Optimal Design of Bistable Nonlinear Energy Sink Based on Global Vibration Control

: Although a linear vibration absorber (LVA) or nonlinear energy sink (NES) can effectively mitigate the vibration of the main system in harmonic excitation, the amplitude of the absorber can be very large. Using the single-objective differential evolutionary (DE) algorithm, this paper pioneers the global control of the main system and a bistable NES to achieve decent vibration mitigation effects and decrease the global response of the system. Using the multi-objective DE algorithms and comparing the optimization results of the LVA, this paper proposes multi-objective and multi-parameter design criteria of a bistable NES. The mass ratio, the displacement amplitude, and the mechanical energy are optimization goals. The results show that the maximum amplitude of the main system and the absorber can be controlled at the same level, and the global control strategy does not change the resonance frequency of the main system. Compared with the LVA, the bistable NES has similar vibration mitigation effects with the variation of the mass ratio in the multi-objective optimization. However, the bistable NES can achieve better control over a larger spring stiffness range. Therefore, through single- and multi-objective optimization design, this paper proves the superiority of the bistable NES in the vibration mitigation. Meanwhile, this paper provides an optimization design method for global vibration control.


Introduction
Vibration widely exists in mechanical systems and brings many hazards, so vibration mitigation has always been a hot issue for researchers and engineers. Applying vibration absorbers is an effective means of vibration mitigation. Traditional vibration absorbers, namely tuned mass dampers or linear vibration absorbers (LVAs), are widely used in the vibration mitigation of various systems, like multistory buildings [1], rectangular plates [2], and airfoil models [3]. The LVA is a simple mass-damping-spring structure, but it can only mitigate vibration in a narrow frequency band and may generate new resonance peaks. To cover these shortages of the LVA, Vakakis and Gendelman added strongly nonlinear stiffness into the absorber and realized 1:1 transient resonance at any frequency and irreversible targeted energy transfer from the main system to the absorber in the impulse excitation [4]; Chen et al. realized the multimode vibration mitigation and did not change the resonance frequencies of the composite plate by using the absorber with zero linear stiffness and cubic nonlinear stiffness in the harmonic excitation [5]. This kind of absorber with strong nonlinearity is called a nonlinear energy sink (NES). At present, the NES has been used in the mitigation of relaxation oscillation of moving belt systems [6], shock vibration of fluid-conveying pipes [7], transverse vibration of flexible bladed rotor systems [8], and stochastic vibration of two-story shear buildings [9].
Since the LVA and the NES have better energy robustness and frequency robustness in vibration mitigation respectively, researchers designed adjustable or hybrid absorber structures to realize both these two kinds of robustness. Habib and Kerschen used the equal-peak method to derive a tuning rule of a nonlinear vibration absorber (NVA) and show better performance than the LVA in harmonic excitation [10]. Habib and Romeo designed a tuned bistable NES with characteristics of the LVA in small amplitude and used this absorber to mitigate the vibration of impulsively excited linear systems in a broad energy range [11]. Benacchio designed a magnetic vibration absorber that can realize the NVA with positive linear stiffness, the monostable NES with zero linear stiffness, or the bistable NES with negative linear stiffness [12]. To broaden the initial energy range of effective vibration mitigation, Farid and Genderman chose proper parameter values to let a pendulum absorber or a hybrid vibro-impact NES act as the LVA in small energies and the rotational NES or the vibro-impact NES in large energies [13,14]. Taghipour et al. combined the LVA and the monostable NES to raise the vibration mitigation efficiency and broaden the stability range of a nonlinear rotor system [15]. Wang et al. also combined the LVA, the monostable NES, and impact surfaces to enhance both energy robustness and frequency robustness of the absorber in seismic response mitigation [16]. The above studies mainly focused on the improvement in the vibration mitigation effect of the absorbers. However, the absorbers with decent vibration mitigation effects may have large amplitudes, which will shorten their service life [16][17][18]. At present, the amplitude control of the absorbers is realized by adding protective devices, e.g. impact surfaces [18][19][20] or piecewise spring devices [21]. This paper will present a new control strategy, i.e. using the optimization algorithm to mitigate both the amplitudes of the main system and the absorber.
In the design of the NES, multiple vibration mitigation measures need considering, such as the amplitude of the main system, energy dissipation rate of the NES, energy decaying duration of the whole system, and transmissibility of the system [22][23][24][25][26][27][28]. Besides, other indexes, like the stability range of the response [29] and the economic factors of the structure [30], should be in consideration.
