{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 "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 }} {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 "" }}{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 factor of about 20. If this package is unavailable, then the first command afte r " }{TEXT 19 7 "restart" }{TEXT -1 33 " should be omitted, and the fl ag " }{TEXT 19 10 "method=dna" }{TEXT -1 42 " should be removed (or re placed, e.g., by " }{TEXT 19 12 "method=rkf45" }{TEXT -1 2 ")." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 835 " How do we turn this into a display of the glass rim? We pick a radius \+ and aim for oscillations that represent at least a 10 percent amplitu de (to visualize the motion). We should think of the problem as that o f a linear chain with periodic boundary, namely the first and (N+1)st \+ masses are identical. We make the assumption of a linear wave equation for transverse waves: the masses are allowed to move only vertically, i.e., perpendicularly to the circumference of the rim. A strong coupl ing constant is compatible with the fact that we have a relatively rig id object with a speed of sound comparable to metals (about 5.6 km/sec .). We also add some friction to provide us with a reasonable steady-s tate regime on resonance. We will explore the value of the resonant fr equency 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 nearest-neighbour coupling and without the driving force. Later the driving force is ad ded for those mass points that are exposed to the varying air pressure from the horn (a modulation as a function of polar angle phi is intro duced, as the component perpendicular to the rim decreases with increa sing phi)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We graph a beam of rays along the chosen polar angles for the \+ N masses to display the approximation of purely transverse motion for \+ the mass points." }}{PARA 0 "" 0 "" {TEXT -1 128 "The units are arbitr ary and dimensionless. No attempt is made in this qualitative approach to connect with real-life parameters." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "restart; libname:=\"C:\\\\MyFiles\\\\maple\\\\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 "f or 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(phi_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\},\{solset\},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*sin(phi)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "Nt:=500: t:='t': dt:=0.1: for it from 1 to N t 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[ir]:=plot([op(conv(s,ph i_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]:=plots[display]([PL[it],seq(P L2[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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "For N=30 we find interesting res ults for:" }}{PARA 0 "" 0 "" {TEXT -1 64 "m=1, k=20 b=1/4 (or b=1/8) \+ f=1.5/3.1 'triangle flip' (3 crests)" }}{PARA 0 "" 0 "" {TEXT -1 37 "f =1.0/3.14 near the desired resonance" }}{PARA 0 "" 0 "" {TEXT -1 71 "f =2.5/3.14 a multiple crest approx. (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 the monopole (and a lso dipole) vibrations are suppressed. In the calculation a low drivin g frequency produces a 'breathing' mode where the rim expands and cont racts. One could suppress this mode by demanding that the circumferenc e of the rim has to remain within certain bounds. The dipole oscillati on is probably prohibited by the vertical forces (we are just simulati ng a cross section through the glass)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 146 "To connect these results with the wa ve equation we would need to understand the scaling of the coupling co nstant k with the number of mass points." }}{PARA 0 "" 0 "" {TEXT -1 489 "The coupling constant K (coefficient of Laplace operator) in the wave equation is related to the propagation speed of sound waves in g lass (4500 km/sec). This is simulated by coupling the neighboring mass es with a constant k. The nearest-mass coupling represents a finite-di fference approximation to the second derivative, and the scaling shoul d be given by k/dx^2=K, where dx is the distance between two masses al ong the circumference. Thus, for fixed length L=2 Pi R, dx = L/N = R d phi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 11 " Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 19 "Vary the frequency " } {TEXT 19 1 "f" }{TEXT -1 70 " such that different stationary shapes ar e 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 the springs ar e 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 fun ction of time. Explore the phase relation between the two motions as t he frequency varies across the resonance. Also note how the steady-sta te amplitude changes as one tunes the driving frequency to resonance. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 15 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }