EN40: Dynamics and Vibrations

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EN40: Dyamics a Vibratios School of Egieerig Brow Uiversity Solutios to Differetial Equatios of Motio for Vibratig Systems Here, we summarize the solutios to the most importat ifferetial equatios of

1 EN40: Dyamics a Vibratios School of Egieerig Brow Uiversity Solutios to Differetial Equatios of Motio for Vibratig Systems Here, we summarize the solutios to the most importat ifferetial equatios of motio that we ecouter whe aalyzig sigle egree of freeom liear systems. The solutios are erive here if you are curious, but we wo t ask you to erive the solutios i exams. ASE I: ASE II ASE III ASE IV 1 x + x= 1 x x= α 1 x ς + + x= 1 x ς + + x = + KF() t with Ft ( ) = F 0 sit ASE V 1 x ς ς y + + x= + K y + with y = Y 0 sit ASE VI 1 x ς K y x + + = with y = Y0 sit

2 SOLUTION 1: The equatio 1 with iitial coitios has solutio or, equivaletly x + x= x= x = v t = ( φ) x= + X0 si t+ 1 ( x0 ) X0 ( x0 ) v0 / ta = + φ = v0 v xt ( ) = + ( x 0 0 )cost+ sit SOLUTION The equatio with iitial coitios 1 x x= α has solutio x= x = v t = 1 v0 1 v ( ) ( 0 x t = + x0 ) exp t ( x0 ) exp t + α + + α ( α ) ( α )

3 SOLUTION 3 The equatio with iitial coitios has the followig solutios: 1 x ς + + x= x= x = v t = ase I: Overampe System ς > 1 v0 + ( ς )( 0 ) 0 ( )( 0 ) ( ) exp( ) + x v + ς exp( ) x xt = + ςt exp( ) where = ς 1 ase II: ritically Dampe System ς = 1 x( t) = + ( x ) + v + ( x ) t exp( t) { [ ] } ase III: Uerampe System ς < 1 v0 + ς ( 0 ) ( ) exp( ) ( 0 )cos x xt = + ςt x + si where = 1 ς The graphs below show x(t) for two types of iitial coitio: the first graph shows results with v =, while the seco graph shows results with x 0 = 0. Both results are for = Graphs of solutios to ODE goverig free vibratio of a ampe sprig-mass system

4 SOLUTION 4: 1 x ς + + x = + KF() t with Ft ( ) = F 0 sit a iitial coitios x= x0 = v0 t = 0 has solutio of the form xt () = + xh() t + xp() t where the steay state solutio (or particular itegral) is xp( t) = X0si ( t + φ) X0 = KF0M ( /, ζ) 1 M ( /, ζ) = φ = ta ( 1 / ) + ( ς / ) ς / 1 1/ 1 / while the trasiet solutio (or homogeeous solutio, or complemetary solutio) is: ase I: Overampe System ς > 1 h h h h v0 + ( ς ) 0 0 ( ) ( ) exp( ) + x v + ς exp( ) x x 0 h t = ςt exp( ) where = ς 1 ase II: ritically Dampe System ς = 1 h h h { } xh( t) = x + v + x t exp( t) ase III: Uerampe System ς < 1 where = 1 ς h h h v ( ) exp( ) 0 + ς 0 0 cos x xh t = ςt x + si I all three preceig cases, we have set h x = x x (0) = x X siφ 0 0 p 0 0 h p v0 = v0 = v0 X0cos t= 0 Observe that for large time, the trasiet solutio always ecays to zero. φ

5 The graphs below plot the amplitue of the steay state vibratio a the steay state phase lea. (a) (b) Steay state respose of a force sprig mass system (a) amplitue a (b) phase SOLUTION 5 The equatio a iitial coitios has solutio of the form 1 x ς ς y + + x= + K y + x= x = v t = xt () = xh() t + xp() t with yt ( ) = Y0 sit where the steay state solutio (or particular itegral) is x ( t) = X si t + φ X = KY M ( /, ζ) p ( ) { 1+ ( ς / ) } 1/ ς / M ( /, ) ta ζ = φ = 1/ 1 (1 4 ς ) / ( 1 / ) + ( ς / ) while the trasiet solutio (or homogeeous solutio) is: ase I: Overampe System ς > 1 h h h h v0 + ( ς ) 0 0 ( ) ( ) exp( ) + x v + ς exp( ) x x 0 h t = ςt exp( )

6 where = ς 1 ase II: ritically Dampe System ς = 1 h h h { } xh( t) = x + v + x t exp( t) ase III: Uerampe System ς < 1 where = 1 ς h h h v ( ) exp( ) 0 + ς 0 0 cos x xh t = ςt x + si I all three preceig cases, we have set h x = x x (0) = x X siφ 0 0 p 0 0 h p v0 = v0 = v0 X0cos t= 0 Observe that for large time, the trasiet solutio always ecays to zero. The graphs below show the steay state amplitue a phase φ (a) (b) Steay state respose of a base excite sprig mass system (a) Amplitue a (b) Phase

7 SOLUTION 6 The equatio 1 x ς K y x + + = with y = Y0 sit a iitial coitios x= x0 = v0 t = 0 has solutio of the form xt () = x () t + x () t where the steay state solutio (or particular itegral) is x ( t) = X si t + φ X = KY M ( /, ζ) p ( ) h / 1 / ( /, ) ς M ta ζ = φ = 1/ 1 / ( 1 / ) + ( ς / ) p while the trasiet solutio (or homogeeous solutio) is: ase I: Overampe System ς > 1 h h h h v0 + ( ς ) 0 0 ( ) ( ) exp( ) + x v + ς exp( ) x x 0 h t = ςt exp( ) where = ς 1 ase II: ritically Dampe System ς = 1 h h h { } xh( t) = x + v + x t exp( t) ase III: Uerampe System ς < 1 where = 1 ς I all three preceig cases, we have set h h h v ( ) exp( ) 0 + ς 0 0 cos x xh t = ςt x + si

8 h x0 = x0 xp (0) = x0 X0si h p v0 = v0 = v0 X0cosφ t= 0 Observe that for large time, the trasiet solutio always ecays to zero. The graphs below show the steay state amplitue a phase lea for ase 6. φ (a) (b) Steay state respose of a rotor excite sprig mass system (a) Amplitue; (b) Phase

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