Dynamic Beam - Physics
The dynamic beam solution provides the beam shape function as a function of both the position along the beam and the time. The dynamic beam solution is solved from Euler-Bernoulli beam equation and the corresponding boundary conditions.
Dynamic beam solution
The Euler-Bernoulli beam equation, ignoring the axial stress term, is
Similar to the static beam equation, assuming no damping, the dynamic beam equation is:
When the beam is in resonance, the solution is:
where the Shape Function Y(x) can be expressed as:
Boundary conditions of the cantilever beam
Take cantilever for example, the boundary conditions are:
- BC 1: ;
- BC 2: ;
- BC 3: ; and
- BC 4:
Thus the solution of the shape function is:
The following static and dynamic analyses calculate the resonance frequency of the beam element. A MATLAB code is also used to find the resonance frequency.
- Beam geometry and design parameters are shown in Mechanical Beam Model, Section 2D Mechanical Beam Validation
From dynamic beam equation
Infinite number of solutions and they are ones that satisfies:
Solve for zeros of equation:
And plug into the solution of the dynamic beam equation
A MATLAB code was used to calculate βL in Equation 1 and resonance frequency (Equation 2).
The first ten resonance modes are calculated as follows:
Tab 1 First ten resonance modes
MATLAB code for calculating resonance frequency
%(c) Copyright 1999-2010 Carnegie Mellon University % All Rights Reserved. function omiga = Cantilever_resonance_mode(w,l,E,rho) y = zeros(1,200); for range = 1:1:200; y(range) = fzero(’cos(x)*cosh(x)+1’,range); end z(1) = y(1); j = 2; for i = 1:1:199 if ((y(i+1) - y(i)) > 1e-5) z(j) = y(i+1); j = j+1; end end %Caculate the vibrational modes based on the solutions z omiga = z.^2.*w./(l^2).*sqrt(E/12/rho)/2/pi; end
The content of this article was contributed by the following Serendi-CDI members:
1. Congzhong Guo, Carnegie Mellon University, November 15, 2011.
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