Vibration attenuation using a nonlinear dynamic vibration absorber with negative stiffness

Abstract

To suppress resonance and broaden the bandwidth for vibration absorption, a nonlinear dynamic vibration absorber (NDVA) with negative stiffness is developed in this paper. The magnetic negative stiffness spring (MNSS), the generation mechanism of negative stiffness, is composed of three magnet rings arranged in attraction, and employed to connect with mechanical spring in parallel. The magnetic force and its corresponding stiffness of the MNSS are investigated firstly. Attaching the NDVA to main mass, the dynamic equations of the coupled system is established, and the averaging method is also applied to the dynamic responses. With the aim of excellent vibration absorption performance of the NDVA, an iterative algorithm is proposed to perform the parameter optimization with the purpose of avoiding time consumption. Finally, the results show that the optimization algorithm can significantly improve the efficiency of parameter selection for the NDVA, resulting in the remarkable vibration attenuation for primary system and a larger bandwidth for vibration absorption.

Keywords

Nonlinear dynamic vibration absorber negative stiffness vibration absorption optimization algorithm

1. Introduction

Linear dynamic vibration absorbers (DVAs) such as traditional tuned mass dampers (TMDs) have been the common auxiliary devices to mitigate undesirable vibration in mechanical systems. However, linear absorbers can work effectively over a narrow frequency range around the resonance, which makes it difficult to absorb low frequency vibration of primary systems and hard to widen the bandwidth for vibration absorption. To overcome these dilemmas, the NDVA based on negative stiffness mechanism is an alternative solution in the field of vibration absorption.

Up to now, the negative stiffness mechanism is widely employed in the field of low frequency vibration isolation due to its high-static-low-dynamic stiffness property [1,2]. There are many ways to obtain the negative stiffness characteristics, such as oblique springs [5] and Euler buckled beam/plate [6]. However, the study of negative stiffness in the design of low-frequency DVAs is still seldom reported. Acar [3] analytically and experimentally studied an adaptive-passive DVA with negative stiffness mechanism, and found that the amplitude of system could be effectively attenuated. Shen [4] then investigated the optimal parameter selection of a DVA with negative stiffness. These studies demonstrate that the negative stiffness is benefit for vibration absorption, whereas the stiffness nonlinearity caused by negative stiffness might bring unwanted effects, such as degrading the vibration absorption of the absorber and introducing the complex dynamic behaviors. Hence, it must be careful to select structural parameters through suitable algorithms to avoid these dilemmas. Li [7] proposed a design method of a nonlinear TMD based on the jump frequency, and the vibration reduction performance is slightly improved compared to linear design method. Jordanov [8] applied a numerical algorithm to the optimal design of linear and nonlinear DVAs in terms of the objective functions. Fallahpasand [9] then optimized a nonlinear pendulum-type TMD using the H_∞ and H₂ methods. As for these methods investigated above, either the vibration absorption is not enough excellent, or the parameter selection algorithm is extremely time-consuming, which means that it is necessary to develop an efficient algorithm to overcome those shortcomings.

The aim of this paper is to present a novel NDVA with MNSS and develop an algorithm to obtain optimal parameters for achieving excellent vibration absorption without the huge loss of time consumption. The section arrangement in this article can be given as follows. The configuration and its characteristic of the proposed NDVA are presented in Section 2. In Section 3, the dynamic equations are established, and the steady-state response is also obtained in virtue of averaging method. The next section presents the proposed algorithm about its principle and procedure of parameter selection. Finally, some conclusions are given in Section 5.

