Mass-Spring Systems and Resonance

Transcription

1 Mass-Spring Sysems and Resonance Comparing he effecs of damping coefficiens An ineresing problem is o compare he he effec of differen values of he damping coefficien c on he resuling moion of he mass on he spring. Consider he following problem: A kg mass is aached o a spring wih spring consan 8 newons/meer. Deermine he equaion of moion which resuls if he iniial displacemen is y(0) = 6, he iniial velociy is 0, here is no forcing, and he fricion coefficien is (a) n = 0 (b) n= 0 (c) n = In any case he differenial equaion is y''() + ny'() + 8 y() = 0 y(0) = 6 y'(0) = 0 ClearAll@"Global` "D; Off@General::spell, General::spellD

2 MassSpring[].nb soleq = DSolve[ { y''[] +0 y'[] + 8 y[] == 0, y[0] == 6, y'[0] == 0}, y[],] ::y@d 6 CosB F>> y = soleq[[,,]] 6 CosB F soleq = DSolve[ { y''[] +0 y'[] + 8 y[] == 0, y[0] == 6, y'[0] == 0}, y[],] ::y@d 3 CosB 3 3 F + SinB F >> y = soleq[[,,]] 3 CosB 3 3 F + SinB F ComplexExpand[y] 6 CosB 3 F + SinB 3 F

3 MassSpring[].nb 3 soleq3 = DSolve[ { y''[] + y'[] + 8 y[] == 0, y[0] == 6, y'[0] == 0}, y[],] ::y@d >> y3 = soleq3[[,,]]

4 4 MassSpring[].nb Plo[ {y,y,y3}, {,0,}, PloRange -> { -8,8}, PloSyle -> {GrayLevel[0], GrayLevel[.], GrayLevel[.4], GrayLevel[.6]}] 3 4 Resonance as a resul of damping and forcing We consider a forced mass-spring oscillaor sysem wih fricion or damping. The mass is aken o be kg. The spring consan is aken o be 6 newons/meer. The damping can be varied. The frequency of he forcing erm can also be varied. We wan o look a he response of he sysem o he his forcing under a variey of damping condiions. The equaion is y''[] + n y'[] + 6 y[] = Sin[ a ] y[0] = 0 y'[0] = 0

5 MassSpring[].nb The forcing frequency coefficien is a. The damping coefficien is n. Clear[y,] homeq = *y''[] + ν y'[] + 6 y[] == 0 6y@D + 0 DSolve[homeq, y[], ] 4 ::y@d ν 48+ν C@D + 4 ν+ 48+ν C@D>> y0 = y[] /. % 4 : ν 48+ν C@D + 4 ν+ 48+ν C@D> y0 = Firs[%] 4 ν 48+ν C@D + 4 ν+ 48+ν C@D yp = A Cos[a ] + B Sin[ a ] A Cos@a D + B Sin@a D D[yp, {,}] + ν D[yp, ] + 6 yp a A Cos@a D a B Sin@a D + ν Ha B Cos@a D a A Sin@a DL + 6 HA Cos@a D + B Sin@a DL

6 6 MassSpring[].nb Collec[ %, {Cos[a ], Sin[a ]}] I6 A a A + abνm Cos@a D + I6 B a B aaνm Sin@a D uceqns = { 6 A - a^ A + a B ν == 0, 6 B - a^ B - a A ν == } 96 A a A + abν 0, 6 B a B aaν = coeffs = Solve[uceqns, {A, B}] ::A aν 36 a + a 4 + a ν,b I 6 + a M 36 a + a 4 + a ν >> yp = Firs[ A Cos[ a ] + B Sin[ a ] /. coeffs] aν Cos@a D 36 a + a 4 + a ν I 6 + a M Sin@a D 36 a + a 4 + a ν Having solved for he paricular soluion, I now pu ha paricular soluion in ampliude-phase form. F0 = Firs[Sqr[ A^ + B^] /. coeffs]//simplify 36 + a 4 + a I +ν M This hen is he ampliude of he seady sae response o he inpu forcing erm. Now we wan o plo he ampliude of he seadysae response as a funcion of he inpu frequency and he damping coefficien.

