Vibration Reduction of a Timoshenko Beam with Multiple Parallel Nonlinear Energy Sinks

Abstract

A nonlinear energy sink is a promising device to reduce structural vibrations. In this work, the vibration reduction performances of multiple parallel nonlinear energy sinks attached to a short beam are investigated based on the Timoshenko beam theory. The dynamic equations of a vibration reduction system subjected to a harmonic excitation are established. The frequency responses are analyzed based on Galerkin discretization and the harmonic balance method, and the accuracy is verified by the Runge–Kutta method. An optimization method based on the genetic algorithm is proposed for the number, location, and cubic stiffness of the nonlinear energy sinks. The study reveals that, with the same total mass, multiple parallel nonlinear energy sinks can achieve a larger vibration reduction ratio than a single nonlinear energy sink. The parameter influences of the nonlinear energy sinks are revealed, and unstable responses with large cubic stiffness are presented. The optimal locations of the multiple parallel nonlinear energy sinks are related to low-order modal shapes. A larger reduction ratio on the resonant amplitude can be achieved compared to a uniform distribution of the nonlinear energy sinks. The optimal locations and cubic stiffness are related to the number of nonlinear energy sinks. In the studied case, the optimal number of nonlinear energy sinks was two.

1. Introduction

Passive vibration reduction has been widely used in civil engineering [1,2,3], spacecraft [4,5,6], and mechanical systems [7,8]. Traditional linear dynamic vibration absorbers (DVA) are effective within a specific narrow frequency band [9,10]. The concept of a nonlinear energy sink (NES) was firstly proposed by Vakakis [11] in 2001. Compared to the DVA, an NES has the advantages of a broad vibration suppression band, small additional mass, and not changing the natural frequency of the primary system. NES has been widely used for the vibration reduction of beams [12,13], plates [14,15,16], and shells [17,18,19].

Many researchers have applied the NESs to continuous beam structures and investigated their vibration reduction performance. Georgiades and Vakakis [20] utilized an NES to dissipate the vibration energy of a simply supported linear beam subjected to an impact excitation. Kani et al. [21] investigated a nonlinear simply supported beam coupled with an NES; a harmonic excitation was used. Liu et al. [22] investigated the vibration mitigation of the nonlinear principal resonance response of a beam-like lattice structure with an NES. Chouvion [23] investigated the nonlinear vibration of a beam with an NES under nonideal boundary supports. Zhang et al. [24] analyzed the effectiveness of an NES coupled to an axially moving string that was subjected to transversal wind loading. Chen et al. [25] investigated the effect of an NES attaching to a laminated composited beam subjected to harmonic excitation. It has been proved that NESs are effective for beams with different structures, excitations, and boundary conditions. Recent research shows that applying multiple parallel NESs can further improve vibration reduction performance. Vaurigaud et al. [26] studied the vibration response of a main oscillator coupled with multiple parallel NESs and optimized the damping ratio of the NESs to guarantee that energy pumping would occur. The vibration absorption rate of parallel NESs was significantly improved and was better than that of one NES. Savadkoohi et al. [27] experimentally studied the vibration reduction performance of a 4-DOF main oscillator coupled with two parallel NESs and demonstrated the efficiency of the parallel NESs. Zhang et al. [28] attached the NESs to axially moving beams and compared the vibration reduction efficiencies of one NES and parallel NESs. The results indicated that the parallel NESs with less total attached mass could obtain better vibration reduction. Chen et al. [29] compared the vibration absorption of single and two parallel NESs of equal mass and found that by tuning the nonlinear stiffness and damping ratio of NESs, the parallel NESs could eliminate the higher branches of the system more effectively. Li et al. [30] applied novel lever parallel NESs to a beam, and the results showed that the vibration absorption performance of lever-type parallel nonlinear energy sinks was better than lever-type single nonlinear energy sinks under the same attached mass. Most of the existing research focuses on the vibration reduction of long beams, with the modeling method based on the Euler beam theory. However, the method is unsuitable for short beams, especially with length/thickness ratios below 5. The Euler beam theory, without considering shear deformation, induces large errors in predicting the dynamic responses of short beams with NESs. There is a lack of research on the theoretical analysis of a short beam attached to multiple parallel NESs.

