# Bifurcation Characteristics of Fundamental and Subharmonic Impact Motions of a Mechanical Vibration System with Motion Limiting Constraints on a Two-Parameter Plane

Authors: Shijun Wang; Guanwei Luo
Journal: Shock and Vibration
 (2021)

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## Abstract

A two-degree-of-freedom periodically forced system with multiple gaps and rigid constraints is studied. Multiple types of impact vibrations occur at each rigid constraint and interact with each other, which results in the emergence of some complex transitions in the system. Through the cosimulation of the key parameters gap valueδ between the two masses and the excitation force frequency ω, the types, existence areas, and bifurcation regularities of the periodic and subharmonic motions can be obtained on the (ω, δ)-parameter plane. In the corresponding three-dimensional surface diagram of the maximum impact velocity, the distribution law of the maximum impact velocity at each constraint can be obtained. The transition laws of fundamental impact motions in the low-frequency parameter domain are studied, and two types of transition regions in the transitions of adjacent fundamental impact motions are found: tongue-like regions and hysteresis regions. Moreover, these two types of transition regions show some atypical partitioning and deformation due to the combined effects of impact vibrations at each constraint. By combining the two-parameter plane diagram and the three-dimensional surface diagram, the effect of changing the gap values between each mass and the fixed constraint and the damping coefficient ζ on the dynamic characteristics of the system is studied. Combining the existence areas of periodic motions and the distribution of maximum impact velocity can provide guidance for the reasonable selection of system parameters.


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## Body

## 1. Introduction

In the mechanical system, there are inevitably gaps and constraints between components and between the components and the fixed boundaries due to the needs of installation and the limitations of machining accuracy. The existence of gaps and constraints makes the mechanical system witness impact vibration when working, which will increase the noise level of the mechanical system and accelerate the wear of components. Due to the existence of constraints and gaps, the system becomes a typical nonsmooth system showing rich dynamic characteristics.People want to reveal the inherent laws of impact vibration systems and use this to guide the effective use of impact vibration characteristics and regularize their potential hazards. A large number of researches in this field can be found in available monographs. Shaw and Holmes [1] considered a one-degree-of-freedom nonlinear oscillator and found period-doubling bifurcation, through which the system can lead to chaos. Aidanpää and Gupta [2] found the characteristics of period doubling, jumping, chaos, and so forth in the two-degree-of-freedom forced vibration system. Peterka and Vacik [3] discussed the law of the transition from periodic motion to chaos and characterized the periodic-impact motion by p/n. As a strong nonlinear factor, dry friction often exists in shock vibration systems, and the coupling of such strong nonlinear factors as gap, constraint, and dry friction was considered in [4–7]. Multi-degree-of-freedom piecewise smooth mechanical system is a multiparameter high-dimensional nonlinear system; the codimension-two bifurcations have been studied in [8–13]. Some experiments were also used to verify the results of numerical simulation of impact vibration systems in [14–16].Due to the existence of constraints and gaps, the impact vibration system has bifurcation types that are not available in smooth systems, such as grazing bifurcation. Nordmark [17] studied the singularities caused by grazing impact and found that the motion after the grazing bifurcation may be nonperiodic. Whiston [18] used the singularity theory to study the shredding mechanism of the globally stable manifold caused by the discontinuity of the impact map. Foale and Bishop [19, 20] defined different types of grazing bifurcation. The discontinuous geometric method of impact mapping was used to further study the singularity and grazing bifurcation of piecewise smooth dynamic systems, as reported in [13, 21–27].Complex chattering behavior has appeared in many impact vibration systems, which has attracted widespread attention from scholars. Chattering is a phenomenon in which there are an infinite number of impacts within one excitation force cycle [28]. Toulemonde and Gontier [29] analyzed the phenomenon of sticking periodic motions of impact oscillators and identified rising bifurcation. Wagg [30] discussed the chattering and sticking behavior that occurs under low-frequency excitation force conditions. Nordmark and Piiroinen [31] introduced the local discontinuous map based on the impact chatting accumulation and used the map to analyze the stability of the limit cycle with chatting.In the previous studies of nonlinear dynamics, the periodic motion mode and bifurcation characteristics of the system were mainly studied through single-parameter bifurcation diagrams. Then, most dynamic systems belong to multiparameter systems, and many parameters affect the dynamic characteristics of the system together. Therefore, how to get the existence areas of the periodic and subharmonic motions and the bifurcation characteristics through the double-parameter cosimulation has become a hot spot for scholars. In [12, 22, 32, 33], the double-parameter bifurcations of one-degree-of-freedom impact vibration system were studied. Luo [34, 35] studied the mode types and distribution areas of the periodic motions of the two-degree-of-freedom impact vibration system on the double-parameter plane.The aim of this paper is to get the types, existence areas, and bifurcation regularities of the periodic and subharmonic motions of the two-degree-of-freedom impact vibration system with multiple constraints and gaps through double-parameter cosimulation. Discover the reasonable matching relationship between the vibroimpact dynamics of the system and the parameters. The paper is organized as follows. In Section2, the physical model of a two-degree-of-freedom periodically forced system with multiple gaps and multiple rigid constraints is introduced. In Section 3, through the double-parameter numerical simulation, the types, existence areas, and bifurcation regularities of periodic motions are obtained on the ω,δ-parameter plane, and the distribution of the maximum impact velocity at each constraint is studied. In Section 4, the effects of changing the gaps δ1=δ2 between the two masses and the foundation supports and the damping coefficient ζ on the system dynamics are studied. Conclusions are obtained in Section 5.


