Dynamic Modeling and Nonlinear Oscillations of a Rotating Pendulum with a Spinning Tip Mass.pptx

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1. 1 REPORT Dynamic Modeling and Nonlinear Oscillations of a Rotating Pendulum with a Spinning Tip Mass Research from Al-Solihat & Al Janaideh -Journal of Sound and Vibration, 2023 NGUYEN VO NGOC THACH 2025/06

2. 2 Rotating pendulums are important models in nonlinear dynamics research. Previous studies often neglect the spin of the tip mass. Research Question: How does the tip mass spin (axis, speed, direction) affect the dynamics of a rotating pendulum? Novelty: First investigation of a rotating pendulum model with a spinning tip mass. Introduction & Motivation Fig. 1. a. Schematic diagram of a rotating pendulum with a spinning tip mass: physical description of system

3. Pendulum: massless rigid rod, length L, oscillates with angle θ(t). Main rotation: Pendulum rotates around the vertical axis with prescribed angular velocity ϕ˙​(t). Tip mass (m): Can spin around an arbitrary axis fixed to the rod with constant angular velocity ω (ω x​ , ω y ​, ω z ​). System has 1 primary degree of freedom: θ(t). System Model Fig. 1. b System kinematics 3

4. Research Objectives 4 Develop an accurate dynamic model for the system. Investigate the influence of tip mass spin parameters: + Spin axis (J 1 ​<==> ω x ​, J 3 <==>​ ω z​ ) + Spin direction (sign of J 1 ​, J 3 ​) Analyze: + Dynamic stability & bifurcation (when ϕ˙​=Ω is constant). + Frequency response (when ϕ˙​(t) is sinusoidal). Fig. 1. b System kinematics

5. Modeling 4 1. Determines the coordinates of the tip mass center (C) Fig. 1. b System kinematics Transforms coordinates from XYZ (rotated by ϕ) to the inertial frame AxAyAz Transforms coordinates from xyz (rotated (by θ ) to the XYZ frame Tip mass position r C (Eq.1) (Eq.2) (Eq.3)

6. Modeling 4 2. Determines linear velocity of the tip mass center (C) Fig. 1. b System kinematics + Rod angular velocity ω C ​ (Eq.4) Rotation around the vertical Az​ axis with velocity ϕ˙ Oscillation of the rod by angle θ around the Y-axis + Tip mass linear velocity v C Using Eq.3 and Eq.4 (Eq.5)

7. 4 3. Determines tip mass angular velocity ω s​ (in xyz frame) and energy Fig. 1. b System kinematics + Tip mass angular velocity ω s​ (in xyz frame) (Eq.6) The angular velocity of the frame itself (xyz) The intrinsic spin of the tip mass ω + Kinetic Energy T: (Eq.7) + Potential Energy V: (Eq.8) Rayleigh Dissipation Function Γ

8. Modeling: Use Lagrange's equation to derive the equation of motion. Research Methodology 5 Bifurcation analysis: + Find equilibrium points θ 0​ . + Examine stability. + Construct bifurcation diagrams and phase plane portraits. Frequency Response Analysis: + Solve the equation of motion using numerical integration (Runge-Kutta). + Employ the arc-length continuation method. + Construct frequency response diagrams. Fig. 1. b System kinematics

9. Investigate the effect of J 1 ​ (spin about x-axis) and J 3 ​ (spin about z-axis) on the equilibrium angle θ ​ as rotation speed Ω varies. Baseline case (no spin, J 1 ​=J 3​ =0): Pitchfork bifurcation at Ω=1. Results: Bifurcation (Constant ϕ˙​=Ω) 6 Fig. 2. Bifurcation diagram of a rotating pendulum with constant Ω and fixed tip mass (ω 3 =ω 1 =J 3 =J 1 =0)

10. Preserves bifurcation type (Pitchfork). Changes the critical bifurcation speed Ω b ​: + J 3 ​>0 (positive spin): Increases Ω b​ (more stable), decreases ∣θ ​∣. Results: Effect of Z-axis Spin (J 3 ​) on Bifurcation 7 Fig. 4. Effect of ω 3 (positive direction) on the bifurcation diagram of a rotating pendulum with at I =0.01 and ω 1 =0.

