| In a practical application,the motion of overhead cranes is usually subjected to uncertain system parameters.Such as uncertain frictions,payload masses,cable lengths and air resistance.Many crane control now treats the payload swing as a swing of a single pendulum.But many types of payloads and cranes in practice constitute double-pendulum dynamics.The double-pendulum effects can cause most cranes to fail to function properly.Therefore,this paper is to study a double-pendulum and weakly dissipated hybrid system.We mainly discuss the existence of periodic solution for the double-pendulum and weakly dissipated hybrid system.When the non homogeneous term1)represents an external periodic force,the hybrid system have a periodic solution.First,we can easily prove the uniqueness of the solution.The main difficulty in this paper is related to the weak dissipation term in the hybrid system,which makes the hybrid system unable to guarantee a uniform decay rate of the energy.When1)satisfies some regularity conditions,we can add a perturbation argument in the system.Then,we can prove the corresponding energy of the perturbed system.According to the fixed point argument,we can prove that the system with perturbation has a unique periodic solution.When1)∈~1(0,),we can use the obtained periodic solution of the perturbed hybrid system to prove that the periodic solution is bounded.Finally,the uniform estimate allows us to deduce the existence of a periodic solution of the initial hybrid system.In the last part of the paper we present some numerical simulations,using the parameter perturbations based on the theoretical results to approximate periodic solutions of double-pendulum and weakly dissipated hybrid systems. |