## 2. Mechanical Model

A two-degree-of-freedom periodically forced system with multiple gaps and rigid constraints is depicted in Figure1, where the masses Mi (i = 1, 2) displacements are represented by Xi (i = 1, 2). The masses Mi (i = 1, 2) are connected to the supporting foundation by linear springs Ki (i = 1, 2) and linear dampers Ci (i = 1, 2). The masses Mi (i = 1, 2) are excited by harmonic force of amplitude Pi (i = 1, 2), frequency Ω, and phase angle τ. The clearance between the mass M1 and the impact surface A1 is B1. The clearance between the mass M2 and the impact surface A2 is B2. The clearance between M1 and M2 is B3. The stop is assumed to be rigid and the coefficient of recovery after a rigid collision is R. The speed before and after the impact of mass Mi (i = 1, 2) is represented by X˙i− and X˙i+ (i = 1, 2), respectively.Figure 1
Physical model of a two-degree-of-freedom periodically forced system with multiple rigid constraints.In the study, we introduced the following dimensionless parameters:(1)t=TK1M1,xi=XiKiP1+P2,μm=M2M2+M1,μk=K2K2+K1,μc=C2C2+C1,ζ=C12K1M1,δj=BjK1P1+P2,f=P2P2+P1,i=1,2,j=1,2,3.It can be found that some parameters in the system have a certain value range through dimensionless parameters, such asμm∈0,1,μk∈0,1,μc∈0,1,f∈0,1. When the system has no impact on each rigid constrains, the motion of the impact system could be described by the nondimensional equations:(2)x¨1+2ζx˙1+x1=1−fsinωt+τμm1−μmx¨2+2ζμc1−μcx˙2+μk1−μkx2=fsinωt+τ,x1>−δ1,x2<δ2,x1−x2<δ3.When the massMi (i = 1, 2) has an impact with the fixed constraints, the velocity of mass Mi (i = 1, 2) is changed immediately.(3)x˙1+=−Rx˙1−,x1=−δ1,x˙2+=−Rx˙2−,x2=δ2.The instantaneous velocity of massM1 and mass M2 before and after collision can be determined by momentum conservation.(4)m1−mx˙1++x˙2+=m1−mx˙1−+x˙2−,R=x˙2+−x˙1+x˙1−−x˙2−,x1−x2=δ3,where x˙i− and x˙i+ (i=1,2) represent the velocities before and after the impact, respectively.For the periodically forced impact system, the massMi (i = 1, 2) may stay on the constraint surfaces with a speed of zero for a period of time under the resultant force and this phenomenon is called sticking. The impact vibration system in Figure 1 contains multiple gaps and rigid constraints, so the system may exhibit sticking phenomenon on multiple rigid constraint surfaces in a certain motion time section. From the conditions of the occurrence of the sticking phenomenon, the sticking conditions of the mass Mi (i = 1, 2) on each rigid constraint surfaces can be analyzed:(1)
The conditions for massM1 sticking to the surface of constraint A1 are x1=−δ1 and x˙1=0, and the resultant force of the harmonic excitation force, spring, and damping force on the mass M1 is satisfied:(5)F1t=1−fsinωt+τ−x1−2ζx˙1≤0.UntilF1t changes direction, the sticking motion is ended.(2)
The conditions for massM2 sticking to the surface of constraint A2 are x2=δ2 and x˙2=0, and the resultant force of the harmonic excitation force, spring, and damping force on the mass M2 is satisfied:(6)F2t=fsinωt+τ−2ζμc1−μcx˙2−μk1−μkx2≥0.UntilF2t changes direction, the sticking motion is ended.(3)
The conditions for massM1 and mass M2 sticking to each other at constraint A12 are x1−x2=δ3 and x˙1−x˙2=0. Whether there is sticking between the two masses depends on the interaction force between the two masses; here, the interaction force is set to N(t). At this time, the resultant forces of mass M1 and mass M2 can be expressed, respectively, as(7)f1t=1−fsinωt+τ−2ζx˙1−x1−Nt,f2t=fsinωt+τ−2ζμc1−μcx˙2−μk1−μkx2+Nt.According tof1t=1−μm/μmf2t, N(t) can be obtained:(8)Nt=μm1−fsinωt+τ−2ζx˙1−x1−1−μm⋅fsinωt+τ−2ζμc1−μcx˙2−μk1−μkx2.WhenNt>0, the two masses stick together and move synchronously. Until the interaction force disappears, that is, Nt=0, the sticking motion is ended.Here, the symbolp/n is used to describe the types of periodic-impact motions at each rigid constraint, where n represents the ratio of the vibration period of the system to the period of the excitation force (n = 1, 2, 3, ...), and p represents the number of impacts at each rigid constraint (p = 0, 1, 2, 3, ...). When n = 1, that is, when the vibration period of the system and the excitation force period are equal, there are fundamental impact motions p/1 (p ≥ 1). When n ≥ 2 and p ≥ 1, there are subharmonic impact motions p/n (n ≥ 2, p ≥ 1).(9)σn=x1,x˙1,x2,x˙2,t∈R4×T|x1=x1min,modt=2π/ωσA1=x1,x˙1,x2,x˙2,t∈R4×T|x1=−δ1,x˙1=x˙1−,σA12=x1,x˙1,x2,x˙2,t∈R4×T|x1−x2=δ3,x˙1=x˙1−,x˙2=x˙2−,σA2=x1,x˙1,x2,x˙2,t∈R4×T|x2=δ2,x˙2=x˙2−.The Poincaré sectionσn is selected by selecting the minimum displacement of the mass M1 in the period of the excitation force and it could determine the number. The Poincaré sections σA1, σA12, and σA2 could determine the number of p on each of the rigid constraints, respectively.Generally, the impact mapping of the system corresponding to the sectionsσA1, σA12, and σA2 can be expressed as(10)Xi+1=fXi,μ,Where, corresponding to sections σA1 and σA12, Xi=x˙1+i,x2i,x˙2+i,τiT and Xi+1=x˙1+i+1,x2i+1,x˙2+i+1,τi+1T; corresponding to section σA2, Xi=x˙1+i,x1i,x˙2+i,τiT and Xi+1=x˙1+i+1,x1i+1,x˙2+i+1,τi+1T; X∈R4 and μ are the system parameters, where μ∈Rm, m = 8.