The parameters of NESs are essential to vibration reduction performance. Georgiades and Vakakis [20] carried out a sensitivity analysis to obtain the optimal nonlinear stiffness of NESs. Chen et al. [25] optimized the parameters of NESs with a sensitivity analysis. Mamaghani et al. [31] numerically investigated the effects of the damping ratio of NESs, as well as the magnitude of the external force to the beam, and obtained the optimized parameters of NESs. Kani et al. [32] optimized the parameters of NESs with both sensitivity analysis and the particle swarm optimization (PSO) method and obtained the best combination of mass and damping ratio of NESs. These studies indicate that parameter optimization is essential for improving the vibration reduction performance of NESs. Moreover, research has shown that optimization for the NESs’ locations can effectively improve their vibration reduction performance. Khazaee et al. [33] investigated the stochastic optimization of multiple parallel NESs when they are attached to a simply supported pipe; both parameters and locations were taken into account. The results showed that stochastic optimization had better robustness than deterministic optimization. Georgiades and Vakakis [34] performed parametric studies by varying the parameters and location of the NES; the results showed that the optimal position for the NES attachments is at the antinodes of the linear modes of the plate. Bab et al. [35] investigated the effects of position and damping of the NES on a rotating beam, and the results showed that the optimal position for connecting the NES to a rotating beam is at the beam tip. In the above research, the number of NESs is predetermined. It remains unknown whether there is an optimal number of NESs, and the relationship between the optimal location and number of NESs is not clear. Multi-variable optimization, including the number, location, and parameters of NESs, is expected to achieve a better design.

In this work, a short beam coupled with multiple parallel NESs is investigated, and a model is established based on Timoshenko beam theory. The parameter influences of the mass ratio, the damping ratio, cubic stiffness, and the locations of the NESs on vibration reduction performance are revealed. An optimization method is proposed based on a genetic algorithm for the multiple parallel NESs. The main contribution of this work lies in two aspects. (i) A continuum model is established for a beam coupled with multiple parallel NESs based on the Timoshenko beam theory, considering shear deformation; thus, the model applies to short beams. (ii) The number, locations, and parameters of the NESs are optimized simultaneously, and some new design guidelines for vibration reduction are provided based on the optimization results.

2. Methods

2.1. Modeling

A short beam coupled with multiple parallel NESs is shown in Figure 1. The length, thickness, and width of the beam are denoted by L, H, and B, respectively. Generally, a short beam is defined as the ratio L/H within the range of 2.5~5 [36], and the shear deformation cannot be ignored for short beams. There are N nonlinear energy sinks on the beam. The mass, cubic stiffness, and damping of each NES are denoted by m, k₃, and c, respectively. d_i is the distance of the ith nonlinear energy sink away from the left end of the beam. a is the place where a harmonic excitation F_e = F cos(2πft) is imposed, where F and f are the amplitude and frequency of the external harmonic excitation, respectively.

In this section, the equations of motion are derived from the Timoshenko theory. It takes the effects of shear deformations and rotary inertias into account, and it is more applicable to short beams, deep beams, and laminated beams.

According to the d’Alembert principle, the force and moment equilibrium equations can be written as:

$$\begin{array}{l}\rho A\left(x\right)\frac{{\partial }^{2}w\left(x,t\right)}{\partial t^2}=-\frac{\partial Q\left(x,t\right)}{\partial x}+q\left(x,t\right)+F\left(x^{*},t\right)\\ \rho I\left(x\right)\frac{{\partial }^2\theta \left(x,t\right)}{\partial t^2}=\frac{\partial M\left(x,t\right)}{\partial x}-Q\left(x,t\right)\end{array}\}$$

(1)

where ρ is the density of the beam material, A(x) is the area of the beam section, I(x) is the rotary inertial of the beam section, q (x, t) is a continuous distributed force on the beam, F (x*, t) is a concentrated force, imposed at x = x*, M (x, t) is the bending moment, and Q (x, t) is the shearing force.

The fundamental functions of the beam in material mechanics can be written as:

$$\begin{array}{l}M\left(x,t\right)=EI\left(x\right)\frac{\partial \theta \left(x,t\right)}{\partial x}\\ \theta \left(x,t\right)-\frac{\partial w\left(x,t\right)}{\partial x}=\frac{Q\left(x,t\right)}{\gamma A\left(x\right)G}\end{array}\}$$

(2)

where γ is a constant, determined by the section shape of the beam, and G is the shearing modulus, determined by the beam material.