2. Modelling the NDVA

2.1. Schematic of the NDVA

As illustrated in Fig. 1, the proposed NDVA comprises a MNSS, two mechanical springs, two magnet retainers and other subsidiary parts, such as a chamber, guide pillars et al.. The MNSS consists of three magnet rings arranged in attraction, where the outer two magnet rings are fixed with magnet retainers at appropriate height to the central one mounted into the chamber. This central magnet ring vibrates along the path guided by pillars, when subjected to external excitation which is transmitted from primary system through rigid rod. The separation from the two outer magnet rings to the central one can be tuned through screwing the magnet retainers back and forth so as to obtain different characteristics of MNSS. Hence, the MNSS can be harnessed to liberally reduce dynamic stiffness by canceling positive stiffness offered by the two mechanical springs, which provides static enough supporting force balancing the weight of central magnet ring with only a small static deformation. At a result, the natural frequency of the NDVA shifts in large frequency range, which makes it easy for the NDVA to select proper structural parameters to dramatically reduce vibration of primary system. To investigate its performance of low-frequency vibration suppression, the first is to make the characteristic of the MNSS clear, which will be conducted in the following subsection.

Fig. 1.

Schematic of the NDVA.

2.2. Characteristic of the MNSS

Figure 2 displays the layout of the MNSS. The magnetic force applied to the central magnet ring is derived theoretically based on the current model and the corresponding stiffness can be obtained as well, which can be expressed as [10]. $\begin{eqnarray}\displaystyle k_{mag} & = & \displaystyle \displaystyle \frac{{\mu}_{0}M_{1}M_{2}}{4{\pi}}\mathop{\sum }_{i=1}^{2}\mathop{\sum }_{j=1}^{2}(-1)^{i+j}\int _{0}^{2{\pi}}\int _{0}^{2{\pi}}\int _{l+h_{1}/2}^{l+h_{1}/2+h_{2}}({\Phi}_{2}-{\Phi}_{1})dz_{2}d{\varphi}_{2}d{\varphi}_{1}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\displaystyle \frac{{\mu}_{0}M_{1}M_{2}}{4{\pi}}\mathop{\sum }_{i=1}^{2}\mathop{\sum }_{j=1}^{2}(-1)^{i+j}\displaystyle \int _{0}^{2{\pi}}\int _{0}^{2{\pi}}\int _{-h_{1}/2-l-h_{2}}^{-h_{1}/2-l}({\Phi}_{2}-{\Phi}_{1})dz_{2}d{\varphi}_{2}d{\varphi}_{1}\end{eqnarray}$ (1) where $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}{\Phi}_{1}=\displaystyle \frac{(z_{2}-z(j))\cos ({\varphi}_{2}-{\varphi}_{1})r(i)r_{in2}}{\left|r_{in2}^{2}+r^{2}(i)-2r_{in2}r(i)\cos ({\varphi}_{2}-{\varphi}_{1})+(z_{2}-z(j))^{2}\right|^{3/2}},\\[24.0pt] {\Phi}_{2}=\displaystyle \frac{(z_{2}-z(j))\cos ({\varphi}_{2}-{\varphi}_{1})r(i)r_{out2}}{\left|r_{out2}^{2}+r^{2}(i)-2r_{out2}r(i)\cos ({\varphi}_{2}-{\varphi}_{1})+(z_{2}-z(j))^{2}\right|^{3/2}}\end{array} & & \displaystyle \nonumber\end{eqnarray}$ and z is the displacement of the central magnet ring, z (1) = z − h ₁∕2, z (2) = z − h ₂∕2, z ₁, 𝜑₁, z ₂, 𝜑₂ denote the locations of current element surface of central and two outer magnet rings, respectively. Employing the geometrical parameters listed in Table 1, the stiffness of the MNSS with respect to z is shown in Fig. 3, which can be approximated with a quadratic polynomial as $\begin{eqnarray}\displaystyle k_{mag}=k_{11}+k_{33}z^{2} & & \displaystyle\end{eqnarray}$ (2) where k ₁₁ and k ₃₃ denote the linear and nonlinear coefficients, respectively. The effects of seperation l on k ₁₁ and k ₃₃ are illustarted in Fig. 4, which can be denoted in exponential format as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle k_{11}=f_{1}(l)=-16.87-7058.82e^{-\frac{l}{0.01057}}-13048.76e^{-\frac{l}{0.00388}}\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle k_{33}=f_{3}(l)=-2.92\times 10^{8}+8.03\times 10^{6}(1-e^{-\frac{l}{0.02864}})+2.85\times 10^{8}(1-e^{-\frac{l}{0.00521}}).\end{eqnarray}$ (4) Taking the positive stiffness k _p of two mechanical springs into account, the total restoring force offered by the NDVA can be given by $\begin{eqnarray}\displaystyle F_{r}=\displaystyle \int _{0}^{z}(k_{mag}+k_{p})dz=k_{1}z+k_{3}z^{3} & & \displaystyle\end{eqnarray}$ (5) where k ₁ = k _p + k ₁₁, k ₃ = k ₃₃∕3.