## 3. Dynamical Characteristics of the Two-Degree-of-Freedom Periodically Forced System with Multiple Rigid Constraints

In this section, we mainly study the mode types, existence areas, and bifurcation regularities of the periodic motions on the two-parameter plane composed of the exciting force frequencyω and the gap value δ, as well as the corresponding distribution law of the maximum impact velocity at each constraint. In model 1, the gap can be divided into two types, one is the gap between the mass Mi (i = 1, 2) and the fixed constraints and the other is the gap between M1 and M2. Consider here that the gaps δ1 and δ2 are taken to have fixed values, and δ3 is taken as a variable parameter and recorded as δ. Here μm=0.5, μk=0.5, μc=0.5, ζ=0.1, R=0.8, and δ1=δ2=1.0 are selected as the basis parameters. The exciting force frequency ω and the gap δ between the two masses are set as bifurcation parameters. Through double-parameter numerical simulation, the mode types and occurrence areas of periodic motions can be get on the ω,δ-parameter plane as depicted in Figure 2. The maximum impact velocity of the system on each constraint surface can be obtained in the ω,δ,x˙max-three-dimensional surface program as depicted in Figure 3. On the ω,δ-parameter plane, different types of periodic motions are represented by different colors and marked with p/n. CIS/1 represents the sticking motions. In addition, the unmarked dark gray areas are mainly quasi-periodic motions, chaotic motions, and unidentified periodic-impact motions (mainly including n or p is too large and existence areas are extremely small). Moreover, GB, Gr, Gb, SN, S, and PB represent the grazing, real grazing, bare grazing, saddle-node, sliding, and period doubling bifurcation, respectively.Figure 2
Periodic motions on theω,δ-parameter plane associated with each constraint. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 3
Three-dimensional surface diagrams of the maximum impact velocity of the system associated with each constraint. (a) ConstraintA1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure2 shows the existence types and distribution areas of the fundamental impact motions and subharmonic motions on the ω,δ-parameter plane associated with each constraint. The corresponding three-dimensional surface diagrams of the maximum impact velocity can be observed in Figure 3. As shown in Figure 2(a), on the ω,δ-parameter plane associated with constraint A1, in the frequency domain of ω≤1.4 the system mainly exhibits 1/1 motion, and there is a small amount of 2/1 motion in the lower frequency domain. In the frequency domain of ω>1.4 the system mainly exhibits impactless motion 0/1. Subharmonic motions mainly exist in the middle frequency domain, mainly 2/2, 1/3, 3/3, 4/4, and 4/4 motions. The corresponding three-dimensional surface diagrams of the maximum impact velocity are shown in Figure 3(a); the peaks of the maximum impact velocity mainly exist in the middle frequency domain where the motion form is mainly 1/1. In the high-frequency domain, the peaks of the maximum impact velocity also exist in the existence domain which mainly contains subharmonic motions 1/3 and 2/4. In the low-frequency domain, the maximum impact velocity is relatively small. As shown in Figure 2(c), on the ω,δ-parameter plane associated with the constraint A2, periodic-impact motions mainly exist in the middle frequency domain. The fundamental motion is mainly 1/1 motion, and the rest are mainly subharmonic motions, including 1/2, 1/2, 1/3, 2/3, and 2/4 motions. The corresponding three-dimensional surface diagrams of the maximum impact velocity are shown in Figure 3(c); the peak values of the maximum impact velocity exist in the regions of 1/1, 1/2, and 1/3 motions. Figure 2(b) shows the ω,δ-parameter plane associated with constraint A12, which presents richer dynamic characteristics on the entire parameter plane. In the domain of large δ and high ω, it is mainly impactless motion 0/1. The dark gray areas and subharmonic motions (mainly 3/3, 4/3, and 4/4 motions) mainly exist in the domain of middle ω. In the parameter domain located at the lower left corner of Figure 2(b), the types of periodic motions are more complicated. It mainly presented as the fundamental impact motions p/1 (p ≥ 1), the incomplete chatting motions p˜/1, and the sticking motions p¯/1, as observed in Figure 4. The fundamental impact motions p/1 (p ≥ 1) present band-like distribution in the domain of low ω, and as the number of impacts p increases, the width of the band-like region gradually narrows. As the exciting force frequency ω decreases, p/1 motion will have a grazing bifurcation, and the number of impacts p increases by one, and it transitions to (p+1)/1 motion. When p keeps increasing, the system will enter incomplete chatting motions p˜/1. At this time, the number of impacts p is very large, but the minimum impact speed is still not zero, and the masses M1 and M2 do not stick together and move synchronously. When the number of impacts p increases sufficiently and the minimum impact velocity decreases to zero, the system will enter the sticking motions p¯/1. At this time, the interaction force between the two masses N(t) > 0, and the two masses are sticking together for synchronous movement. The three-dimensional surface diagrams of the maximum impact velocity corresponding to Figure 2(b) are shown in Figure 3(b); the fundamental impact motions regions near ω=0.7 and ω=1.2 have larger maximum impact velocity; in the middle frequency domain, the peaks of maximum impact velocity appear in the subharmonic motions 3/3 and 4/4 regions.Figure 4
Local enlargement of Figure2(b).In order to further study the bifurcation characteristics of the system, a single-parameter bifurcation diagram that crosses theω,δ-parameter plane along a horizontal scan can be taken. The global bifurcation diagrams of the system when the gap value δ=0.1 is taken can be seen in Figure 5, and the relation between single-parameter bifurcation diagrams and periodic motions on the ω,δ-parameter plane can be observed. In the middle frequency domain, impact motions occur in all three constraints. For example, at the point ω,δ=0.85,0.1, the number of cycles n is two, the number of impacts p at constraint A1 is two, the number of impacts p at constraint A12 is six, and the number of impacts p at constraint A2 is one, as shown in Figure 6. In the low-frequency range, impact motions mainly occur at constraints A1 and A12. The impact motions at constraint A1 are mainly fundamental 1/1 and 2/1 motions, while the impact motions at constraint A12 are more complicated, which include p/1 (p ≥ 1), p˜/1, and p¯/1 motions. Figure 7 shows a partial enlarged bifurcation diagram of the low-frequency part of Figure 5(b), in which the red part is the sticking motions. It can be seen that, with the decrease of ω, the 2/1 motion passes through the bare-grazing bifurcation point G2/1b (ω=0.668) and enters the region containing the subharmonic motion 10/4 and then transfers to the 3/1 motion. The phase diagram at G2/1b is shown in Figure 8(a); it can be seen that the impact velocity at grazing bifurcation point is zero. The phase diagram at G3/1r (ω=0.543) is shown in Figure 8(b), and 3/1 motion transfers to 4/1 motion. The phase diagram at G4/1r (ω=0.505) is shown in Figure 8(c), and 4/1 motion transfers to 5/1 motion. So far, it can be found that, with the decrease of ω in the low-frequency region, the p/1 motion passes through real grazing bifurcation, and the number of impacts p increases once and directly enters (p + 1)/1 motion. When passing through bare-grazing bifurcation, the system will enter an unstable region that may include subharmonic motions, quasi-periodic motions, and chaotic motions, and then it will enter (p + 1)/1 motion. When ω is further reduced, the number of impacts p increases one by one, and the system enters p˜/1. The time series of the system when ω=0.3619 is shown in Figure 9. At this time, ω is very close to the sliding bifurcation point, and the system is in the incomplete chatting motion. It can be seen that the relative displacement x1−x2 is infinitely close to δ=0.1 with the increase of ωt in Figures 9(a) and 9(b). The relative velocity x˙1−x˙2 is infinitely close to zero with the increase of ωt in Figures 9(c) and 9(d). When ω continues to decrease, the system enters the sticking motions through the sliding bifurcation. Time series of sticking motions with ω=0.35 are shown in Figure 10. It can be seen that, after the system has undergone a chattering-impact sequence in which the relative displacement x1−x2 and the relative velocity x˙1−x˙2 are gradually reduced, x˙1−x˙2 is reduced to zero, and x1−x2 is equal to the gap value δ=0.1. The interaction force between the two masses Nt>0, and the two masses are sticking together and moving synchronously. In the low-frequency domain, as ω decreases, the transition law of the system on the ω,δ-parameter plane associated with constraint A12 can be summarized as(11)ω↓:p¯1←SBif⋯←GBifp˜1←GBif⋯←GBifp+11←GBifp1←GBif⋯←GBif41←GBif31←GBif21←GBif11.Figure 5
