{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Syste m" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 15 "Vibrating Glass" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 630 "The wine glass shattered by an amplified audio signal which oscillates at the natura l frequency of the glass represents a powerful demonstration of the re sonance phenomenon. The pattern of oscillation of the walls of the win e glass shortly before break-up can be made visible by recording a vid eo with the help of a strobe light. In this worksheet we perform a rat her simple mechanical simulation which should demonstrate that the obs erved shape of the vibration: the rim of the glass changes from circul ar to elliptic shape, then back to circular, and then to an elliptic d eformation in the perpendicular direction, and so forth." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 281 "We consider a sys tem of masses coupled by springs that undergo forced damped motion. In a periodic setup (last mass couples to the first) this system could r epresent the rim of a glass set in motion by an airwave. We allow near est-neioghbour coupling with a strong spring constant " }{TEXT 257 1 " k" }{TEXT -1 95 ", and also allow for an optional spring by which the \+ mass is bound to the equilibrium position." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Our problem in Maple is to set up a set of differential equations for " }{TEXT 19 6 "dsolve" }{TEXT -1 21 ". We make use of the " }{TEXT 19 3 "dna" }{TEXT -1 67 " \+ package (a C-compiled differential equation solver integrated int " } {TEXT 19 6 "dsolve" }{TEXT -1 113 ") to speed up computations by a fac tor of about 20. If this package is unavailable, then the first comman d after " }{TEXT 19 7 "restart" }{TEXT -1 33 " should be omitted, and \+ the flag " }{TEXT 19 10 "method=dna" }{TEXT -1 42 " should be removed \+ (or replaced, e.g., by " }{TEXT 19 12 "method=rkf45" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 254 "The development of the dna 'plug-in' i s based on a collaboration of Drs E.S. Cheb-Terrab and A.Wittkopf, the required zip-file for Windows (or Linux) implementations can be downl oaded from: http://lie.uwaterloo.ca/dna.htm. For the authors' pages lo ok at:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{URLLINK 17 "http://lie.uwat erloo.ca/ecterrab" 4 "http://lie.uwaterloo.ca/dna.htm" "" }{TEXT -1 7 ", or " }{URLLINK 17 "http://www.cecm.sfu.ca/~wittkopf/" 4 "http://w ww.cecm.sfu.ca/~wittkopf/" "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 835 "How do we turn this into a display of the glass rim? W e pick a radius and aim for oscillations that represent at least a 10 percent amplitude (to visualize the motion). We should think of the p roblem as that of a linear chain with periodic boundary, namely the fi rst and (N+1)st masses are identical. We make the assumption of a line ar wave equation for transverse waves: the masses are allowed to move \+ only vertically, i.e., perpendicularly to the circumference of the rim . A strong coupling constant is compatible with the fact that we have \+ a relatively rigid object with a speed of sound comparable to metals ( about 5.6 km/sec.). We also add some friction to provide us with a rea sonable steady-state regime on resonance. We will explore the value of the resonant frequency and the shape of the glass vibrations by trial and error." