Ch6-2(4-5)c - 2009-04-22 1 MAE351 Mechanical Vibrations...

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2009-04-22 1 1 MAE351 Mechanical Vibrations Lecture 26 (Chap. 6.4-6.5) 2 6.4 Torsional Vibrations from calculus dx x d from solid mechanics G: shear modulus J : polar moment of cross-section Nt’2 nd lt h l t d dx t x J dx 2 2 ) , ( Newton’s 2 law on the element dx : Substituting the above expressions yields: Eq. of motion when , G and J are constant: t x 2 2 ) , ( ) , ( t t x J x t x GJ x
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2009-04-22 2 3 Initial and boundary conditions Two spatial(boundary) and two time(initial) conditions are required. See Table 6.4 for a list of possible boundary conditions As an example, for a clamped-free rod: Boundary conditions: 0 ) , ( and 0 ) , 0 ( t l G t x See Table 6.5 for solutions of various conditions ) ( ) 0 , ( and ) ( ) 0 , ( 0 0 x x x x t Initial conditions: 4 Ex. 6.4.1 Torsional vibration of a grinding shaft with two disks at each end (Grinding tool) • Top end of the shaft is connected to a pulley (x=0) J 1 includes collective inertia of drive belt, pulley and motor Inertia of pulley, drive belt and motor Shaft of stiffness GJ, length l and mass di t Grinding head inertia density
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2009-04-22 3 5 Use the torque balance at top and bottom to get the boundary conditions at the top at the bottom x x t x J x t x GJ t t x J x t x GJ 2 2 2 0 2 2 1 0 ) , ( ) , ( ) , ( ) , ( l x l x t 6 Again use separation of variables t T x t x ) ( ) ( ) , ( T T x x G c t T t T G x x  0 0 ) ( ) ( ) ( ) ( 2 2 2 2 G c t t ) ( ) ( , ) ( ) (
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2009-04-22 4 7 Boundary condition at 0 x 0 ) ( ) 0 ( ) ( ) 0 ( 1  T GJ t T J t T GJ ) 0 ( ) 0 ( ) ( ) ( ) 0 ( ) ( 1 2 2 2 1   J J c t T t J Similarly, the boundary condition at x= l yields: 8 1 2 1 2 2 1 0 ) 0 ( sin cos ) ( ) 0 ( cos sin ) ( x a x a x a x a x a x a x   2 1 1 2 2 1 1 2 2 1 2 1 1 2 cos sin sin cos ) ( ) ( ) 0 ( ) 0 ( l a l a J J l a l a l J J l l x a J J a J J   CHARACTERISTIC EQUATION 2 2 2 1 2 1 ) ( ) ( ) )( ( ) tan( Jl l J J l J J Jl l
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2009-04-22 5 9 has 0 as its first solution.
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Ch6-2(4-5)c - 2009-04-22 1 MAE351 Mechanical Vibrations...

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