Bifurcation diagrams of the system whenδ=0.1. (a) Impact bifurcation diagram at constraint A1; (b) impact bifurcation diagram at constraint A12; (c) impact bifurcation diagram at constraint A2; (d) periodic bifurcation diagrams.
(a)(b)(c)(d)Figure 6
Phase plane portraits of periodic motion whenδ=0.1 and ω=0.85. (a) Phase plane portraits of M1, 2/2 motion; (b) phase plane portraits of relative displacement and relative velocity of M1 and M2, 6/2 motion; (c) phase plane portraits of M2, 1/2 motion.
(a)(b)(c)Figure 7
Bifurcation diagrams of the low-frequency region of the system at constraintA12. (a) Impact bifurcation diagram; (b) periodic bifurcation diagram.
(a)(b)Figure 8
Phase diagrams at grazing bifurcation point. (a)G2/1b, ω=0.668; (b) G3/1r, ω=0.543; (c) G3/1r, ω=0.505.
(a)(b)(c)Figure 9
Time series near the sliding bifurcation point whenω=0.3619. (a) Time series of x1−x2; (b) local enlargement of Figure 9(a); (c) time series of x˙1−x˙2; (d) local enlargement of Figure 9(c).
(a)(b)(c)(d)Figure 10
Time series of sticking motion whenω=0.35. (a) Time series of x1−x2; (b) local enlargement of Figure 10(a); (c) time series of x˙1−x˙2; (d) local enlargement of Figure 10(c).
(a)(b)(c)(d)From theω,δ-parameter plane associated with constraint A12, it can be found that the fundamental impact motions p/1 (p ≥ 1) are in bands distribution. It can also be found that there are transition regions in the adjacent fundamental impact motions in Figures 2(b), 4, 11(a), and 11(b). This transition regions look like the shape of the tongue, so these regions can be named as tongue-like regions, marked as TRp/1∩p+1/1. When the numerical simulation was performed with decreasing ω and increasing ω, respectively, and depicted on the same ω,δ-parameter plane, another type of transition regions, the hysteresis regions, is found, marked as HRp. Due to the difference of initial values, there is coexistence of p/1 and (p+1)/1 in HRp, as shown in the dark gray areas in Figure 11(c).Figure 11
The partial enlargement of Figure2(b). (a) Tongue-like region: TR0/1∩1/1; (b) tongue-like region: TRp/1∩p+1/1p≥2; (c) tongue-like region: TRp/1∩p+1/1p≥4 and hysteresis region HRpp≥4.
(a)(b)(c)As shown in Figure11(a), the tongue-like region TR0/1∩1/1 between the impactless motion and the fundamental 1/1 motion contains complex subharmonic motions (2/3, 2/2, 1/3, 1/4, 2/4, 4/4, and 4/8 motions), and 2/4 motion occupies the largest area. Figure 12 shows that the single-parameter bifurcation diagrams cross the tongue-like region TR0/1∩1/1 longitudinally when ω=0.9; the transition law with decreasing of δ is as follows: 0/1 motion ⟶ G0/1b ⟶ 1/4 motion ⟶ chaos ⟶ 1/3 motion ⟶ 2/4 motion ⟶ chaos ⟶ 1/1 motion. Figure 11(b) depicts the band-like regions of p/1 (p ≥ 2) motions and the tongue-like regions TRp/1∩p+1/1p≥2 in the adjacent fundamental p/1 and (p + 1)/1 motions. Figure 11(c) is a partial enlarged view of Figure 11(b), and the hysteresis regions HRp are drawn by numerical simulation through decreasing and increasing of ω (represented by ω↔). At this time, there are two types of transition regions: tongue-like regions and hysteresis regions. Due to the mutual influence of complex impact motions caused by multiple gaps and constraints, the hysteresis regions only appear on the right side of the tongue-like regions, and those on the left side are replaced by the subharmonic motions embedded in the tongue-like region. It can be found that the upper boundary of the tongue-like region TRp/1∩p+1/1 is bare-grazing bifurcation line Gp/1b of p/1 motion, and the lower boundary is the saddle-node bifurcation line SN1+p/1 of (p + 1)/1 motion. Figures 13(a) and 13(b) depict the single-parameter bifurcation diagrams crossing tongue-like regions of Figure 11(b) along a vertical scan for ω=0.455; p/1 motion passes through the bare-grazing bifurcation into the tongue-like region including subharmonic motions, quasi-periodic motions, and chaotic motions and then enters (p + 1)/1 motion. Figures 13(c) and 13(d) are partial enlargement of Figures 13(a) and 13(b); the 5/1 motion crossing G5/1b at δ=0.19811 (the phase diagram of G5/1b is depicted in Figure 14) enters subharmonic 13/3 motion and then enters 18/4 motion, which is the main part of the tongue-like region, and then it enters chaotic motion and finally withdraws to 6/1 motion. It can be further observed in Figures 11(b) and 11(c) that the main subharmonic motions in the domains of TR3/1∩4/1, TR4/1∩5/1, TR5/1∩6/1, TR6/1∩7/1, TR7/1∩8/1, TR8/1∩9/1, and TR9/1∩10/1 are 14/4, 18/4, 22/4, 26/4, 30/4, 34/4, and 38/4 motions, respectively. The main subharmonic motions contained in the tongue-like regions on this parameter domain can be summarized as np+2/n (p = 1, 2, ...; n = 2, 3, ...). It can be seen in Figure 4 that, with the decrease of ω, the boundaries of the tongue-like regions are further destroyed. At this time, the main subharmonic motions in the tongue-like regions TR2/1∩3/1, TR3/1∩4/1, TR4/1∩5/1, TR5/1∩6/1, and TR6/1∩7/1 are 5/2, 7/2, 9/2, 11/2, and 13/2, respectively. Therefore, the main form of subharmonic motions contained in the tongue-like regions in this parameter domain is np+1/n (p = 1, 2, ...; n = 2, 3, ...).Figure 12
Bifurcation diagrams that cross the tongue-like regionTR0/1∩1/1 longitudinally at ω=0.9. (a) Impact bifurcation diagram; (b) periodic bifurcation diagram.
(a)(b)Figure 13
Bifurcation diagrams that cross the tongue-like regionTRp/1∩p+1/1p≥4 longitudinally at ω=0.455. (a) Impact bifurcation diagram; (b) periodic bifurcation diagram; (c) local enlargement of Figure 13(a); (d) local enlargement of Figure 13(b).
(a)(b)(c)(d)Figure 14
Phase diagrams at grazing bifurcation pointG5/1b when δ=0.19811 and ω=0.668. (a) Overall; (b) local enlargement of Figure 14(a).
(a)(b)In Figure11(c), it can be seen that hysteresis region HRp exists between the upper bound SN1+p/1 and the lower bound Gp/1r. Figure 15 is a single-parameter bifurcation diagram synthesized by two numerical simulation modes of increasing ω and decreasing ω when δ=0.09 crosses the hysteresis region HR5 laterally. With the decrease of ω, 5/1 motion enters 6/1 motion through G5/1r; with the increase of ω, 6/1 motion enters 5/1 motion through SN6/1. It can be seen that the 5/1 motion and the 6/1 motion coexist in the hysteresis region HR5 (5/1 motion is represented by black, and 6/1 motion is represented by red). Take the point ω=0.4853 in the hysteresis regions HR5 in Figure 15 to draw the phase diagrams. When the initial values X0=τ0,x10,x˙10,x20,x˙20=0.0,−0.097587,0.690129,−0.187587,−0.089877 are used for numerical simulation, the system is in 5/1 motion, as shown in Figures 16(a) and 16(b); when the initial values X0=τ0,x10,x˙10,x20,x˙20=0.0,=−0.080625,0.705641,−0.170624,−0.068216 are used for numerical simulation, the system is in 5/1 motion, as shown in Figure 16(c).Figure 15
Bifurcation diagrams that cross the hysteresis regionHR5 with the forward and backward of ω when δ=0.09.Figure 16
Phase diagram of 5/1 motion and 6/1 motion with different initial values coexisting in the hysteresis domainHR5 when δ=0.09 and ω=0.4853. (a) 5/1 motion; (b) local enlargement of Figure 16(a); (c) 6/1 motion.
(a)(b)(c)


