V. Modal Frequencies of a Cantilever Beam

To find the first three natural frequencies of vibration for a cantilever beam with a uniform mass.

Reference

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

Also compared with ANSYS^® Finite Element software.

Problem

Find the first three natural frequencies, f₁, f₂ and f₃ of the cantilever beam.

Model for dynamic beam no. 4

E = 30,000 ksi

I = 1.3333 in⁴

A = 4 in²

w = 1.124 lb/in

L = 80 in

Comparison

表 1. Comparison of results
Result Type	Theory	ANSYS	STAAD.Pro	Difference
Frequency, f₁ (Hz)	10.247	10.247	10.237	negligible
Frequency, f₂ (Hz)	64.221	64.197	63.974	negligible
Frequency, f₃ (Hz)	179.82	180.14	178.672	<1%

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\08 Dynamic Analysis\Modal Frequencies of a Cantilever Beam.STD is typically installed with the program.

STAAD PLANE : FREQUENCIES OF A CANTILEVERED BEAM
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 4 0 0; 3 8 0 0; 4 12 0 0; 5 16 0 0; 6 20 0 0; 7 24 0 0;
8 28 0 0; 9 32 0 0; 10 36 0 0; 11 40 0 0; 12 44 0 0; 13 48 0 0;
14 52 0 0; 15 56 0 0; 16 60 0 0; 17 64 0 0; 18 68 0 0; 19 72 0 0;
20 76 0 0; 21 80 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 5 6; 6 6 7; 7 7 8; 8 8 9; 9 9 10;
10 10 11; 11 11 12; 12 12 13; 13 13 14; 14 14 15; 15 15 16; 16 16 17;
17 17 18; 18 18 19; 19 19 20; 20 20 21;
MEMBER PROPERTY AMERICAN
1 TO 20 PRIS AX 4 IZ 1.3333
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.290909
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 FIXED
CUT OFF FREQUENCY 200
CUT OFF MODE SHAPE 3
LOAD 1 UNIFORM MASS FOR MODAL ANALYSIS
MEMBER LOAD
1 TO 20 UNI GY 1.124
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH

STAAD Output

               CALCULATED FREQUENCIES FOR LOAD CASE       1
       MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
         1                      10.237                  0.09768 
         2                      63.974                  0.01563 
         3                     178.672                  0.00560 
            MODAL WEIGHT (MODAL MASS TIMES g) IN POUN         GENERALIZED
      MODE           X             Y             Z              WEIGHT
         1       0.000000E+00  5.507472E+01  0.000000E+00    2.252359E+01
         2       0.000000E+00  1.693755E+01  0.000000E+00    2.278794E+01
         3       0.000000E+00  5.820203E+00  0.000000E+00    2.323735E+01
 MASS PARTICIPATION FACTORS 
                     MASS  PARTICIPATION FACTORS IN PERCENT
                     --------------------------------------
           MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z
             1     0.00  62.82   0.00    0.000   62.819    0.000
             2     0.00  19.32   0.00    0.000   82.138    0.000
             3     0.00   6.64   0.00    0.000   88.777    0.000