Flexible shaft

Plane flexible shaft

Bending waves are transversal waves that can propagate in limited media with non- vanishing shear stress , for example in beams (application case:  e.g. triangles ) and in plates (application case: e.g.  bells ). In contrast to expansion waves , the periodic deflection of the medium takes place perpendicular ("transversal") to the direction of propagation, so that the wave is also described as a periodic change in the radius of curvature .

Wave equation

bar

According to the Euler-Bernoulli theory, the wave equation of a flexural wave on a beam is first order :

${\displaystyle {\frac {\partial ^{2}z}{\partial t^{2}}}+{\frac {E\cdot I}{\rho \cdot A}}\cdot {\frac {\partial ^{4}z}{\partial x^{4}}}=0}$

With

• ${\displaystyle z({\vec {x}},t)}$ the transverse deflection (in the figure: z vertical, x horizontal)
• ${\displaystyle t}$the time
• ${\displaystyle E}$the modulus of elasticity
• ${\displaystyle I}$the area moment of inertia
  • ${\displaystyle E\cdot I}$the bending stiffness
• ${\displaystyle \rho }$the density of the bar
• ${\displaystyle A}$the beam cross-sectional area .

For one dimension (position variable ) results from the harmonic approach ${\displaystyle x}$

${\displaystyle z=z_{0}\cdot e^{i(\omega t+kx)}}$

With

• the amplitude ${\displaystyle z_{0}}$
• the Euler's number ${\displaystyle e}$
• the imaginary unit ${\displaystyle i}$
• the angular frequency ${\displaystyle \omega =2\pi \cdot f}$
• the circular wavenumber ${\displaystyle k}$

the dispersion relation :

${\displaystyle k^{2}={\sqrt {\frac {\rho \cdot A}{E\cdot I}}}\cdot \omega .}$

The phase velocity is therefore strongly dependent on the frequency (and thus also on ): ${\displaystyle c={\frac {\omega }{k}}}$ ${\displaystyle f}$ ${\displaystyle \omega }$

${\displaystyle c={\sqrt {\omega }}\cdot {\sqrt[{4\,}]{\frac {E\cdot I}{\rho \cdot A}}}}$.

plate

The corresponding equation for a flexible wave on a plate is:

${\displaystyle {\frac {\partial ^{2}z}{\partial t^{2}}}+{\frac {E\,h^{2}}{12\rho (1-\mu ^{2})}}\,\nabla ^{4}z=0}$

with the additional designations

• ${\displaystyle h}$ the height of the plate
• ${\displaystyle \mu }$the Poisson's ratio
• ${\displaystyle \nabla }$the Nabla operator .

This equation leads to the dispersion relation

${\displaystyle k^{2}={\frac {\sqrt {12}}{h}}\cdot {\sqrt {\frac {\rho \cdot (1-\mu ^{2})}{E}}}\cdot \omega }$

and the phase velocity:

${\displaystyle c={\sqrt {\omega }}\cdot {\sqrt[{4\,}]{\frac {h^{2}\cdot E}{12\cdot \rho \cdot (1-\mu ^{2})}}}.}$

Group speed

In both cases the group velocity is just twice as great as the phase velocity: ${\displaystyle c_{g}={\frac {d\omega }{dk}}}$

${\displaystyle c_{g}=2\cdot c}$.

literature

• Michael Möser: Technical acoustics . Springer ( google books ).