From my understanding, the equation that models the transverse vibration of a beam is (Euler Bernoulli):(adsbygoogle = window.adsbygoogle || []).push({});

[tex]u_{tt} = - \frac{EI}{A \rho} \cdot u_{xxxx} [/tex]

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?

[tex]u_{tt} = \frac{T}{A \rho} \cdot u_{xx} - \frac{EI}{A \rho} \cdot u_{xxxx} [/tex]

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):

[tex]u_{tt} = \frac{T}{A \rho} \cdot u_{xx} [/tex]

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.

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# Beam Theory Constants Question

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