Dynamic Beam - Physics
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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.
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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:
Analytical calculation
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
Dynamic analysis
From dynamic beam equation
Infinite number of solutions and they are ones that satisfies:
(1) |
Solve for zeros of equation:
And plug into the solution of the dynamic beam equation
(2) |
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
Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
βL | 1.875 | 4.694 | 7.855 | 10.996 | 14.137 | 17.279 | 20.420 | 23.562 | 26.704 | 29.845 |
f/ Hz | 1.639E+05 | 1.027E+06 | 2.877E+06 | 5.638E+06 | 9.319E+06 | 1.392E+07 | 1.944E+07 | 2.589E+07 | 3.325E+07 | 4.153E+07 |
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
Contributors
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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