Analyzing the relationship between shear force and deformation, the total slope of the beam deflection curve can be written as:

$$\partial w\left(x,t\right)/\partial x=\theta \left(x,t\right)-\psi \left(x,t\right)$$

(3)

where Ψ (x, t) is the shearing angle, also called slope loss.

Substituting Equations (2) and (3) into Equation (1), the vibration equation of a beam under external forces can be written as:

$$\begin{array}{l}EI\frac{{\partial }^{4}w\left(x,t\right)}{\partial x^4}+\rho A\frac{{\partial }^{2}w\left(x,t\right)}{\partial t^2}+C\frac{\partial w\left(x,t\right)}{\partial t}-\rho I\left(1+\frac{E}{\gamma G}\right)\frac{{\partial }^{4}w\left(x,t\right)}{\partial x^2\partial t^2}\\ +\frac{{\rho }^{2}I{\partial }^{4}w\left(x,t\right)}{\gamma G\partial t^4}=q\left(x,t\right)+\frac{\rho I}{\gamma AG}\frac{{\partial }^{2}q\left(x,t\right)}{\partial t^2}-\frac{EI}{\gamma AG}\frac{{\partial }^{2}q\left(x,t\right)}{\partial x^2}\\ +F\left(x^{*},t\right)+\frac{\rho I}{\gamma AG}\frac{{\partial }^{2}F\left(x^{*},t\right)}{\partial t^2}\end{array}$$

(4)

where C is the structure damping coefficient of the beam. When the beam vibrates freely, the free vibration function can be written as:

$$\begin{array}{l}EI\frac{{\partial }^{4}w\left(x,t\right)}{\partial x^4}+\rho A\frac{{\partial }^{2}w\left(x,t\right)}{\partial t^2}+C\frac{\partial w\left(x,t\right)}{\partial t}\\ -\rho I\left(1+\frac{E}{\gamma G}\right)\frac{{\partial }^{4}w\left(x,t\right)}{\partial x^2\partial t^2}+\frac{{\rho }^{2}I{\partial }^{4}w\left(x,t\right)}{\gamma G\partial t^4}=0\end{array}$$

(5)

where the 4th and 5th term in Equation (5) consider the shearing deformation and the effect of the moment of inertia, respectively.

After attaching N NESs to the beam, the coupled vibration equations of the system can be written as:

$$\begin{array}{l}EI\frac{{\partial }^{4}w\left(x,t\right)}{\partial x^4}+\rho A\frac{{\partial }^{2}w\left(x,t\right)}{\partial t^2}+C\frac{\partial w\left(x,t\right)}{\partial t}-\rho I\left(1+\frac{E}{\gamma G}\right)\frac{{\partial }^{4}w\left(x,t\right)}{\partial x^2\partial t^2}\\ +\frac{{\rho }^{2}I{\partial }^{4}w\left(x,t\right)}{\gamma G\partial t^4}=q\left(x,t\right)+\frac{\rho I}{\gamma AG}\frac{{\partial }^{2}q\left(x,t\right)}{\partial t^2}-\frac{EI}{\gamma AG}\frac{{\partial }^{2}q\left(x,t\right)}{\partial x^2}+F\left(x^{*},t\right)\\ +\frac{\rho I}{\gamma AG}\frac{{\partial }^{2}F\left(x^{*},t\right)}{\partial t^2}+{\displaystyle \sum _{i=1}^N\left(k_i{\left(w\left(d_i,t\right)-z_i\left(t\right)\right)}^3+c_i\left(\frac{\partial w\left(d_i,t\right)}{\partial t}-\frac{\mathrm{d}{z}_i\left(t\right)}{\mathrm{d}t}\right)\right)}\\ {\displaystyle \sum _{i=1}^N\left(m_i\frac{{\mathrm{d}}^2{z}_i\left(t\right)}{\mathrm{d}{t}^2}+k_i{\left(z_i\left(t\right)-w\left(d_i,t\right)\right)}^3+c_i\left(\frac{\mathrm{d}{z}_i\left(t\right)}{\mathrm{d}t}-\frac{\partial w\left(d_i,t\right)}{\partial t}\right)\right)}=0\end{array}$$

(6)

where m_i, k_i, and c_i are the parameters of the ith NES on the beam.