Fig. 2.

Layout of the MNSS.

Table 1

Parameters of magnet rings

Parameters	Values
The inner radii of central magnet ring and outer two ones r _in1∕r _in2	4/8 mm
The outer radii of central magnet ring and outer two ones r _out1∕r _out2	12/16 mm
The height of central magnet ring and outer two ones h ₁∕h ₂	8/8 mm
The separation between central magnet ring and outer two ones l	10 mm

Fig. 3.

Stiffness of the MNSS with respect to displacement z.

Fig. 4.

Stiffness coefficients k ₁₁ and k ₃₃ with respect to separation l: (a) k ₁₁; (b) k ₃₃.

3. Analysis of dynamics

3.1. Dynamic equations

As shown in Fig. 5, attaching the NDVA to the primary system which is supported by k _s, c _s and excitated with Fcos(ωt), the dynamic equations of the coupled system can be expressed as follows $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}m_{s}\ddot{z}_{s}+c_{s}{\dot{z}}_{s}+k_{s}z_{s}+c({\dot{z}}_{s}-{\dot{z}})+k_{1}(z_{s}-z)+k_{3}(z_{s}-z)^{3}=F\cos {\omega}t\\ m\ddot{z}+c({\dot{z}}-{\dot{z}}_{s})+k_{1}(z-z_{s})+k_{3}(z-z_{s})^{3}=0\end{array}\right. & & \displaystyle\end{eqnarray}$ (6) where c represents the viscous coefficient of the NDVA, m _s, c _s, k _s denote the mass, viscous damping and stiffness of the primary system, respectively. By introducing the dimensionless parameters, the Eq. (6) can be transferred into matrix form as $\begin{eqnarray}\displaystyle \boldsymbol{M}\ddot{\boldsymbol{X}}+\boldsymbol{C}\dot{\boldsymbol{X}}+\boldsymbol{K}\boldsymbol{X}=\boldsymbol{f}. & & \displaystyle\end{eqnarray}$ (7) Where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{M}=\left[\begin{array}{@{}cc@{}}{\lambda}^{2}(1+{\mu}) & {\lambda}^{2}{\mu}\\ {\lambda}^{2} & {\lambda}^{2}\end{array}\right],\quad \boldsymbol{C}=\left[\begin{array}{@{}cc@{}}2\text{Z}{\lambda} & 0\\ 0 & 2{\zeta}{\lambda}{\omega}_{\boldsymbol{o }}\end{array}\right],\quad \boldsymbol{K}=\left[\begin{array}{@{}cc@{}}1 & 0\\ 0 & {\omega}_{\boldsymbol{ o}}^{2}\end{array}\right],\quad \boldsymbol{f}=\left[\begin{array}{@{}c@{}}\cos {\tau}\\ -\frac{{\kappa}}{{\mu}}y^{3}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\mu}=\frac{m}{m_{s}},\quad Y_{0}=\frac{F}{k_{1}},\quad y_{s}=\frac{x_{1}}{X_{0}},\quad y=\frac{x_{1}-x_{2}}{X_{0}},\quad {\omega}_{n}=\sqrt{\frac{k_{1}}{m}},\quad {\Omega}_{n}=\sqrt{\frac{k_{s}}{m_{s}}},\quad {\zeta}=\frac{c}{2m{\omega}_{\text{n}}},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \text{Z}=\frac{c_{s}}{2m_{s}{\Omega}_{n}},\quad {\lambda}=\frac{{\omega}}{{\Omega}_{n}},\quad {\omega}_{o}=\frac{{\omega}_{n}}{{\Omega}_{n}},\quad {\kappa}=\frac{k_{3}Y_{0}^{2}}{k_{1}},\quad {\omega}t={\tau}.\nonumber\end{eqnarray}$

3.2. Steady-state response

With the application of averaging method, the steady-state response of Eq. (7) is assumed as

Fig. 5.

