Analysis of a parametrically excited 2DOF oscillator with nonlinear restoring magnetic force and rotating rectangular beam

Mathematical Model Experimental Setup Complex Averaging Method Dynamical Analysis For m1 ≠ m2 Acknowledgement: This research is funded by the National Science Center, Poland, under program PRELUDIUM 23 No. DEC-2024/53/N/ST8/00400, and Muhammad Junaid-U-Rehman, is a PhD student at the IDS, TUL, Poland. Muhammad Junaid-U-Rehman¹, Grzegorz Kudra¹, Krystian Polczyński¹, Kevin Dekemele², Jan Awrejcewicz¹ Email: [email protected], Grzegorz.kud[email protected]z.pl, [email protected]. Fig. 1. 1. neodymium magnets; 2. clamps for magnets; 3. movable bearing block 4. movable clamp of the flexible beam; 5. flexible beam; 6. manual brakes; 7. movable cart; 8. Hall sensor of the movable cart; 9. profile rail with magnetic tape; 10. fixed cart; 11. Hall sensor of the fixed cart; 12. profile rail; 13. magnetic ruler; 14. fixed clamp of the flexible beam, 15. fixed bearing block; 16. flexible clutch; 17. stepper motor with built-in encoder Dimensional equations of motion for 2DOF system A set of 2nd order coupled ordinary differential equations of the supposed system can be written as 𝑢1′′+ 2 𝜁1 𝑢1′+𝜎1 𝑓0𝑢1′+ 𝑝+ 𝑞 𝑐𝑜𝑠2𝜔𝜏 𝑢1−𝑢2+1−𝑝𝑢1+𝜅13 𝑢13+𝜅15 𝑢15=0, (3) 𝑢2′′ + 2 𝜁2 𝑢2′+𝜎2 𝑓0𝑢2′+ 𝜇𝑝+ 𝑞 𝑐𝑜𝑠2𝜔𝜏 𝑢2−𝑢1+𝜅21 𝑢2+𝜅23 𝑢23+𝜅25 𝑢25=0. (4) 𝑓0ሶ𝑥𝑖=൞1, if 𝑥𝑖′>0, ∈ [−1, 1], if 𝑥𝑖′=0, −1, if 𝑥𝑖′<0. Fig. 2. Stiffness characteristics for full nonlinear model and its approximation by a 3rd and 5th degree polynomial 𝑚1ሷ𝑥1+𝑐1ሶ𝑥1+𝑇1 𝑓0ሶ𝑥1+𝐾𝑡 𝑥1−𝑥2+𝐹𝑠1𝑥1=0 (1) 𝑚2ሷ𝑥2+𝑐2ሶ𝑥2+𝑇2 𝑓0ሶ𝑥2+𝐾𝑡 𝑥2−𝑥1+𝐹𝑠2𝑥2=0 (2) To derive the amplitude-frequency response, the CxA method is employed. The timedependent complex variables representing the modulated amplitudes, 𝐴1, 𝐴2∈ℂ : 2ζ1ωA1+2iωA1′+A1−pA2+σ1f0+q2A1−A2+3κ13A12A1+10κ15A13A12−ω2A1=0, 2iζ2ωA2+2iωA2 ′+κ21A2−μpA1+μpA2+σ2g0+μq 2A2−A1+3κ23A2 2A2 +10κ25A2 3A2 2−ω2A2=0. 2𝐴1𝜏𝑒𝑖𝜔𝜏=𝑢1𝜏 −𝑖𝑢′𝜏 𝜔, 2𝐴2𝜏𝑒𝑖𝜔𝜏=𝑢2𝜏 −𝑖𝑣′𝜏 𝜔. (6) Fig.5: Bifurcation plot of the local maxima of 𝑢1 for coordinates 𝑢1 (a) and 𝑢2 (b), the largest Lyapunov exponent λ1 (c), and results from CxA method: A comparison analysis. Dynamical Analysis For m1 = m2 ❖Junaid-U-Rehman, M, et al. (2024). Analytical, numerical and experimental observation of isolated branches of periodic orbits in 1DOF mechanical parametric oscillator. JSV, 584 , 118454. ❖Witkowski, K, et al. (2021). Modeling and dynamics analysis of a forced 2DOF mechanical oscillator with magnetic springs, MSSP. 148-107138. Refer ences ¹Department of Automation, Biomechanics, and Mechatronics, Lodz University of Technology, Stefanowski st., 90-537, Łódź, Poland Analysis of a parametrically excited 2DOF oscillator with nonlinear restoring magnetic force and rotating rectangular beam ²Department of Electromechanical Systems and Metal Engineering, Ghent University, Technologiepark 46, Ghent, Belgium The set-valued function 𝑓0ሶ𝑥𝑖 is used to