Abstract
The nonlinear energy sink (NES) is an effective method to weaken or eliminate the line spectra feature of ship machinery equipment. However, due to the strong nonlinearity of the NES, there are multistable attractors in the NES vibration absorption system. Moreover, when the excitation characteristics or working conditions of the mechanical equipment change, the NES may migrate from the small-amplitude motion state to the large-amplitude motion state, resulting in a high-amplitude solitary branch response and thereby lowering its vibration absorption efficiency. In this study, the small-amplitude control of the NES nonlinear vibration absorption system has been studied by using the open-loop-plus-nonlinear-closed-loop (OLPNCL) method. First, the dynamic equation of the NES nonlinear vibration-absorbing system was formulated, and its global behavior was analyzed. The regulations of global features and coexisting attractors were found out. Second, the migration between different attractors was carried out under the OLPNCL control, which can ensure that the system always works at the lowest line spectrum intensity and the best overall vibration absorption performance. Finally, the simulation results verify the feasibility and effectiveness of the OLPNCL method and achieve effective vibration absorption in the low-frequency range and line spectrum control under variable operating conditions.
1. Introduction
With continual interest in expanding the performance envelope of engineering systems, nonlinear components are increasingly utilized in real-world applications. A recent trend in the technical literature is to exploit nonlinear dynamical phenomena instead of avoiding them [1–3]. Nonlinearity is also more and more utilized for vibration mitigation. The applications of nonlinear vibration absorbers to eliminate undesirable vibration of linear structures have been examined by numerous researchers [4–6]. Recent developments in passive nonlinear vibration absorbers include the NES [7, 8] and nonlinear viscous dampers [9, 10]. Actually, in view of narrow bandwidth, the linear tuned mass damper or dynamic vibration absorber can be ineffective when the primary structure exhibits frequency–energy-dependent nonlinear oscillations [11]. Many researchers addressed the vibration mitigation of nonlinear mechanical systems using nonlinear dynamical absorbers [12–14].
As it is known, the NES is a candidate for nonlinear vibration absorbers for vibration mitigation of nonlinear mechanical structures. An NES is commonly composed of a small mass with respect to the primary structure, a viscous damper, and a spring with zero or quasi-zero linear stiffness but strong nonlinear stiffness. Thanks to the essential nonlinearity, the kinetic energy of the primary structure can be transferred, localized, and diminished within the NES over a broad frequency spectrum, which is known as targeted energy transfer (TET) [15]. It has gained considerable interest in a range of applications [6], including vibration attenuation in civil engineering [16, 17], vibration suppression in mechanical engineering [18, 19], vibration mitigation in aeronautics and astronautics [20, 21], and energy harvesting with NESs [22, 23].
However, Dudkowski et al. [24] studied hidden attractors in dynamic systems and found that hidden attractors have small basins of attractions, and the system evolution on them is very sensitive towards external perturbations (noise), initial conditions, and small changes in the system’s parameters. Typically, even small perturbations can lead to unexpected switches to a different attractor. Pisarchik and Hramov [25] pointed out that multistable systems are very sensitive to parameter perturbations that can cause qualitatively different behavior, especially near the bifurcation point, where a tiny change in a control parameter may result in the emergence of a large number of attractors. Meanwhile, the amplitude of coexistent attractors varies greatly, so the performance of vibration isolation highly depends on initial conditions and system parameters, which means that it is possible to jeopardize the quality of vibration isolation and even amplify the vibration. A vibration system with multiple interference sources is highly prone to mitigating from a small-amplitude to a large-amplitude motion state, thus lowering the vibration absorption efficiency of the NES. Therefore, a control method is urgently needed to quickly transform the motion of other attractors to the minimum vibration of the foundation. The global control method is developed from open loop and closed loop to open-plus-closed-loop and open-plus-nonlinear-closed-loop (OPNCL) [26]. The former three methods have some shortcomings, such as overstrict initial value limit and overcomplicated basins of migrating range determination. Starosvetsky and Gendelman [27] designed an NES with a properly tuned piecewise-quadratic damping element, which allowed the complete elimination of undesirable periodic regimes under certain parameter conditions. Eason [28] added an adaptive length pendulum (ALP) device to the NES structure and adjusted the pendulum length through the semiactive control technology to make the system out of the regions of solitary branch response. Liu et al. [11] designed an NES structure with adjustable parameters using two inclined spring-damper elements, which could remove the high-amplitude solitary branch response by adjusting the initial inclination angle of the spring-damper elements. Zang et al. [29] proposed a lever-type NES, which could suppress the solitary branch response by adjusting the fulcrum position of the lever. But so far, the above-mentioned control methods still have some technical challenges, such as high control energy consumption and strict requirements for the stability of the control track. If we hope to realize that the large-amplitude periodic attractor migrates to the small-amplitude chaotic attractor accurately and quickly by smaller control energy and fewer control numbers, some further research still needs to be carried out.