Equivalent mechanical model.

\begin{eqnarray}\displaystyle \boldsymbol{X}=\boldsymbol{u}\cos ({\tau})+\boldsymbol{v}\sin ({\tau}) & & \displaystyle\end{eqnarray}

(8) where u = [u _s u]^T and v = [v _s v]^T vary slowly with respect to τ. The slow varying assumption can be given by

\begin{eqnarray}\displaystyle \boldsymbol{u}^{\prime }\cos ({\tau})+\boldsymbol{v}^{\prime }\sin ({\tau})=0. & & \displaystyle\end{eqnarray}

(9) Substituting Eqs. (8) and (9) into Eq. (7) yields

\begin{eqnarray}\displaystyle (\boldsymbol{M}\boldsymbol{v}^{\prime }-\boldsymbol{M}\boldsymbol{u}+\boldsymbol{C}\boldsymbol{v}+\boldsymbol{K}\boldsymbol{u})\cos ({\tau})-(\boldsymbol{M}\boldsymbol{u}^{\prime }+\boldsymbol{M}\boldsymbol{v}+\boldsymbol{C}\boldsymbol{u}-\boldsymbol{K}\boldsymbol{v})\sin ({\tau})=\boldsymbol{f}(\boldsymbol{u},\boldsymbol{v},{\tau}). & & \displaystyle\end{eqnarray}

(10) Following the priciple of averaging method, the steady-state response u and v are incorrelate with time τ. By integrating and averaging Eq. (10) from 0 to 2π, we can get

\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-10.0pt}\boldsymbol{M}\boldsymbol{u}^{\prime }=-\frac{1}{2}(\boldsymbol{M}-\boldsymbol{K})\boldsymbol{v}-\frac{1}{2}\boldsymbol{C}\boldsymbol{u}-\frac{1}{2}\left(\begin{array}{@{}c@{}}0\\ -\frac{3}{4}\frac{{\kappa}}{{\mu}}Y^{2}v\end{array}\right),\quad \boldsymbol{M}\boldsymbol{v}^{\prime }=\frac{1}{2}(\boldsymbol{M}-\boldsymbol{K})\boldsymbol{u}-\frac{1}{2}\boldsymbol{C}\boldsymbol{v}+\frac{1}{2}\left(\begin{array}{@{}c@{}}1\\ -\frac{3}{4}\frac{{\kappa}}{{\mu}}Y^{2}u\end{array}\right)\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}

(11) where the definition has been used:

Y_{s}^{2}=u_{s}^{2}+v_{s}^{2}

, Y ² = u ² + v ². In order to obtain the steady-state solution of Eq. (11), the following conditions should be satisfied

\begin{eqnarray}\displaystyle \boldsymbol{u}^{\prime }=0,\quad \boldsymbol{v}^{\prime }=0. & & \displaystyle\end{eqnarray}

(12) Substituting Eq. (12) into Eq. (11), the amplitude-frequency relationship can be obtained as

\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Y^{2}\left(\left(2{\zeta}{\lambda}{\omega}_{0}({\lambda}^{2}(1+{\mu})-1)-2Z{\lambda}({\omega}_{0}^{2}-{\lambda}^{2})-\frac{3}{2}\frac{{\kappa}}{{\mu}}Z{\lambda}Y^{2}\right)^{2}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \left.+\left(\begin{array}{@{}l@{}}4Z{\zeta}{\lambda}^{2}{\omega}_{0}+{\lambda}^{4}{\mu}+({\lambda}^{2}(1+{\mu})-1)({\omega}_{0}^{2}-{\lambda}^{2})\\ +\left(\frac{3}{4}\frac{{\kappa}}{{\mu}}({\lambda}^{2}(1+{\mu})-1)Y^{2}\right)\\ \end{array}\right)^{2}\right)={\lambda}^{4}\end{eqnarray}

