Beam Theory Constants Question

[thegreenlaser]

From my understanding, the equation that models the transverse vibration of a beam is (Euler Bernoulli):
u_{tt} = - \frac{EI}{A \rho} \cdot u_{xxxx}
where E is Young's modulus, I is the 2nd moment of area, A is the cross-sectional area, and rho is the density of the beam. This equation, however, doesn't take axial load into account. Now, what I'm wondering is, does the following equation accurately model the beam with an axial load?

u_{tt} = \frac{T}{A \rho} \cdot u_{xx} - \frac{EI}{A \rho} \cdot u_{xxxx}

The coefficient of the second spatial derivative term is straight from the 1D wave equation describing an ideal string (with T being the applied tension):
u_{tt} = \frac{T}{A \rho} \cdot u_{xx}In the case of a vibrating stiff string, is it valid to combine the two equations (Euler-Bernoulli and the 1D wave equation for a string) in this way? I'm using this equation to model a guitar string, and I'm fairly certain that something is wrong with the coefficients (it seems that the stiffness parameter is too large). I've gone over my math and my coding a number of times, so I'm pretty sure that there's something fundamentally wrong with my derivation of the constants rather than a little algebraic mistake. I have to admit that I'm a little bit out of my league with the beam equation, since I've only taken a 1st year statics class and done some reading in the last few days. I'm hoping someone with a little more expertise on beam mechanics can help me out. My suspicion is that there's some assumptions involved in the derivations of the two equations that make them 'incompatible,' or at least such that I can't just sum them in this way.

 

[AlephZero]

Your equation looks right, but the devil is probably in the detail of the notation!

See section 4.2.3 of this, for a worked example:
http://freeit.free.fr/Knovel/Structural Vibration - Analysis and Damping/45806_04.pdf

 

[thegreenlaser]

Your equation looks right, but the devil is probably in the detail of the notation!

See section 4.2.3 of this, for a worked example:
http://freeit.free.fr/Knovel/Structural Vibration - Analysis and Damping/45806_04.pdf

Thanks for that link. It appears from that worked example that I'm doing the right thing, at least as far as a uniform circular bar sort of case. With that (hopefully) settled, my next question would be if my error lies in my treatment of composite materials. Using some educated guess-work, I came up with the following stiffness parameter for a composite material (in this case, material 1 is a cylindrical 'core' and material 2 is a cylindrical 'shell' encasing the core):

\frac{EI}{A\rho} = \frac{E_1 I_1 + E_2 I_2}{A_1 \rho_1 + A_2 \rho_2}

As I said though, this was really just educated guess-work and seeing if it reduces to the right thing in all the special cases I could think of.

Though now that I think about it, I wonder if my assumption that the outer wrapping (which is really a tight helical spring) acts like a cylinder is correct. I did even go so far as to find the average 2nd moment of area and average cross-sectional area of the wrapping, which did reduce the stiffness, but I'm guessing that the fact that a spring is sort of 'disjointed' allows it to bend a lot easier, and probably makes treating it as a uniform cylinder a really inaccurate assumption.

 

[AlephZero]

Though now that I think about it, I wonder if my assumption that the outer wrapping (which is really a tight helical spring) acts like a cylinder is correct.

No, that assumption is wrong. If a continuous hollow cylinder bends, then (like any beam bending situation) there are axial tension and compression forces on opposite sides of the cylinder. That is the main cause of the stiffness of the cylinder. For your coiled wrapping, those forces don't exist. Instead of stretching and compressing the material along the length of the "beam", the gaps between the coils open and close slightly instead.

The basic reason for the wrapped construction is to add mass to the string without adding any stiffness when it bends or when it is stretched. Try using the E and I values for the core on its own, and the total mass of the core plus the wrapping.

 

[thegreenlaser]

No, that assumption is wrong. If a continuous hollow cylinder bends, then (like any beam bending situation) there are axial tension and compression forces on opposite sides of the cylinder. That is the main cause of the stiffness of the cylinder. For your coiled wrapping, those forces don't exist. Instead of stretching and compressing the material along the length of the "beam", the gaps between the coils open and close slightly instead.

The basic reason for the wrapped construction is to add mass to the string without adding any stiffness when it bends or when it is stretched. Try using the E and I values for the core on its own, and the total mass of the core plus the wrapping.

Aha! I presumed that the spring would have negligible effect after I did a little test with some weights. I tried only accounting for the core, and while it was markedly better, it was still just a little bit off. What I never thought about was the fact that the mass term from the wrapping should remain in there. I just tried that and it's at least very close to correct now. Once I finish my chapter on discrete Fourier transforms I can do some frequency analysis and find out for sure. Thanks a bunch for the help!