Subharmonic Bifurcation for a Non-smooth Double Pendulum with Unilateral Impact

A unilateral impact double pendulum model with hinge links is constructed to detect subharmonic bifurcation for the high dimensional non-smooth system. The non-smooth and nonlinear coupled factors lead a barrier for high dimensional conventional nonlinear techniques. By introducing reversible transformation and energy time scale transformation, the system is expressed as a smooth decoupling form of energy coordinates. Thus, the concept of subharmonic Melnikov function is extended to high-dimensional nonsmooth systems, and the influence of impact recovery coefficient on the existence of subharmonic periodic orbits of double pendulum is revealed. The efficiency of the theoretical results is verified by phase portraits, time process portraits and Poincaré section.


Introduction
In practical engineering, there are many factors such as impact, gap and so on. Generally, the impact of manipulator and robot leg with ground can be simplified as the non-smooth double pendulum [1,2]. Although the smooth double pendulum is a classical nonlinear model [3,4], the cross coupling of non-smooth and high-dimensional

System Modelling and Preliminary Analysis
Consider a two-degree-of-freedom nonlinear impact oscillator, which can be modeled by a smooth hinged double pendulum with a rigid wall fixed to the base, as shown in Fig. 1. The horizontal harmonic excitation A cos(Ωt) is applied to the base. On the right, the position of the rigid wall is 0 ; on the left, the position of the horizontal rigid wall is − 2 .
It is assumed that the pendulum and walls are rigid. When the pendulum and rigid wall impact, the vector field of the system changes suddenly under the action of instantaneous pulse, and the jump discontinuous point appears on the phase portraits. Assume that the pendulum length, the pendulum mass, and the damping at the hinge joint are represented by l 1 , l 2 , m 1 , m 2 , c 1 , c 2 , respectively. The quality of the rod is ignored. Considering that the two pendulum can only swing at a small angle in the same vertical plane.
The pendulum angle 1 and 2 are taken as the generalized coordinate and the zero potential energy point at m 1 , m 2 level. Hence, the differential equations of motion of the system are where ̇− is the velocity just before the instant of impact and ̇+ indicates the rebound velocity. −(1 − ) represents an impact recovery coefficient and ≤ 1 is indicative of the energy loss upon impact. Let The impact double pendulum model can be taken as the model of the manipulator. In the design of the manipulator, we hope to design reasonable physical or geometric parameters to make the system appear periodic motion. However, the impact results in strong nonlinear and non-smoothness of the system. So it is difficult to calculate the condition of periodic orbits directly. The actual motion angle of the manipulator in engineering practice is very small, so it is meaningful to study the impact periodic orbits of the small angle motion of the system theoretically. The system is addressed as follows (4) , Denote x 1 , x 2 , y 1 , y 2 = 1 , � 1 , 2 , � and the state-space R 4 is split into two open, disjoint subsets S + and S − by the switching manifolds L + and L − . The subsets S + and S − and the switching manifolds L + and L − can be formulated as which satisfy S = S + × S − × L + × L − ∪ (0, 0, 0, 0) ∈ R 2 × R 2 , and the switching manifolds L + and L − are denoted by Let k i → k i , a → a, → , system (6) is rewritten to the first order form where f (x 1 , x 2 , y 1 , y 2 , ) = k 1 x 2 − k 2 y 2 + a 2 cos( ), g(x 1 , x 2 , y 1 , y 2 , ) = −k 3 x 2 + k 4 y 2 ,(x 1 , x 2 , y 1 , y 2 ) ∈ S + ∪ S − , and the impact laws satisfy

Let
The matrix expression of system (7a) and (7b) is When = 0 , the unperturbed system yields Clearly, for system (9) in the region (1) and (2)  Consider a singular point (0, 0, 0, 0) . Its Jacobian matrix is which leads to According to the physical meaning of the actual engineering parameters > 0, ≥ 0 , the parameters ( , ) are divided into three regions as shown in Fig. 3.
Region Q1,(1 − ) 2 + 4 > 0, > 1 , the singularity is a saddle-center point. The central manifold theorem can be used to reduce the dimension and then to study the properties of the system, which will be discussed in the subsequent study.
Region Q2,(1 − ) 2 + 4 > 0, = 1 , the system has a double zero root and a pair of conjugate pure virtual roots, which is a residual dimension 3 case or Hopf bifurcation; it will be discussed in a subsequent study.
We are interested in region Q3. In the following, we will study which of periodic solution persist for perturbed non-smooth double pendulum system (7a, 7b).