In order to express equations of motion in dimensionless form, the following variables are defined:

$$\begin{array}{l}{\overline{w}}_i\left(x,t\right)=\frac{w_i\left(x,t\right)}{L},\overline{x}=\frac{x}{L},{\overline{z}}_i=\frac{z}{L},\overline{t}=\frac{t}{L^2}\sqrt{\frac{EI}{\rho A}},\\ \overline{C}=\frac{C}{\sqrt{\rho AEI}},\overline{c_i}=\frac{c_i}{\sqrt{\rho AEI}},p=\left(1+\frac{E}{\gamma G}\right)\frac{I}{A{L}^2},\\ q=\frac{E{I}^2}{\gamma \mathrm{A}G{L}^4},\overline{K_i}=\frac{k_i{L}^6}{EI},r_i=\frac{I{\overline{K}}_i{L}^2}{\gamma A^{2}G},s=\frac{E{I}^2}{\gamma A^{2}G{L}^2},\\ \mu =\frac{L^3}{EI},\eta =\frac{I}{\gamma A^{2}G{L}^2},{\epsilon }_i=\frac{m_i}{\rho A},\overline{f}=\frac{f}{{\omega }_0}\end{array}$$

(7)

and thus, the equations of motion for the Timoshenko theory in dimensionless form are obtained as:

$$\begin{array}{l}\frac{{\partial }^2\overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial {\overline{t}}^2}+\frac{{\partial }^4\overline{w}\left(\overline{x},\overline{t}\right)}{\partial {\overline{x}}^4}+\overline{C}\frac{\partial \overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial \overline{t}}-\\ p\frac{{\partial }^4\overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial {\overline{x}}^2\partial {\overline{t}}^2}+\frac{q}{\gamma G}\frac{{\partial }^4\overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial {\overline{t}}^4}+{\displaystyle \sum _{i=1}^N\left(\begin{array}{l}\overline{K_i}{\left(\overline{w}\left({\overline{x}}_i,\overline{t}\right)-{\overline{z}}_i\left(\overline{t}\right)\right)}^3+\\ \left(\frac{{\partial }^2}{\partial {\overline{t}}^2}{r}_i{\left(\overline{w}\left({\overline{x}}_i,\overline{t}\right)-{\overline{z}}_i\left(\overline{t}\right)\right)}^3\right)\\ +{\overline{c}}_i\left(\frac{\partial \overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial \overline{t}}-\frac{\mathrm{d}\overline{z_i}\left(\overline{t}\right)}{\mathrm{d}\overline{t}}\right)\end{array}\right)}\\ +s\left(\frac{{\partial }^3\overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial {\overline{t}}^3}-\frac{{\mathrm{d}}^3\overline{z_i}\left(\overline{t}\right)}{\mathrm{d}{\overline{t}}^3}\right)=\mu \overline{F}\left(\overline{x^{*}},\overline{t}\right)+\eta \frac{{\partial }^2}{\partial {\overline{t}}^2}\overline{F}\left(\overline{x^{*}},\overline{t}\right)\\ {\displaystyle \sum _{i=1}^N{\epsilon }_i\frac{{\mathrm{d}}^2\overline{z}\left(t\right)}{\mathrm{d}{\overline{t}}^2}}+{\displaystyle \sum _{i=1}^N\left({\overline{K}}_i\left(\overline{z_i}\left(\overline{t}\right)-\overline{w}\left({\overline{x}}_i,\overline{t}\right)\right)+\overline{c_i}\left(\frac{\mathrm{d}\overline{z_i}\left(\overline{t}\right)}{\mathrm{d}\overline{t}}-\frac{\partial \overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial \overline{t}}\right)\right)}=0\end{array}$$

(8)

