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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.
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 single-degree-of-freedom impact oscillators, for example, [1–3]. 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, [4–8]. Aidanpaa and Gupta  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  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  researched the symmetric period motion and bifurcations of a two-degree-of-freedom vibroimpact system. Luo  developed a method to investigate the symmetry of solutions in nonsmooth dynamical systems and obtained all possible stable and unstable motions. Luo et al.  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 , 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, [10–13]. 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  analyzed the codimension-one and codimension-two grazing bifurcations in impact microactuators. Thota et al.  investigated the distribution of such codimension-two grazing bifurcations in single-degree-of-freedom impact oscillators and inquired into the possible dynamical characteristics of the system response on neighborhoods of such bifurcation points. Csaba and Champneys  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  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.  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.
2. Double Grazing Periodic Motion in Three-Degree-of-Freedom Impact Oscillator
2.1. Mechanical Model
The mechanical model for a three-degree-of-freedom vibrator with masses , , and is shown in Figure 1. Displacements of masses , , and are represented by , , and , respectively. The masses are connected to linear springs with stiffness , stiffness , and stiffness . The excitations on the three masses are harmonic with amplitudes , , and , respectively. The excitation frequency and the phase are the same for three masses. Mass moves between rigid stops and . When the displacement of mass is or , mass will hit rigid stop or . The impact is described by a coefficient of restitution , and it is assumed that the duration of impact is negligible compared to the period of the force. Damping in the mechanical model is assumed as proportional damping of the Rayleigh type, which in this case implies .
Between consecutive impacts, for , the differential equations of motion are
When the impact occurs, for , the velocity of the impacting mass is changed according to the impact law, and the impact equations of mass are given bywhere − and + denote the values just before and after impact, respectively.
Equations (1) and (2) are rewritten in nondimensional form for :where a dot denotes differentiation with the nondimensional time . Let , , and . The nondimensional quantities , , , , , , , , and have been introduced, where .
Equation (3) is amenable to analytical treatment due to the special relation between stiffness and damping. Let represent the canonical model matrix of (3). , , and denote the eigenfrequencies of the system as impacts do not occur. Taking as a transition matrix, the motion equation (3) under the change of variables iswhere , , is an unit matrix of degree , and are diagonal matrixes, and , , , and . The general solutions of (3) are given bywhere denotes the time when mass collides with constraint or , are the elements of the canonical modal matrix , , , and are the constants of integration which are determined by the initial conditions and modal parameters of the system, and are the amplitude parameter, , and
2.2. The Condition for Existence of Grazing Periodic Motion
If oscillator 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 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 aswhere
Substituting (11) into the initial conditions and the periodic conditions yieldswhereThus, ifwe have , , and . Hence , , and . For simplicity, assume that the parameters are chosen such that the integral constants , , , , , and are vanishing. Inserting integral constants into (6) givesas the grazing periodic motion sets off from the grazing point on constraint oras the grazing periodic motion sets off from the grazing point on constraint . Then it follows thatwhere and . Denote the acceleration of oscillator as for the case in which the periodic grazing motion begins from the grazing point on constraint with and . If , we have and .
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 , , , and is shown in Figure 2, where .
In order to verify the existence condition obtained, numerical simulation of the original system equations (3)-(4) will be given. For fixed , , , , , , and , a double grazing period-1 motion is obtained with , as shown in Figure 3. Figures 3(a) and 3(b) are the phase portrait and time history of oscillator . Figure 3(a) shows that oscillator collides with constraints and with zero velocity. It illustrates the validity of the condition for existence of grazing periodic motion.
3. Poincaré Mapping of Double Grazing Periodic Motion in Three-Degree-of-Freedom Impact Oscillator
3.1. Definitions in Geometric Structure
In terms of the state vector , it follows thatwhere mod denotes the phase of the excitation and equals the acceleration of the oscillator as a function of , , and . Denote the corresponding flow function by .
Suppose that the movement of the oscillator is limited by symmetrical rigid constraints placed at corresponding to state-space discontinuity surfaces and (see Figure 4):such that and during the motion of the oscillator and . Let and ; it follows that .
The transversal intersections of a state-space trajectory with and are at points and , respectively, such that and , .
Suppose that trajectory with at and at ; it follows that
Indeed, if , a collision occurs between the oscillator and right constraint , and if , a collision occurs between the oscillator and left constraint . We model the collision as an instantaneous impact with a characteristic coefficient of restitution ; that is, the state immediately after impact relates to that immediately before impact according to the jump map:
In contrast, points of grazing contact of a state-space trajectory with and correspond to points and , respectively, such that
Introduce Poincaré surfaces and :such that and correspond to a transversal intersection of a state-space trajectory with surfaces and , respectively. Since , it follows that is a local minimum in the value of (i.e., the distance to ) along state-space trajectories of the system. By transversality, it follows that nearby trajectories achieve locally unique points of intersection with corresponding to local minima in the value of . By the same as above, since , the trajectories nearby the grazing one achieve locally unique points of intersection with corresponding to local maxima in the value of .
Now suppose that and are symmetrical grazing contact points of a periodic trajectory of the system with and , such that and . Ignoring the effects of the jump map and using the transversality, it is possible to define local Poincaré maps and , such that and .
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 . Here, two discontinuity-mappings and are introduced on a neighborhood of points and , such that surface is invariant under (i.e., ) and surface is invariant under (i.e., ); therefore, Poincaré mapping associated with surface 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 and are obtained as follows:where and .
