Existence and Stability of the Periodic Orbits Induced by Grazing Bifurcation in a Cantilever Beam System with Single Rigid Impacting Constraint

Existence and Stability of the Periodic Orbits Induced by Grazing Bifurcation in a Cantilever Beam System with Single Rigid Impacting Constraint Run Liua, Yuan Yueb,∗, Jianhua Xiee a, b Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China e School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China Abstract Grazing which can induce many nonclassical bifurcations, is a special dynamic phenomenon in some non-smooth dynamical systems such as vibro-impact systems with clearance. In this paper, the existence and stability of the periodic orbits induced by the grazing bifurcation in a cantilever beam system with impacts are uncovered. Firstly, the Poincaré mapping of the system is obtained by the discontinuous mapping method. Secondly, the periodic orbits are determined by means of shooting method, and Jacobian matrix in the case of non-impact is obtained subsequently. Thirdly, for various impacting patterns, a combination of inhomogeneous equations and inequations is obtained to determine the existence of period orbits after grazing. Furthermore, the stability criterion of the grazing-induced periodic orbits is given. Numerical results verify the effectiveness of theoretical analysis. What’s more, we also give a conjecture about the relationship between eigenvalues and the type of periodic orbits when eigenvalues are imaginary numbers.


Introduction
Because of the existence of collision, vibro-impact system is strongly nonlinear and non-smooth，and has various nonclassical bifurcations caused by grazing. Shaw [1] considered a class of elastically constrained impact vibration systems, and found that the zero-velocity impact will cause the dynamics characteristics of the real primary system. Xu [16] studied the grazing bifurcation problem of a class of 2-DOF impact systems with clearance, and proposed a discrete time feedback method to control the stability of periodic orbits induced by the grazing bifurcation. Jiang [17] studied the grazing bifurcation of impact oscillators under elastic and rigid constraints by using the path tracking method, and calculated the bifurcation points under different stiffness and recovery coefficients. Brzeski [18] studied the grazing bifurcation problem of a kind of church pendulum model with clearance, and revealed a new grazing bifurcation phenomenon different from the period adding motion, that is, " impact adding" phenomenon. Yin [19] considered the single-degree of freedom impulse oscillator, and used the time-discrete control method to control the unstable orbit after the grazing bifurcation. In Ref. [20], GPU parallel computing technology is used to study the two-parameter dynamics problem in the neighborhood of grazing bifurcation points.
There is important significance to study the dynamics of beam structures with impact. Emans [21] first introduced the nonlinear restoring force to establish the nonlinear elastic impact model, and studied the chaotic phenomenon by numerical simulation. Wang [22] carried out global dynamics analysis on the situation of the one-side elastic impact and bilateral impact of the system. Huangfu [23] conducted a numerical analysis of unilateral grazing bifurcation and verified that the local discontinuous mapping is also effective when it contains nonlinear terms. This paper focuses on the existence and stability of the periodic orbits induced by grazing in a cantilever beam system with impacts. In Sect. 2, the mathematical model and the dynamical equations are represented briefly. In Sect. 3, the discontinuous mapping method is used to established the Poincaré compound mapping, and a combination of inhomogeneous equations and inequations is obtained to determine the existence and stability of the periodic orbits induced by grazing bifurcations. In addition, in the case that the eigenvalues are imaginary numbers, a conjecture about the relationship between eigenvalues and the type of periodic orbits is given. In Sect. 4, numerical simulations verify the effectiveness of the theoretical analysis. Conclusions are given in Sect.5.

Cantilever Beam system with impacts
The cantilever beam system with impacts is shown in Fig.1. The mass M is subjected to harmonic excitation, and the cantilever beam is simplified into two plate springs neglecting of the mass. The length of plate springs is L. The bending stiffness is EI, where E is elastic modulus, and I is rotational inertia. The Cartesian coordinate was established as shown in Fig.1. The cantilever beam is assumed to undergo small amplitude vibration. When there is no impact, the differential equation of motion of the mass M is: The collision is described by Newton impact model: where r is Newton's coefficient of restitution of impact( (0,1) r ∈ ), which X +  and X −  denote the velocities after and before the impact, respectively. In Eq.(1), the elastic restoring force ( ) a X is described as a nonlinear form [21] 3 3 5