Existence of Periodic Solution of a Perturbed System with 1:2 Internal Resonance
We shall study the subharmnic periodic orbits with 1:2 internal resonance in the non-smooth double pendulum system. The parameter can be taken = 3 2 , = 1 3 . where x 1 , x 2 , y 1 , y 2 ∈ S + ∪ S − and the impact law is satisfied In order to detect the existence of impact periodic solutions of high-dimensional perturbed systems, we need to decouple the system and the following reversible transform is given.

Reversible Transform
The vector symbol is given and the matrix can be constructed Let When = 0 , the unperturbed system (10a, 10b) becomes where Γ + and Γ − represent new impact surface after transform. System (12) is a decoupled two-degree-of-freedom nonlinear piecewise Hamiltonian system, i.e., system (12) can be rewritten in the following form Similarly, the perturbed system (8) can be rewritten in the following form H 1 (X) and H 2 (Y) are the piecewise Hamiltonian functions of the unperturbed system, which are smooth in the corresponding zones S + and S − ; F ± (X, Y, , , ) and G ± (X, Y, , , ) are T-periodic function in t . Moreover, H 1 (X) and H 2 (Y) represent the energy of the unperturbed system (15). Obviously, the impact law matrix (14) becomes a non-diagonal matrix, which leads to a new obstacle for the high-dimensional non-smooth system. In order to overcome the obstacle, the energy-time scale transform will be introduced.
On the one hand, the energy perturbation of the system (22) is Δh 1 . And the energy of the system is invariant under scale transform. So, Fortunately, in ref. [12], the energy change of impact periodic solution of single degree of freedom impact system has been calculated. So, we get Let H + 1Δ − H − 1Δ be the perturbation of the energy before and after the impact. It is the algebraic sum of the input energy of the external forcing and the energy dissipated by the damping. Because the impacts are instantaneous t + (25) Δh 1 = Δh 11 , by numerical simulations to verify the theoretical analysis. The amplitude and impact recovery coefficient of the system are taken to meet the parameter range of the periodic orbits, and the single side impact period of the system is found. Figure 5 shows the phase portrait and time history portraits of period motion for upper pendulum. Figure 6 shows the phase portrait and time history portraits of period motion for lower pendulum.

Conclusions
In this paper, the subharmonic Melnikov method of non-smooth double pendulum is significantly improved by using energy-time scale transform and Poincaré map.
In particular, an invertible transformation is introduced to decouple the unilateral constrained double pendulum system. In order to solve the obstacle of nonsmooth term coupling, energy time scale transformation is introduced, and the system is expressed as a smooth form of energy coordinates, which provides a new idea for the research of high-dimensional nonsmooth systems. Furthermore, based on the Poincaré map, the subharmonic Melnikov function is extended to the high dimensional non-smooth system. This method for detecting subharmonic dynamic  responses of two-degree-of-freedom forced nonlinear oscillators can be well used to determine the existence of subharmonic periodic orbits, which shows that an important step has been taken in the study of complex dynamics of the high-dimensional non-smooth system.
Author Contributions GXY and ZG study conceptualization and writing the manuscript. TRL formal analysis and writing the manuscript.
Funding This work is supported by the National Natural Science Foundation of China (No. 12072203, 11872253), the Hundred Excellent Innovative Talents in Hebei Province (No.SLRC2019037), the "333 talent project" in Hebei Province (No.A202005007).

Data Availability
The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

Conflict of interest
The authors declare that they have no conflict of interest.
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