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I’m trying to solve a dynamic problem that involves a cantilever beam fixed onto a support that experiences a time history load on the free end of the beam as shown below:

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Cross section Beam is rectangular beam with 1m width and height.

The objective is to obtain the natural frequency of the beam. I believe this is the equation to use:

\omega_n = \frac{\alpha_n}{L^2} \sqrt{\frac{EI}{m}}

Please correct me if my approach is incorrect.

The paramenters are the following:

\alpha_n = 3.57

L (Length) = 5m

E = 205 KN/mm^2

I = 1m^2

m = 2kg (0.204N)

So the natural frequency \omega_n should be

\omega_n = 45.72 Hz

Now, when I run it through Staad Pro (haven’t tried AnSys yet I heard its powerful tool). With the same parameters where I use a Time History Load of -0.204 N onto the end node of the beam and applied the same parameters plus the self loads of 1 throughout the system, my results indicate that its 22.684 Hz for the 1st Mode. Could you guys tell me where I am going wrong all this time is it the analytical solution my Staad Pro or both?!

Appreciate your help!

The script of the Staad Pro is defined below.

STAAD SPACE

START JOB INFORMATION

ENGINEER DATE 06-Jul-12

END JOB INFORMATION

INPUT WIDTH 79

UNIT METER NEWTON

JOINT COORDINATES

1 0 0 0; 2 5 0 0;

MEMBER INCIDENCES

1 1 2;

DEFINE MATERIAL START

ISOTROPIC STEEL

E 2.05e+011

POISSON 0.3

DENSITY 76819.5

ALPHA 1.2e-005

DAMP 0.03

TYPE STEEL

STRENGTH FY 2.532e+008 FU 4.078e+008 RY 1.5 RT 1.2

END DEFINE MATERIAL

MEMBER PROPERTY

1 PRIS YD 1 ZD 1

CONSTANTS

MATERIAL STEEL ALL

SUPPORTS

1 FIXED

DEFINE TIME HISTORY DT 0.0013888

TYPE 1 FORCE

1 -0.204 2 0

ARRIVAL TIME

1 2

DAMPING 0

LOAD 1 LOADTYPE None TITLE LOAD CASE 1

TIME LOAD

2 FY 1 1 1.000000

CALCULATE RAYLEIGH FREQUENCY

SELFWEIGHT X 1 LIST ALL

SELFWEIGHT Y 1 LIST ALL

SELFWEIGHT Z 1 LIST ALL

PERFORM ANALYSIS PRINT ALL

FINISH

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# Dynamic Analysis Cantilever Beam (Staad Pro vs Analysis)

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