describe the Coulomb friction law Non-Dimensional equations of motion for 2DOF system By converting Eqs. (1) & (2) into non-dimensional form, we establish a nondimensional time variable, 𝜏=𝜔𝑛𝑡, and non-dimensional displacement of the system 𝑢𝑖=𝑥𝑖 𝛿1, using these assumptions and we have: ζi=ci 2miωn , σi=Ti mi ωn 2δ1, p=kξ+kη 2m1ωn 2, q=kξ−kη 2m1ωn 2 , κij=δ1j−1 miωn 2kij, μ=m1 m2. (5) Where, Substituting the Eq. (6) into Eqs. (3) & (4), we get the following simplified form: Using the polar form A1=12a1eiα1, A2=12a2eiα2 into Eq. (7), and supposing A1′= A2 ′= 0, for steady case, then separating the real and imaginary part, which get the following equations: (7) 𝑞2𝑎1𝑐𝑜𝑠2𝛼1−𝑞2𝑎2𝑐𝑜𝑠𝛼2+𝛼1+34𝜅13𝑎13+58𝜅15𝑎15+1−𝜔2𝑎1 −𝑝𝑎2𝑐𝑜𝑠𝛼2−𝛼1=0, 2𝜔𝜁1𝑎1−𝑝𝑎2𝑠𝑖𝑛𝛼2−𝛼1−𝑞2𝑎1𝑠𝑖𝑛2𝛼1+𝑞2𝑎2𝑠𝑖𝑛𝛼2+𝛼1+4𝜋𝜎1=0 𝑞2𝜇𝑎2𝑐𝑜𝑠2𝛼2−𝑞2𝜇𝑎1𝑐𝑜𝑠𝛼1+𝛼2+34𝜅23𝑎23+58𝜅25𝑎25+𝜅21+𝜇𝑝−𝜔2𝑎2 −𝜇𝑝𝑎1𝑐𝑜𝑠𝛼1−𝛼2=0 2𝜔𝜁2𝑎2−𝜇𝑝𝑎1𝑠𝑖𝑛𝛼1−𝛼2−𝑞2𝜇𝑎2𝑠𝑖𝑛2𝛼2+𝑞2𝜇𝑎1𝑠𝑖𝑛𝛼1+𝛼2+4𝜋𝜎2=0 (8) (9) (a)(b) (b)(a) Fig.3: Comparing the CxA results with a RK sweep simulation for m1 (a) and m2 (b) where σ1=0. Fig.4: Comparing the CxA results with a RK sweep simulation for m1 (a) and m2 (b) where σ1=0.009106. (a) (b) (c) (a)(b) (b)(a) Fig.6: Comparing the CxA results with a RK sweep simulation for m1 (a) and m2 (b) where σ1=0. Fig.7: Comparing the CxA results with a RK sweep simulation for m1 (a) and m2 (b) where σ1=0.009106. Fig.8: Bifurcation plot of the local maxima of 𝑢1 for coordinates 𝑢1 (a) and 𝑢2 (b), the local maxima of 𝑢2 (c), and the largest Lyapunov exponent λ1 (d) as the excitation frequency 𝜔 ranges from 1.11 to 1.0 (for backward) and 1.11 to 1.15 (for forward) and the results from the CxA method: A comparison analysis. (d) (c) (b) (a) (a) (b) (c) (d) (e) Fig.9: The time-domain plots (a–b) show steady-state oscillations for 𝑢1 and 𝑢2, ω = 1.09, The phase-space plots and singlepoints in Poincaré maps (c-d) display closed-loop trajectories, (e) indicating periodic behaviour. (a) (b) (e) (c) (d) Fig.10: The time-domain plots (a-b) show steady-state oscillations for 𝑢1 and 𝑢2, ω = 1.11, The phase-space plots (c-d) display an uncertain trajectory, and Poincaré maps and (e) indicate quasi periodicity. ➢A unique setup 2DOF parametric oscillator with dry friction successfully analysed analytically and numerically. We discussed different dynamics of the system like periodic, quasi-periodic, and chaotic. ➢Nonlinear Insight-Highlights the impact of magnetic stiffness and dry friction on system behavior. ➢Practical Relevance-Provides guidance for designing vibration control and energy harvesting systems. ➢Future Potential : Establishes a foundation for extending to multi-DOF and actively controlled structures. Fig.11: The time-domain plots (a-b) show steady-state oscillations for 𝑢1 and 𝑢2, ω = 1.125, The phase-space plots (c-d) display an uncertain trajectory, and Poincaré section and Lyapunov exponent λ1 (e) indicates a chaotic regime. (a)(b)(e) (c)(d) Conclusion