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Created by: ruebenko on 03-14-2019 13:00:58
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Bell Acoustics

Georgios Varnavides, Wendi Kong,Monica Shifflet, Emily Thai, Hannah von OldenburgDecember 2016

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

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

    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)]]
    Out[5]=
    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}
    ·0,0,-
    Yν
    (1-2ν)(1+ν)
    ,{0,0,0},-
    Y
    2(1+ν)
    ,0,0.
    ∇
    {x,y,z}
    w[x,y,z]+
    ∇
    {x,y,z}
    ·0,-
    Yν
    (1-2ν)(1+ν)
    ,0,-
    Y
    2(1+ν)
    ,0,0,{0,0,0}.
    ∇
    {x,y,z}
    v[x,y,z]+
    ∇
    {x,y,z}
    ·-
    Y(1-ν)
    (1-2ν)(1+ν)
    ,0,0,0,-
    Y
    2(1+ν)
    ,0,0,0,-
    Y
    2(1+ν)
    .
    ∇
    {x,y,z}
    u[x,y,z]
    ∇
    {x,y,z}
    ·{0,0,0},0,0,-
    Yν
    (1-2ν)(1+ν)
    ,0,-
    Y
    2(1+ν)
    ,0.
    ∇
    {x,y,z}
    w[x,y,z]+
    ∇
    {x,y,z}
    ·0,-
    Y
    2(1+ν)
    ,0,-
    Yν
    (1-2ν)(1+ν)
    ,0,0,{0,0,0}.
    ∇
    {x,y,z}
    u[x,y,z]+
    ∇
    {x,y,z}
    ·-
    Y
    2(1+ν)
    ,0,0,0,-
    Y(1-ν)
    (1-2ν)(1+ν)
    ,0,0,0,-
    Y
    2(1+ν)
    .
    ∇
    {x,y,z}
    v[x,y,z]
    ∇
    {x,y,z}
    ·{0,0,0},0,0,-
    Y
    2(1+ν)
    ,0,-
    Yν
    (1-2ν)(1+ν)
    ,0.
    ∇
    {x,y,z}
    v[x,y,z]+
    ∇
    {x,y,z}
    ·0,0,-
    Y
    2(1+ν)
    ,{0,0,0},-
    Yν
    (1-2ν)(1+ν)
    ,0,0.
    ∇
    {x,y,z}
    u[x,y,z]+
    ∇
    {x,y,z}
    ·-
    Y
    2(1+ν)
    ,0,0,0,-
    Y
    2(1+ν)
    ,0,0,0,-
    Y(1-ν)
    (1-2ν)(1+ν)
    .
    ∇
    {x,y,z}
    w[x,y,z]

    Cylinder Eigenfrequencies

    Materials Properties

    Visualize the various material moduli.

    Minimization

    Experimental Results

    Acoustic Comparison

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