(13)

\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle ({\lambda}^{2}Y_{s})^{2}=(2{\zeta}{\lambda}{\omega}_{0}Y)^{2}+\left(\frac{3}{4}\frac{{\kappa}}{{\mu}}Y^{3}+({\omega}_{0}^{2}-{\lambda}^{2})Y\right)^{2}.\end{eqnarray}

(14)

4. Optimization and result

As discussed in Section 1, there are many drawbacks of optimization algorithms in existing studies for nonlinear absorber, such as the poor performance of vibration absorption and high time consumption. Therefore, it is desirable to develop a new approach with low time-consumption to obtain optimal structural parameters for vibration absorption enhancement. It is known that most of complex nonlinear dynamics including bifurcations, instabilities and jumps exist in the district of multiple responses near resonance, but optimal frequency-amplitude is a curve of single solution. Hence, the relationship between the number of solutions and structural parameters should be derived to seek for the critical condition distinguishing single solution and multiple solutions, which can be utilized to determine the optimal parameters. So the following will perform the derivation of critical condition in Section 4.1, and the parameter optimization in Section 4.2.

4.1. Derivation of critical condition

Rewriting Eq. (13) as $\begin{eqnarray}\displaystyle {\alpha}_{1}s+{\alpha}_{2}s^{2}+{\alpha}_{3}s^{3}+{\alpha}_{4}=0 & & \displaystyle\end{eqnarray}$ (15) where s = Y ² and 𝛼₁, 𝛼₂, 𝛼₃, 𝛼₄ are given in Appendix. Differentiating Eq. (15) with respect to z yields $\begin{eqnarray}\displaystyle {\alpha}_{1}+2{\alpha}_{2}s+3{\alpha}_{3}s^{2}=0. & & \displaystyle\end{eqnarray}$ (16) By eliminating s from Eqs (13) and (14), the existence of multiple solutions should satisfy the critical condition $\begin{eqnarray}\displaystyle 27{\alpha}_{3}^{2}{\alpha}_{4}^{2}+4{\alpha}_{2}^{3}{\alpha}_{4}+4{\alpha}_{1}^{3}{\alpha}_{3}-18{\alpha}_{1}{\alpha}_{2}{\alpha}_{3}{\alpha}_{4}-{\alpha}_{1}^{2}{\alpha}_{2}^{2}=0. & & \displaystyle\end{eqnarray}$ (17) Replacing 𝛼₂, 𝛼₃ for $\widetilde{{\alpha}_{2}},\widetilde{{\alpha}_{3}}$ (details in Appendix) for simple manipulations, the relationship between the number of solutions and distinct parameters can be determined. Especially, with fixed 𝜇, ω₀ and ζ, the plane of 𝜅 and 𝜆 (𝜆 > 0, 𝜅 ), mapping the zone of different numbers of solutions should be divided into two domains by N⁺ and N⁻, which are shown in Fig. 6 and expressed as $\begin{eqnarray}\displaystyle N^{\pm }=\left\{\left({\lambda},{\kappa}\right)\left|\begin{array}{@{}l@{}}{\lambda}>0,54\tilde{{\alpha}}_{3}^{2}{\alpha}_{4}^{2}{\kappa}-{\sigma}{\mu}(18{\alpha}_{1}\tilde{{\alpha}}_{2}\tilde{{\alpha}}_{3}{\alpha}_{4}-4\tilde{{\alpha}}_{2}^{3}{\alpha}_{4})\\ \mp \sqrt{(4{\alpha}_{4}^{3}d-18{\alpha}_{1}\tilde{{\alpha}}_{2}\tilde{{\alpha}}_{3}{\alpha}_{4})^{2}-108\tilde{{\alpha}}_{3}^{2}{\alpha}_{4}^{2}\left(4{\alpha}_{1}^{3}\tilde{{\alpha}}_{3}-\tilde{{\alpha}}_{2}^{2}{\alpha}_{1}^{2}\right)}=0.\end{array}\right.\right\} & & \displaystyle\end{eqnarray}$ (18) As given in Fig. 6(a), the meaning of domains can be interpreted as that domain 1 denotes the zone of single solution and domain 2, three solutions. When to Employ those parameters listed in Table 1, the graph as shown in Fig. 6(b) indicates that |𝜅| must be less than |𝜅_min| if to avoid multiple solutions. Hence, Taking Eq. (4) and Section 3 into account, the sufficient condition is acquired that separation l should satisfy $\begin{eqnarray}\displaystyle l\geq l_{cr} & & \displaystyle\end{eqnarray}$ (19) where $l_{cr}=f_{3}^{-1}(3{\kappa}_{min}({\mu},{\omega}_{0},{\zeta})k_{1}^{3}/F^{2})$ .

