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
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