By eliminating the effects of shear deformation and rotary inertial from Equation (8), the equations of motion according to Euler–Bernoulli theory are obtained as follows:

$$\begin{array}{l}\frac{{\partial }^2\overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial {\overline{t}}^2}+\frac{{\partial }^4\overline{w}\left(\overline{x},\overline{t}\right)}{\partial {\overline{x}}^4}+\overline{C}\frac{\partial \overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial \overline{t}}+{\displaystyle \sum _{i=1}^N\left(\begin{array}{l}\overline{K_i}{\left(\overline{w}\left({\overline{x}}_i,\overline{t}\right)-{\overline{z}}_i\left(\overline{t}\right)\right)}^3\\ +{\overline{c}}_i\left(\frac{\partial \overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial \overline{t}}-\frac{\mathrm{d}\overline{z_i}\left(\overline{t}\right)}{\mathrm{d}\overline{t}}\right)\end{array}\right)}\\ =\mu \overline{F}\left(\overline{x^{*}},\overline{t}\right)\\ {\displaystyle \sum _{i=1}^N{\epsilon }_i\frac{{\mathrm{d}}^2\overline{z_i}\left(t\right)}{\mathrm{d}{\overline{t}}^2}}+{\displaystyle \sum _{i=1}^N\left({\overline{K}}_i\left(\overline{z_i}\left(\overline{t}\right)-\overline{w}\left({\overline{x}}_i,\overline{t}\right)\right)+\overline{c_i}\left(\frac{\mathrm{d}\overline{z_i}\left(\overline{t}\right)}{\mathrm{d}\overline{t}}-\frac{\partial \overline{w}\left({\overline{x}}_i,\overline{t}\right)}{\partial \overline{t}}\right)\right)}=0\end{array}$$

(9)

In order to study the steady-state response of the system, the Galerkin method combined with the harmonic balance method is used. Considering the geometric boundary conditions of simply supported ends, the $\overline{w}\left(\overline{x},\overline{t}\right)$, the vertical displacement can be assumed as:

$$\overline{w}\left(\overline{x},\overline{t}\right)={\displaystyle \sum _{j=1}^M\sin \left(\frac{j\pi \overline{x}}{\overline{L}}\right)}\left(A_j\sin \left(2\pi \overline{f}\overline{t}\right)+B_j\cos \left(2\pi \overline{f}\overline{t}\right)\right)$$

(10)

where M indicates the adequate truncated order mode shapes. For this work, the first four mode shapes are considered, and thus, M is set as 4.

In order to solve the dimensionless nonlinear equations of motion in Equation (8), Equation (10) is substituted into Equation (8), and 4 + N discrete nonlinear equations of the coupled system are obtained. Equaling the harmonic coefficients of sine and cosine at both ends, respectively, a system of 8 + 2N nonlinear equations is obtained. Then, the Newton–Rapson method, combined with the arc-length method, is used to obtain the approximate analytical solutions. Finally, the amplitude frequency curves are obtained for the fixed values of system parameters and varying the excitation frequency.

2.2. Optimization

The performance of NESs is affected by the parameters, number, and locations of the NESs. The optimization objective is to maximize NES efficiency at the first resonance frequency. In order to compare the vibration reduction performance of NESs, the optimization problem can be described as:

$$\begin{array}{l}\max {\eta }_{\mathrm{NES}}=\frac{{\displaystyle {\int }_0^{\overline{t}}\left({\displaystyle \sum _{i=1}^N{c}_i\left(\overline{w}\left({\overline{x}|}_{\overline{x}={\overline{d}}_i},\overline{t}\right)-{\dot{z}}_i\right)}\right)}}{{\displaystyle {\int }_0^{\overline{t}}F\left(t\right)\overline{w}\left({\overline{x}|}_{\overline{x}={\overline{x}}^{*}},\overline{t}\right)}}\\ \mathrm{s}.\mathrm{t}. \{\begin{array}{l}0\le N\le 5, N\in N^{+},\\ {10}^5\le k_3\le {10}^9{ \mathrm{N}/\mathrm{m}}^3,\\ 0\le {\overline{d}}_i\le 1, {\overline{d}}_i=0.05\times N^{+}\end{array}\end{array}$$

(11)

where the NESs’ efficiency [20] is selected as the indicator, denoted by η_NES, and N⁺ refers to the positive integer set. To obtain an acceptable result, the responses of the beam must be stable, and unstable nonlinear phenomena are not supposed to occur.

