Codimension-Two Grazing Bifurcations in Three-Degree-of-Freedom Impact Oscillator with Symmetrical Constraints

This paper investigates the codimension-two grazing bifurcations of a three-degree-of-freedom vibroimpact system with symmetrical rigid stops since little research can be found on this important issue. The criterion for existence of double grazing periodic motion is presented. Using the classical discontinuity mapping method, the Poincaré mapping of double grazing periodic motion is obtained. Based on it, the sufficient condition of codimension-two bifurcation of double grazing periodic motion is formulated, which is simplified further using the Jacobian matrix of smooth Poincaré mapping. At the end, the existence regions of different types of periodic-impact motions in the vicinity of the codimension-two grazing bifurcation point are displayed numerically by unfolding diagram and phase diagrams.


Introduction
Impacting phenomena exist in a large number of mechanical systems.Because the collision introduces essential nonlinearity and discontinuity, the vibroimpact systems can exhibit rich and complicated dynamical behavior.There is rich literature on the analysis of the dynamics for impact oscillator systems.The early work mainly focuses on the singledegree-of-freedom impact oscillators, for example, [1][2][3].For multidegree-of-freedom vibroimpact systems, detailed studies of dynamics (including stability and bifurcations) using numerical simulations and qualitative analyses were carried out in decades, for example, [4][5][6][7][8].Aidanpaa and Gupta [4] analyzed a two-degree-of-freedom vibroimpact system and obtained the expression of periodic motion, which is too complex to analyze the dynamical behavior.Leine [5] presented an asymptotic approximation method for the critical restitution coefficient of a parametrically excited impact oscillator and described its dynamics by a unilaterally constrained Hill's equation.Yue and Xie [6] researched the symmetric period -2 motion and bifurcations of a two-degree-of-freedom vibroimpact system.Luo [7] developed a method to investigate the symmetry of solutions in nonsmooth dynamical systems and obtained all possible stable and unstable motions.Luo et al. [8] considered multiperformance, multiprocess coupling, and multiparameter simulation analysis for dynamics of a two-degree-of-freedom periodically forced system with a clearance represented by two symmetric rigid stops.
A special situation arises when an impact with zero velocity occurs, namely, grazing impact.Grazing impact gives a nondifferentiable Poincaré mapping, which is important for the bifurcation when stable nonimpact motion changes to impact motion.The pioneer work in this field was done by Nordmark [9], who developed systematic method that is so-called discontinuity-mapping approach to investigate grazing dynamics and its attendant bifurcations, providing the results which laid the foundation for many subsequent studies, for example, [10][11][12][13].Li et al. [14,15] investigated the existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibroimpact system with unilateral constraint and symmetric constraints, respectively.Dankowicz and Zhao [16] analyzed the codimension-one and codimension-two grazing bifurcations in impact microactuators.Thota et al. [17] investigated the distribution of such codimension-two grazing bifurcations in single-degree-offreedom impact oscillators and inquired into the possible dynamical characteristics of the system response on neighborhoods of such bifurcation points.Csaba and Champneys [18] analyzed nonsmooth bifurcations in both one and two parameters for a simple mechanical model of a pressure relief valve which is an autonomous impact oscillator.Dankowicz and Katzenbach [19] collected four distinct instances of grazing contact of a periodic trajectory in a hybrid dynamical system under a common abstract framework and established selected general properties of the associated near-grazing dynamics.Mason et al. [20] analyzed a model of a periodically forced impact oscillator with two discontinuity surfaces and provided new insights into the extremely rich dynamical behavior including codimension-one, codimension-two, and codimension-three bifurcations by discontinuity-geometry methodology.
Despite that much work has been carried out to analyze nonsmooth codimension-two bifurcation of impact system, little work has been reported on the analysis of such bifurcation in multidegree-freedom with two discontinuity surfaces.In this paper, we investigated codimension-two grazing bifurcations in three-degree-of-freedom impact oscillator with symmetrical constraints.This paper is organized as follows.A three-degree-of-freedom vibroimpact system with proportional damping property is considered and an existing criterion of double grazing period- motion is proposed in Section 2. The Poincaré mapping is obtained by combination of discontinuity map and smooth Poincaré mapping of double grazing periodic motion in Section 3. The Poincaré mapping will be used to analyze the sufficient conditions of stability of double grazing periodic trajectories and codimension-two grazing bifurcations in Section 4. Using the above result, the dynamical features near critical points of grazing codimension-two bifurcation are displayed by numerical simulation in Section 5. Finally, some conclusions are drawn in Section 6.

