- #1

FEAnalyst

- 342

- 144

- TL;DR Summary
- How to use the formula for stress in cantilever subjected to harmonic force at the free end?

Hi,

in the book titled "Formulas for Dynamics, Acoustics and Vibrations" by R.D. Blevins, I've found a formula that can be used to calculate the bending stress in a cantilever beam subjected to harmonic force applied at the free end. The formula looks like this: $$\sigma=\frac{F_{0}Ec}{m \omega_{1}^{2}L} \frac{\pi^{2}}{4L^{2}} \frac{2.1^{2}(1.78)(1-cos{\pi x_{0}}/2L)}{\sqrt{(1-f^{2}/f_{i}^{2})^{2}+(2 \xi_{i} f / f_{i})^{2}}} cos{\frac{\pi x}{2L}} cos{(\omega t - \varphi)}$$ Quite complex but I managed to calculate the first part of this equation (let's call it ##S##) and I'm left with just the last term: $$\sigma=S \cdot cos{(\omega t - \phi)}$$ where: ##\omega## - frequency, ##t## - time and ##\varphi## - phase angle. I think that I can calculate ##\omega## using the frequency of excitation (##\omega=2 \pi f##). The thing is that I don't know what to substitute for time - this problem is supposed to be steady state and I want to use the formula to verify the numerical analysis done in frequency, not time, domain. I also don't know what to do with the phase angle. In the numerical simulation I only define the force magnitude. I do realize that loads in such analyses are given by ##F= F_{mag} \cdot sin(\omega t+ \varphi) ## but for the simulation I don't have to specify ##\varphi##.

Does anyone know how to use this formula (what to do with the last term)?

in the book titled "Formulas for Dynamics, Acoustics and Vibrations" by R.D. Blevins, I've found a formula that can be used to calculate the bending stress in a cantilever beam subjected to harmonic force applied at the free end. The formula looks like this: $$\sigma=\frac{F_{0}Ec}{m \omega_{1}^{2}L} \frac{\pi^{2}}{4L^{2}} \frac{2.1^{2}(1.78)(1-cos{\pi x_{0}}/2L)}{\sqrt{(1-f^{2}/f_{i}^{2})^{2}+(2 \xi_{i} f / f_{i})^{2}}} cos{\frac{\pi x}{2L}} cos{(\omega t - \varphi)}$$ Quite complex but I managed to calculate the first part of this equation (let's call it ##S##) and I'm left with just the last term: $$\sigma=S \cdot cos{(\omega t - \phi)}$$ where: ##\omega## - frequency, ##t## - time and ##\varphi## - phase angle. I think that I can calculate ##\omega## using the frequency of excitation (##\omega=2 \pi f##). The thing is that I don't know what to substitute for time - this problem is supposed to be steady state and I want to use the formula to verify the numerical analysis done in frequency, not time, domain. I also don't know what to do with the phase angle. In the numerical simulation I only define the force magnitude. I do realize that loads in such analyses are given by ##F= F_{mag} \cdot sin(\omega t+ \varphi) ## but for the simulation I don't have to specify ##\varphi##.

Does anyone know how to use this formula (what to do with the last term)?