Finding the optimal values of k₃ and the locations for NESs is a global searching problem. Exhausting all the possible combinations is a time-consuming process. In this study, a genetic algorithm is employed to solve the problem. The flowchart of the optimization approach is shown in Figure 2. The first step is to code for the k₃ and the locations for the NESs. When optimizing the k₃ for an NES, the chromosomes are coded with a real coding method. The number in the chromosomes array refers to the real value of k₃. When optimizing the locations for NESs, the binary coding method is used, and the number “1” and “0” in the strings indicate whether there is a nonlinear energy sink mounted there. Five hundred chromosomes are generated as the first generation to guarantee its diversity. After calculating the fitness function η_NES of the first generation, the best chromosomes are selected. Then, the “cross” and “mutation” operators are performed. The mutation operator here is set at 0.5, which can prevent the algorithm from being trapped into a locally optimal solution. The maximum number of iterations Gen_max is set at 20 to guarantee that the algorithm will have good quality as well as convergence.

3. Analysis Results

In this section, the difference between these two types of beam theories is discussed. The frequency responses are demonstrated. The influences of the NES parameters on vibration reduction performance are revealed. The optimizations of k₃ and the configuration of multiple parallel NESs are carried out with the genetic algorithm.

3.1. Modal Analysis

The performance of NESs is affected by the parameters, number, and locations of the NESs. Modal analysis is performed to present the differences between Timoshenko beam theory and Euler beam theory. A finite element beam is established with ABAQUS 6.14 to verify the accuracy of the first four resonance frequencies, and a B21 beam element is used. For a short beam [36], the ratio of length L to height H is from 2.5 to 5. Here, the constant parameters for the short beam and the nonlinear energy sinks are fixed to L = 1 m, H = 0.2 m, B = 0.2 m, E = 2.1 GPa, ρ = 7.8 × 10³ kg/m³, a = 0.2 m, and F = 1 × 10⁵ N. The first four modal frequencies of the beam based on each theory are listed in

Table 1. Comparison of the first four modal frequencies between two beam theories.

Order	Timoshenko Theory (Error)	Euler Theory (Error)
1st	43.94 Hz (0.06%)	46.92 Hz (6.71%)
2nd	152.75 Hz (0.24%)	187.72 Hz (22.89%)
3rd	292.15 Hz (0.34%)	423.33 Hz (44.90%)
4th	440.24 Hz (0.35%)	747.96 Hz (69.90%)

. Compared with the finite element results, it is shown that as the order of modal frequency increases, the errors based on the Euler theory are obviously enlarged, while the frequency errors of each mode based on Timoshenko beam theory are within 1%. Thus, the response results in the following sections are solutions based on Timoshenko beam theory. It should be noted that the proposed method, based on Timoshenko beam theory, also applies to long beams with L/H > 5, and the results are similar to those based on Euler beam theory.

The first four mode shapes are shown in Figure 3; the ratios of the first mode peak and the rest mode peak are nearly equal to 8:1. Thus, in the following section, the responses of the first mode are mainly considered, and the measuring point is set at x = 0.6 m, which does not only focus on the response of the first mode but also takes the rest mode into consideration. In the following figures, $\overline{W}$ represents the amplitude of the steady-state response of the measuring point on the beam.

3.2. Vibration Reduction Performance

The main purpose of introducing NESs is to reduce the vibration amplitude of the beam. In this work, the m, c, and k₃ of each NES are the same. Equation (8) is solved with a semi-analytical method, and the performance of the three NESs is shown in Figure 4. To validate its accuracy, a fourth-order Runge–Kutta method is employed, and the results are compared in Figure 4. It is shown that a good agreement occurs when comparing the numerical simulation results and the semi-analytical results. In the following analysis, the approximate solutions based on the semi-analytical method will be used. When attaching three NESs to a beam, significant vibration reduction is obtained. The drop percent of the response amplitude at the first resonance frequency is up to 47.42%. It demonstrates the high efficiency of NESs.

3.3. Parameter Analysis

To further improve the performance of NESs, the effects of NES parameters are investigated by varying the value of m, c, and k₃, respectively. The NESs are mounted randomly on a beam in this section. The parameter influence of the mass ratio ε is demonstrated in Figure 5. The system parameters are fixed to ζ = 0.1, k₃ = 10⁷ N/m³, d₁ = 0.5 m, d₂ = 0.75 m, d₃ = 0.8 m. As ε increases from 0.05 to 0.2, better performances at the first two resonance frequencies are obtained. However, there is usually a limit to the total attached mass.

