# V. First Modal Frequency of a Cantilever Beam

To calculate the Natural frequency of vibration for a rectangular cantilever beam with a mass at the free end.

## Reference

Hand calculation using known formulas.

## Problem

Find the natural frequency of vibration, ω, of the cantilever beam.

### Cantilever beam with mass at free end

 E = 30,000 ksi

 h = 12.0 in

 b = 6.0 in

 P = 10.0 k

 L = 120 in

## Hand Calculations

Stiffness at free end:

k = 3EI/L3 = 45 k/in

Mass

m = w/g = 10.0 k / (386.4 k-sec2/in) = 0.02588 k-sec2/in

Circular frequency:

$ω = k m = 45 0.02588 = 41.7 ⁢ rad/sec$

= 6.637 cycles/sec

## Comparison

Table 1. Comparison of results
Result Type Theory STAAD.Pro Difference
Frequency, f (Hz) 6.637 6.633 none

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

STAAD PLANE A RECTANGULAR CANTILEVER BEAM WITH A MASS AT THE FREE END
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
INPUT WIDTH 72
SET SHEAR
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 5 0 0; 3 10 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3;
UNIT INCHES KIP
MEMBER PROPERTY AMERICAN
1 2 PRIS YD 12 ZD 6
UNIT FEET KIP
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 4.32e+06
POISSON 0.290909
END DEFINE MATERIAL
UNIT INCHES KIP
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 FIXED
CUT OFF MODE SHAPE 1
UNIT FEET KIP
2 CON Y -10 5
1 2 UNI GY -0.001
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
*PRINT MODE SHAPES
FINISH


               CALCULATED FREQUENCIES FOR LOAD CASE       1