11. Preserves bifurcation type (Pitchfork). Changes the critical bifurcation speed Ω b ​: + J 3 ​<0 (negative spin): Decreases Ω b ​ (less stable), increases |θ ​|. Results: Effect of Z-axis Spin (J 3 ​) on Bifurcation 8 Fig. 5. Effect of ω 3 , J 3 (negative direction) on the bifurcation diagram of a rotating pendulum with I =0.01 and ω 1 =J 1 =0.

12. Preserves bifurcation type (Pitchfork). Inversion phenomenon: If J 3 ​≤−2, the pendulum can stabilize in an inverted state (θ ​=± π ) over a range of Ω. Results: Effect of Z-axis Spin (J 3 ​) on Bifurcation 9 Fig. 6. Bifurcation diagrams of a rotating pendulum when ω 3 ≤−200 (J 3 ≤−2) at I =0.01 and ω 1 =0 for: (a) J 3 =−2 and (b) J 3 =−2.5.

13. Breaks the Pitchfork bifurcation. Emergence of 3 distinct branches: + One stable/unstable branch (depending on sign of J 1 ​). + One unstable/stable branch (depending on sign of J 1 ​). + One saddle-node bifurcation branch. Magnitude ∣J 1 ​∣ affects the onset speed (Ω) of the saddle-node bifurcation. Sign of J 1 ​ (spin direction) determines the stability of the branches. Results: Effect of X-axis Spin (J 1 ​) on Bifurcation 10 Fig. 8. Effect of ω 1 (positive direction) on the bifurcation diagram of a rotating pendulum at I =0.01 and ω 3 =0 for: (a) ω 1 =100 (J 1 =1) and (b) ω 1 =200 (J 1 =2).

14. Breaks the Pitchfork bifurcation. Emergence of 3 distinct branches: + One stable/unstable branch (depending on sign of J 1 ​). + One unstable/stable branch (depending on sign of J 1 ​). + One saddle-node bifurcation branch. Magnitude ∣J 1 ​∣ affects the onset speed (Ω) of the saddle-node bifurcation. Sign of J 1 ​ (spin direction) determines the stability of the branches. Results: Effect of X-axis Spin (J 1 ​) on Bifurcation 11 Fig. 9. Effect of ω 1 (negative direction) on the bifurcation diagram of a rotating pendulum at I =0.01 and ω 3 =0 for: (a) ω 1 =−100 (J 1 =−1) and (b) ω 1 =−200 (J 1 =−2).

15. Qualitative behavior similar to the J1​-only case (still 3 branches, including saddle-node). However, the relative position and stability of the branches depend on both the signs and magnitudes of J1​ and J3​. Results: Combined Spin (J 1 ​ & J 3 ​) on Bifurcation 12 Fig. 12. Bifurcation diagram of θ0 as function of Ω for a combined tip mass spin at J1=1 for: (a) J3=2 and (b) J3=−2. Fig. 13. Bifurcation diagram of θ0 as function of Ω for a combined tip mass spin at J1=−1 for: (a) J3=2 and (b) J3=−2.

16. Examine oscillation amplitude θ max ​ versus excitation frequency Ψ. Special Case: + No spin or spin only about Z-axis (J 1 ​=0): No steady-state response, oscillations decay to 0 (underdamped). Results: Frequency Response (Sinusoidal ϕ˙​(t)) 13 Fig. 14. Time response of the pendulum excited by sinusoidal ϕ′ at I =0.01, ζ=0.03, ϕ = π /18 rad and Ψ =0.8 for: (a) tip mass spinning along the x with J 1 =1 (J 3 =0) and (b) fixed tip mass (J 1 =J 3 =0).

17. Steady-state periodic response occurs. Exhibits nonlinear softening effect. Response amplitude increases as: + Magnitude ∣J 1 ​∣ increases; Excitation amplitude ϕ increases. Damping ratio ζ reduces amplitude near resonance. Important: Spin direction (sign of J 1 ​) does not affect response amplitude when J 3​ =0. Effect of X-axis Spin (J 1 ​) on Frequency Response 14 Fig. 16. Influence of the magnitude of angular excitation (ϕ ) on the frequency response of the system at I =0.01, ζ=0.01, ω 1 =5 (J 1 = 0.05), ω 3 =0 (J 3 = 0), ϕ =10°. Fig. 15. Influence of ω 1 (J 1 ) on the frequency response of the system at I =0.01, ζ=0.01, ω 3 =0 (J 3 =0), and ϕ = π /18 (10°).