Double Grazing Periodic Motion in
Three-Degree-of-Freedom Impact Oscillator Between consecutive impacts, for | 1 | < , the differential equations of motion are ) sin (Ω + ) . ( When the impact occurs, for | 1 | = , the velocity of the impacting mass is changed according to the impact law, and the impact equations of mass  1 are given by where − and + denote the values just before and after impact, respectively.Equations ( 1) and (2) are rewritten in nondimensional form for | 1 | < : where a dot (⋅) denotes differentiation with the nondimensional time .Let  1 ̸ = 0,  1 ̸ = 0, and Figure 1: Schematic of the there-degree-of-freedom impact oscillator with symmetrical constraints.
The nondimensional quantities  3) is amenable to analytical treatment due to the special relation between stiffness and damping.Let Ψ represent the canonical model matrix of (3). 1 ,  2 , and  3 denote the eigenfrequencies of the system as impacts do not occur.Taking Ψ as a transition matrix, the motion equation (3) under the change of variables  = Ψ is where  = ( 1 ,  2 ,  3 )  ,  = ( 1 ,  2 ,  3 )  ,  is an unit matrix of degree 3 × 3,  and Λ are diagonal matrixes, and +   sin ( + ) +   cos ( + )) , ( = 1, 2, 3) , (6) where  0 denotes the time when mass  1 collides with constraint  or ,   are the elements of the canonical modal matrix Ψ,   =    2  ,   = √ 2  −  2  ,   and   are the constants of integration which are determined by the initial conditions and modal parameters of the system,   and   are the amplitude parameter,  = 1, 2, 3, and

The Condition for Existence of Grazing Periodic Motion. If
oscillator  1 impacts each rigid constraint with zero velocity and the direction of the acceleration is opposite to the motion, then we say that the system is undergoing grazing motion.A grazing period motion may be denoted by - which means that oscillator  1 grazes with each constraint for  times in  periodic external excitation force.In the following, we will derive an existence condition of grazing motion with period , where  is the period of external excitation.Assume that the grazing periodic motion begins from the grazing point on constraint .
The initial conditions of grazing period- motion are The periodic conditions of grazing period- motion are If the grazing periodic motion begins from the grazing point on constraint , similar to the case above, the initial conditions and the periodic conditions are Substituting the above condition into the general solutions of ( 6), we can obtain the expression of   and   as where Substituting ( 11) into the initial conditions and the periodic conditions yields where Thus, if we have  1 = 0,  2 = 0, and  3 = 0. Hence  1 = 0,  2 = 0, and  3 = 0.For simplicity, assume that the parameters are chosen such that the integral constants  1 ,  2 ,  3 ,  1 ,  2 , and  3 are vanishing.Inserting integral constants into (6) gives as the grazing periodic motion sets off from the grazing point on constraint  or as the grazing periodic motion sets off from the grazing point on constraint .Then it follows that where  1 =  1  11 +  2  12 +  3  13 and  2 =  1  11 +  2  12 +  3  13 .Denote the acceleration of oscillator  1 as  1 (or  2 ) for the case in which the periodic grazing motion begins from the grazing point on constraint (or ) Based on the analysis above, if there exists a double grazing periodic trajectory in the system with initial condition and periodic conditions, then system parameters must satisfy the following condition: Using the condition for existence of double grazing periodic motion for the three-degree-of-freedom impact system, the curve of points in the (, ) parameters space corresponding to the existence of a double grazing periodic trajectory with  2 =  3 = 5,  2 =  3 = 10,   = 0.05, and  = 0.8 is shown in Figure 2, where  = 1, 2, 3.
In order to verify the existence condition obtained, numerical simulation of the original system equations ( 3)-(4) will be given.For fixed  2 =  3 = 5,  2 =  3 = 10,   = 0.05,  = 0.8,  10 = 1,  20 = 0, and  30 = 0, a double grazing period-1 motion is obtained with  = 0.1,  = 1.44159 as shown in Figure 3. Figures 3(a) and 3(b) are the phase portrait and time history of oscillator  1 .Figure 3(a) shows that oscillator  1 collides with constraints  and  with zero velocity.It illustrates the validity of the condition for existence of grazing periodic motion.

Poincaré Mapping of Double
Grazing Periodic Motion in Three-Degree-of-Freedom Impact Oscillator where  =  mod 2 denotes the phase of the excitation and   equals the acceleration of the oscillator as a function of   , V  , and .Denote the corresponding flow function by Φ(, ).

