# 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

Table 1. Comparison of results
Result Type Theory ANSYS STAAD.Pro Difference
Frequency, f1 (Hz) 10.247 10.247 10.237 none
Frequency, f2 (Hz) 64.221 64.197 63.974 none
Frequency, f3 (Hz) 179.82 180.14 178.672 <1%

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\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
1 TO 20 UNI GY 1.124
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH
``````

```               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
```