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Figure 7: NRMSD for axial and vertical displacement for different number of shape functions and different number of bodies.

Figure 7 NRMSD for axial and vertical displacement for different number of shape functions and different number of bodies.

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Figure 1: Definition of the inertial frame and the body frame, with their representative vectors. 2.1.2 Influence of deformation degrees of freedom
Figure 1: Definition of the inertial frame and the body frame, with their representative vectors. 2.1.2 Influence of deformation degrees of freedom
Figure 2: Cantilever beam. The blue dotted line describes the location of the beam’s elastic axis and the vector sp the position of a point P perpendicular to the elastic axis. 2.2.1 Shape function matrix 2.2 Structural Model
Figure 2: Cantilever beam. The blue dotted line describes the location of the beam’s elastic axis and the vector sp the position of a point P perpendicular to the elastic axis. 2.2.1 Shape function matrix 2.2 Structural Model
And then one can obtain the resulting flexible displacement at any point P described by sp = As discussed in [8] one can describe the strain forces in terms that depend only on the flexible
And then one can obtain the resulting flexible displacement at any point P described by sp = As discussed in [8] one can describe the strain forces in terms that depend only on the flexible
Table 1: Properties of each of the beams that make up the double pendulum.
Table 1: Properties of each of the beams that make up the double pendulum.
To verify the numerical implementation of the dynamic and structural model, a simulation of a flexible double pendulum under the effect of gravity in the initial position illustrated in Fig. 3 was performed. To perform the simulation, the software Simscape Multibody@and with the Flexible Multibody Rayleigh-Ritz model (FMBRR), developed in this work, was used. Sim- scape Multibody is a multibody system simulation software that uses a graphical interface of block diagrams and Euler-Bernoulli beam finite elements for flexible beam modeling.
To verify the numerical implementation of the dynamic and structural model, a simulation of a flexible double pendulum under the effect of gravity in the initial position illustrated in Fig. 3 was performed. To perform the simulation, the software Simscape Multibody@and with the Flexible Multibody Rayleigh-Ritz model (FMBRR), developed in this work, was used. Sim- scape Multibody is a multibody system simulation software that uses a graphical interface of block diagrams and Euler-Bernoulli beam finite elements for flexible beam modeling.
Figure 4: Comparison between the results obtained by the model and by the software Simscape Multibody©. In Fig. 4 it can be seen that the results obtained by the model of the present work are very similar to the results obtained by the software Simscape Multibody©, even with different struc- tural methods. So, it can be said that the dynamic-structural model is coherent and properly implemented.
Figure 4: Comparison between the results obtained by the model and by the software Simscape Multibody©. In Fig. 4 it can be seen that the results obtained by the model of the present work are very similar to the results obtained by the software Simscape Multibody©, even with different struc- tural methods. So, it can be said that the dynamic-structural model is coherent and properly implemented.
Figure 5: Clamped beam defined by 2 bodies j and k with a Ff’, force at the free end. Table 2: Beam Properties.
Figure 5: Clamped beam defined by 2 bodies j and k with a Ff’, force at the free end. Table 2: Beam Properties.
Figure 6: Comparison between analytically obtained modal shapes and polynomial shape functions, in dashed line. The Ist shape function in blue, 2nd in black, 3rd in red and 4th in green. 3.2.1 Static results
Figure 6: Comparison between analytically obtained modal shapes and polynomial shape functions, in dashed line. The Ist shape function in blue, 2nd in black, 3rd in red and 4th in green. 3.2.1 Static results
Figure 8: Axial and vertical displacement at the free end of the beam as a function of the force applied F’,.
Figure 8: Axial and vertical displacement at the free end of the beam as a function of the force applied F’,.
Figure 9: NRMSD for axial and vertical displacement for different number of bodies.
Figure 9: NRMSD for axial and vertical displacement for different number of bodies.
Figure 10: Axial and vertical displacement at the free end and error in position in y and z for 4 bodies simulation. Comparison with AMS and IMS.
Figure 10: Axial and vertical displacement at the free end and error in position in y and z for 4 bodies simulation. Comparison with AMS and IMS.
Figure 11: Axial and vertical displacement at the free end of the beam as a function of time. nonlinearities, that is, the magnitude of the vertical displacement is much greater than the mag- nitude of the axial displacement. Then, it is observed that the result for the vertical displacement does not change with the increase in the number of bodies, while for the axial displacement a convergence is observed with the increase in the number of bodies using the PSF basis. Fur- thermore, there is once again a difficulty in convergence using the ASF basis.
Figure 11: Axial and vertical displacement at the free end of the beam as a function of time. nonlinearities, that is, the magnitude of the vertical displacement is much greater than the mag- nitude of the axial displacement. Then, it is observed that the result for the vertical displacement does not change with the increase in the number of bodies, while for the axial displacement a convergence is observed with the increase in the number of bodies using the PSF basis. Fur- thermore, there is once again a difficulty in convergence using the ASF basis.
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Flexible Multibody DynamicsRayleigh Ritz
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