5.4.5.3. A cantilever beam

Two cantilever beam models shown in Figure 5.22 and Figure 5.23 have a fixed-free end condition with ten lumped masses. One is modeled by using ten beam force elements and the other is modeled by using one flexible body of RecurDyn. The flexible beam model is originally generated in ANSYS. The material properties and geometric entity conditions of the beam are shown in Table 5.7.

Figure 5.22 Beam model using RecurDyn/Beam element

Figure 5.23 Beam model using RecurDyn/RFlex body element

Table 5.7 The material properties and geometric entity conditions of beam

Length	0.4 m
Mass	3.9888 Kg
Young’s modulus	1x109 N/m2
Inertia of area	1.215 x 10-8 m4
Area	0.0018 m2
Damping ratio	0.0

In Ref. 5, the analytic natural frequencies of these beams are computed as:

(5.41)\[\omega_1=1.875^2\sqrt{\frac{EI}{\rho{AL^4}}}, \omega_2=4.694^2\sqrt{\frac{EI}{\rho{AL^4}}}, \omega_3=7.855^2\sqrt{\frac{EI}{\rho{AL^4}}}\]

Substituting the values in Table 5.7, the natural frequencies become

\(\omega_1=1.875^2\sqrt{\frac{EI}{\rho{AL^4}}}=24.2537=>f_n=3.8601\)

\(\omega_2=4.694^2\sqrt{\frac{EI}{\rho{AL^4}}}=152.0085=>f_n=24.1929\)

\(\omega_3=7.855^2\sqrt{\frac{EI}{\rho{AL^4}}}=425.6714=>f_n=367.7477\)

The eigenvalues of this beam model are validated against the analytic solution in Table 5.8.

Table 5.8 Eigenvalues of the cantilever beam model

Mode Number	Undamped Natural Frequency (Hz)
	Beam element	Flexible Body	Analytic solution
1	3.84002E+00	3.84259E+00	3.8426
2	2.37455E+01	2.38154E+01	23.8154
3	6.55744E+01	6.60152E+01	66.0152
4	1.26483E+02	1.28016E+02	128.016

In addition, RecurDyn can show the mode shapes of the beam model through 3D animation, as shown in Figure 5.24 and Figure 5.25.

Figure 5.24 Mode shapes of the beam model through 3D animation

The mode shapes of the model using RecurDyn/Beam element

Figure 5.25 Mode shapes of the beam model

The mode shapes of the model are using RecurDyn/RFlex.