In addition, multiple parameters need to be optimized in different NES structures, e.g. the mass ratio and clearance of the vibro-impact NES [20,31,32] and the mass ratio, damping, spring stiffness, geometry parameter, and location of the bistable NES [7,33]. In the multi-objective and multi-parameter optimization of the NES, intelligent optimization algorithms can be used to describe the Pareto front (distribution of non-dominated solutions in the target space). For example, Raj and Santhosh combined a weighting approach and the genetic algorithm to demonstrate the Pareto front of displacement amplitude and voltage response of a harmonically excited nonlinear oscillator coupled with an NVA and an energy harvester [34]. Khazaee used the non-dominated sorting genetic algorithm to display the Pareto front of the expectation and variance of the energy dissipation rate of the NES with parameter uncertainties [7,35]. In fact, the intelligent optimization algorithm can not only show the characteristics of non-dominated solutions in the target space but show the characteristics of tuning parameters. The latter will help to guide the design of the NES.
However, there is little research on the extraction of the design criteria of the NES through the multi-objective intelligent optimization algorithm. On the other hand, the multi-objective design of the NES can refer to that of the LVA. For example, in Wang's study, the optimized single-sided vibro-impact track NES requires smaller damping versus the LVA [30]. In the authors' previous study, the multi-objective optimization results of the LVA and the bistable NES show high similarity in the impulsive excitation [17]. What about the similarities and disparities of the multi-objective optimization results of the LVA and the bistable NES in the harmonic excitation?
This paper will answer this question and propose new multi-objective and multi-parameter design criteria of the bistable NES.
According to the above discussion, this paper is structured as follows. Section 2 describes the models of a linear main system couple with an LVA or an NES. Sections 3 and 4 firstly propose the single-objective optimization issues on the amplitude control of the main system or the whole system in single-or sweep-frequency excitation conditions and then solve these issues by the differential evolutionary (DE) algorithm and the parameter sensitivity analysis. And the characteristics of the frequency responses of the optimal situations are analyzed. In Sections 5 and 6, the multi-objective optimization issues are proposed, and these issues are solved by the multi-objective DE algorithm. The characteristics of the Pareto fronts and the optimized parameters of the LVA and the NES are illustrated. Finally, Section 7 addresses the conclusions.    [36,37]. M1, C1, and K1 are the mass, damping, and spring stiffness of the LO. M2, C2, and KS are the mass, damping, and spring stiffness of the two absorbers.
L is the free length of the springs of the absorbers. L1 and L2 are the characteristic lengths of the springs of the NES in two directions. By adjusting the distance of the two fixes ends (2L1), the two symmetric springs of the NES can be unstretched or stretched. X1 and X2 are the displacements of the LO and the absorber, respectively. F0 and Ω are the amplitude and frequency of external harmonic force, respectively. By utilizing Newton's law, the dynamic equations of the LO-LVA and the LO-NES systems can be derived The dimensionless forms of Eqs. (1) and (2) are 1  1 1  1  2  1  2  s  1  2  2  0  2  2  1  1  2  2   2  2  2  1  s  1  2  2  2  2  1  1  2  2   1  2 1 cos 0, where the dot (•) is the derivative compared to t . The expressions of dimensionless variables and   parameters are   1  1  2  1  2  1  0  0  1  2  1  2  1  1  1 0   S  0  2  2  2  s  0  2  2  1 0  1 0  1  1  0  0   ,  ,  , In the reduced system, the LO only contains the damping parameter c1, while the LVA and the NES have three (ε, c2, and ks) and four (ε, c2, ks, and l1) independent parameters, respectively. They correspond to the mass ratio, damping, spring stiffness, and geometry parameter of the absorber.
When |x1-x2+l2|<l1, Eq. (4) can be converted into the dimensionless polynomial equations by using the Taylor series: 3  s  1  1 1  1  2  1  2  s  1  2  2  1  2  2  0  3  11   3  s  2  2  2  1  s  1  2  2  1  2  2  3  11   1  2 1 cos 0, where Eq. (6) keeps the first two terms of the Taylor series. From Eq. (4), it can be seen that the geometry parameter l1 determines the property of the NES (0<l1<1 for the bistable NES with negative linear stiffness, l1=1 for the monostable NES with zero linear stiffness, and l1>1 for the NVA with positive linear stiffness). The potential energy surfaces reflecting the characteristics of these three NVAs can be found in Ref. [17]. For the bistable NES, the smaller l1 is, the deeper the Substituting Eq. (7) into Eq. (3), and setting the coefficients of sin(ωt) and cos(ωt) as 0, the expressions of the amplitudes B1~B4 can be obtained, and the steady-state displacement amplitudes of the LO and the LVA (ALO and ALVA) can be computed by Then the optimization of the LO-LVA system can be easily conducted. However, the exact solutions are hard to be computed in the LO-NES system. In previous studies, the steady-state responses of the LO-NES system can be approximately analyzed by the slow invariant manifold [26], the complexification-averaging method [38], and the harmonic balance method [39]. Yet the time history and amplitudes of aperiodic responses, e.g. strongly modulated response and chaos, cannot be well predicted by these methods, while the Runge-Kutta method (a common way of numerical simulation) can. Therefore, this paper will use the results of the Runge-Kutta method to optimize the LO-NES system.
In this paper, the damping value of the LO c1 will be constantly set as 0.2. The parameter optimization ranges of the absorber are 10 -7 ≤ε≤0.02, 10 -7 ≤c2≤10, 10 -7 ≤ks≤20, and 10 -7 ≤l1≤2. The results of the Runge-Kutta method will be computed in 100 periods, and the last 20-period results will be left for computing the steady-state amplitudes. The time step is 1/200 of a period. In all the excitation conditions, the initial values of displacement and velocity are 0, and the range and step size of the excitation frequency ω are 0.8~1.2 and 0.02, respectively.