## 4. The Correlation between Dynamics and System Parameters

This section mainly studies the relationship between system dynamics and system parameters. The effect of changing the fixed gap valueδ1=δ2 and damping coefficient ζ on dynamics is studied, respectively. In the following analysis, only the parameters that need to be studied are changed; other parameters are the same as the basis parameters.


### 4.1. The Correlation between Dynamic Performance and Fixed Gap Valueδ1=δ2

In the analysis of basis parameters in the previous chapter, the gapsδ1=δ2=1.0 are considered. Here, the larger gap value δ1=δ2=1.5 and the smaller gap value δ1=δ2=0.5 are used for analysis.When taking a larger gap valueδ1=δ2=1.5, Figure 17 shows the existence types and distribution areas of periodic motions on the ω,δ-parameter plane associated with each constraint. The corresponding three-dimensional surface diagrams of the maximum impact velocity can be observed in Figure 18. As for low ω, there is no impact vibration at constraint A1 and constraint A2; the impact vibration of the system is mainly represented as the impact between mass M1 and mass M2 at constraint A12, and its mode types are fundamental impact motions, the incomplete chatting motions, and sticking motions. Figure 19 is a partial enlarged view of Figure 17(b) in the low ω and small δ domain; impact vibration only occurs at constraint A12. The band-like region of fundamental p/1 motion is more complete, and the boundaries of tongue-like region TRp/1∩p+1/1 are also more obvious. For example, the upper boundary of TR2/1∩3/1 is the curve G2/1b, and the lower boundary is the curve SN3/1; the subharmonic motions in TR2/1∩3/1 mainly appear as 5/2 motion. The upper boundary of TR3/1∩4/1 is the curve G3/1b, and the lower boundary is the curve PD4/1; the subharmonic motions in TR3/1∩4/1 mainly appear as 7/2 motion. It can be found that the upper boundary of TRp/1∩p+1/1 is the curve Gp/1b, and the lower boundary is the curve SN1+p/1 or the possible PD1+p/1, where the main form of subharmonic motions is (np + 1)/n (p = 1, 2, …; n = 2, 3, …). On the ω,δ-parameter plane associated with constraint A1, impact vibrations occur only in the vertical-strip region in the middle frequency domain, which mainly presents 1/1 motion, as well as a small amount of subharmonic motions (mainly 2/2, 3/3, 4/4, and 5/5 motions); see Figure 17(b). On the ω,δ-parameter plane associated with constraint A2, there is a small amount of 1/1 motion in the middle frequency domain.Figure 17
Periodic motions on theω,δ-parameter plane associated with each constraint when δ1=δ2=1.5. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 18
Three-dimensional surface diagrams of the maximum impact velocity of the system associated with each constraint whenδ1=δ2=1.5. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 19
Local enlargement of Figure17(b).As shown in Figure19, compared with the basis parameters, the maximum impact velocity at constraints A1 and A2 is reduced, and the maximum impact velocity at constraint A12 is increased. At constraint A1, the peak of the maximum impact velocity appears near the vertical-strip region with ω = 1.0. At constraint A12, the peak of the maximum impact velocity exists in the domain of 1/1 motion. At constraint A2, the peak of the maximum impact velocity appears in the existence area of 1/1 motion and the existence area of subharmonic motion 1/2.When taking a smaller gap valueδ1=δ2=0.5, Figure 20 shows the existence types and distribution areas of periodic-impact motions on the ω,δ-parameter plane associated with each constraint. As for low ω, there are impact vibrations at each constraint. At constraint A1, the system mainly presents fundamental impact motions, the incomplete chatting motions, and the sticking motions which are mainly distributed like several vertical strips. At constraint A2, the system mainly presented as 1/1 motion, as well as a small amount of 2/1 motion. On the ω,δ-parameter plane associated with constraint A12, due to the complex impact vibrations occurring at constraint A1, the band-like region of fundamental p/1 motion is deformed and segmented in low-frequency domain. Figure 21 is a partial enlarged view of Figure 20(b) in the low ω and small δ domain, the band-like region of fundamental p/1 motion is disrupted, the existence areas and boundaries of the tongue-like regions are no longer clear, and more unrecognized gray areas are embedded in the domain of small δ. For example, the subharmonic motion 7/2 that appears during the transition of 3/1 motion and 4/1 motion appears as an interval, and its shape no longer appears as a tongue; the subharmonic motion 9/2 that appears during the transition of 4/1 motion and 5/1 motion also showed a similar distribution law. It can be seen that although the band-like region of fundamental p/1 motion is divided, the subharmonic motions between the adjacent p/1 and (p + 1)/1 motions still conform to the law of (np + 1)/n (p = 1, 2, …; n = 2, 3, …). More unrecognized gray areas are embedded in the high-frequency domain of the system.Figure 20
Periodic motions on theω,δ-parameter plane associated with each constraint when δ1=δ2=0.5. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 21
Local enlargement of Figure20(b).The three-dimensional surface diagrams of the maximum impact velocity can be observed in Figure22. The maximum impact velocity at constraint A1 increases compared to the basis parameters, and its distribution area is roughly the same as that of the basis parameters. The peaks of the maximum impact velocity appear in the existence areas of 1/1 and 2/1 motions in the large δ domain. At constraint A12, the maximum impact velocity reduces compared to the basis parameters, and the peak appears in the subharmonic 4/2 motion region distributed in the domain of small δ and low ω. At constraint A2, the maximum impact velocity is increased compared with the basis parameters, and the peak appears in the fundamental 1/1 motion region distributed in the domain of middle ω.Figure 22
Three-dimensional surface diagrams of the maximum impact velocity of the system associated with each constraint whenδ1=δ2=0.5. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)By comparison with the basis parameters, for largeδ1=δ2, the system has no impact vibration at constraint A1 and constraint A2 at low frequency, and impact vibration only occurs at constraint A12. On the ω,δ-parameter plane associated with constraint A12, the band-like regions of fundamental p/1 motions and the tongue-like regions TRp/1∩p+1/1 have complete boundaries. The maximum impact velocity increases at constraint A12 and decreases at constraint A1 and constraint A2. With the decrease of δ1=δ2, the impact vibration at constraint A1 and constraint A2 gradually intensified. At constraint A1, there is a transition from p/1 motion to p¯/1 motion. At constraint A12, the mode types and distribution areas of the periodic-impact vibrations have not changed much. Due to the interaction of the impact vibrations of each constraint, the band-like regions of fundamental p/1 motions are cut by the gray unidentified motions; the boundaries of the tongue-like regions TRp/1∩p+1/1 are also destroyed, but the laws of the main subharmonic motions mode types remain the same. The system embeds a large amount of gray unrecognized motions in the middle frequency domain, which reduces the existence areas of periodic motions. The distribution domains of the peaks of the maximum impact velocity have little change. The maximum impact velocity increases at constraints A1 and A2 and decreases at constraint A12.