In this paper, a migration control algorithm based on the deterministic relationship between different attractors and their basins of attraction is introduced, which can quickly migrate the mechanical equipment response to a small-amplitude attractor when it is at a large-amplitude attractor or maintains it at a small-amplitude attractor when it has already been so. The proposed method can improve the stability of the NES and enhance its engineering application value.
The rest of this paper is organized as follows: In Section 2, the dynamic model of the NES vibration-absorbing system is established, and the global properties of the system are obtained using the cell mapping method [30–32]. In Section 3, the migration control method for multiple attractors in nonlinear systems is introduced in detail. Finally, in Section 4, simulation calculations are carried out for the migration control between different periodic attractors, periodic attractors, and chaotic attractors, and the effectiveness of the migration control method is verified.
2. Dynamic Modeling and Global Behavior Analysis of the NES Vibration Absorption System
2.1. Dynamic Modeling
The dynamics model of a general dynamic vibration absorber is built for the mechanical equipment coupled with an NES, as shown in Figure 1.
The main system is composed of , , and , among which is the mechanical equipment to be damped, which is connected to the rigid base through the linear spring with stiffness and damper . The NES, comprising mass , stiffness , and damper , is installed on the upper layer of the mechanical equipment and coupled with the mechanical equipment. When the external excitation signal is applied to the mechanical equipment, the vertical displacements generated by the mechanical equipment and the NES are and , respectively. The NES satisfies the cubic stiffness characteristic, and the restoring force of the nonlinear spring is .
The dynamics equations of the system are
The periodic operation of the ship hydraulic pump, air compressor, and other mechanical equipment will generate mechanical vibration, and the vibration will excite the hull through the transmission path such as the floating raft and the base to form the line spectrum component in the underwater radiated noise. Therefore, we focus on the study of the vibration suppression effect of nonlinear energy wells under harmonic excitation.
The excitation form follows the equitation , in which is the amplitude of the harmonic excitation and is the frequency of excitation.
The dimension of length is introduced, given by , and corresponds to the static deformation of the linear spring under the action of the mass , and dimensionless transformations are performed:
The dimensionless dynamics equations of the system under the condition of harmonic excitation can be obtained asHere,
Equations in (3) can be rewritten in the form of a space state equation set aswhere is the state vector of the system and is the polynomial.
2.2. Global Bifurcation Analysis
An important prerequisite for realizing the small-amplitude control method of the attractor is an in-depth and comprehensive global bifurcation analysis of the dynamical system. Only in this way, the various attractors in the system can be accurately controlled according to their dynamic characteristics, so as to achieve the ideal control effect. In this study, we focus on the influence of the excitation force amplitude on the global bifurcation behavior and the global bifurcation behavior of mechanical equipment and the NES system, which can provide prior information for the migration control of attractors.
