# TORSIONAL VIBRATION 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 equations used for vibration of unifrom shaft calculations are given below the torsional vibration calculator.

### 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] $${ f }_{ 1 }=\frac { 1 }{ 2\pi } \sqrt { \frac { GK }{ (J+{ J }_{ s }/3)l } }$$

 Symbol Parameter 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