The parameter influence of the damping ratio ζ is demonstrated in Figure 6. The system parameters are fixed to k₃ = 10⁷ N/m³, ε = 0.1, d₁ = 0.5 m, d₂ = 0.75 m, and d₃ = 0.8 m. As the damping ratio ζ increases from 0.05 to 0.15, the amplitude of the beam decreases in all frequency ranges. However, due to material limits, there is usually a limit to damping ratios as well.

The parameter influence of cubic stiffness k₃ is shown in Figure 7. The system parameters are fixed to ε = 0.1, ζ = 0.1, d₁ = 0.5 m, d₂ = 0.75 m, and d₃ = 0.8 m. With cubic stiffness ranges from 100 to 10⁷ N/m³, the resonant amplitudes are almost the same. This indicates that it is not possible to improve vibration reduction performance by adjusting cubic stiffness within this range. With the cubic stiffness of 10⁸ N/m³, a larger reduction is achieved for the first resonant amplitude, together with a slight change of resonant frequency. With the cubic stiffness of 10⁹ N/m³, an unstable response is exhibited due to strong nonlinearity.

To obtain reliable vibration reduction performance, in the following section, all the optimization results are based on the stable response of the NESs. However, trying to find the optimal value of k₃ with the method of exhaustion is very time-consuming. To obtain the optimal value in a short time, the genetic algorithm is used to solve this problem. The optimization function is the energy dissipated portion η_NES. The searching range of k₃ is set at k₃ = [10⁵, 10⁹] N/m³, where the steady-state results can be guaranteed. The constant parameters of the genetic algorithm are fixed at P(N) = 500, Gen_max = 20. After iterating for 20 generations, the optimal k₃ value for three NESs is founded at k₃ = 9 × 10⁷ N/m³.

4. Optimization Results

In this section, the total attached mass of NESs is a constant, and the optimal number and distribution of NESs are investigated.

The effect of NES distribution on vibration reduction is considered. For three NESs, the system parameters are fixed to ε = 0.1, ζ = 0.1 and k₃ = 9 × 10⁷ N/m³. The constant parameters of the genetic algorithm are fixed to P(N) = 500, Gen_max = 20. The optimal locations of three NESs, obtained with the genetic algorithm, are mounted at d₁ = 0.5 m, d₂ = 0.6 m, d₃ = 0.75 m. To demonstrate the effect of the optimal distribution, a uniform distribution of three NESs is set as a reference group, i.e., d₁ = 0.25 m, d₂ = 0.5 m, d₃ = 0.75 m. The optimization process of the genetic algorithm for three NESs is listed in

Table 2. The distribution optimization process of the genetic algorithm.

Iteration	di (m)	The 1st Peak	Reduction	The 2nd Peak	Reduction
0	/	0.077	/	0.0062	/
1	[0.35, 0.75, 0.8]	0.044	42.86%	0.0058	7.05%
2	[0.45, 0.5, 0.8]	0.042	45.45%	0.0058	7.05%
3	[0.5, 0.6, 0.75]	0.040	48.05%	0.0057	8.06%
4	[0.5, 0.6, 0.75]	0.040	48.05%	0.0057	8.06%

. The optimal results are achieved with three iterations. The vibration reduction performances are listed in

Table 3. Comparison of the performance between different distributions of parallel NESs.

Distribution	di (m)	The 1st Peak	Reduction	The 2nd Peak	Reduction
No NES	/	0.077	/	0.0062	/
Optimal	[0.5, 0.6, 0.75]	0.040	48.05%	0.0057	8.06%
Uniform	[0.25, 0.5, 0.75]	0.044	42.86%	0.0056	9.67%

. To verify the convergence of the optimal results, an optimization with initial population P(N) = 1000 was carried out, and the same optimal results were obtained. The performances of optimal distribution and uniform distribution are shown in Figure 8. Compared with the uniform distribution, the optimal distribution decreased by 5.19% more at the first resonance. This not only demonstrates the effect of distribution on performance but also proves the efficiency of the genetic algorithm.