18. Steady-state periodic response occurs. Exhibits nonlinear softening effect. Response amplitude increases as: + Magnitude ∣J 1 ​∣ increases; Excitation amplitude ϕ ​ increases. Damping ratio ζ reduces amplitude near resonance. Important: Spin direction (sign of J 1 ​) does not affect response amplitude when J 3​ =0. Effect of X-axis Spin (J 1 ​) on Frequency Response 15 Fig. 17. Influence of the damping ratio (ζ) on the frequency response of the system at I =0.01, ω 1 =5 (J 1 = 0.05), ω 3 =0 (J 3 = 0), ϕ =10°.

19. Increases response amplitude compared to the J 1 ​-only case. Excites 1/2-order superharmonic resonance (at Ψ ≈1/2). This does not occur when J 1 ​=0 or J 3 ​=0. Relative spin direction matters: +Response is larger when J 1​ and J 3 ​ have the same sign (J 1 ​J 3 ​>0). +Response is smaller (but softening effect is stronger) when J 1 ​ and J 3​ have opposite signs (J 1 ​J 3 ​<0) Effect of Combined Spin J 1 ​,J 3 ​ on Frequency Response Fig. 18. Influence of the combined tip mass spin on the frequency response of the system at I =0.01, ζ=0.01, ϕ =10° and J 1 =0.03. The three frequency response diagrams are plotted for J 3 =0, J 3 =0.3 and J 3 =0.6.

20. Effect of Combined Spin J1​,J3​ on Frequency Response Fig. 19. Influence of the combined tip mass spin on the frequency response of the system at I =0.01, ζ=0.01, ϕ =10° and J 1 =0.03. The three frequency response diagrams are plotted for J 3 =0, J 3 =−0.3 and J 3 =−0.6. Increases response amplitude compared to the J 1 ​-only case. Excites 1/2-order superharmonic resonance (at Ψ ≈1/2). This does not occur when J 1 ​=0 or J 3 ​=0. Relative spin direction matters: +Response is larger when J 1​ and J 3 ​ have the same sign (J 1 ​J 3 ​>0). +Response is smaller (but softening effect is stronger) when J 1 ​ and J 3​ have opposite signs (J 1 ​J 3 ​<0)

21. Effect of Combined Spin J1​,J3​ on Frequency Response Fig. 20. Influence of the tip mass spin directions on the frequency response of the system at I =0.01, ζ=0.01, ϕ o =10° and J 1 =0.03. The two frequency response diagrams are plotted for J 3 =0.6 (J 1 J 3 >0) and J 3 =−0.6 (J 1 J 3 <0). Increases response amplitude compared to the J 1 ​-only case. Excites 1/2-order superharmonic resonance (at Ψ ≈1/2). This does not occur when J 1 ​=0 or J 3 ​=0. Relative spin direction matters: +Response is larger when J 1​ and J 3 ​ have the same sign (J 1 ​J 3 ​>0). +Response is smaller (but softening effect is stronger) when J 1 ​ and J 3​ have opposite signs (J 1 ​J 3 ​<0)

22. Tip mass spin is an essential factor, fundamentally altering the nonlinear dynamics of the rotating pendulum. Spin axis and direction have a decisive influence on: + Bifurcation type and stable equilibria. + Existence and characteristics of the frequency response. Novel complex nonlinear phenomena (changes in bifurcation type, inversion, superharmonic resonance) were discovered due to the interaction between the main rotation and the tip mass spin. Key Conclusions

23. Contributions: + New and more comprehensive dynamic model. + Deep insight into the gyroscopic effects from the tip mass. Future Work: + Investigate the effect of varying spin velocity ω. + Study chaotic regimes. + Consider other factors (rod elasticity, more complex damping). + Potential applications (governors, vibration absorbers...). Contributions & Future Work

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