### 4.2. The Correlation between Dynamic Performance and Damping Coefficientζ

The damping coefficientζ is the precoefficient of the velocity term in the differential equation of the system, which affects the impact absorption characteristics of the dynamic system. The damping coefficient is ζ = 0.1 in the basis parameters; the larger value ζ = 0.2 and the smaller value ζ = 0.05 are selected to analyze its influence on the dynamics.When a larger damping coefficientζ = 0.2 is taken, the mode types and existence regions of the periodic motions on the ω,δ-parameter plane associated with each constraint are depicted in Figure 23. The impact vibrations of the system only occur at constraint A1 and constraint A12, and there is no impact vibration at constraint A2. Figure 23(a) is the ω,δ-parameter plane associated with constraint A1; similar to the basis parameter, it is mainly 1/1 motion in the domain of ω ≤ 1.4, and the domains of 2/1, 2/2, 3/3, and 4/4 motions are significantly reduced. There is subharmonic motion 4/2 in the middle frequency domain. The parameter domain of ω > 1.4 is impactless motion 0/1. The ω,δ-parameter plane associated with constraint A12 is shown in Figure 23(b), the existence domain of the fundamental impact motions p/1 (p ≥ 1) increases, and the embedded subharmonic motions and unrecognized gray areas decrease significantly. The existence domain of the fundamental impact motions p/1 (p ≥ 1) extends to the domain of large ω, its bulge forms a semicircular domain, and Figure 24 is a partial enlarged view of it. In the semicircular parameter domain, the transition process with decreasing of ω can be summarized as(12)ω↓:41←SN Bif51←SN Bif⋯←SN Bifp1←SN Bifp+11←SN Bifp˜1←Rising Bifp¯1←Sliding Bifp˜1←GBif…←GBifp+11←GBifp1←GBif…31←GBif21.Figure 23
Periodic motions on theω,δ-parameter plane associated with each constraint when ζ = 0.2. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 24
Local enlargement of Figure23(b).Tongue-like regions appear between adjacentp/1 and (p + 1)/1 motions in the semicircular parameter domain. The tongue-like region TR3/1∩3/1 contains the main forms of subharmonic motions as 2/5, 7/3, and 8/3 motions. The tongue-like region TR4/1∩3/1 contains the main forms of subharmonic motions as 10/3 and 11/3 motions. It can be found that the main forms of subharmonic motions in the tongue-like regions TRp/1∩p+1/1 are (np + 1)/n and (np + 2)/n (p = 1, 2, …; n = 2, 3, …).The three-dimensional surface diagrams of the maximum impact velocity can be observed in Figure25. The maximum impact velocities at constraint A1 and constraint A12 are both reduced compared to the basis parameters. At constraint A1, the peaks of the maximum impact velocity are similar to those in basis parameters, appearing in the domain of 1/1 motion. At constraint A12, the peaks of the maximum impact velocity appear in the region of 1/1 motion in the small δ domain and the region of subharmonic motion 4/4.Figure 25
Three-dimensional surface diagrams of the maximum impact velocity of the system associated with each constraint whenζ = 0.2. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)When a smaller damping coefficientζ = 0.05 is taken, periodic motions on the ω,δ-parameter plane associated with each constraint are depicted in Figure 26. It can be seen that, on the ω,δ-parameter plane associated with each constraint, the types and distribution areas of periodic motions are roughly the same as those in the basis parameters, but the existence regions of subharmonic motions and the unidentified gray areas increase and are embedded to the domains of fundamental impact motions. The distribution and transition laws of the fundamental impact motions p/1 (p ≥ 1) on ω,δ-parameter plane associated with constraint A12 are mainly analyzed. Figure 27 is a partial enlarged view of Figure 26(b) in the low ω and small δ domain; compared with the basis parameters, the wave-like undulations of the p/1 (p ≥ 1) motions band-like regions increase. The area of tongue-like regions increases, but, due to the embedding of the gray unrecognized area, the boundaries of the tongue-like regions are destroyed. The tongue-like regions in the parameter domain of ω∈0.4,0.5 are similar to those in Figure 11(b) with the basis parameter, but the area of the tongue-like regions increases. The subharmonic motions in tongue-like regions TR3/1∩4/1 and TR3/1∩4/1 are 10/4 and 14/4, respectively. It can be seen that the main subharmonic motions in the tongue-like regions in this parameter domain are still (np + 2)/n (p = 1, 2, …; n = 2, 3, …). In the parameter domain of ω∈0.3,0.4, the shape of the tongue-like regions is seriously deformed, and its boundaries are no longer clear due to the embedding of more gray areas. The subharmonic motions in tongue-like regions TR2/1∩3/1 and TR3/1∩4/1 are 5/2 and 7/2, respectively. It can be seen that the main subharmonic motions in the tongue-like regions in this parameter domain are (np + 1)/n (p = 1, 2, …; n = 2, 3, …).Figure 26
Periodic motions on theω,δ-parameter plane associated with each constraint when ζ = 0.05. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 27
Local enlargement of Figure26(b).In three-dimensional surface diagrams of the maximum impact velocity shown in Figure28, the maximum impact velocity associated with each constraint has increased compared to the basis parameters. At constraint A1, peaks of the maximum impact velocity appear in the areas where unrecognized gray area and the subharmonic motion 2/2 exist in the domain of middle ω and large δ. At constraint A12, peaks of the maximum impact velocity appear in the regions of periodic-impact motions 3/1, 2/1, and 1/1, and subharmonic motion 4/2 exists in the small δ domain. At constraint A2, peaks of the maximum impact velocity appear in the areas where 1/1 motion exists in the domain of small δ, as well as the subharmonic motion 2/2 and the gray unrecognized area in the domain of large δ.Figure 28
Three-dimensional surface diagrams of the maximum impact velocity of the system associated with each constraint whenζ = 0.05. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)By comparison with the basis parameters, changing the damping coefficientζ has a greater influence on the mode types and existence areas of the periodic-impact motions. The large ζ makes the impact vibrations only occur in constraint A1 and constraint A12. At constraint A1, the medium low ω domain is mainly 1/1 motion, and the medium high ω domain is mainly impactless motion 0/1. At constraint A12, the band-shaped regions of the fundamental impact motions extend to the domain of large δ, and the unrecognized gray areas are sharply reduced. The area of the tongue-like regions decreased during the transition process, and its pattern types also changed. The maximum impact velocity at each constraint is smaller, and the number of peaks is reduced. The peaks only appear in the existence domain of 1/1 motion. When the damping coefficient ζ decreases, the impact vibration at constraint A1 and constraint A2 is enhanced, and the domain of the fundamental impact motions is embedded with a large number of subharmonic motions and gray unidentified areas. At constraint A12, the wave-like undulations of the p/1 (p ≥ 1) motions band-like regions increase, and the gray area divides the band-like regions, and its integrity is destroyed. The boundaries of the tongue-like regions are no longer complete, but the mode types of the subharmonic motions are consistent with the reference parameters. The maximum impact velocity at each constraint increases; the number of peaks increases, mainly distributed in the occurrence domains of 1/1 motion and subharmonic motions, as well as the unidentified gray areas.