In our study, the 4th Runge–Kutta method is applied, and the simulation software is MATLAB/Simulink. With the integration time set as 1000 periods, the response value within 500 periods, after the system motion state stabilizes, is used as the data source. The excitation force amplitude is controlled to change within a large parameter range, , with a step length of , and the following parameters are set: , , , , and . As shown in Figure 2, the global bifurcation diagram and the maximum Lyapunov exponent spectrum of the NES system is in the parameter range of . It can be seen from Figure 2, regardless of the forward continuation or backward continuation, the system exhibits very complex dynamic characteristics when the amplitude of the excitation force changes within a large parameter range. Periodic motion, quasiperiodic motion, and chaotic motion all exist in the system, in which stable periodic motion takes the dominant position, while quasiperiodic motion and chaotic motion are mainly located in the amplitude range of certain excitation forces. Therefore, by changing the excitation amplitude, the NES vibration absorption system can realize the migration of different motion states. In addition, multistable attractors coexist in the system with multiple parameter ranges, such as there are two period-1 attractors that coexist when , one period-1 attractor and one quasiperiodic attractor when , one period-3 attractor and one chaotic attractor when , and one period-2 attractor and one chaotic attractor when .
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The coexistence of multistable attractors is determined by the multivalued nature of the nonlinear system, which enables each attractor to have its corresponding basin of attraction. Therefore, when the excitation force amplitude is in a typical interval, the cell mapping method should be used to further determine the corresponding relationship between the attractors and their basins of attraction in the NES system.
In the typical interval I, the excitation force amplitude is set to 0.68. There are two stable period-1 attractors in the system response, whose phase trajectories are shown in Figure 3. As illustrated, attractor has a larger amplitude than attractor A2. According to the cell mapping method, if the analysis plane and are selected, with the initial conditions of the fixed variables and both set to be 0, then it can be divided into 100 × 100 discrete cells (totaling 40,000) as the primitive ones. The number of parallel computing , and the calculation results are shown in Figure 4(a). In the figure, the position of attractor is marked by “,” and the red areas are its basins of attraction; the position of attractor A2 is marked by “,” and the cyan areas represent its basins of attraction. It can be seen that the area of basins of attraction occupies 86.39% of the entire analysis plane, while for basins of attraction, the value is 13.61%. However, if the analysis plane selected are and , with the initial conditions of the fixed variables and still set to be 0, the calculation result is shown in Figure 4(b). It can be known that in this analysis plane, basins only occupy 3.05% of the total area, with the remaining 96.95% occupied by basins.
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In the typical interval II, the excitation force amplitude is set to 19.5. There is a stable period-3 attractor and a stable chaotic attractor in the system response, whose phase trajectories are shown in Figure 5. If the analysis plane and are selected, with the initial conditions of the fixed variables and both set to 0, then it can be divided into 200×200 discrete cells (totaling 40,000) as the primitive ones. The number of parallel computing , and the calculation results are shown in Figure 6(a). In the figure, the “” represents the position of the period-3 attractor, while the green area represents the position of the chaotic attractor, with the red and cyan areas corresponding to the basins of attraction of the period-3 attractor and the chaotic attractor, respectively. As illustrated in Figure 6(a), in the analysis plane, the system response is dominated by chaotic motion; the chaotic attractor’s basins of attraction occupy 72.98% of the area of the entire plane, while the period-3 attractor’s basins only occupy 27.02%. Additionally, the two attractors are interwoven, and the overall behavior is complicated. If the analysis plane and are selected, with the fixed variables and , the calculation results are shown in Figure 6(b). In the analysis plane, the chaotic attractor’s basins of attraction occupy 72.98% of the area of the entire plane and the period-3 attractor’s basins only occupy 26.76%. Moreover, the fractal boundary of the two attractors’ basins of attraction in the analysis plane is not as obvious as in Figure 6(a), which suggests that the system response is sensitive to the initial condition changes of NESs.
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3. The Migration Control Method of the Attractor
3.1. Basic Theory of the Migration Control Method
Multivalued nonlinear systems generally exist and belong to multiple attractor subsystems, and each attractor has its own attraction domain. The transport control that realizes the conversion between different attractors is called migration control.