Discontinuity-Mapping.
When considering the effects of the jump map associated with the impact surface, the dynamics of the impact oscillator under perturbations in initial conditions away from the grazing periodic trajectory may be analyzed using the discontinuity-mapping approach originally introduced by Nordmark [9].Here, two discontinuitymappings  1 and  2 are introduced on a neighborhood of points  * 1 and  * 2 , such that surface  1 is invariant under  1 (i.e.,  ∈  1 ,  1 () ∈  1 ) and surface  2 is invariant under  2 (i.e.,  ∈  2 ,  2 () ∈  2 ); therefore, Poincaré mapping  associated with surface  1 for the flow near the grazing trajectory including the effects of the jump map can be written as According to the discontinuity-mapping approach, the discontinuity-mappings of  1 and  2 are obtained as follows: where ))( * 2 ).Since  * 1 satisfies ℎ  1 ( * 1 ) = 0, such that (29) is written as

The Poincaré
consequently, From the above analysis, the mapping  2smooth () ∘  2 () is written as

Stability at Grazing and Codimension-Two Grazing Bifurcation
The stability of the grazing periodic trajectory when ignoring the effects of the constraint is determined by the eigenvalues of its Jacobian matrix.In contrast, the Jacobian matrix of the grazing periodic trajectory in the absence of the constraint is discontinuous and becomes singular; the stability properties of the grazing periodic trajectory in the presence of the bilateral constraint are determined by Poincaré mapping .
If the points near the grazing point which start from either the impact side or nonimpacting side are trapped close to the grazing point after iterating the mapping equations ( 31)-( 32), the grazing periodic trajectory is stability.For an impact point  in the vicinity of the grazing point  * 1 , which satisfies that is, that is, it means an impact point impacts discontinuity surface  1 again and the impact will be perpetuated, which results in a large stretching in a direction given by the image of vector  1 under Jacobians  1smooth, and  2smooth, , and the trajectory is unstable.According to the above analysis meaning an impact point impacts discontinuity surface  1 again and the impact will be perpetuated; the grazing periodic trajectory is unstable.By just changing for  ≤ , it happens that an impact is followed by nonimpacting for some iterations but eventually impacts discontinuity surface  1 again and the impact will be perpetuated, and the grazing periodic trajectory is unstable.Thus, if ℎ  2  ( * 2 ) 1smooth, ( * 1 )( 2smooth, ( * 2 ) 1smooth, ( * 1 )) (−1)  1 < 0 and ℎ  1  ( * 1 )( 2smooth, ( * 2 ) 1smooth, ( * 1 ))   1 < 0 for any  ≥ 1, 1 ≤  ≤ , stability is lost.
In the same way, if for any  ≥ 0,  ≥ , it happens that an impact is followed by nonimpacting for some iterations but eventually impacts discontinuity surface  2 again and the impact will be perpetuated, and the grazing periodic trajectory is unstable.Moreover, if for any  ≥ 0,  ≥ , it happens that an impact is followed by nonimpacting for some iterations but eventually impacts discontinuity surfaces  1 and  2 again and the impact will be perpetuated, and the grazing periodic trajectory is unstable.
According to above analysis, the codimension-two grazing bifurcation points (the definition of such points is seen in [17]) correspond to for all  ≥ 0 and  ≥ 0, respectively.Let   express the codimension-two grazing bifurcation points;   can be written in four cases; that is, for  = 0, 1, 2, . ... In the following, take the third case, for example, the codimension-two grazing bifurcation criterion   is simplified as far as possible and the more analytic expressions are obtained. Let Since  1smooth, ( * 1 )( * 1 ) = 0, it follows that Thus, According to the relevant definitions in Section 3 and the implicit function theorem, it is straightforward to show that where  is unit matrix.

Numerical Simulations
Taking the sencond case derived in the above section as an example, the codimension-two grazing bifurcation points are corresponding to   = −( + 1) * 1 1 Φ  12 = 0.The curve of points in the (, ) parameters space corresponding to the existence of a grazing periodic trajectory with  2 =  3 = 5.0,  2 =  3 = 10.0,  1 =  2 =  3 = 0.05, and  = 0.8 is shown in Figure 5. Here, points for which  1 = 0 are indicated by the asterisk ( * ).As shown in Figure 5, with  decreases, codimension-two grazing bifurcation points appear to have an increased tangency with the grazing curve.
For example, under case of  10 = 1,  20 = 0, and  30 = 0, taking  = 0.28109,  = 1.92036 along the grazing curve shown in Figure 5, then existence condition of double grazing periodic motion and  1 = 0 are both satisfied using the formulae derived from previous sections, which means that the point is a codimension-two grazing bifurcation point.Simulating with the above parameters, a grazing periodic trajectory is obtained as shown in Figure 6, which may cause complicated dynamical behaviors with the change of some parameters.

Figure 3 :
Figure 3: Double grazing period-1 motions of the system: (a) the phase portraits of oscillator  1 ; (b) the time response of oscillator  1 .

Figure 4 :
Figure 4: A schematic of the flow with impacting.