Abstract
The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.
1. Introduction
Grazing bifurcation, one type of discontinuity-induced bifurcations, has been extensively studied in vibroimpact system as it has complex dynamics and is widely encountered in many engineering examples. On analysis of the dynamics near grazing in a general class of impact oscillator systems, a classical concept of analysis is the so-called discontinuity-mapping approach initially conceived of by Nordmark [1, 2] (see [3, 4] for an overview). The analysis is usually carried out by finding an appropriate local map describing the system dynamics in neighborhood of the grazing event. The local map can then be combined with an analytic Poincaré map to give the so-called grazing normal form whose dynamics can be shown to be topologically equivalent to those of the underlying flow. The grazing normal form derived by the discontinuity-mapping approach is used to analyze the local dynamics in the vicinity of a grazing trajectory. As shown in [5–8], the normal form map of the rigid impact oscillator contains a square-root term causing a singularity in the first derivative, which results in an abrupt loss of the stability. And different bifurcation scenarios associated with switching between impacting motions and nonimpacting motions near grazing were also described. In particular, the discontinuity-mapping approach was used by Fredriksson and Nordmark [9] to establish conditions for the persistence or disappearance of a local attractor in the vicinity of a grazing periodic trajectory. In addition, some conditions for the persistence of a local attractor in the immediate vicinity of quasiperiodic grazing trajectories in an impacting dynamical system were formulated by Thota and Dankowicz [10]. When the stability conditions of grazing bifurcation are degenerate, the codimension-two would occur, which is always a hot topic. Kowalczyk et al. [11] proposed a strategy for classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems and studied their nonsmooth transitions. Foale [12] analyzed the results of a special codimension-two grazing bifurcation in a single-degree-of-freedom impact oscillator by using the impact surface as a Poincaré section. In addition, the study also shows that the bifurcation of the saddle node bifurcation and the flip bifurcation can meet at a certain codimension-two grazing points in the parameter plane. Thota et al. [13] studied the distribution of codimension-two grazing bifurcation point according to the discontinuous mapping in the single-degree-of-freedom collision oscillator and discussed the possible dynamic characteristics of the system response near this bifurcation point. Xu et al. [14] studied the codimension-two grazing bifurcation of n-degree-of-freedom vibrators with bilateral constraints and obtained and simplified the existence conditions of codimension-two grazing bifurcation. Similar phenomena about codimension-two grazing bifurcation can also be found in Refs. [15, 16]. Yin et al. [17] discussed the important role of some degenerated grazing bifurcation points in the transition between saddle node bifurcation and period doubling bifurcation. Dankowicz and Zhao [18, 19] studied the grazing bifurcations of the codimension-one and the codimension-two of a class of impact microactuators.
The loss of local attractors near grazing bifurcation may arise catastrophic changes of system response and lead to codimension-two or more complicated bifurcation; therefore, controlling near-grazing dynamics becomes necessary and significant. Dankowicz and Jerrelind [20] used the linear feedback control method to control the grazing bifurcation of the piece smooth dynamic system, so that the system has local attractors near the grazing orbit. Dankowicz and Svahn [21] presented for the existence of event-driven control strategies that guarantee the local persistence of system attractors with at most low-velocity contact in vibro impacting oscillators. Misra and Dankowicz [22] developed a rigorous control paradigm for regulating the near-grazing bifurcation behavior of limit cycles in piecewise-smooth dynamical systems. Yin et al. [23] analyzed the stability for near-grazing period-one impact motion to suppress grazing-induced instabilities. The bounded eigenvalues are further confined to the unit circle, and the continuous transition between the nonimpact motion and the controlled impact motion is obtained. Xu et al. [24] discussed the control problem of near-grazing dynamics in a two-degree-of-freedom vibroimpact system with a clearance.
Based on the concept of controlling the persistence of local attractors near the grazing trajectory in impact oscillator with unilateral constraints mentioned in [20–24], this paper aims to control the stability of grazing bifurcation or control the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Compared with impact oscillator with unilateral constraint, the instability problems near grazing trajectory become more complex for impact oscillator with symmetrical constraints as mentioned in [17]. The stability of double grazing motion bifurcation in the system is lost in three ways, and the existence conditions of the codimension-two grazing bifurcation occur in four different cases accordingly. For this complex unstable problem, analytic expressions of stability criterion are obtained in this paper. Based on the stability criterion, the stability control strategy of the persistence of near-grazing attractors is proposed. Furthermore, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. This paper is organized as follows. In Section 2, a two-degree-of-freedom vibroimpact system with symmetrical constraints is introduced. In Section 3, near-grazing bifurcation dynamics are analyzed. In Section 4, the discrete-in-time feedback control method is designed to maintain the persistence of the local attractor of double grazing period motion. In Section 5, numerical simulation is used to verify the feasibility of the control method. Finally, the conclusion is given in Section 6.
