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Created by: ruebenko on 03-14-2019 13:00:58
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Bell Acoustics
Bell Acoustics
Georgios Varnavides, Wendi Kong,Monica Shifflet, Emily Thai, Hannah von OldenburgDecember 2016
Introduction
Introduction
The familiar bell sound is produced by superposition of eigenfrequencies - usually given with respect to the prime (2nd eigenfrequency).
Out[39]=
We can simulate these ‘ideal rations’ inside a decay envelope and produce the following sound.
Model the sounds of a bell.
In[17]:=
Clear[sineBank,audioData,sound]primeFrequency=200;ratios={0.5,1,1.2,1.5,2.,2.5,2.667,3,4};decay=Table[If[t<.003,100t,(1-t/4)^3],{t,0,4-1/44100,1/44100}];sineBank["ideal"]=Mean[(AudioGenerator[{"Sin",#},4]&/@(primeFrequencyratios))];audioData["ideal"]=First[AudioData[sineBank["ideal"]]]decay;sound["ideal"]=ListPlay[audioData["ideal"],SampleRate44100,PlayRangeAll]
Problem statement
Problem statement
There are two convolved effects in bell acoustics. Namely the bell geometry and the material stiffness.Can we reproduce the sound of a bell using arbitrary geometries?In particular holding geometry constant (for an easy to fabricate closed-cap cylinder), can we tune materials properties to achieve an aesthetically pleasing bell sound?Motivation:
Bell casting is a costly process
Traditional bells are usually casted slightly thicker than necessary. They are then turned on a lathe while being tuned simultaneously. This is a wasteful process
Scientific curiosity
Obtaining eigenfrequencies
Obtaining eigenfrequencies
Importing the bell STL file
Importing the bell STL file
Import the geometric model and generate a boundary mesh.
In[1]:=
Needs["NDSolve`FEM`"]stlBell=Import[FileNameJoin[{NotebookDirectory[],"SupportFiles","bell.stl"}],"ElementMesh"];
Visualize the boundary mesh with boundary regions colored.
In[3]:=
rr=Most[Range[0,1,1/(Length[stlBell["BoundaryElementMarkerUnion"]])]];colors=ColorData["BrightBands"][#]&/@rr;stlBell["Wireframe"["MeshElementStyle"(Directive[FaceForm[#],EdgeForm[]]&/@colors)]]
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Generate the full mesh.
In[6]:=
emesh=ToElementMesh[stlBell,"MeshOrder"1,"MaxCellMeasure".0001]
Out[6]=
ElementMesh[{{-2.00457,2.00457},{-2.00457,2.00457},{0.0651961,3.9267}},{TetrahedronElement[<66510>]}]
setup a 3D stress operator.
In[7]:=
stressOperator[Y_,ν_]:={Inactive[Div][({{0,0,-((Yν)/((1-2ν)(1+ν)))},{0,0,0},{-(Y/(2(1+ν))),0,0}}.Inactive[Grad][w[x,y,z],{x,y,z}]),{x,y,z}]+Inactive[Div][({{0,-((Yν)/((1-2ν)(1+ν))),0},{-(Y/(2(1+ν))),0,0},{0,0,0}}.Inactive[Grad][v[x,y,z],{x,y,z}]),{x,y,z}]+Inactive[Div][({{-((Y(1-ν))/((1-2ν)(1+ν))),0,0},{0,-(Y/(2(1+ν))),0},{0,0,-(Y/(2(1+ν)))}}.Inactive[Grad][u[x,y,z],{x,y,z}]),{x,y,z}],Inactive[Div][({{0,0,0},{0,0,-((Yν)/((1-2ν)(1+ν)))},{0,-(Y/(2(1+ν))),0}}.Inactive[Grad][w[x,y,z],{x,y,z}]),{x,y,z}]+Inactive[Div][({{0,-(Y/(2(1+ν))),0},{-((Yν)/((1-2ν)(1+ν))),0,0},{0,0,0}}.Inactive[Grad][u[x,y,z],{x,y,z}]),{x,y,z}]+Inactive[Div][({{-(Y/(2(1+ν))),0,0},{0,-((Y(1-ν))/((1-2ν)(1+ν))),0},{0,0,-(Y/(2(1+ν)))}}.Inactive[Grad][v[x,y,z],{x,y,z}]),{x,y,z}],Inactive[Div][({{0,0,0},{0,0,-(Y/(2(1+ν)))},{0,-((Yν)/((1-2ν)(1+ν))),0}}.Inactive[Grad][v[x,y,z],{x,y,z}]),{x,y,z}]+Inactive[Div][({{0,0,-(Y/(2(1+ν)))},{0,0,0},{-((Yν)/((1-2ν)(1+ν))),0,0}}.Inactive[Grad][u[x,y,z],{x,y,z}]),{x,y,z}]+Inactive[Div][({{-(Y/(2(1+ν))),0,0},{0,-(Y/(2(1+ν))),0},{0,0,-((Y(1-ν))/((1-2ν)(1+ν)))}}.Inactive[Grad][w[x,y,z],{x,y,z}]),{x,y,z}]}
Inspect the stress operator.
In[8]:=
stressOperator[Y,ν]//MatrixForm
Out[8]//MatrixForm=
∇ {x,y,z} Yν (1-2ν)(1+ν) Y 2(1+ν) ∇ {x,y,z} ∇ {x,y,z} Yν (1-2ν)(1+ν) Y 2(1+ν) ∇ {x,y,z} ∇ {x,y,z} Y(1-ν) (1-2ν)(1+ν) Y 2(1+ν) Y 2(1+ν) ∇ {x,y,z} |
∇ {x,y,z} Yν (1-2ν)(1+ν) Y 2(1+ν) ∇ {x,y,z} ∇ {x,y,z} Y 2(1+ν) Yν (1-2ν)(1+ν) ∇ {x,y,z} ∇ {x,y,z} Y 2(1+ν) Y(1-ν) (1-2ν)(1+ν) Y 2(1+ν) ∇ {x,y,z} |
∇ {x,y,z} Y 2(1+ν) Yν (1-2ν)(1+ν) ∇ {x,y,z} ∇ {x,y,z} Y 2(1+ν) Yν (1-2ν)(1+ν) ∇ {x,y,z} ∇ {x,y,z} Y 2(1+ν) Y 2(1+ν) Y(1-ν) (1-2ν)(1+ν) ∇ {x,y,z} |
Cylinder Eigenfrequencies
Cylinder Eigenfrequencies
Materials Properties
Materials Properties
Visualize the various material moduli.
Minimization
Minimization
Experimental Results
Experimental Results
Acoustic Comparison
Acoustic Comparison
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