Transverse Waves in a Bar

• Shear waves
  
  
  • Velocity of shear waves

          where           ...   shear elastic modulus

• Torsional waves
  
  • Velocity of torsional waves

       where      J  ...  polar moment of inertia,   
                                         C ... constant depending on the cross-section geometry

• Bending waves
  
  • More complex than the preceding mode
    both longitudinal and transverse components are coupled
    
  • Situation on a differential angular segment of a beam:

Internal stress in the x direction (linearly distributed):

Total internal bending moment (M)
for a rectangular cross-section

      where I ... moment of inertia of the beam cross-section
                                                                            for a rectangular cross-section:

Radius of gyration of an area (g):

Radius of curvature (R)

•  Is not in general constant
• Generally, for MEMS devices, displacements u_z are limited to small values and
  

Approximate relation for the estimation of the radius R:

• Velocity of bending waves
  supposing that the wavelength is large compared to transverse dimensions of the bar
  satisfied in many MEMS applications

 

 Boundary conditions for flexural waves in a bar:

Frequencies of flexural modes

 

Quotients q_n=2n/c_B=2L/l_n depend merely on the manner of fixing the bar
and on the order of the harmonic

Values of these quotients for the first few modes and for different beam supports are given in the following table