## 4.1. The Correlation between Dynamic Performance and Fixed Gap Valueδ1=δ2

In the analysis of basis parameters in the previous chapter, the gapsδ1=δ2=1.0 are considered. Here, the larger gap value δ1=δ2=1.5 and the smaller gap value δ1=δ2=0.5 are used for analysis.When taking a larger gap valueδ1=δ2=1.5, Figure 17 shows the existence types and distribution areas of periodic motions on the ω,δ-parameter plane associated with each constraint. The corresponding three-dimensional surface diagrams of the maximum impact velocity can be observed in Figure 18. As for low ω, there is no impact vibration at constraint A1 and constraint A2; the impact vibration of the system is mainly represented as the impact between mass M1 and mass M2 at constraint A12, and its mode types are fundamental impact motions, the incomplete chatting motions, and sticking motions. Figure 19 is a partial enlarged view of Figure 17(b) in the low ω and small δ domain; impact vibration only occurs at constraint A12. The band-like region of fundamental p/1 motion is more complete, and the boundaries of tongue-like region TRp/1∩p+1/1 are also more obvious. For example, the upper boundary of TR2/1∩3/1 is the curve G2/1b, and the lower boundary is the curve SN3/1; the subharmonic motions in TR2/1∩3/1 mainly appear as 5/2 motion. The upper boundary of TR3/1∩4/1 is the curve G3/1b, and the lower boundary is the curve PD4/1; the subharmonic motions in TR3/1∩4/1 mainly appear as 7/2 motion. It can be found that the upper boundary of TRp/1∩p+1/1 is the curve Gp/1b, and the lower boundary is the curve SN1+p/1 or the possible PD1+p/1, where the main form of subharmonic motions is (np + 1)/n (p = 1, 2, …; n = 2, 3, …). On the ω,δ-parameter plane associated with constraint A1, impact vibrations occur only in the vertical-strip region in the middle frequency domain, which mainly presents 1/1 motion, as well as a small amount of subharmonic motions (mainly 2/2, 3/3, 4/4, and 5/5 motions); see Figure 17(b). On the ω,δ-parameter plane associated with constraint A2, there is a small amount of 1/1 motion in the middle frequency domain.Figure 17
Periodic motions on theω,δ-parameter plane associated with each constraint when δ1=δ2=1.5. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 18
Three-dimensional surface diagrams of the maximum impact velocity of the system associated with each constraint whenδ1=δ2=1.5. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 19
Local enlargement of Figure17(b).As shown in Figure19, compared with the basis parameters, the maximum impact velocity at constraints A1 and A2 is reduced, and the maximum impact velocity at constraint A12 is increased. At constraint A1, the peak of the maximum impact velocity appears near the vertical-strip region with ω = 1.0. At constraint A12, the peak of the maximum impact velocity exists in the domain of 1/1 motion. At constraint A2, the peak of the maximum impact velocity appears in the existence area of 1/1 motion and the existence area of subharmonic motion 1/2.When taking a smaller gap valueδ1=δ2=0.5, Figure 20 shows the existence types and distribution areas of periodic-impact motions on the ω,δ-parameter plane associated with each constraint. As for low ω, there are impact vibrations at each constraint. At constraint A1, the system mainly presents fundamental impact motions, the incomplete chatting motions, and the sticking motions which are mainly distributed like several vertical strips. At constraint A2, the system mainly presented as 1/1 motion, as well as a small amount of 2/1 motion. On the ω,δ-parameter plane associated with constraint A12, due to the complex impact vibrations occurring at constraint A1, the band-like region of fundamental p/1 motion is deformed and segmented in low-frequency domain. Figure 21 is a partial enlarged view of Figure 20(b) in the low ω and small δ domain, the band-like region of fundamental p/1 motion is disrupted, the existence areas and boundaries of the tongue-like regions are no longer clear, and more unrecognized gray areas are embedded in the domain of small δ. For example, the subharmonic motion 7/2 that appears during the transition of 3/1 motion and 4/1 motion appears as an interval, and its shape no longer appears as a tongue; the subharmonic motion 9/2 that appears during the transition of 4/1 motion and 5/1 motion also showed a similar distribution law. It can be seen that although the band-like region of fundamental p/1 motion is divided, the subharmonic motions between the adjacent p/1 and (p + 1)/1 motions still conform to the law of (np + 1)/n (p = 1, 2, …; n = 2, 3, …). More unrecognized gray areas are embedded in the high-frequency domain of the system.Figure 20
Periodic motions on theω,δ-parameter plane associated with each constraint when δ1=δ2=0.5. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 21
Local enlargement of Figure20(b).The three-dimensional surface diagrams of the maximum impact velocity can be observed in Figure22. The maximum impact velocity at constraint A1 increases compared to the basis parameters, and its distribution area is roughly the same as that of the basis parameters. The peaks of the maximum impact velocity appear in the existence areas of 1/1 and 2/1 motions in the large δ domain. At constraint A12, the maximum impact velocity reduces compared to the basis parameters, and the peak appears in the subharmonic 4/2 motion region distributed in the domain of small δ and low ω. At constraint A2, the maximum impact velocity is increased compared with the basis parameters, and the peak appears in the fundamental 1/1 motion region distributed in the domain of middle ω.Figure 22
Three-dimensional surface diagrams of the maximum impact velocity of the system associated with each constraint whenδ1=δ2=0.5. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)By comparison with the basis parameters, for largeδ1=δ2, the system has no impact vibration at constraint A1 and constraint A2 at low frequency, and impact vibration only occurs at constraint A12. On the ω,δ-parameter plane associated with constraint A12, the band-like regions of fundamental p/1 motions and the tongue-like regions TRp/1∩p+1/1 have complete boundaries. The maximum impact velocity increases at constraint A12 and decreases at constraint A1 and constraint A2. With the decrease of δ1=δ2, the impact vibration at constraint A1 and constraint A2 gradually intensified. At constraint A1, there is a transition from p/1 motion to p¯/1 motion. At constraint A12, the mode types and distribution areas of the periodic-impact vibrations have not changed much. Due to the interaction of the impact vibrations of each constraint, the band-like regions of fundamental p/1 motions are cut by the gray unidentified motions; the boundaries of the tongue-like regions TRp/1∩p+1/1 are also destroyed, but the laws of the main subharmonic motions mode types remain the same. The system embeds a large amount of gray unrecognized motions in the middle frequency domain, which reduces the existence areas of periodic motions. The distribution domains of the peaks of the maximum impact velocity have little change. The maximum impact velocity increases at constraints A1 and A2 and decreases at constraint A12.


