To find the natural frequencies of vibration for a simply supported beam.
Reference
Roark’s Formulas for Stress and Strain, Warren C. Young, McGraw Hill, 6th edition.
Problem
Find the first five flexural natural frequencies of the simple beam. Neglect shear deformation and rotary inertia.
Model for dynamic beam no. 7
Hand Calculations
Weight
wweight = Ax * density = 2.0 (0.1) = 0.2 lb/in
wmass = 0.2 /(386.4) = 0.000518
From Table 36, Item 1b of the reference:
Table 1. Modal stiffness and natural frequencies
Mode |
kc
|
Frequency (Hz) |
1 |
9.87 |
445.7 |
2 |
39.5 |
1,783.7 |
3 |
88.8 |
4,010.0 |
4 |
158 |
7,134.9 |
5 |
247 |
11,154 |
Comparison
Table 2. Comparison of results
Result Type |
Theory |
STAAD.Pro
|
Difference |
Frequency, f1 (Hz) |
445.7 |
445.495 |
none |
Frequency, f2 (Hz) |
1,783.7 |
1,781.968 |
none |
Frequency, f3 (Hz) |
4,010.0 |
4,009.310 |
none |
Frequency, f4 (Hz) |
7,134.9 |
7,127.074 |
none |
Frequency, f5 (Hz) |
11,154 |
11,133.978 |
none |
STAAD Output
CALCULATED FREQUENCIES FOR LOAD CASE 1
MODE FREQUENCY(CYCLES/SEC) PERIOD(SEC)
1 445.495 0.00224
2 1781.968 0.00056
3 4009.310 0.00025
4 7127.074 0.00014
5 11133.978 0.00009
MODAL WEIGHT (MODAL MASS TIMES g) IN POUN GENERALIZED
MODE X Y Z WEIGHT
1 0.000000E+00 3.228953E+00 0.000000E+00 2.000000E+00
2 0.000000E+00 5.965927E-28 0.000000E+00 2.000000E+00
3 0.000000E+00 3.469944E-01 0.000000E+00 2.000000E+00
4 0.000000E+00 7.262741E-31 0.000000E+00 2.211146E+00
5 0.000000E+00 1.165685E-01 0.000000E+00 2.000000E+00
MASS PARTICIPATION FACTORS
MASS PARTICIPATION FACTORS IN PERCENT
--------------------------------------
MODE X Y Z SUMM-X SUMM-Y SUMM-Z
1 0.00 84.97 0.00 0.000 84.972 0.000
2 0.00 0.00 0.00 0.000 84.972 0.000
3 0.00 9.13 0.00 0.000 94.104 0.000
4 0.00 0.00 0.00 0.000 94.104 0.000
5 0.00 3.07 0.00 0.000 97.171 0.000