Spring stiffness of a cantilever beam with uneven loading

In summary: Your Name] In summary, the conversation discusses a problem with finding the spring stiffness and natural frequency of a wedge shaped petal valve. The equation being used assumes a constant cross section, but the valve has a varying cross section. The conversation suggests using the equation for a tapered beam to find the correct value for k, and using the actual mass distribution of the valve instead of the equivalent mass in the calculation. It is also important to use the correct values for E and I for the material being used.
  • #1
motthomas
2
0
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 I've 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 I've 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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  • #2


Hi Thomas,

I can definitely help you with your problem. First of all, it is important to note that the equation you are using, k=3EI/l^3, is for a cantilever beam with a constant cross section. In your case, the cross section is not constant, so the equation will not give you an accurate result.

To find the spring stiffness k for your wedge shaped valve, you will need to use the equation for a tapered beam, which takes into account the varying cross section. This equation is k=3EI/(l^3(1+2a)), where a is the ratio of the width at the free end to the width at the fixed end. In your case, a=25/8=3.125.

Once you have the correct value for k, you can calculate the natural frequency using the equation fn=(1/2pi)sqrt(k/Meq). However, instead of using the equivalent mass Meq, I would recommend finding the actual mass distribution of your valve and using that in the equation. This will give you a more accurate result.

Additionally, make sure you are using the correct values for E (Young's modulus) and I (area moment of inertia) for the material you are using. These values can vary for different materials, so double check to ensure you are using the correct ones for steel with a thickness of 0.006".

I hope this helps and good luck with your calculations! Let me know if you have any further questions.


 

1. What is the definition of a cantilever beam?

A cantilever beam is a structural element that is fixed at one end and free at the other, allowing it to support a load.

2. How does uneven loading affect the spring stiffness of a cantilever beam?

Uneven loading causes the beam to bend, resulting in a decrease in spring stiffness. The stiffer side of the beam will resist the load more than the weaker side, causing an uneven distribution of stress and a decrease in overall stiffness.

3. What is the formula for calculating spring stiffness of a cantilever beam with uneven loading?

The formula for spring stiffness of a cantilever beam with uneven loading is given by k = 3EI/L², where k is the spring stiffness, E is the modulus of elasticity, I is the moment of inertia, and L is the length of the beam.

4. How does the modulus of elasticity affect the spring stiffness of a cantilever beam?

The modulus of elasticity is a measure of a material's stiffness and is directly proportional to the spring stiffness of a cantilever beam. A higher modulus of elasticity results in a stiffer beam, while a lower modulus of elasticity results in a more flexible beam.

5. What are some factors that can affect the spring stiffness of a cantilever beam with uneven loading?

Some factors that can affect the spring stiffness of a cantilever beam with uneven loading include the material properties of the beam, such as modulus of elasticity and moment of inertia, the length and cross-sectional shape of the beam, and the magnitude and distribution of the applied load.

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