1d wave equation - physicsforums.com
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1D Wave EquationThe derivation and solution for modeling a fixed-free cantilever
beam
Nick BrunoMAT 661
Spring 2010
If time… Extension to voltage in polyvinylidene fluoride (PVDF) strip
IntroductionThe lateral response of a vibrating cantilever beam is studied using the
wave equation by:
Deriving a fourth order wave equation considering a free body diagram.
Solving the equation using the technique of separation of variables with a Fourier expansion.
Finding orthogonal eigenfunctions and producing eigenvalues using root finding.
Applying initial conditions which are used to determine coefficients which yield a complete system time response.
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Introduction
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PZT - Lead zirconate titanate (MFC)
Derivation
A one dimensional beam will be considered with an applied distributed vertical load as seen below:
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Derivation
Summing all the forces in the z direction and moments about the origin “O” results in the following equations:
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We also know:
Derivation
Making substitutions with the latter yields:
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Which simplifies to a fourth order PDE for a uniform beam…
orwhere
Free vibration
Separation of VariablesLetting the response equal the product of independent
functions, separating the variables, and setting them equal to a positive separation constant yields the following:
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Application of Boundary Conditions
The boundary conditions for a cantilever beam are defined by:
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No Displacement @ x =0 No Slope @ x =0 No Bending
Moment @ x = LNo Shear Force @ x = L
These translate to the following for our eigenvalue problem:
We will use these
Application of Boundary Conditions
This results in a system of equations that can be used to determine the eigenvalues and a relation between coefficients C1 and C2
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In order for a non-trivial solution to be found
so equating zero with the determinant of the 2X2 matrix seen will yield eigenvalues for the system
Application of Boundary Conditions
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Application of Boundary Conditions
The relation between C1 and C2 can be found as follows:
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Application of Initial ConditionsRemember from before…
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The chosen initial conditions for the beam are as follows:
whereδ= 0.004 [m] = Initial displacement at the tipL = 0.107 [m] = Length of the beamthus,
Orthogonality?Before solving for An we must first check orthogonality of the
eigenfunctions.
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Check
Example Solution
Use inner products
No weight function!
Assuming convergence
Summary of Data
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Summary of Data
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Summary of Data
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Summary of Data
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u
ut utt
Summary of Data
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Summary of Data
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V * %e
Comparable
Sodano et al. 2003