Consider the following nonautonomous continuous nonlinear dynamical system:where and is the polynomial degree of . We assume that there exists at least one nonzero parameter, for example, . is a switching function, and , which will mitigate violent reactions when control action is added. is the starting time, and when , the controller does not work, and when , the controller begins to work. is a control function, associated with and , where is the system response and is the target trajectory function. When the solutions of the system are transported to , it can be obtained as follows:where defines the error between and . In the process of transport control, the target trajectory can be a dynamic behavior with any topological properties, but it should be limited to a certain target domain , and and ; is the set of all attractor basins. In addition, transporting basins are defined as a set of continuous initial states that the system can be transported to the target trajectory that satisfy the condition . Once the initial state is determined and the control is applied, the system can run stably on the target trajectory , and neither further monitoring of the dynamic behavior of the system nor feedback information to stabilize the behavior is required. If a trajectory from an attractor A1 is transported to another attractor A2, the only requirement is that the starting state of the transporting trajectory should be chosen in the attractor basin of A1 and the ending state of the transporting trajectory should be chosen in the attractor basin of A2. The system trajectory enters the vicinity of the target trajectory to start the transporting control so that the system can run along the target trajectory . After entering the attractor basin of A2, the controller is turned off and the system can run stably on the attractor A2.
3.2. The Open-Loop-Plus-Nonlinear-Closed-Loop Control Method
Because the open-loop-plus-linear-closed-loop (OLPLCL) control method has a lower control effect for the system of the high-order polynomial. Therefore, we improve the linear-closed-loop part as nonlinear feedback of “the nonlinear function of the difference between the target orbit variable and the controlled system variable” and propose an open-loop-plus-nonlinear-closed-loop (OLPNCL) control method. The method has been successfully used to achieve generalized chaotic synchronization between the driving system and the response system [33].
Using the OLPNCL control method, equation (7) can be completed, and the control term iswhere the matrix is given by and is constant matrix, whose eigenvalues have negative real parts. is the nonlinear closed-loop control action, and its ith element is given by the following equation:
Substituting equations (8) and (9) into (6) of the control system and results in , we get is expanded in the error (5) for the small as shown in the following nonlinear approximation:
It is seen that has pth degree polynomials for general dynamics, and in the OLPNCL control, which gives the following error equation:
The right side of equation (12) does not contain the target function as the coefficient of . If is taken as , it is found that the error in equation (12) obeys . Since all eigenvalues of have negative real parts, the Lyapunov direct method is used to prove the asymptotic stability of the zero solution of this equation. Therefore, is not being empty sets and any of the basins of transporting associated with and is global.
3.3. The Migration Control of Coexisting Attractors in the NES System
The dimensionless dynamic equation of the dynamic vibration absorption system is shown in equation (5), considering the goal function as and the error equation as . It can be found that the equation is similar to equation (6), when equation (5) is added to the OLPNCL action:where
The constant matrix is taken as a diagonal matrix of eigenvalues with negative real part:
When , the error equation can be obtained as
After applying OLPNCL control to the NES system, error can quickly converge to 0, and its transmission domain is global. Therefore, for any smooth target orbit function, as long as can connect different attractors’ basins of attraction, OLPNCL control can make the response of mechanical equipment migrate from a large-amplitude to a small-amplitude attractor.
4. Numerical Simulation and Analysis
4.1. Migration Control of Different Periodic Attractors
When the parameters are set as , , , , , and , the system has a large-amplitude attractor with period-1 motion and a small-amplitude attractor with period-1 motion. Figure 7 shows the time history diagrams of different attractors when the initial conditions are (1,0,0,0) (red curve) and (0,0,0,0) (blue curve). In the figure, attractor A1 has a higher amplitude than attractor A2.
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When OLPNCL control is applied to the system (5) and matrix , target orbit functions , , , and . The positions of the target orbits in the phase space are shown in Figure 8, where the red areas are basins of attraction, the blue areas are basins of attraction, and the solid green lines represent the target orbits. As illustrated, in and analysis planes, both the selected target orbits are channels connecting the two attractors’ basins. The duration of control applied is set to greater than 300 s (the system has stabilized). When the system response is in basins, the control is applied, given by function . When the system response migrates to the target orbit and gets into basins, the control is removed, given by the function .
