An improved non-smooth coordinate transformation for analyzing bilateral vibro-impact systems with stochastic...

Area of Science:

• Mechanical Engineering
• Nonlinear Dynamics
• Stochastic Systems

Background:

• Vibro-impact systems present analytical challenges due to their non-smooth nature.
• Non-smooth coordinate transformations are crucial for simplifying these systems by creating continuous trajectories.
• Existing transformations require enhancement for complex stochastic scenarios.

Purpose of the Study:

• To introduce and derive an improved non-smooth coordinate transformation method.
• To extend this method for analyzing bilateral vibro-impact systems under stochastic excitations.
• To validate the method's effectiveness for complex systems with asymmetric barriers.

Main Methods:

• Detailed derivation of the improved non-smooth coordinate transformation.
• Application of the transformation to convert non-smooth systems into continuous and periodic trajectory systems.
• Validation using Fokker-Planck equation analysis and comparison with Monte-Carlo simulations.

Main Results:

• The improved transformation successfully converts non-smooth vibro-impact systems into a form with continuous and periodic trajectories.
• Validation against Monte-Carlo simulations shows good agreement for stationary probability density functions.
• The method is effective for systems with asymmetric bilateral or unilateral barriers and distinct restitution coefficients.

Conclusions:

• The improved non-smooth coordinate transformation method provides an effective tool for analyzing stochastic responses and bifurcations in complex vibro-impact systems.
• This method simplifies the analysis of systems previously intractable with classical methods.
• The approach is versatile, applicable to various barrier configurations and excitation types.