7 MassSpring[].nb 7 Plo3D[ F0, {ν, 0, 6}, {a,,4}, PloRange -> {0, }] Noice ha somehing funny is happening near a =, and n = 0. The error messages sugges ha he graph "his" inifiniy near here. Noice oo ha he naural frequency of he undamped oscillaor is! This sounds suspiciously like he dicionary definiion given above! Les plo he response as a funcion of forcing frequnecy for a damping coefficien near n = 0.

8 8 MassSpring[].nb Plo[F0 /. ν -> 0., {a,,3}] The responce ges large near a =, and n = 0.. Le's acually find he full soluion o he iniial value problem. y =(y0 + yp) /. {a ->, ν -> 0.} H L C@D + H L C@D Cos@ D Sin@ D ComplexExpand[ y] 0.06 C@D Cos@.7309 D C@D Cos@.7309 D Cos@ D + I 0.06 C@D Sin@.7309 D C@D Sin@.7309 DM Sin@ D ic =( y /. -> 0) == C@D + C@D 0

9 MassSpring[].nb 9 ic = (D[y,] /. -> 0) == H L C@D H L C@D 0 iceqns = { ic, ic} C@D + C@D 0,.883 H L C@D H L C@D 0< Solve[ iceqns, {C[], C[]}] 88C@D , C@D << y = y /. % 9H L H L + H L H L Cos@ D Sin@ D= ComplexExpand[ Firs[%]] Cos@.7309 D Cos@ D Sin@.7309 D + I Cos@.7309 D Sin@.7309 DM Sin@ D rialsoln =Expand[%] I M 0.06 Cos@.7309 D Cos@ D I M 0.06 Sin@.7309 D Sin@ D

10 0 MassSpring[].nb Plo[ rialsoln,{, 0, 40},PloPoins -> 0 ] So he soluion has been amplified! The seady sae ampliude is abou double ha of he inpu! Now le's see wha happpens when here is no fricion. Forced, Undamped Moion Le's see wha happpens when here is no fricion. The equaion has changed considerably, so we should properly resolve from he beginning. Clear[z,] hde = z''[] + 6 z[] == 0 6z@D + 0

11 MassSpring[].nb DSolve[hde, z[], ] ::z@d C@D CosB 6 F + C@D SinB 6 F>> z0 = %[[,,]] C@D CosB 6 F + C@D SinB 6 F zp = C Cos[ ] + D Sin[ ] C Cos@ D + D Sin@ D D[ zp, {,} ] + 6 zp 4 D Cos@ D 4CCos@ D 4 C Sin@ D 4DSin@ D + 6 HC Cos@ D + D Sin@ DL Simplify[%] HH D+ CL Cos@ D + H C+ DL Sin@ DL zp = (-/) Cos[ ] Cos@ D D[ zp, {,}] + 6 zp Cos@ D + Sin@ D z = z0 + zp Cos@ D + C@D CosB 6 F + C@D SinB 6 F

12 MassSpring[].nb ic = (z /. -> 0) ==0 C@D 0 ic = (D[z,] /. -> 0) == C@D 0 Solve[ {ic, ic}, {C[], C[]}] ::C@D 0, C@D 6 >> z = Firs[Firs[z /. %]] Cos@ D Plo[ z, {, 0, 40}, PloPoins -> 0]

13 MassSpring[].nb 3 Noice ha growing facor of in he paricualr soluion. No wonder ha he graphing complained abou an infinie oupu response ampli ude when he fricion coefficien b = 0.! This is he case of pure resonance, he previous graph was he case of damped resonance. Exercise Given he iniial value problem x''( ) + x() = cos w + 3sim w x(0) = 0 x'(0) = 0 Find a value of w such ha he soluion has beas, and graph he soluion. Find a value such ha he soluion is resonance, and graph he soluion. Find a value of w such ha he soluion is he superposiion of wo periodic funcions, he raio of whose periods is irraional. Graph he soluion. Is he soluion periodic?