2. Mechanical Model and Double Grazing Periodic Motion
Figure 1 shows the schematic model of a two-degree-of-freedom impact oscillator with a clearance. Masses and are connected to linear viscous dampers and by linear springs with stiffness and , respectively. The harmonic forces of the amplitude and are applied to the masses and , respectively, and the harmonic force is applied only to the mass in the horizontal direction. The mass moves between the symmetrical rigid stops and . When the mass strikes rigid stop or , the motion becomes a nonlinear motion and the impact is described by the recovery factor . Assuming that the damping in the mechanical model is the Rayleigh type proportional damping, it can be known that .
The governing equation is described by
When , the collision occurs. At this time, the collision equation is as follows:where and represent the first and second derivatives of with respect to time , respectively. indicates the instantaneous speed at which the mass approaches the rigid stop or . indicates the instantaneous speed at which the mass leaves the rigid stop or .
Introduce the nondimensional quantities as follows:
According to equations (1)–(3), the system can be transformed into nondimensional forms.where and represent the first and second derivatives of with respect to the nondimensional time , respectively.
Suppose be the canonical modal matrix of equation (4), and coordinate transformation of equation (4). Let 4.
(4)where and represent the eigenfrequencies of the system, , , is the unit matrix, and are diagonal matrices, , , and . The general solution of equation (4) is given bywhere , represents the time at which the mass collides with the rigid stop or ; we set the mass to collide with the rigid stop when . is an element of the canonical modal matrix , , and . The initial conditions and modal parameters of the system determine the integral constants and . and are amplitude parameters, and the expression is given by
According to the initial conditions and periodic conditions of the double grazing periodic-n motion, the existence conditions of two-degree-of-freedom impact oscillator double grazing periodic-n motion is as follows:where and .
In addition, in order to ensure that the impacting cycle of the mass does not adhere to rigid stop, the acceleration of the mass satisfies and the acceleration of the mass satisfies .
3. Near-Grazing Dynamics
3.1. The Stability Criterion of Double Grazing Bifurcation
Let be a state vector, such that , and it follows thatwhere represents the acceleration of the oscillator as a function of . Define as the local flow function associated with . Suppose that the movement of the oscillator is limited by symmetrical rigid constraints placed at corresponding to state space discontinuity surfaces and , where
The oscillator moves between two rigid stops when and . When or , the oscillator collides with the rigid stop. In addition, let , .
When the oscillator collides with the constraint, we establish a function with a characteristic restitution coefficient to represent the jump map, i.e., represents the instantaneous state after the collision and before the collision, where
The state space trajectory and the grazing contact points of and are, respectively, corresponding to points and , so that
Choose a constant phase angle as a Poincaré section, and it has the following form: , where , . The modulus of is , . Since the periodic trajectory of the system is symmetrical, we set another Poincaré section as .
With as the bifurcation parameter, we define the flow maps and by the evolution of the smooth flow on the Poincaré sections and ; the expressions of and are as follows:where and .where and .
The critical bifurcation value can be obtained from expression (9). Since the vector field and the smooth flow function do not contain the bifurcation parameter , the values of the matrices .
We use the discontinuous mapping method introduced by Nordmark to analyze the system; the two discontinuous maps and are introduced into the neighborhood of points and such that the surface is invariant under ,i.e., . The same surface is invariant under , i.e., . According to the discontinuous mapping method, the discontinuous mapping of and is expressed bywherewhere
According to (13)–(17), the composite maps and are written as
We will discuss the stability of double grazing bifurcation using the Poincaré map . Starting from the vicinity of the grazing point, whether it is at the impact side or the nonimpact side, if it is still close to the grazing point after the iterative mapping , then the local attractor near the grazing trajectory exists; we refer to such scenario as a continuous grazing bifurcation or we say that the grazing bifurcation is stable. Otherwise, we refer to scenario as a discontinuous grazing bifurcation.
When point x starts from the impact side near the grazing point , . Here, we can think of the approximation as . Similarly, we consider the approximation as .
Ifit means is located at the nonimpact side near the grazing point .
After the iteration of the mapping , ifthen,
It means the 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 the vector under the Jacobian and . Therefore, the local attractor near the grazing trajectory does not exist. According to the analysis, the stability criterion of grazing bifurcation under which a local attractor persists near a grazing trajectory is formulated as follows:andfor all positive integer .
According the same method, another stability criterion of grazing bifurcation under which a local attractor persists near a grazing trajectory is formulated as follows:andfor all positive integer .