Single-objective functions and differential evolutionary algorithm
Since real structures may be excited in fixed or varied frequency, this paper will study the optimization issue of the LO-LVA and LO-NES systems in single-and sweep-frequency excitation conditions. As the main purpose of adding vibration absorber is to reduce the amplitude of the main system as much as possible, the single-objective function (SOF) will be set as min(ALO) and min(max(ALO)) in single-and sweep-frequency excitation conditions, respectively. Although the introduction of the absorber can mitigate the vibration of the main system, the amplitude of the absorber may be quite large. Therefore, by considering the maximum amplitude of the whole system, the SOF will also be set as min(max(ALO, ALVA or ANES)), where ANES denotes the displacement amplitude of the NES. In brief, this paper will study four types of SOFs in Table 1.
The decay rate of the target amplitude relative to the original LO amplitude (without the absorber) can be calculated in each SOF. Then the amplitude decay rate can be used to evaluate the vibration mitigation effects of the LVA and the NES in different objective and excitation conditions. The DE algorithm is often used to search for the optimal solution of nonlinear and non-differentiable problems and has the advantage of fast convergence and high robustness [40].
Here, the DE algorithm is used to optimize the parameters of the LVA and the NES in four single-objective optimization situations. The procedure of the single-objective DE algorithm is as follows: 8 (i) Initialization: The initial population [u1(i, 0), u2(i, 0),…, uD(i, 0)] denoting the values of ε, c2, ks, and l1 are randomly generated within the thresholds. D is the dimension of tuning parameters.
For the LVA, D=3; for the NES, D=4. i is the population number. 1≤i≤M, where M is the population size.
(ii) Mutation: In the jth iteration, three different members (number r1, r2, and r3) except the ith member are randomly selected in the population to generate the ith mutant member. The expression where F is the scale factor and satisfies the uniform distribution on the interval [-2,2]. Small and large absolute values of F are beneficial to searching the local and global optima, respectively.
(iii) Crossover: The mixed member w(i, j) is obtained by mixing the original member u(i, j) and where r4 is a random number uniformly distributed on the interval [0,1]. cr is the crossover probability. The greater cr is, the more information of the original member will be replaced by the mutant member. To replace the information of the original member as much as possible, cr is set as 0.9. Ir is a random integer of 1~D and is used to make the mixed member and the original member differ in at least one dimension.
(iv) Selection: Comparing the values of the single-objective function of the original member u(i, j) and the mixed member w(i, j), the optimal one will be left as the original member u(i, j+1) in the next iteration. If j is smaller than the maximum iteration jmax, then j=j+1 and back to Step (ii).
Otherwise, the trial is over.
Since the single-objective DE algorithm converges to the global optimal solution with a certain probability [41,42], 20 trials are carried out on the same optimization problem to search for the optimal result. Through convergence verification, the population size M and the maximum iteration

