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Solution of equations of motion We know from experience that: Substituting the above equation to eq. of motion, we obtain two eqs w.r.t. A, B : The above two equations are satisfied for every t

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Trivial solution A, B=0. In order to have a nontrivial solution: First natural frequency: The displacement for the first natural frequency is: This vector is called mode shape. Constant cannot be determined. Usually assume first entry is 1. Therefore, mode shape is:

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Thus, both masses move in phase and the have the same amplitudes.

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Second natural frequency: Mode shape for above frequency: Usually assume first entry is 1. Therefore, mode shape is: The two masses move in opposite directions.

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Displacements of two masses are sums of displacements in the two modes:

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General expression for vibration of the two-degree of freedom system

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Observations: System motion is superposition of two harmonic (sinusoidal) motions with frequencies 1 and 2. Participation of each mode depends on initial conditions. Four unknowns (c 1, c 2, 1, 2 ) can be found using four initial conditions. Can find specific initial conditions so that only one mode is excited.

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How to solve a free vibration problem involving a two degree of freedom system 1) Write equations of motion for free vibration (no external force or moment) 2) Assume displacements are sinusoidal waves, and plug in equations of motion: Obtain equation: where 3) Solve det[C(ω)]=0, obtain two natural frequencies, ω 1, ω 2. 4) Solve, Assume A=1, and obtain first mode shape. Obtain second mode shape in a similar manner.

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5) Find constants from initial conditions. Free vibration response is:

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ME 440 Intermediate Vibrations Tu, March 10, 2009 Chapter 4: Vibration Under General Forcing Conditions © Dan Negrut, 2009 ME440, UW-Madison.

ME 440 Intermediate Vibrations Tu, March 10, 2009 Chapter 4: Vibration Under General Forcing Conditions © Dan Negrut, 2009 ME440, UW-Madison.

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