3.2. Codimension-Two Grazing Bifurcation
The local attractor near grazing trajectory is lost in there ways.
Case 1. Ifi.e., , andi.e., , then it means that the point from the impact side near the grazing point will impact the discontinuous surface again after the iteration of the mapping and the impact will continue, which results in a large stretching in a direction given by the image of the vector under the Jacobian and ; therefore, the local near-grazing attractor will lose, where discontinuous grazing bifurcation occurs.
If the impact point is followed by nonimpacting for some iterations but eventually impacts discontinuous surface , the impact will be perpetuated which lead to the instability of grazing bifurcation or the loss of near-grazing attrators.
Therefore,for any , the stability of grazing bifurcation will be lost, where and are positive integers.
Case 2. When the point x satisfying is from the impact side near the grazing point , if and , it means that the impact point will impact with the discontinuous surface again and the impact will continue, and the grazing bifurcation is discontinuous.
If the impact point is followed by nonimpacting for some iterations but eventually impacts discontinuous surface , the impact will be perpetuated. Finally, the stability of grazing bifurcation will also be lost.
That is,for any , the stability of grazing bifurcation will be lost.
Case 3. When the point x satisfying is from the impact side near the grazing point , if and , this means that the impact point will impact with the discontinuous surfaces and again and the impact will continue, and the grazing bifurcation is discontinuous. If the impact point is followed by non-impacting for some iterations but eventually impacts discontinuous surface and the impact will be perpetuated. Finally, the stability of grazing bifurcation will also be lost.
That is,for any , the stability of grazing bifurcation will be lost.
According to the above analysis, the conditions of codimension-two grazing bifurcation are obtained as follows (the definition of such points is seen in Ref. [10]):where .
4. Controlling the Persistence of Near-Grazing Attractors
As the lose of stability for grazing bifurcation may arise catastrophic changes of system response and codimension-two even more complicated bifurcation, it is necessary to control the stability of grazing bifurcation by controlling the persistence of the local near-grazing attractor. We use discrete-in-time feedback control to stabilize the grazing bifurcation. Two constant phase angles are defined as Poincaré sections, and two discrete-in-time feedback controllers on Poincaré sections are designed. After that, we obtain a new compound map and then control the stability of the grazing bifurcation by controlling the parameters on the controller.
We define the other two constant phase angles as Poincaré sections and , where and .
According to and , we define the mappings and , and the resulting expansion is as follows:where is the fixed point of the grazing orbit on the Poincaré section , , and .
In the same way, according to and , we define the mappings and , and the expansion is as follows:where is the fixed point of the grazing orbit on the Poincaré section , , and .
Discrete-in-time feedback controllers are designed on the Poincaré sections and , respectively, as follows:
We can calculate the expressions of maps and which have the following forms:where
The combining the map and , obtain a controlled map as , expansion as follows:where .
Similarly, combining the map with the map , the map under control , is obtained and its expansion has the following forms:where .
We can calculate the expressions of as follows:where and the symbol Δ refers to a nontrivial coefficient.
Combining (39) and (15), we construct a composite map of ; the expressions are as follows:where .
Similarly, we can combine (40) and (17) to construct a composite map of ; the expression is as follows:where .
According to the analysis in Section 3, there are three cases in which the grazing bifurcation is unstable, where discontinuous grazing bifurcation occurs. For Case 1, the impact point will impact with the discontinuous surface again after some iterations and the impact will be perpetuated. Based on the stability criteria (24) and (25), we control on the controller to ensure that and , for positive integer , such that the grazing bifurcation will be continuous.
For Case 2, the impact point will impact with the discontinuous surface again after some iterations and the impact will be perpetuated. Based on the stability criteria (26) and (27), we control on the controller so that and for positive integer . The grazing bifurcation will be continuous.
For Case 3, the impact point will impact with the discontinuous surfaces and again, and after some iterations, the impact will be perpetuated. Based on the stability criteria (24) and (25) or (26) and (27), the grazing bifurcation will be continuous by controlling parameters to ensure the local attractor near grazing bifurcation persists.
For the codimension-two grazing bifurcation point, the persistence of near-grazing attractors is also controlled by controlling based on the stability criteria (24) and (25) or (26) and (27). For example, if , we might let and by controlling .
If , we might let and by controlling .
5. Numerical Experiments
For this system, we take suitable parameters as an example. is supposed as the bifurcation parameter; the critical value for grazing bifurcation is obtained from formula (9). According to the analysis of near-grazing dynamics in Section 3, under the set of fixed parameters, and . The impact point will impact with the discontinuous surfaces again and the impact will be perpetuated, which satisfies the instability conditions in Case 1. The bifurcation diagram as shown in Figure 2(a) is obtained through numerical simulation by discontinuity maps (19) and (20). It is clear that the grazing bifurcation is discontinuous because the discontinuous transition occurs at the bifurcation point . In addition, we take as the Poincaré section; the bifurcation diagram as shown in Figure 2(b) is obtained through direct numerical simulation by (4) and (5), and it is clear that the discontinuous jump also occurs at the grazing bifurcation point .