Numerical examples of single-objective optimization 4.1 Results of the DE algorithm in single-frequency excitation conditions
In this subsection, the results of SOF-1 and SOF-2 of the LO-NES system will be first presented. Then the comparison between the LVA and the NES will be conducted. Fig. 3 shows the objective results of the LO-NES system. For the SOF-1 in Fig. 3  Corresponding to the optimization results of amplitude decay rate in Fig. 3, Fig. 4 and Table 2 present the results of four optimized parameters of the NES. It can be seen that all the values of the optimized mass ratio ε are close to its upper limit (10%), which indicates better vibration mitigation effects prefer larger mass of the NES. For the values of the optimized damping c2, differences can be found in the SOF-1 and the SOF-2. That is, the values of c2 are very close to 0 in the SOF-1.
Combined with the results of Fig. 3(a) and Fig. 3(c), the NES exhibits anti-resonance characteristics similar to those of the undamped LVA in the SOF-1. In the SOF-2, most values of c2 are confined in the range 0.01~1, and the value of c2 will not be affected by f0 but will increase when ω increases.
For the optimized spring stiffness ks, most values are confined in the range 0.01~10. For the optimized geometry parameter l1, most values are larger than 1, which corresponds to the NVA with positive linear stiffness. These tuning parameter results will help to the practical design of the NES in single-frequency excitation conditions.  In the SOF-1, the optimal NES is similar to the undamped LVA. What about the similarities and disparities of the optimal NES and the optimal LVA in the SOF-2? The results in Table 3 show that the optimal NES and the optimal LVA have almost the same values of the mass ratio ε, the damping c2, and the amplitude decay rate RA,2 (vibration mitigation effect) in different excitation conditions. Besides, both the NVA with positive linear stiffness (l1>1) and the bistable NES (l1<1) can achieve the optimal control of the LO and the absorber, and the differences in the optimized values of the geometry parameter l1 will cause those of the NES spring stiffness ks.   Table 4 and Table 5. Decent vibration mitigation effects can be witnessed in the SOF-3 and the SOF-4. Since the SOF-4 includes the control of the amplitude of the absorber, the optimal amplitude decay rate in Table 5 is not up to that of the SOF-3 in Table 4. However, the value of RA,4

Results of the DE algorithm in sweep-frequency excitation conditions
in Table 5 is very close to the value of RA,2 with ω=1 in Table 3, which indicates the similar global control effects in both single-and sweep-frequency excitation conditions. Besides, for the optimal NES and the optimal LVA, the similarities can be found in the mass ratio, the damping, and the vibration mitigation effect in Table 4 and Table 5. Yet all the results of the geometry parameter l1 are larger than 1, which may cause some misunderstanding that only the NVA with positive linear stiffness can achieve the optimal vibration mitigation effect in the sweep-frequency excitation condition. Since the optimal parameter combinations of the NES are not unique, and the DE algorithm easily converges to certain positions (like l1>1) but misses some other important information (like the bistable situation with l1<1) in the parameter space, other approaches are needed to capture this important information.