Existence and stability of the grazing-induced periodic orbits
The impact clearance is selected as the bifurcation parameter, and the position constraints is given by Q d µ = + , where µ is the small perturbation. When there is no impact, the system belongs to a smooth dynamic system. If the mass contacts with the impact surface at zero speed, grazing takes place, which may induce various nonclassical bifurcations. The periodic orbit of the system can be determined by the shooting method numerically. In order to study the stability and dynamics of the periodic orbits after grazing, a local Poincaré section is introduced. Then the periodic orbit of the original system corresponds the fixed point of the Poincaré map, and the dynamics of the periodic orbit is equivalent to that of the fixed point of the Poincaré map. The equations of motion are smooth slightly beyond the impact surface. However, when the grazing takes place, it is non-differentiable. The existence and stability of the grazing-induced periodic orbits can be further revealed by introducing a set of inhomogeneous equations and inequations [24] .

1 Establishing the Poincaré compound mapping by using the discontinuous mapping method
Introducing the time Poincaré cross section (modT), the left side of Eq.(4) is a function of It is assumed that when G t t = , the grazing happens, and at this moment Then we introduce the time Poincaré cross section to obtain a simpler impact 6 mapping form g . Let the section time t be close to G t ( G t t ≠ ). In the vicinity of G t , the time t can be slightly smaller or slightly larger than G t .We use the flow of the system (4) to convert the time to In this way, the Poincaré section approaches to G t t = , and then the low-speed impact must occur either before or after G t .
State coordinates on the Poincaré section are chosen as: When there is not impact, the Poincaré mapping of system (4) is smooth at 0 = x and 0 µ = ,and can be described by the following function: . However, the accurate analytical form of the Poincaré mapping cannot be obtained.
Although there are some methods that can be used to obtain the approximate solutions, they rely generally on the presupposition of small perturbation of system parameters, and are only suitable for weakly nonlinear systems. First, according to the shooting method, we can obtain the periodic orbit and its Jacobi matrix A : Next, the zero-time discontinuous mapping g is introduced to denote the process of the lowspeed impacts, then the Poincare mapping of the whole system can be expressed as    (7) Assuming that when the parameter µ changes, the equilibrium point crosses through C I A f (9) If eµ is small and positive, ( ) x µ  is a non-impact equilibrium point for the whole composite mapping P . That is, if h is positive, g is an identity mapping. In the case that h is negative and the impact surface is not considered, the local zero-time discontinuous mapping near the grazing point at time x . In the case that some trajectories do not cross Σ locally(i.e., In contrast, the discontinuity mapping will be non-trivial for the trajectory that passes through Σ ,and the trajectory hits Σ at the point * x by the map (2)).By extending the smooth flow field ( ) x under the action of the flow map φ ,or equally backward form * 2 x .
The same trajectory starting from * 2 x can also be continued backward under the action of φ for a time δ so that it passes through the point * 3 x at the time G t . So if * ( ) 0 h x < , the mapping process can be expressed as The local discontinuous mapping expression near the grazing bifurcation point can be written The following derivation is given for zero-time discontinuous mapping.
First, we consider process which takes time δ . The first order Taylor expansion of the flow function φ with respect to time can be expressed as (13) Second, considering the process Third, considering the process Substituting Eqs. (13) and (14) into Eq. (15),we obtain . Therefore, the expression of the zero-time discontinuous mapping of the grazing point can be written as x B x (18)

2 Determining the periodic orbit
The period orbit of the original system is equivalent to the period fixed point of the composite mapping. The fixed point of n -period should satisfy: . However ,due to the existence of different expressions of g and singularities at 0 h = , the solving process is very difficult. Therefore, first it is necessary to distinguish between 0 h > and 0 h < , where 0 h < represents a low-speed impact, and h 0 > represents no impact.
Non-impact periodic orbits meet (19) where the restriction is   (20) and the linear stability is (21) In order to simplify the analysis process, we use the term 1 n -period motion to represent one low-speed impact occurs in the n-period orbit. That is, 1 n − non-impact iterations are followed by one impact iteration. In this way, the square root singularity of Eq. (17) can be ignored. Then we and we can solve y as follows: Now we linearize this Poincaré map to determine the stability of the periodic orbits. The linearizing expression is: (25) When µ changes, the equilibrium point crosses the cross section. That is, holds when 0 µ → . It can be concluded that the first term of Equation (25) is: det( ) 0 − ≠ I A hold, the orbit will lose its stable state. What's more, when 0 µ → , from the perspective of stability, the low-period orbit will lose its stability and translates to the high-period orbit, which will be discussed further below.

