Ch1-3(4-6)c - MAE351 Mechanical Vibrations Lecture 3(Chap...

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1 MAE351 Mechanical Vibrations Lecture 3 (Chap. 1.4-1.6) 1 Modeling and Energy Methods (Section1.4) Modeling: An art or process of writing an equation or Energy Method: An alternative way to determine the equation of motion systems of equation to describe the motion of a physical device, done mostly by Newton’s 2 nd law 2 An alternative way to calculate the natural frequency of a system Useful if the forces or torques acting on the object or mechanical part are difficult to determine
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2 Potential & Kinetic Energy The potential energy of mechanical systems U is often stored in “springs” x 0 x=0 (remember that for a spring F=kx ) 0 0 0 0 x kx x F U x x spring d d The kinetic energy of mechanical systems T is due M k Mass Spring 3 to the motion of the “mass” in the system T T rot trans Conservation of Energy For a simple, conservative (i.e., no damper), mass spring system, the total mechanical energy must be conserved: U T or constant conserved : At two different times t 1 and t 2 , the increase in potential energy must be equal to a decrease in kinetic energy 4 (or vice versa ). max max T U T T U U and 1 2 2 1
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3 Deriving the equation of motion k x x=0 0 2 2 d d U T d d ) ( M Mass Spring 5  0 0 kx x m x kx x m x dt dt time all for zero be cannot Since Natural frequency using If the solution is given by A sin( t+ ), then the maximum potential and kinetic energies can be used to calculate the max max UT A m kA A m T kA U n 2 2 2 2 1 1 2 1 2 1 ) ( max max equal be must values two these Since natural frequency of the system as 6 m k n n 2 2 2 ) (
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4 Example Compute the natural frequency of this roller fixed in place by a spring. Assume that it is a conservative system (i.e., no losses) and rolls without slipping. r m,J x(t) k 7 T T trans rot and Solution (continued) 2 2 Rot 2 1 r x J T r x r x r m,J x(t) k max max 2 max max 2 2 2 max max Thus 1 at happens of max value The ) ( 2 1 ) ( 2 1 at happens of max value The U T kA U A x U A m r A J T A v T n n n 8
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Ch1-3(4-6)c - MAE351 Mechanical Vibrations Lecture 3(Chap...

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