Results of the parameter sensitivity analysis
In Subsection 4.1 and 4.2, the parameters and amplitude decay rates of the optimal LVA and the optimal NES are obtained by the DE algorithm in different objective and excitation conditions. However, it is very difficult to achieve the optimal vibration mitigation effect due to the influence of parameter uncertainty and manufacturing error in practical engineering. So the parameter range to achieve decent vibration mitigation effects needs to be found out. Combined with the above DE algorithm results and the parameter sensitivity analysis, this task can be well accomplished, and more differences in the parameter design of the absorbers may appear. According to the results in Table 3, Table 4, and Table 5, this subsection will present the influences of the absorber parameters on the vibration mitigation effect in three SOFs and two excitation strength conditions (f0=0.005 or 0.1). The mass ratio ε of the absorbers will be set as 0.1 (the upper limit value).  Table 3, Table 4, and Table 5. On the other hand, the parameter sensitivity analysis can more comprehensively reflect the optimal parameter range of the NES than the DE algorithm, especially the bistable NES.  In Fig. 5 and Fig. 6, the effective combinations of the spring stiffness ks and the geometry parameter l1 are searched in the optimal damping condition. These results show that the NES can achieve decent vibration mitigation effects within a wide range of the geometry parameter. Then, among the bistable NES, the monostable NES (l1=1), or the NVA with positive linear stiffness, which type of the NVA has a wider effective parameter range? Taking the monostable NES as a reference, the bistable NES with l1=0.99 and the NVA with l1=1.01 are selected to compute the results of amplitude decay rates in different SOF and excitation conditions, and the effective ranges of the damping c2 and the spring stiffness ks are found, as shown in Fig. 7, Fig. 8, Fig. 9, and Table   6. It can be seen that, in most cases, the vibration mitigation effect of the NES is decided by both the damping and the spring stiffness, while that of the monostable NES in Fig. 7(b), Fig. 8(b), and Fig. 9(b) is basically independent of its spring stiffness in small excitation (f0=0.005). In this situation, the monostable NES has optimal stiffness robustness. Since the dynamic equation of the monostable NES can be converted into the form of zero linear stiffness and cubic nonlinear stiffness by Taylor series in small excitation, the monostable NES does not need tuning stiffness to achieve the optimal vibration mitigation. However, the vibration mitigation effect of the optimal monostable NES (the upper limit value of the amplitude decay rate) is inferior to the optimal LVA, the optimal bistable NES, and the optimal NVA with positive linear stiffness. In large excitation, the solution of the dynamic equation is not in the convergence domain of Taylor series because of the large response of the system. At this time, the monostable NES does not have the characteristics of zero linear stiffness, and tuning stiffness is needed to achieve optimal vibration mitigation. That is also why the results of parameter analysis of the monostable NES in large excitation are very close to those of the bistable NES and the NVA with positive linear stiffness. In addition, the results in Fig.   7, Fig. 8, Fig. 9, and Table 6 indicate the optimal stiffness robustness of the bistable NES among the three types of the NVA in large excitation. Since the NES has a linear damping structure, the NES also needs tuning damping to realize the optimal vibration mitigation, and the effective range of the damping c2 of the three NVAs is almost the same as that of the LVA.    Table 6 Upper limit values of effective spring stiffness of the NES in different SOF, excitation strength, and geometry conditions in Fig. 7, Fig. 8, and Fig. 9.