(a)
(b)
(c)
(d)
(e)
(f)
We use the control method in Section 4 to stabilize the unstable grazing bifurcation phenomena. When the parameters of and are fixed, the ranges of parameters and are calculated according to control strategy, which is named two-dimensional control region of and . Here, we select as the fixed parameters; a two-dimensional control region graph of and is obtained as shown in Figure 3(a), where is axis, is axis, and the blue region is the region of control parameters for ensuring the stability of the grazing bifurcation.
(a)
(b)
(c)
We select to control the system and draw Figure 4(a) through the discontinuity maps (40) and (41). At the same time, we take as the Poincaré section and obtain Figure 4(b) by direct numerical simulation. As shown in Figures 4(a) and 4(b), there exists the local attractor near grazing bifurcation.
(a)
(b)
(c)
(d)
(e)
(f)
Take the set of parameters , as an example. , is obtained from equation (9). Without control, and . are obtained under the above fixed parameters. It satisfies the instability conditions in case 1 according to expression (30). The bifurcation diagram as shown in Figure 2(c) is obtained through numerical simulation by discontinuity maps (19) and (20). It is clear that the grazing bifurcation is discontinuous because the discontinuous transition occurs at the bifurcation point . In addition, we take as the Poincaré section, and the bifurcation diagram as shown in Figure 2(d) is obtained through direct numerical simulation by (4) and (5); it is clear that the discontinuous jump also occurs at the grazing bifurcation point .
Based on the stability criterion mentioned in Section 4, we select as the fixed parameters; a two-dimensional control region graph of and is obtained as shown in Figure 3(b), where is axis, is axis, and the blue region is the region of control parameters for ensuring the stability of the grazing bifurcation. Then, we select to control the system, and the bifurcation diagram is obtained as shown in Figure 4(c) by the discontinuity maps (40) and (41). In addition, we take as the Poincaré section and obtain Figure 4(d) by direct numerical simulation. As shown in Figures 4(c) and 4(d), there exists the local attractor near grazing bifurcation.
Fixed parameters . the critical value, such that . It corresponds to the codimension-two grazing bifurcation. We obtain the bifurcation diagram Figure 2(e) through numerical simulation by the discontinuity maps (19) and (20). As shown in Figure 2(e), the system is in a complex chaotic state near the grazing point. Then, we take as the Poincaré section and obtain Figure 2(f) by direct numerical simulation.
Based on the control strategy mentioned in Section 4, we select as the fixed parameters; a two-dimensional control region graph of and is obtained as shown in Figure 3(c), where is axis, is axis, and the blue region is the region of control parameters for suppressing the chaos. Then, we select to control the system, and the bifurcation diagram is obtained as shown in Figure 4(e) by the discontinuity maps (40) and (41). In addition, we take as the Poincaré section and obtain Figure 4(f) by direct numerical simulation.
Comparing Figure 2 with Figure 4, it is shown that the results obtained by mappings are in good agreement with the direct numerical simulation, and the control effect is further verified.
6. Conclusions
The discontinuous grazing bifurcation is often accompanied by a jump phenomenon. In addition, complex chaotic regions occur near the codimension-two bifurcation points. How to avoid the dramatic change of system response caused by jump phenomenon of grazing bifurcation or the codimension-two bifurcation point is an urgent requirement for the control mechanism of discontinuous grazing bifurcation. In this paper, a discrete-in-time feedback control strategy is used to control the near-grazing dynamics of double grazing period motion in a two-degree-of-freedom vibroimpact system with symmetrical constraints. Compared to unilateral constrained systems, the stability criterion of double grazing period motion becomes more complex. Based on the stability criterion, the control strategy is designed to control the near-grazing dynamics and the two parameters’ control region is obtained. For the case of discontinuous grazing bifurcation, we take two jumping phenomena (jumping from nonimpact periodic motion to and impact periodic motion) as examples to stabilize grazing bifurcation by controlling the parameters on the controller. In addition, the chaos dynamics near codimension-two grazing bifurcation point is controlled by using this strategy. Finally, the feasibility of the control strategy is illustrated by comparing the numerical simulation of composite mapping with the direct numerical simulation of Poincare section.
Data Availability
The simulation data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the project sponsored by the National Natural Science Foundation of China (nos. 11602059 and 11872154) and Guangxi Natural Science Foundation (no. 2018JJA110046).