Euler-Bernoulli Beams

• a first approximation: neglects effects of rotational lateral inertia. Plane subsections remain plane and parallel to the normal.
• Relevant quantities:
  • $\rho=$ density
  • $A=$ cross-subsectional area
  • $E=$ Young's modulus
  • $I=$ moment of inertia
• Get a set of equations:
  

  $$\frac{\partial w}{\partial t} = \frac{1}{\rho A}\frac{\partial^{2}q}{\partial x^{2}}$$

  $$\frac{\partial q}{\partial t} = -EI\frac{\partial^{2}w}{\partial x^{2}}$$

  
  

  In constant parameter case:

  $$\frac{\partial^{2}w}{\partial t^{2}} = -b^{2}\frac{\partial^{4}w}{\partial x^{4}}, \qquad b = \sqrt{\frac{EI}{\rho A}}$$ (2)

  
  

• Phase velocity unbounded