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Under OLPNCL control, the system response phase trajectories are shown in Figure 9. In the figures, the red solid line represents the trajectory of the system before the control is applied, the black solid line represents the transient trajectory during control application, the green solid line represents the trajectory of the system along the target orbit, the blue solid line represents the system trajectory after the control is removed, and the black arrow represents the changing direction of the system’s trajectory before and after the control is applied. Figure 9 shows the change curves of the system’s response amplitude with time during the control process. When , the vibrations are large-amplitude; when , it is the transient process running along the target orbit; when , the vibrations are small-amplitude. As illustrated by Figures 9 and 10, the vibration responses of the mechanical equipment and NES both can quickly migrate from a large amplitude to a small-amplitude motion state after a short transient process, thereby reducing the vibration amplitude of the mechanical equipment.
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4.2. Migration Control of Periodic and Chaos Attractors
When the system is under the conditions that , , , , , and , the period-3 attractor (A1) is found to coexist with the chaotic attractor (A2). Figure 11 presents the time history diagrams of different attractors under the initial conditions of (−1.25, 1, 0, 0) (red curve) and (−1, 1, 0, 0) (blue curve), respectively. As illustrated, A1 has a higher average amplitude than A2.
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The following conditions are set: matrix and target orbits , , , and . Figure 12 shows the position of the target orbit in the phase space, in which the red areas are the period-3 attractor’s basins of attraction, the blue areas are the chaotic attractor’s basins of attraction, and the green curve is the target orbit. It can be seen that the target orbit connects the two attractors’ basins of attraction in both the two analysis planes. Figures 13 and 14 present the phase trajectories and time history of the system response during the control process. As illustrated, the OLPNCL method enables the migration from a large-amplitude period 3 attractor to a relatively small-amplitude chaotic attractor.
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Figure 15 shows the power spectra of mechanical equipment responses before and after control is applied. As illustrated, when the mechanical equipment is in a state of chaotic motion, the line spectrum components are significantly reduced. Except for the characteristic line spectrum, frequency bands exhibit continuous spectrum characteristics. At the same time, the intensity of the characteristic line spectrum before and after the control which is applied decreases from 38.44 dB to 30.61 dB. Therefore, under this parameter, the OLPNCL method can not only lower the intensity of line spectra but also reconstruct their structure, thereby achieving the dual purpose of reducing the intensity of line spectra while hiding their characteristics.
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5. Conclusion
The dynamics model of the NES vibration absorber used for mechanical equipment was built and the global behavior of the model was analyzed by the cell mapping method. The regulations of global features and coexisting multistable attractors were found out, and the OLPNCL migration control algorithm was introduced, achieving migration control of multistable attractors. The main conclusions are as follows:(1)The NES system exhibited rich dynamic characteristics in a large parameter range, and the coexistence of multiple stable attractors occurred in multiple typical parameter intervals, with complicated global behaviors. In addition, the coexistence of attractors might occur between period and period , period and chaotic attractor , and different attractors were found to have varying amplitudes, which provided prior information for attractor migration control.(2)The transmission domain of the OLPNCL method was global in the phase space, and the target orbit could be given by an arbitrary vector function connecting two different basins of attraction. Besides, the transmission accuracy was not limited by the polynomial degree of the control equation, and the OLPNCL method was particularly suitable for complex nonlinear systems with the highest polynomial degree .(3)With the OLPNCL method, the migration control of two different types of coexisting attractors was achieved, making the system continuously and stably to operate on a small-amplitude attractor. In particular, when the system response migrated from a large-amplitude periodic attractor to a small-amplitude chaotic attractor, the response characteristic line spectrum intensity of mechanical equipment after the application of the control was reduced from 38.44 dB to 30.61 dB, and the number of line spectra was also significantly lowered. Therefore, the OLPNCL method could not only reduce the intensity of line spectra but also restructure them by taking advantage of the system’s chaotic response, thereby achieving the dual purpose of reducing the intensity of line spectra while concealing their characteristics.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The study was supported by National Natural Science Foundation of China (Grant no. 51579242), Natural Science Foundation of Hubei Province, China (Grant no. 2018CFB182), and Natural Science Foundation of Hubei Province, China (Grant no. 2022CFB405).