V. Rayleigh Natural Frequency of a Cantilever Beam

To calculate the Natural frequency of vibration using the Rayleigh method for a light cantilever beam with a mass at the free end.

Reference

Thomson, W.T., Vibration Theory and Applications, Prentice-Hall, Inc., 1965.

Problem

Find the natural frequency of vibration, f, of a mass, m, attached to the end of a light cantilever beam of length, L, and flexural stiffness, EI.

Model for dynamic beam no. 2

E = 30,000 ksi

I = 1.3333 in⁴

m = 0.1 lb-sec²/in

L = 30 in

Comparison

Table 1. Comparison of results
Result Type	Theory	STAAD.Pro	Difference
Frequency, f (Hz)	33.553	33.5365	none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\08 Dynamic Analysis\Rayleigh Natural Frequency of a Cantilever Beam.STD is typically installed with the program.

STAAD PLANE NATURAL FREQUENCY OF A CANTILEVERED MASS
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
*  REFERENCE:  THOMSON, W.T., "VIBRATION THEORY AND APPLICATIONS",
*      PRENTICE HALL INC., ENGLEWOODS, NEW JERSEY, 1965
*
INPUT WIDTH 72
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 30 0 0;
MEMBER INCIDENCES
1 1 2;
MEMBER PROPERTY AMERICAN
1 PRIS IZ 1.33333
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.290909
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 FIXED
LOAD 1 NATURAL FREQUENCY
JOINT LOAD
2 FY -38.64
CALCULATE RAYLEIGH FREQUENCY
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS
FINISH

STAAD Output

   **********************************************************
   *                                                        *
   * RAYLEIGH FREQUENCY FOR LOADING     1 =   33.53649 CPS  *
   * MAX DEFLECTION =  0.00870 INCH GLO Y,  AT JOINT     2  *
   *                                                        *
   **********************************************************