Objective function f0
Lower limit of RA

Frequency responses of the LO with the optimized absorber
The above three subsections present the similarities and disparities of the effective parameter combinations of the LVA and the NES in ε=0.1 and different objective and excitation conditions.
So what are the similarities and disparities of their dynamic responses? This subsection will study this issue. Since the system in sweep-frequency excitation contains more dynamic behaviors than the system in single-frequency excitation, this subsection will focus on the frequency responses in the SOF-3 and the SOF-4. Since the results in Fig. 6 imply the poor performance of the bistable NES in a large range of geometry parameter l1 in large excitation, the authors seek for the explanation by comparing the frequency responses and the optimized parameter values of the optimal bistable deep-well NES (l1=0.9), the optimal bistable shallow-well NES (l1=0.99), and the optimal monostable NES (l1=1).
The results in Fig. 10(d) show that the optimal damping value of the bistable deep-well NES differs from those of the bistable shallow-well NES and the monostable NES, and the bistable deep-well NES system has six kinds of responses in the whole frequency domain in large excitation (f0=0.1).
Numbers 1~6 in Fig. 10    affected by the nonlinear stiffness, the frequency responses of the optimal bistable shallow-well NES (l1=0.99) and the optimal monostable NES (l1=1) are smaller than that of the optimal LVA and also maintain the single-peak characteristics in large excitation, which indicates the possible better 20 performance of the NES than the LVA in the global control. Although the optimal bistable deep-well NES (l1=0.9) does not produce aperiodic responses in Fig. 12(d), its vibration mitigation effect still cannot reach that of the optimal bistable shallow-well NES and the optimal monostable NES in Fig. 12(e) and (f) due to the difference of the optimal damping values.

Multi-objective functions and differential evolutionary algorithm
Sections 3 and 4 use the steady-state displacement amplitude to optimize the parameters of the absorbers. To fully display the vibration mitigation effect of the absorbers, researchers often examine multiple indexes, such as transmissibility and energy indexes [17,27,28]. Besides, not only mitigation effects but economic factors need considering in the design of the absorbers [30].
For example, Wang et al. took the displacement, damping force, and restoring force of the absorbers as the indexes of economic factors [30]. In fact, the economic factors also depend on the mass of the absorber; that is, manufacturing and maintenance costs will increase as the mass of the absorber increases. On the other hand, the previous study [17] and the above single-objective optimization results show that the larger the mass of the absorber, the more conducive to vibration mitigation.
Therefore, the mass of the absorber is a very important index, and the parameter design of the absorber may differ in different requirements of the mass and the vibration mitigation effect.
Therefore, the following content of this paper will present the multi-objective and multi-parameter design of the absorbers by taking the mass ratio ε and multiple vibration mitigation indexes as objective functions and using the multi-objective DE algorithm.
Combined with the amplitude index, the mechanical energy and the input energy of the system will also be adopted for evaluating the vibration mitigation performance.
where tend=200π/ω and dt=π/(100ω where tstart=160π/ω. Combining the mass ratio, the three mitigation performance measures, and the four SOFs in Table 1, this paper will study the optimization issues in four multi-objective functions (MOFs), as shown in Table 7.  The external archive and the non-dominated sorting are two common ways to deal with the multi-objective optimization problem [43,44]. Accordingly, the flow charts of the multi-objective DE algorithms based on the external archive and the non-dominated sorting are shown in Fig. 13 and   (13) For convenience, the upper limit of the population size of the external archive or the non-dominated sorting group is set as the original population size M.  Besides, to accelerate the convergence of the EADE and NSDE algorithms, the upper limit of each vibration mitigation index (the original LO amplitude, mechanical energy, and input energy) is set, and the radial basis function (RBF) network is introduced. The RBF network is a very effective parameter identification method and composed of an input layer, a hidden layer, and an output layer.
The characteristics and training process of the RBF network can be referred to Ref. [45]. In the training process of the current population, the variables of the input layer are the mass ratio ε and the target amplitude A, and the variables of the output layer are the three parameters of the LVA or the four parameters of the NES. As the training process is completed, the mass ratio and the target amplitude of the current population are firstly adjusted and then set as the input layer, and the adjusted parameter values of the absorber are calculated by the trained RBF network. The adjustment strategy of the mass ratio ε* and the target amplitude A* is where n1 and n2 are the random numbers uniformly distributed in the interval of (0.9,1). Since there may be some deviations in the results of the RBF network, the adjusted parameter values of the absorber are returned to the dynamic equation (3)  6 Numerical examples of multi-objective optimization 6

