Spring stiffness of a cantilever beam with uneven loading

motthomas

Hi all!

Im stuck with a bit of a problem. I have a wedge shaped petal valve which I need to find the spring stiffness k and hence the natural frequency of vibration for.

In all the texts Ive looked up so far, the equations have always assumed the beam is of constant cross section or has a point mass located at the tip of the beam.

The valve I am trying to model is wedge shaped with the smallest section being cantilevered. The cantilevered end is 8mm across, the end is 25mm wide and the valve is 32mm long.

The k equation Ive been using so far is from Rao: k=3EI/l^3. Again this assumes a constant cross section. Since, through small deflections, most of the bending will occur at the narrow end, I have found k for a rectangular valve the same cross section as the root of my valve and the same length.

Then to deal with the extra mass of the rest of the valve minus this rectangular section, I found the position of the centroid for this extra mass, found the mass of the extra section, found the moment this mass produces around the root and calculated the equivalent mass which would produce the same moment if it were placed at the tip.

Using Raos equation for equivalent mass of a beam with tip load: Meq=M+0.23m, where M is mass at tip and m is mass of beam.

My natural frequency was then calculated by fn=(1/2pi)sqrt(k/Meq). The problem is that my fn comes out a bit on the high side for a valve made from steel with 0.006" thickness. Its coming out at 66Hz but I was expecting the frequency to be closer to 40Hz.

Can anyone think if I am making a mistake in my calculations? Is my modelling as accurate as it can be?

Thanks,
Thomas

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