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Vibrating Glass
The wine glass shattered by an amplified audio signal which oscillates at the natural frequency of the glass represents a powerful demonstration of the resonance phenomenon. The pattern of oscillation of the walls of the wine glass shortly before break-up can be made visible by recording a video with the help of a strobe light. In this worksheet we perform a rather simple mechanical simulation which should demonstrate that the observed shape of the vibration: the rim of the glass changes from circular to elliptic shape, then back to circular, and then to an elliptic deformation in the perpendicular direction, and so forth.
We consider a system of masses coupled by springs that undergo forced damped motion. In a periodic setup (last mass couples to the first) this system could represent the rim of a glass set in motion by an airwave. We allow nearest-neioghbour coupling with a strong spring constant k , and also allow for an optional spring by which the mass is bound to the equilibrium position.
Our problem in Maple is to set up a set of differential equations for dsolve . We make use of the dna package (a C-compiled differential equation solver integrated int dsolve ) to speed up computations by a factor of about 20. If this package is unavailable, then the first command after restart should be omitted, and the flag method=dna should be removed (or replaced, e.g., by method=rkf45 ).
The development of the dna 'plug-in' is based on a collaboration of Drs E.S. Cheb-Terrab and A.Wittkopf, the required zip-file for Windows (or Linux) implementations can be downloaded from: http://lie.uwaterloo.ca/dna.htm. For the authors' pages look at:
http://lie.uwaterloo.ca/ecterrab, or http://www.cecm.sfu.ca/~wittkopf/
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 amplitude (to visualize the motion). We should think of the problem as that of 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 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 reasonable 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.
First the differential equations are set up with the nearest-neighbour coupling and without the driving force. Later the driving force is added 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 introduced, as the component perpendicular to the rim decreases with increasing phi).
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.
The units are arbitrary and dimensionless. No attempt is made in this qualitative approach to connect with real-life parameters.
> restart; libname:="C:\\MyFiles\\maple\\maple6\\dna","C:\\Program Files\\Maple 6\\lib":
> N:=30: m:=1: k:=20: b:=1/8:
> R0:=1:
> for i from 1 to N do
> if i=1 then
> iM1:=N: iP1:=2:
> elif i=N then iP1:=1: iM1:=i-1:
> else iP1:=i+1: iM1:=i-1:
> end if:
> 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):
> IC||i:=y||i(0)=0,D(y||i)(0)=0:
> end do: phi_n:=n->evalf((n-1)*2*Pi/N):
> f:=0.95/3.14: # close to the natural frequency?
> for j from 1 to 4 do:
> DE||j:=lhs(DE||j)=rhs(DE||j)+cos(phi_n(j))/10*sin(2*Pi*f*t);
> 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:
> DEset:=seq(DE||i,i=1..N):
> ICset:=seq(IC||i,i=1..N):
> solset:=seq(y||i(t),i=1..N):
> sol:=dsolve({DEset,ICset},{solset},numeric,output=listprocedure,method=dna):
> for i from 1 to N do: Y||i:=subs(sol,y||i(t)): od:
> conv:=(r,phi)->[r*cos(phi),r*sin(phi)]:
> 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:
> for ir from 1 to N do: PL2[ir]:=plot([op(conv(s,phi_n(ir))),s=0..1.2*R0],color=blue): od:
> for it from 1 to Nt do: PLc[it]:=plots[display]([PL[it],seq(PL2[ir],ir=1..N)],scaling=constrained): od:
> plots[display](seq(PLc[it],it=1..Nt),insequence=true,scaling=constrained);
>
For N=30 we find interesting results for:
m=1, k=20 b=1/4 (or b=1/8) f=1.5/3.1 'triangle flip' (3 crests)
f=1.0/3.14 near the desired resonance
f=2.5/3.14 a multiple crest approx. (8 crests around the circumference)
What remains unexplained by the model is why in nature the monopole (and also dipole) vibrations are suppressed. In the calculation a low driving frequency produces a 'breathing' mode where the rim expands and contracts. One could suppress this mode by demanding that the circumference of the rim has to remain within certain bounds. The dipole oscillation is probably prohibited by the vertical forces (we are just simulating a cross section through the glass).
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.
The coupling constant K (coefficient of Laplace operator) in the wave equation is related to the propagation speed of sound waves in glass (4500 km/sec). This is simulated by coupling the neighboring masses with a constant k. The nearest-mass coupling represents a finite-difference approximation to the second derivative, and the scaling should be given by k/dx^2=K, where dx is the distance between two masses along the circumference. Thus, for fixed length L=2 Pi R, dx = L/N = R dphi.
Exercise 1:
Vary the frequency f such that different stationary shapes are obtained for the vibration.
Exercise 2:
Vary the strength of the coupling k : describe your observations of the vibrating motion as the springs are chosen to be softer.
Exercise 3:
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 two motions as the frequency varies across the resonance. Also note how the steady-state amplitude changes as one tunes the driving frequency to resonance.
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