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

\frac{\mathrm{d}^2}{\mathrm{d} x^2} \left[ EI \frac{\mathrm{d}^2 y(x)}{\mathrm{d} x^2} \right] = -q(x)

Similar to the static beam equation, assuming no damping, the dynamic beam equation is:

\frac{\partial^2}{\partial x^2} \left[ EI \frac{\partial^2 y(x,t)}{\partial x^2} \right] = -q(x,t) = -m_L(x)\frac{\partial^2 y(x,t)}{\partial t^2}

When the beam is in resonance, the solution is:

y(x,t) = Y(x)\cos{\omega \, t}

where the Shape Function Y(x) can be expressed as:

Y\left(x\right) = A\cos{\beta x} + B\sin{\beta x} + C\cosh{\beta x} + D \sinh{\beta x}


Boundary conditions of the cantilever beam

Take cantilever for example, the boundary conditions are:

Abstract drawing of cantilever
  • BC 1: y \left(0 \right) = 0;
  • BC 2: \theta(0) = \left. \frac{dy}{dx}\right|_{x=0} = 0;
  • BC 3: M(L) = EI \left. \frac{d^2 y}{dx^2} \right|_{x=L} = 0; and
  • BC 4: V(L) = EI \left. \frac{d^3 y}{dx^3} \right|_{x=L} = 0

Thus the solution of the shape function is:

\cos{\beta L} \, \cosh{\beta L} = -1


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.

Dynamic analysis

From dynamic beam equation

\frac{\partial^2}{\partial x^2} \left[ EI \frac{\partial^2 y(x,t)}{\partial x^2} \right] = -q(x,t) = -m_L(x)\frac{\partial^2 y(x,t)}{\partial t^2}

Infinite number of solutions and they are ones that satisfies:

\cos{\beta L} \, \cosh{\beta L} = -1 (1)

Solve for zeros of equation:

\cos{\beta L} \, \cosh{\beta L} + 1 = 0

And plug into the solution of the dynamic beam equation

\beta^4 - \frac{m_L \omega^2}{EI} = 0

 \omega = \beta^2 w \sqrt{\frac{E}{12\rho}} = \frac{\left(\beta L\right)^2 w}{L^2} \sqrt{\frac{E}{12\rho}} (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.

2. add your name, affiliation and date of contribution as the next entry and increment the value of this line by one.

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