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 360 "First the differential equations are set up with the n earest-neighbour coupling and without the driving force. Later the dri ving force is added for those mass points that are exposed to the vary ing air pressure from the horn (a modulation as a function of polar an gle phi is introduced, as the component perpendicular to the rim decre ases with increasing phi)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We graph a beam of rays along the chosen polar a ngles for the N masses to display the approximation of purely transver se motion for the mass points." }}{PARA 0 "" 0 "" {TEXT -1 128 "The un its are arbitrary and dimensionless. No attempt is made in this qualit ative approach to connect with real-life parameters." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "restart; libname:=\"C:\\\\MyFiles\\\\mapl e\\\\maple6\\\\dna\",\"C:\\\\Program Files\\\\Maple 6\\\\lib\":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "N:=30: m:=1: k:=20: b:=1/8: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "R0:=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "if i=1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "iM1:=N: iP1:=2 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "elif i=N then iP1:=1: iM1:=i-1 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "else iP1:=i+1: iM1:=i-1:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "DE||i:=m*diff(y||i(t),t$2)=k*(y||iM1(t)+y||iP1(t)-2*y ||i(t)) - b*diff(y||i(t),t) -0.5*y||i(t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "IC||i:=y||i(0)=0,D(y||i)(0)=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "end do: phi_n:=n->evalf((n-1)*2*Pi/N):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "f:=0.95/3.14: # close to the natural frequency ?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for j from 1 to 4 do:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "DE||j:=lhs(DE||j)=rhs(DE||j)+cos(ph i_n(j))/10*sin(2*Pi*f*t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "if j>1 then k:=N-j+2: DE||k:=lhs(DE||k)=rhs(DE||k)+cos(phi_n(k))/10*sin(2*Pi *f*t); fi: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "DEset:=seq(DE||i, i=1..N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ICset:=seq(IC||i,i=1..N ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solset:=seq(y||i(t),i=1..N): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "sol:=dsolve(\{DEset,ICset\},\{s olset\},numeric,output=listprocedure,method=dna):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "for i from 1 to N do: Y||i:=subs(sol,y||i(t)): od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "conv:=(r,phi)->[r*cos(phi),r*si n(phi)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "Nt:=90: t:='t': dt:=0. 25: for it from 1 to Nt do: ti:=it*dt; i:='i': PL[it]:=plot([seq(conv( R0+Y||i(ti),phi_n(i)),i=1..N),conv(R0+Y1(ti),phi_n(1))],thickness=2): \+ od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "for ir from 1 to N do: PL2[i r]:=plot([op(conv(s,phi_n(ir))),s=0..1.2*R0],color=blue): od:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "for it from 1 to Nt do: PLc[it]:=p lots[display]([PL[it],seq(PL2[ir],ir=1..N)],scaling=constrained): od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plots[display](seq(PLc[it],it=1 ..Nt),insequence=true,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%(ANIMATEG6fp7D-%'CURVESG6%7A7$$ \"3'*******yN[+5!#<$\"\"!F07$$\"3[+++lE5'y*!#=$\"3'******\\L+,3#F47$$ \"3y*****HJ&[R\"*F4$\"3A+++H5;pSF47$$\"3i*****>V&=$4)F4$\"3W*****\\SV+ )eF47$$\"3S+++'Qd9p'F4$\"3c+++=jhJuF47$$\"3#******pO.++&F4$\"3c*****\\ ')f-m)F47$$\"3-+++v*p,4$F4$\"3[******y_c5&*F47$$\"33+++HYGX5F4$\"3A+++ a*=_%**F47$$!3%******Rj%GX5F4FS7$$!31+++U*p,4$F4$\"3))*****H;l0^*F47$$ !3T+++0+++]F4$\"39+++NSDg')F47$$!35+++mgI\"p'F4$\"3E+++_#[9V(F47$$!3w* ****\\%*p,4)F4$\"3'******>__y(eF47$$!3x******zXXN\"*F4$\"31+++BkOnSF47 $$!3q******3gZ\"y*F4$\"3)******>!p6z?F47$$!