## 4.2. The Correlation between Dynamic Performance and Damping Coefficientζ

The damping coefficientζ is the precoefficient of the velocity term in the differential equation of the system, which affects the impact absorption characteristics of the dynamic system. The damping coefficient is ζ = 0.1 in the basis parameters; the larger value ζ = 0.2 and the smaller value ζ = 0.05 are selected to analyze its influence on the dynamics.When a larger damping coefficientζ = 0.2 is taken, the mode types and existence regions of the periodic motions on the ω,δ-parameter plane associated with each constraint are depicted in Figure 23. The impact vibrations of the system only occur at constraint A1 and constraint A12, and there is no impact vibration at constraint A2. Figure 23(a) is the ω,δ-parameter plane associated with constraint A1; similar to the basis parameter, it is mainly 1/1 motion in the domain of ω ≤ 1.4, and the domains of 2/1, 2/2, 3/3, and 4/4 motions are significantly reduced. There is subharmonic motion 4/2 in the middle frequency domain. The parameter domain of ω > 1.4 is impactless motion 0/1. The ω,δ-parameter plane associated with constraint A12 is shown in Figure 23(b), the existence domain of the fundamental impact motions p/1 (p ≥ 1) increases, and the embedded subharmonic motions and unrecognized gray areas decrease significantly. The existence domain of the fundamental impact motions p/1 (p ≥ 1) extends to the domain of large ω, its bulge forms a semicircular domain, and Figure 24 is a partial enlarged view of it. In the semicircular parameter domain, the transition process with decreasing of ω can be summarized as(12)ω↓:41←SN Bif51←SN Bif⋯←SN Bifp1←SN Bifp+11←SN Bifp˜1←Rising Bifp¯1←Sliding Bifp˜1←GBif…←GBifp+11←GBifp1←GBif…31←GBif21.Figure 23
Periodic motions on theω,δ-parameter plane associated with each constraint when ζ = 0.2. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 24
Local enlargement of Figure23(b).Tongue-like regions appear between adjacentp/1 and (p + 1)/1 motions in the semicircular parameter domain. The tongue-like region TR3/1∩3/1 contains the main forms of subharmonic motions as 2/5, 7/3, and 8/3 motions. The tongue-like region TR4/1∩3/1 contains the main forms of subharmonic motions as 10/3 and 11/3 motions. It can be found that the main forms of subharmonic motions in the tongue-like regions TRp/1∩p+1/1 are (np + 1)/n and (np + 2)/n (p = 1, 2, …; n = 2, 3, …).The three-dimensional surface diagrams of the maximum impact velocity can be observed in Figure25. The maximum impact velocities at constraint A1 and constraint A12 are both reduced compared to the basis parameters. At constraint A1, the peaks of the maximum impact velocity are similar to those in basis parameters, appearing in the domain of 1/1 motion. At constraint A12, the peaks of the maximum impact velocity appear in the region of 1/1 motion in the small δ domain and the region of subharmonic motion 4/4.Figure 25
Three-dimensional surface diagrams of the maximum impact velocity of the system associated with each constraint whenζ = 0.2. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)When a smaller damping coefficientζ = 0.05 is taken, periodic motions on the ω,δ-parameter plane associated with each constraint are depicted in Figure 26. It can be seen that, on the ω,δ-parameter plane associated with each constraint, the types and distribution areas of periodic motions are roughly the same as those in the basis parameters, but the existence regions of subharmonic motions and the unidentified gray areas increase and are embedded to the domains of fundamental impact motions. The distribution and transition laws of the fundamental impact motions p/1 (p ≥ 1) on ω,δ-parameter plane associated with constraint A12 are mainly analyzed. Figure 27 is a partial enlarged view of Figure 26(b) in the low ω and small δ domain; compared with the basis parameters, the wave-like undulations of the p/1 (p ≥ 1) motions band-like regions increase. The area of tongue-like regions increases, but, due to the embedding of the gray unrecognized area, the boundaries of the tongue-like regions are destroyed. The tongue-like regions in the parameter domain of ω∈0.4,0.5 are similar to those in Figure 11(b) with the basis parameter, but the area of the tongue-like regions increases. The subharmonic motions in tongue-like regions TR3/1∩4/1 and TR3/1∩4/1 are 10/4 and 14/4, respectively. It can be seen that the main subharmonic motions in the tongue-like regions in this parameter domain are still (np + 2)/n (p = 1, 2, …; n = 2, 3, …). In the parameter domain of ω∈0.3,0.4, the shape of the tongue-like regions is seriously deformed, and its boundaries are no longer clear due to the embedding of more gray areas. The subharmonic motions in tongue-like regions TR2/1∩3/1 and TR3/1∩4/1 are 5/2 and 7/2, respectively. It can be seen that the main subharmonic motions in the tongue-like regions in this parameter domain are (np + 1)/n (p = 1, 2, …; n = 2, 3, …).Figure 26
Periodic motions on theω,δ-parameter plane associated with each constraint when ζ = 0.05. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)Figure 27
Local enlargement of Figure26(b).In three-dimensional surface diagrams of the maximum impact velocity shown in Figure28, the maximum impact velocity associated with each constraint has increased compared to the basis parameters. At constraint A1, peaks of the maximum impact velocity appear in the areas where unrecognized gray area and the subharmonic motion 2/2 exist in the domain of middle ω and large δ. At constraint A12, peaks of the maximum impact velocity appear in the regions of periodic-impact motions 3/1, 2/1, and 1/1, and subharmonic motion 4/2 exists in the small δ domain. At constraint A2, peaks of the maximum impact velocity appear in the areas where 1/1 motion exists in the domain of small δ, as well as the subharmonic motion 2/2 and the gray unrecognized area in the domain of large δ.Figure 28
Three-dimensional surface diagrams of the maximum impact velocity of the system associated with each constraint whenζ = 0.05. (a) Constraint A1; (b) constraint A12; (c) constraint A2.
(a)(b)(c)By comparison with the basis parameters, changing the damping coefficientζ has a greater influence on the mode types and existence areas of the periodic-impact motions. The large ζ makes the impact vibrations only occur in constraint A1 and constraint A12. At constraint A1, the medium low ω domain is mainly 1/1 motion, and the medium high ω domain is mainly impactless motion 0/1. At constraint A12, the band-shaped regions of the fundamental impact motions extend to the domain of large δ, and the unrecognized gray areas are sharply reduced. The area of the tongue-like regions decreased during the transition process, and its pattern types also changed. The maximum impact velocity at each constraint is smaller, and the number of peaks is reduced. The peaks only appear in the existence domain of 1/1 motion. When the damping coefficient ζ decreases, the impact vibration at constraint A1 and constraint A2 is enhanced, and the domain of the fundamental impact motions is embedded with a large number of subharmonic motions and gray unidentified areas. At constraint A12, the wave-like undulations of the p/1 (p ≥ 1) motions band-like regions increase, and the gray area divides the band-like regions, and its integrity is destroyed. The boundaries of the tongue-like regions are no longer complete, but the mode types of the subharmonic motions are consistent with the reference parameters. The maximum impact velocity at each constraint increases; the number of peaks increases, mainly distributed in the occurrence domains of 1/1 motion and subharmonic motions, as well as the unidentified gray areas.


## 5. Conclusions

In this paper, we consider the dynamic characteristics of a two-degree-of-freedom periodically forced system with multiple gaps and multiple rigid constraints. On theω,δ-parameter plane associated with each constraint, the mode types, existence areas, and bifurcation regularities of the periodic motions are obtained. The distribution area of the maximum impact velocity and the correlation between the periodic motions distributions are acquired.(1)
Under the basis parameters, on theω,δ-parameter planes associated with constraints A1 and A2, the mode types of impact vibrations are relatively simple. On the ω,δ-parameter plane associated with constraint A12, the system exhibits rich dynamic characteristics. For low ω, the system presents the fundamental impact motions p/1 (p ≥ 1), the incomplete chatting motions p˜/1, and the sticking motions p¯/1. There are two types of transition regions between adjacent fundamental impact motions p/1 and (p + 1)/1: tongue-like regions TRp/1∩p+1/1 and hysteresis regions HRp. TRp/1∩p+1/1 is bounded by the upper boundary Gp/1b and the lower boundary SN1+p/1. Subharmonic motions dominate in TRp/1∩p+1/1 and their mode types are (np + 1)/n or (np + 2)/n (p = 1, 2, …; n = 2, 3, …). HRp exists between the upper boundary SN1+p/1 and the lower boundary Gp/1r. Due to the different initial values, p/1 and (p + 1)/1 motions coexist in HRp. The peaks of the maximum impact velocity generally appear in the existence domains of 1/1 motion, partial subharmonic motions, and a small part of identified gray areas.(2)
The influence of changing the gap valueδ1=δ2 and the damping coefficient ζ on the system dynamics is separately considered. With the decrease of δ1=δ2, the impact vibrations at each constraint are strengthened and the mutual reaction is increased. At constraint A1, the low-frequency region appears as vertical strips of p/1 (p ≥ 1), p˜/1, and p¯/1 motions. On the ω,δ-parameter plane associated with constraint A12, the band-like domains of p/1 (p ≥ 1) are divided by the embedded gray areas, and the boundaries of the tongue-like regions TRp/1∩p+1/1 are partially destroyed. The maximum impact velocities at constraints A1 and A2 increase, and the maximum impact velocity at constraint A12 decreases. The increase in the number of subharmonic impact vibrations and unidentified gray areas leads to an increase in the peak number of maximum impact velocities.The damping coefficientζ has a great influence on the types and distribution areas of the periodic motions. With the decrease of the damping coefficient ζ, the domains of fundamental impact motions decrease, and the domains of subharmonic motions and the unrecognized gray areas increase sharply. The wave-like undulations of the p/1 (p ≥ 1) motions band-like regions increase, and the area of the tongue-like regions increases. At each constraint, the maximum impact velocity increases and the number of peaks increases.Many mechanical vibration systems can be described by a two-degree-of-freedom periodically forced system with multiple gaps and rigid constraints. In incomplete chatting motions and sticking motions, since a lot of impacts occur in one exciting force cycle, it may accelerate the wear of components and increase noise level. The larger impact velocity mostly appears in the domains of 1/1 motion and partial subharmonic motions, which may aggravate the pitting fatigue in the mechanical system. Therefore, multiparameter cosimulation can better provide theoretical guidance for the parameters design of impact vibration systems.



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*Source: 6634013-2021-03-28.xml*