# TORSIONAL VIBRATION CALCULATION OF A SHAFT

Torsional vibration calculation of a shaft to find natural frequency of a uniform shaft with a concentrated end mass. Shaft is fixed from one end and the other end is free (cantilevered shaft).

### Torsional Vibration Calculator of a Shaft: INPUT PARAMETERS Parameter Value Density of shaft [ps] g/cm^3 kg/m^3 lb/in^3 Shaft outer diameter [do] mm cm m inch ft Shaft inner diameter [di] Shaft length [l] Modulus of rigidity [G] GPa psi*10^6 Mass moment of inertia of end mass [J] g*mm^2 kg*m^2 lb*in^2 lb*ft^2

Note: Use dot "." as decimal separator.

 RESULTS Parameter Value Polar second moment of area of uniform shaft [K] --- mm^4 cm^4 inch^4 ft^4 Mass moment of inertia of uniform shaft [Js] --- g*mm^2 kg*m^2 lb*in^2 lb*ft^2 First torsional natural frequency of the system [f1] --- Hz

### Definitions:

Modulus of rigidity (modulus of elasticity in shear): The rate of change of unit shear stress with respect to unit shear strain for the condition of pure shear within the proportional limit. Typical values Aluminum 6061-T6: 24 GPa, Structural Steel: 79.3 GPa.

Polar Moment of Inertia: A geometric property of cross section. Measure of ability how a beam resists torsion.

### Torsional Vibration Equations:

 Equation Polar second moment of area of a hollow shaft [K] $$K=\frac { 1 }{ 2 } \pi \left( { r }_{ o }^{ 4 }-{ r }_{ i }^{ 4 } \right)$$ Mass moment of inertia of a hollow shaft [Js] $${ J }_{ s }=\frac { 1 }{ 2 } { m }_{ s }\left( { r }_{ i }^{ 2 }+{ r }_{ o }^{ 2 } \right)$$ First torsional natural frequency of the system (Approximately) [f1] $${ f }_{ 1 }=\frac { 1 }{ 2\pi } \sqrt { \frac { GK }{ (J+{ J }_{ s }/3)l } }$$

ms - Shaft mass, J - Mass moment of inertia of the point mass, l - Length of the shaft,  ri- Hollow shaft inner radius, ro - Shaft outer radius

### Reference:

• Young, W. C., Budynas, R. G.(2002). Roark's Formulas for Stress and Strain . 7nd Edition, McGraw-Hill, Chapter 16 , pp 767 - 768