\"\"F0$!3!)*****\\hn?5%!#F7 $$!3[+++0gZ\"y*F4$!33+++?p6z?F47$$!3&)*****4eaa8*F4$!3!******4Umt1%F47 $$!3W+++S*p,4)F4$!3a+++HD&y(eF47$$!39+++`gI\"p'F4$!39+++k#[9V(F47$$!3Z *****p++++&F4$!3/+++MSDg')F47$$!3'******R$*p,4$F4$!38+++m^c5&*F47$$!3- +++:YGX5F4$!3I+++b*=_%**F47$$\"3++++GYGX5F4$!3A+++a*=_%**F47$$\"37+++$ )*p,4$F4$!3M+++w_c5&*F47$$\"30+++#Q.++&F4$!3#******f&)f-m)F47$FB$!3c++ +=jhJuF47$$\"3'******fV&=$4)F4$!39++++M/!)eF47$$\"3_+++A`[R\"*F4$!3A++ +45;pSF47$F2$!3))*****RL+,3#F4F+-%'COLOURG6&%$RGBG$\"#5FbpF/F/-%*THICK NESSG6#\"\"#-F(6$7S7$F/F/7$$\"3h*******\\ech#!#>F/7$$\"3-+++v*G:*[FiuF /7$$\"3u******\\L)4X(FiuF/7$$\"3)******\\MSF+\"F4F/7$$\"3#)****\\Fy:f7 F4F/7$$\"3')****\\d'*)o\\\"F4F/7$$\"3w****\\(>ZIu\"F4F/7$$\"3u****\\xO i(*>F4F/7$$\"3#)****\\FPQ^AF4F/7$$\"3/+++IrS7DF4F/7$$\"3p*****\\o;Bu#F 4F/7$$\"3*********QS6+$F4F/7$$\"3[******\\o-hKF4F/7$$\"3(*******4cZ6NF 4F/7$$\"3S****\\xq!*QPF4F/7$$\"3&********3X$4SF4F/7$$\"3s******f:WQUF4 F/7$$\"3f****\\<_$\\]%F4F/7$$\"3**)*****fs#3u%F4F/7$$\"3!)****\\<#Q'** \\F4F/7$$\"33++]_u3Y_F4F/7$$\"3[*****\\PJK]&F4F/7$$\"3%*****\\n(p$RdF4 F/7$$\"3A*****\\#p2%*fF4F/7$$\"3o****\\xgkeiF4F/7$$\"3g****\\-V&*)['F4 F/7$$\"3.+++&\\$pPnF4F/7$$\"37******>am%*pF4F/7$$\"3?*****\\JigC(F4F/7 $$\"3G****\\PEdR2\"F.F/7$$\"3&*****\\o#R05\"F.F/ 7$$\"3()*****>`9V7\"F.F/7$$\"3)****\\<#Rm\\6F.F/7$$\"3%****\\A_ER<\"F. 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(8 crests around the \+ circumference)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 462 "What remains unexplained by the model is why in nature t he monopole (and also dipole) vibrations are suppressed. In the calcul ation a low driving frequency produces a 'breathing' mode where the ri m expands and contracts. One could suppress this mode by demanding tha t the circumference of the rim has to remain within certain bounds. Th e dipole oscillation is probably prohibited by the vertical forces (we are just simulating a cross section through the glass)." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "To connect these \+ results with the wave equation we would need to understand the scaling of the coupling constant k with the number of mass points." }}{PARA 0 "" 0 "" {TEXT -1 489 "The coupling constant K (coefficient of Laplac e operator) in the wave equation is related to the propagation speed \+ of sound waves in glass (4500 km/sec). This is simulated by coupling t he neighboring masses with a constant k. The nearest-mass coupling rep resents a finite-difference approximation to the second derivative, an d the scaling should be given by k/dx^2=K, where dx is the distance be tween two masses along the circumference. Thus, for fixed length L=2 P i R, dx = L/N = R dphi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 19 "Vary th e frequency " }{TEXT 19 1 "f" }{TEXT -1 70 " such that different stati onary shapes are obtained for the vibration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 2:" }}{PARA 0 "" 0 " " {TEXT -1 34 "Vary the strength of the coupling " }{TEXT 19 1 "k" } {TEXT -1 92 ": describe your observations of the vibrating motion as t he springs are chosen to be softer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 260 11 "Exercise 3:" }}{PARA 0 "" 0 "" {TEXT -1 285 "Graph the motion of the mass corresponding to phi=0 and to phi =Pi/2 as a function of time. Explore the phase relation between the tw o motions as the frequency varies across the resonance. Also note how \+ the steady-state amplitude changes as one tunes the driving frequency \+ to resonance." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "5 25 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }