.ScalableTestSuite.Mechanical.FlexibleBeam.Verification.FlexibleBeamCheck

Information

In the model, FlexibleBeamModelica is verified by an analytical solution in terms of the vibration frequency of the free end of the flexible beam. In this model, vibration of the free end of FlexibleBeamModelica and analytical solution are implemented.

ANALYTICAL SOLUTION:

The partial differential equation describing the motion of the infinitesimal element is:

Assuming a stationary solution of the bending motion in the form:

where α(x) is a function of space alone describing the waveform of the stationary vibration and β(x) is a time dependent vibration amplitude coefficient.

By placing the stationary solution w(x,t) into the PDE and later defining:

The natural frequencies and the modes of vibration of a beam in bending depend on physical parameters such as length, section, material and also boundary conditions. For the cantilever beam, there are four boundary conditions, two for the clamped end x=0 and two for the free end x=l.

The general solution becomes a linear combination of trigonometric equations:

and after placing the boundary conditions into the function α(x), we reduce the function into a new form. Moreover, from the boundary conditions it is obtained: A=C=0 and;

Thus, it can be written in the form:

where B=1/2. Moreover, with the boundary conditions, we obtain the frequency equation for the cantilever beam:

By solving the frequency equation, obtained values for the first 6 modes are given in the table below:

Therefore, for each frequency, there is a characteristic vibration:

An approximation for Ak is given below according to P which is force. It is calculated as if the beam starts vibration at t=0.

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