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5. Conclusions

Figure 35 5. Conclusions

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Ficure 1: The pendulum motion on an ellipse.
Ficure 1: The pendulum motion on an ellipse.
We calculate the velocity of the center c by differentiating (9) and (10). Consider the following parameters and vari- ables [3]: where d is the spring length at relative equilibrium, J denotes the principal moment of inertia for the axis c¢, | is the free length of the spring, yw is a large parameter, K is the force constant of the spring, and ,©, and © are the generalized coordinates. Use (9), (10), (11), and (13) to find the kinetic energy and potential one and then construct the Lagran- gian = kinetic energy—potential one in terms of the gener- alized coordinates as follows:
We calculate the velocity of the center c by differentiating (9) and (10). Consider the following parameters and vari- ables [3]: where d is the spring length at relative equilibrium, J denotes the principal moment of inertia for the axis c¢, | is the free length of the spring, yw is a large parameter, K is the force constant of the spring, and ,©, and © are the generalized coordinates. Use (9), (10), (11), and (13) to find the kinetic energy and potential one and then construct the Lagran- gian = kinetic energy—potential one in terms of the gener- alized coordinates as follows:
Substituting (15) and (16) into (17), we get
Substituting (15) and (16) into (17), we get
Equations (15),(16), and (18) are a quasilinear system of the second order which describes the pendulum equations for this model. We aim to solve this system by a new procedure named the large parameter method [11] to get the approximated periodic solutions. 3.3. Approximate Periodic Solutions. In this section, we search the approximated periodic solutions for the case of nonresonance using the large parameter technique. As- suming the solutions for (15-18), the following is obtained:
Equations (15),(16), and (18) are a quasilinear system of the second order which describes the pendulum equations for this model. We aim to solve this system by a new procedure named the large parameter method [11] to get the approximated periodic solutions. 3.3. Approximate Periodic Solutions. In this section, we search the approximated periodic solutions for the case of nonresonance using the large parameter technique. As- suming the solutions for (15-18), the following is obtained:
TABLE 1: @max Of the first case (v = 0,4 = 5000).
TABLE 1: @max Of the first case (v = 0,4 = 5000).
Ficure 3: The analytical solutions for case (i) in Section 4.1. Figure 2: Description of the pendulum motion for case (i) in Section 4.1.
Ficure 3: The analytical solutions for case (i) in Section 4.1. Figure 2: Description of the pendulum motion for case (i) in Section 4.1.
Figure 4: The numerical solutions for case (i) in Section 4.1.
Figure 4: The numerical solutions for case (i) in Section 4.1.
Figure 6: The analytical solutions for case (ii) in Section 4.1. FiguRE 5: Description of the pendulum motion for case (ii) in Section 4.1.
Figure 6: The analytical solutions for case (ii) in Section 4.1. FiguRE 5: Description of the pendulum motion for case (ii) in Section 4.1.
Figure 8: The analytical solutions for case (iii) in Section 4.1. TABLE 3: max Of the third case (v = 1.5, 4 = 5000).
Figure 8: The analytical solutions for case (iii) in Section 4.1. TABLE 3: max Of the third case (v = 1.5, 4 = 5000).
Figure 7: The numerical solutions for case (ii) in Section 4.1.
Figure 7: The numerical solutions for case (ii) in Section 4.1.
TABLE 4: @max Of the fourth case (QO = 1.5, u = 1250).
TABLE 4: @max Of the fourth case (QO = 1.5, u = 1250).
Figure 9: The numerical solutions for case (iii) in Section 4.1.
Figure 9: The numerical solutions for case (iii) in Section 4.1.
Figure 10: The analytical solutions for case (iv) in Section 4.1.
Figure 10: The analytical solutions for case (iv) in Section 4.1.
Figure 11: The numerical solutions for case (iv) in Section 4.1.
Figure 11: The numerical solutions for case (iv) in Section 4.1.
TABLE 5: Pmax Of the fifth case (OQ. = 2.5, 4 = 1250). Ficure 12: The analytical solutions for case (v) in Section 4.1.
TABLE 5: Pmax Of the fifth case (OQ. = 2.5, 4 = 1250). Ficure 12: The analytical solutions for case (v) in Section 4.1.
Figure 15: The analytical solution © for case (i) in Section 4.2.
Figure 15: The analytical solution © for case (i) in Section 4.2.
Ficure 16: The analytical solution ® for case (i) in Section 4.2.
Ficure 16: The analytical solution ® for case (i) in Section 4.2.
Figure 14: The analytical solution $ for case (i) in Section 4.1.
Figure 14: The analytical solution $ for case (i) in Section 4.1.
Figure 13: The numerical solutions for case (v) in Section 4.1.
Figure 13: The numerical solutions for case (v) in Section 4.1.
Figure 20: The analytical solution f for case (ii) in Section 4.2.
Figure 20: The analytical solution f for case (ii) in Section 4.2.
Ficure 19: The stability phrase ©’ and © for case (i) in Section 4.2.
Ficure 19: The stability phrase ©’ and © for case (i) in Section 4.2.
Figure 18: The stability phrase @’ and @ for case (i) in Section 4.2.
Figure 18: The stability phrase @’ and @ for case (i) in Section 4.2.
Figure 17: The stability phrase f’ and f for case (i) in Section 4.2.
Figure 17: The stability phrase f’ and f for case (i) in Section 4.2.
Figure 21: The analytical solution © for case (ii) in Section 4.2.
Figure 21: The analytical solution © for case (ii) in Section 4.2.
Figure 22: The analytical solution ® for case (ii) in Section 4.2.
Figure 22: The analytical solution ® for case (ii) in Section 4.2.
Ficurs 24: The stability phrase @' and @ for case (ii) in Section 4.2.
Ficurs 24: The stability phrase @' and @ for case (ii) in Section 4.2.
Figure 23: The stability phrase f’ and B for case (ii) in Section 4.2.
Figure 23: The stability phrase f’ and B for case (ii) in Section 4.2.
Figure 26: The analytical solution $ for case (iii) in Section 4.2.
Figure 26: The analytical solution $ for case (iii) in Section 4.2.
Figure 28: The analytical solution ® for case (iii) in Section 4.2.
Figure 28: The analytical solution ® for case (iii) in Section 4.2.
Ficurs 25: The stability phrase ©’ and © for case (ii) in Section 4.2.
Ficurs 25: The stability phrase ©’ and © for case (ii) in Section 4.2.
Figure 27: The analytical solution © for case (iii) in Section 4.2.
Figure 27: The analytical solution © for case (iii) in Section 4.2.
Ficure 32: The analytical solution f for case (iv) in Section 4.2. Figure 31: The stability phrase ©’ and @ for case (iii) in Section 4.2.
Ficure 32: The analytical solution f for case (iv) in Section 4.2. Figure 31: The stability phrase ©’ and @ for case (iii) in Section 4.2.
Ficurs 30: The stability phrase @' and @ for case (iii) in Section 4.2. Figure 29: The stability phrase ' and for case (iii) in Section 4.2.
Ficurs 30: The stability phrase @' and @ for case (iii) in Section 4.2. Figure 29: The stability phrase ' and for case (iii) in Section 4.2.
Frcure 33: The stability phrase f' and for case (iv) in Section 4.2.
Frcure 33: The stability phrase f' and for case (iv) in Section 4.2.
Related topics:
MathematicsApplied MathematicsComputer ScienceNanotechnologyPerturbation MethodsMechanical VibrationsInverted PendulumNumerical Analysis and Computational MathematicsRigid Body DynamicsRotational Motion
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