3 Existence of periodic orbits induced by grazing
In order to make the representation of (22)-(24) clearer, new coordinates ( ) x is a function of q and Q , By formula (9), we have 1 e = − . According to Eq. (7), we obtain 2 2 ( , ) ( ( ) , ) According Eq. (8), we have Then Eq. (22) can be changed as   µ → +, the value of Λ tends to infinity, indicating that the original periodic orbit will lose the stability. If there is still a steady motion after grazing, the original periodic orbit will bifurcate into same high-period one or chaotic one, which is verified by the numerical analysis in section 4.
Through the above transformation, a combination of equations and inequalities are obtained to determine the dynamics near grazing point. In this way, the sufficient conditions for the existence of periodic solutions and the stability criteria of these orbits can be obtained.

4 Determing the type of the periodic motion induced by grazing by the eigenvalues
In this section, we use lemma 2 to discuss in detail of the different eigenvalue cases.
Case I. Eigenvalues are reals.
In this case, supposing Ref. [24] has proved that, in the case of =1 κ , there is a unique periodic 1 n -period motion if (0,1/ 2 ) s n ∈ . Here , we focus on the cases of 1 κ < . After a large number of numerical simulations (see section 4), we have the following conjecture which uncovers the relationship between conjugate the complex eigenvalues and the type of the periodic motions induced by grazing.

Numerical simulation
In this section, the clearance µ is chosen as the bifurcation parameter, the coefficient of impacting restitution is fixed as 0.65 r = .

1 The case of real eigenvalues
In this section, taking the system parameters: =0.7 x represents the displacement of the cantilever beam. Here we can see that the numerical results are consistent with Theorem 1. We discuss the five cases as follows in detail. Table 1 Eigenvalues of period orbits and γ -values  It should be mentioned that the stability criterion Eq.(46) only takes the leading term. Then there is an error caused by the omitted high-order term. In the case that the absolute value of the maximum eigenvalue is small, the stability criterion agrees well with the numerical simulation results, as shown in Fig. 3(a)-(c). But in the case that the absolute value of the maximum eigenvalue is larger, there may be a larger error between theoretical analysis and numerical simulation (see Fig.   3(d)-(f)). However, Eq.(46) can explain qualitatively the phenomenon of period-adding sequence very well. Ref. [25] summarized and classified the dynamics induced by grazing bifurcation. If the eigenvalue is a real number and 1 Fig. 4. It is shown that when µ decrease gradually, the dynamics changes from chaotic motion to the period adding motion with chaotic bands between adjacent periodic motions.
Subsequently, the chaotic bands will disappear and there is only the period-adding cascade when 0 µ → +.

2 The case of conjugate complex eigenvalues
In the case of conjugate complex eigenvalues, we will select different parameter combinations to verify conjecture 1. Taking

Conclusion
Considering a cantilever beam system with a single impacting constraint, we uncover the existence and stability of periodic orbits induced by grazing. The non-smooth dynamic system is transformed into a combination of non-homogeneous equations and inequalities through a given impact mode, thereby avoiding the singularity of the local discontinuous mapping. Subsequently, we obtain the existence conditions and the stability criteria of the periodic orbits induced by grazing bifurcations. This stability criterion explains further the phenomenon that the low-period orbit can only transition to the high-period one as 0 µ → +, which is known as period-adding bifurcation.
In the case that the eigenvalues are real numbers, we give sufficient conditions for the existence of 1 n periodic motions. And, the conditions are further simplified into a simpler algebraic criterion by introducing the Jacobi matrix of the non-impact periodic orbit and its eigenvalues.
Taking the impact clearance as the bifurcation parameter, the numerical simulation is used to verify the effectiveness of this criterion. It should be noted that in the case of 1 >2/3 λ , the numerical simulation shows that chaotic bands may alternate with periodic bands ,till the period-adding cascade appears when the parameter µ decrease and pass through some critical value.
In the case that the eigenvalues is a pair of complex numbers, we give a conjecture about the relationship between the 1 n periodic motion and the eigenvalues based on a large number of numerical simulations. Ref. [24] gives a conjecture which 1 n periodic orbit exists in the case of 0 1/ 2 s n < < . However, this paper show that the there is a 1 n -period motion induced by grazing in the case of (1/ 2 ,1/ 2( 1)) s n n ∈ − . The effectiveness of this conjecture remains to be proved theoretically in the future.

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.

Data Availability Statements
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.