.1 Results of the Pareto fronts
In this subsection, the results of the Pareto fronts will be used to judge the convergence of the EADE and NSDE algorithms and compare the vibration mitigation effects in the NES and the LVA and different mass ratio ε and excitation strength f0 conditions. The results on the three objective planes that reflect the trend of the vibration mitigation effects (decay rates) changing with the mass ratio ε will be presented as the Pareto fronts of the four-objective optimization problem. the effectiveness of the two algorithms, and the higher computational efficiency of the NSDE algorithm are discovered. In the following calculation of sweep-frequency excitation condition, the NSDE algorithm will be adopted, and jmax will be set as 200.   between the global control effect and the mass ratio can also be witnessed in Fig. 16(d), (e), and (f).
In brief, the results of the Pareto fronts in Fig. 15 and Fig. 16 will help to find the solutions satisfying the actual requirements of the mass ratio and vibration mitigation.   Table 8, where l1,max, l1,min, l1,mean, and l1,std denote the maximum, minimum, mean value, and standard deviation of the optimized l1 in different ε. The results in Table 8 show that the optimization values of l1 have little change in different ε. The control of the LO is achieved by the NVA with strong positive linear stiffness (l1>1), while the control of the whole system is achieved by the NES close to the monostable state (l1≈1),

Results of the optimized parameters of the absorbers
i.e. the bistable shallow-well NES and the NVA with weak positive linear stiffness. Since the bistable deep-well NES and the monostable NES do not have optimal vibration mitigation effects in certain excitation conditions of single-objective optimization, they are also absent from the multi-objective optimization. As the single-objective optimization results in Table 3, Table 5, and   From the above analysis, the multi-objective and multi-parameter design procedure of the LVA and the NES can be summarized: (i) Determine the target of vibration mitigation, such as the vibration control of the main system or the whole system.
(ii) Determine the appropriate range of the geometry parameter l1 in the bistable shallow-well NES and the NVA with positive linear stiffness.
(iii) Find the appropriate range of the mass ratio ε by the Pareto fronts and the actual demand.
(iv) For the LVA, the optimal ranges of the damping c2 and the spring stiffness ks will be fixed by the positive linear relations with ε.
(v) For the NES, the optimal range of c2 is the same as that of the LVA, while the optimal range of ks will be adjusted by l1 and excitation strength f0. The closer the NES is to the monostable state or the smaller f0 is, the larger ks is.

Conclusions
This paper proposes a new vibration control strategy in harmonic excitation, namely the global control of the linear main system coupled with the absorber by using the differential evolutionary (DE) algorithm. Compared with the large amplitude of the absorber and the variation of the resonance frequency in the local control of the main system, the global control can diminish the maximum amplitudes of both the main system and the absorber and maintain the resonance frequency of the main system. So the global control strategy has high application value.
This paper also compares the similarities and disparities between the linear vibration absorber (LVA) and the nonlinear energy sink (NES) in vibration mitigation and parameter design in singleand multi-objective optimization. In the single-objective optimization, the optimal LVA and the optimal NES have almost the same vibration mitigation effects, mass ratio, damping, and frequency responses in small excitation. Different from the LVA, the NES has multiple optimal combinations of spring stiffness and geometry parameter; the NES can achieve decent vibration mitigation effects in a wider range of spring stiffness, where the monostable NES and the bistable NES have the best stiffness robustness in small and large excitation conditions, respectively; the optimal NES has better frequency response than the optimal LVA in large excitation.
In the multi-objective optimization, mass ratio, displacement amplitude, mechanical energy, The global control strategy proposed in this paper is implemented in a single-degree-offreedom main system. If this strategy is applied to the multi-degree-of-freedom main system, the vibration mitigation effect and the influence on the resonance frequency need further investigation.
Besides, the linear damping structure of the absorbers has great impacts on vibration mitigation. In future work, the damping structure of the absorbers will be improved for higher damping robustness. Figure 1 Model of an LO coupled with an LVA.            Flow chart of the external archive DE algorithm.

Figure 14
Flow chart of the non-dominated sorting DE algorithm.