The effect of the NES number on vibration reduction is considered. Here, N varies from one to five, and the corresponding k₃ is optimized. The optimal locations for different N NESs are obtained, and the optimal performances of N NESs are shown in Figure 9. From Figure 9, it is found that under the premise of not changing the total attached mass, maximum vibration reduction occurs at N = 2. However, when the number of parallel nonlinear energy sinks continues to increase, the vibration reductions decrease. It is not the case that more is better.

The mounting locations of nonlinear energy sinks and their performances are listed in

Table 4. Comparison of the performance between different numbers of NESs.

N	di (m)	k3 (N/m3)	The 1st Peak	Reduction	The 2nd Peak	Reduction
0	/	/	0.077	/	0.0062	/
1	[0.5]	1.9 × 109	0.044	42.85%	0.0062	0.03%
2	[0.5, 0.75]	4.1 × 108	0.035	54.55%	0.0058	6.54%
3	[0.5, 0.6, 0.75]	9.0 × 107	0.040	48.05%	0.0057	8.06%
4	[0.2, 0.25, 0.6, 0.75]	7.0 × 107	0.041	46.75%	0.0056	8.06%
5	[0.05, 0.5, 0.75, 0.8, 0.95]	4.0 × 107	0.046	40.25%	0.0055	11.29%

. Both the vibration reduction performance at the first resonance and the second resonance are analyzed.

According to the results of the genetic algorithm, the optimal locations and cubic stiffness are different for different numbers of nonlinear energy sinks. When there is only one nonlinear energy sink attached to the beam, the optimal mounting location is at d₁ = 0.5 m, where the peak of the first modal shape occurs. When N increases to two, the optimal mounting locations are d₁ = 0.5 m, d₂ = 0.75 m, where the peaks of the first two modes occur, respectively. When N increases to three, the optimal mounting locations are d₁ = 0.5 m, d₂ = 0.75 m, d₃ = 0.6 m, and d₃ is the location where the performance computed. A conclusion can be drawn that the optimal locations of multiple parallel NESs are related to both low-order modal shapes. The optimal locations for Ns equal to 4 and 5 are obtained with the genetic algorithm, and the performances are compared with other distributions as well. It is found that the optimal number of NESs is not the case of ‘the more, the better’. In this study, the N of the best vibration reduction performance equals to 2, and a further increase in the number of NESs may lead to declining effective performance.

In conclusion, the optimization results provide three guidelines for vibration reduction design with NESs. Firstly, it is effective to divide a single NES into multiple parallel NESs for better vibration reduction performance. However, there is an optimal number of NESs, and more is not better. Secondly, a uniform distribution of NESs is not the best choice, and the optimal locations are related to the modal shape of the beam. Thirdly, it is essential to optimize the number, location, and parameters of the NESs simultaneously to achieve the best performance.

5. Conclusions

A simply supported beam coupled with distributed NESs is investigated, and the responses are investigated based on Timoshenko beam theory. Numerical simulation verified the accuracy of the analytical results based on the harmonic balance method. The effects of the NES parameters on vibration reduction were discussed. The parameters and locations of NESs were optimized with the genetic algorithm method, and several optimized parallel NESs were investigated. The conclusions can be drawn as follows:

(1)

For a short beam coupled with NESs, it is necessary to establish dynamic models with Timoshenko beam theory. The errors of the first four resonance frequencies based on Timoshenko beam theory were within 1%, while those based on Euler beam theory exceeded 69%. With the same total mass, the multiple parallel nonlinear energy sinks achieved better vibration reduction performance than a single nonlinear energy sink, and the first resonant peak was reduced by 54.55%.

(2)

When increasing the mass ratio and damping ratio of NESs, better performance at the first two resonance frequencies can be obtained. When adjusting the cubic stiffness of NESs, effective vibration reduction is obtained within a certain range. Unstable responses occur when the value of cubic stiffness exceeds a threshold.

(3)

The optimal locations of multiple parallel NESs are related to low-order modal shapes, and the optimal locations and cubic stiffness are different for different numbers of nonlinear energy sinks. In the studied case, when the number of parallel NESs equaled two, the best performance was obtained. A further increase in the number of NESs may lead to declining effective performance.