A Simulation of a Turkish Cymbal using the Finite-Element-Method 1. Conditions for a Cymbal to answer musical purposes 2. Spectral Analysis - transient behaviour 3. FEM- Simulation - plate movement - eigenfrequency analysis - eigenmode analysis - boundary conditions - nonlinear mode coupling 4. Conclusions of 3D FEM Simulation 5. Analysis of different hammered bumps 6. Transient Analysis 7. Perspectives Common Characteristics of Cymbals 1. higher tin percentage leads to lower eigenfrequencies 2. handicraft tends to result in higher value sound 3. the higher a diameter, the lower the sound 4. rising thickness leads to higher sound and slower attack, longer sustain 5. a larger bell results in stressed bell sound 6. higher concavity leads to inharmonic behaviour Material - alloy: 80 % copper, 20 % tin - Poisson ration: ν = 0,35 - Density: 8,73 g/m^3 - E-Module: 108900 N/mm^2 - Melting point: 915-1040 °C - heat conductance: 67,2 J/(s*m*K) - expansion coefficient lin.: 17,3 µm/(m*K)=10^-6/K - electrical conductance: 9 Ohm/mm^2 source.: www.deutsches-kupferinstitut.de Spectral Analysis of 16“ Cymbal 50 ms, 12-1000 Hz Spectral Analysis of 16“ Cymbal 40-100 ms, 12-1000 Hz Spectral Analysis of 16“ Cymbal 50 ms, 1 - 3,5 kHz Spectral Analysis of 16“ Cymbal 50 ms, 1 - 3,5 kHz Spectral Analysis of 16“ Cymbal 50 ms, 1 Hz - 2,5 kHz Plate Movement Young's modulus, height ∂2 u ∂t 2 = Acceleration Eh 1 ∂1− shearing strain, linear 4 ∇ u= Poisson ration ∂4 u ∂ x 4 ∂ ∂2 u ∂2 u ∂ x 2 ∂ y 2 x-strain ∂4 u ∂ y 4 y-strain Eigenfrequency Analysis To find the eigenfrequencies and modes of deformation of a cymbal, the eigenfrequencies f in the structural mechanics field are related to the eigenvalues returned by the computational solvers through: f = /2 Analysis of Eigenmodes The frequency f can be expressed as: f m , n=C n∗m2n p Whereas C(n) is a constant factor varied a little by every n. The value p rises with thickness and cross-section of the cymbal. Constraints -x,y and z-directions are constrained at the bell of the cymbal - the bell is assumed to be simply supported, similar to real cymbal stands used with standard drumsets Nonlinear Mode Coupling The Frequency between two Modes can be discibed in general as: c f= 2L - Whereas c is the wavepropagation speed an L the length of the resonator. - phase independence - interference can be destructive or constructive Indicators for Mode Coupling - stiffness and tension restoring forces are rel. small, tension generated by mode displacements has a large effect - hammered bumps are known to generate mode coupling - delay in excitations, irregular amplitude variation leads to acceptance of chaotic behaviour - timbre of the cymbal arises from rel. slow buildup of modes of vibration having high frequencies - the harder the impact, the more significant the nonlinear coupling becomes 3 D Geometrie with FEM-Mesh Eigenmode at 146 Hz Eigenmode at 257 Hz View of an Acoustical Camera View of an Acoustical Camera Eigenmode at 747 Hz Eigenmode at 1086 Hz Eigenmode at 1103 Hz Eigenmode at 1356 Hz Eigenmode at 1502 Hz View through an Acoustical Camera Eigenmode at 1782 Hz Eigenmode at 2578Hz Comparison of Eigenmodes in Hz Simulation Wavelet Analysis 146 257 288 583 747 1086 1356 1502 1782 2078 144 250 254 584 747 1256 1321 3502 5342 6731 2096 2554 3433 3610 5347 6763 Influences by different hammerings Reinforcement of modes by square hammering 681.77 Hz Reinforcement at 2156.06 Hz Hammering effects modes outside the displacement area (496.08 Hz) 2171.62 Hz Reinforcement of nodes by round hammering (690.26 Hz) 2171.62 Hz Transient Simulation Gauß-Impuls used in transient analysis: I =e 2 −.001−t / 10e-7 1.13e-7 s 1.8e-7 s 1e-7 s 1e-5 s Results and Perspectives - investigation of special movements - production techniques can be systemised - formulation of instrument qualities regarding musical requirements - specification of eigenmodes through computing power - large exitation leads to requirement of shell model and nonlinear shearing strain [email protected]
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