3.3. The Poincaré Mapping
Expanding the smooth mapping near , it follows that
Expanding event function near , it follows that
Since satisfies , such that (29) is written asconsequently,
From the above analysis, the mapping is written as
4. 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 , which satisfies , ifthat is,that is,it means an impact point impacts discontinuity surface again and the impact will be perpetuated, which results in a large stretching in a direction given by the image of vector under Jacobians and , and the trajectory is unstable.
According to the above analysismeaning an impact point impacts discontinuity surface again and the impact will be perpetuated; the grazing periodic trajectory is unstable.
By just changingtofor , it happens that an impact is followed by nonimpacting for some iterations but eventually impacts discontinuity surface again and the impact will be perpetuated, and the grazing periodic trajectory is unstable. Thus, if and for any , , stability is lost.
In the same way, iffor any , , it happens that an impact is followed by nonimpacting for some iterations but eventually impacts discontinuity surface again and the impact will be perpetuated, and the grazing periodic trajectory is unstable.
Moreover, iffor any , , it happens that an impact is followed by nonimpacting for some iterations but eventually impacts discontinuity surfaces and 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 ) correspond tofor all and , respectively. Let express the codimension-two grazing bifurcation points; can be written in four cases; that is,for .
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.
Since , it follows thatThus,
According to the relevant definitions in Section 3 and the implicit function theorem, it is straightforward to show thatwhere is unit matrix.
From the semigroup property of the smooth flowit follows by differentiation with respect to and evaluation at , , and thatConsequently,
In the same way, we obtain that
Moreover, it is straightforward to show thatIn the same way, we obtain thatwhere integer superscripts denote matrix powers.
Since , and according to (53), it follows that
In the same way, we can obtain the simplified expressions of the other three cases as follows:
5. Numerical Simulations
Taking the sencond case derived in the above section as an example, the codimension-two grazing bifurcation points are corresponding to . The curve of points in the parameters space corresponding to the existence of a grazing periodic trajectory with , , , and is shown in Figure 5. Here, points for which 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 , , and , taking , along the grazing curve shown in Figure 5, then existence condition of double grazing periodic motion and 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.
When , using with the same parameters , , , and , the curve of points in the parameters space corresponding to the existence of a grazing periodic trajectory is shown in Figure 7. Here, points for which are indicated by the asterisk ().
In order to make the bifurcation behavior in the vicinity of the codimension-two grazing bifurcation point clear, the unfolding diagram is presented in the neighborhood of codimension-two grazing bifurcation point corresponding to by a large number of numerical simulations. For fixed , , , , , , and , it is known from above analysis that a codimension-two grazing bifurcation point corresponds to and . Changing values of parameters and near , . Existence regions of different types of periodic-impact motions of the system, that is, the unfolding diagram, are presented in the parameter plane plotted in Figure 8, where the dash curve represents double grazing motion. As shown in Figure 8, the dynamical behavior in the vicinity of the codimension-two grazing bifurcation point is complex. The parameter space is divided into regions characterized by different types of motions.
For fixed , sample phase portraits of each region obtained through perturbing are shown in Figures 9(a)–9(k). Conveniently, we introduce the symbol to describe different types of periodic-impact motion of the system, where represents the number of the exciting force period in the motion period and and mean the number of impacts occurring at right and left constraints, respectively, when or ; that is, and mean the number of grazings occurring at right and left constraints, respectively, while or ; that is, and symbolize no impact occurring at the right and left constraints. Decreasing from to , the phase portraits are in region (a), in region (b), in region (c), in region (d), in region (e), in region (f), in region (g), in region (h), in region (i), in region (j), and in region (k), respectively.
For fixed , sample phase portraits of each region obtained through perturbing are shown in Figures 9(l)–9(v). Decreasing from to , the phase portraits are in region (l), in region (m), in region (n), in region (o), in region (p), in region (q), in region (r), in region (s), in region (t), in region (u), and in region (v), respectively.
It is difficult to analyze codimension-two bifurcation of grazing periodic motion for multidegree-of-freedom vibroimpact system theoretically, especially with symmetrical constraints. So far, there are some researches that explored the codimension-two bifurcation of grazing periodic motion for dynamical system with unilateral constraint. As far as we know, the work on codimension-two bifurcation of grazing periodic motion of vibratory system with bilateral constraints is not seen. Focusing on the above unsolved problem, this paper presents the existence condition of double grazing motion and obtains its Poincaré mapping using the classical discontinuity-mapping approach. Based on the results, the condition of codimension-two grazing bifurcation is established and simplified. Therefore, it is convenient to find the codimension-two bifurcation points by using the obtained formulae. By numerical simulation, the complex and rich dynamical behaviors near the codimension-two grazing bifurcation point are presented. For example, there exist period-10 impact-8 motion, period-6 impact-4 motion, period-4 impact-4 motion, and so forth in the vicinity of the codimension-two grazing bifurcation point. The detailed dynamical features near critical point of grazing codimension-two bifurcation will be analyzed further in the future works.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the projects sponsored by the National Natural Science Foundation of China (nos. 10972059 and 11372077), the Guangxi Natural Science Foundation (nos. 2013GXNSFAA019017 and 2014GXNSFBA118024), and the Scientific Research Foundation of Guangxi University (no. XBZ120251).
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