Fig. 6.

Partition of the plane of 𝜆 and 𝜅 into two domains (a) 𝜆 > 0; (b) 𝜆 > 0 and 𝜅 < 0 (parameters used here are: m _s = 1 kg, m = 0. 1 kg, c _s = 1 N ⋅ s∕m, c = 1 N ⋅ s∕m, k _s = 1 × 10⁴ N∕m, k _P = 4800 N∕m, l = 10 mm, k ₁₁ = −3750. 7 N∕m, k ₃₃ = −4. 58 × 10⁷ N∕m³, F = 3 N).

4.2. Optimizing and analysis

From Eq. (19), it can be seen that multiple solutions should be avoid as long as the separation l between the central magnet ring and two outer magnet rings exceeds the certain range, while Eq. (19) cannot be utilized to determine optimal separation l owing to the inequality.

Fig. 7.

Optimization procedure.

Therefore, an iterative algorithm is developed to obtain the optimal value of separation l by means of the convergence of parameter to constant. The key of the proposed algorithm is that optimal parameters of minimizing mechanical vibration is equivalent regardless of whether the absorber is linear or not, which means that optimal parameters in the design of linear absorber can be considered as target to obtain the optimal parameters of nonlinear absorber with iteration procedure. Considering the optimal ζ_opt and ω_opt of linear absorber given by [11 ] $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\zeta}_{opt}=\frac{1}{4}\sqrt{\frac{8+9{\mu}-4\sqrt{4+3{\mu}}}{1+{\mu}}}\end{eqnarray}$ (20) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\omega}_{opt}=\frac{{\omega}_{n}^{\prime }}{{\Omega}_{n}}=\frac{2}{1+{\mu}}\sqrt{\frac{2(16+23{\mu}+9{\mu}^{2}+2(2+{\mu})\sqrt{4+3{\mu}})}{3(64+80{\mu}+27{\mu}^{2})}}\end{eqnarray}$ (21) and the backbone curve of duffing oscillator [12] $\begin{eqnarray}\displaystyle {\omega}_{d}^{2}={\omega}_{n}^{2}+\frac{3}{4}\frac{k_{3}}{m}Y^{2} & & \displaystyle\end{eqnarray}$ (22) after replacing ω_d for ω_opt, it yields the body of iteration $\begin{eqnarray}\displaystyle {\omega}_{0}=\sqrt{{\omega}_{opt}^{2}-\frac{3}{4}\frac{k_{3}}{m{\Omega}_{n}^{2}}Y^{2}}. & & \displaystyle\end{eqnarray}$ (23) The procedure to obtain optimal separation l is depicted in Fig. 7. There are four steps. Assigning those parameters that are irrelevant to iteration, |𝜅_min| is first to calculate to obtain l _i as well as k ₃. The following step is to select the park of amplitude from the frequency-amplitude curve in term of present parameters, and designate it to Y. Thirdly, ${\omega}_{0}^{^{\prime }}$ can be determined through substituting l _i k ₃ and Y obtained from above two steps into Eq. (23). In step 4, giving the value of ${\omega}_{0}^{^{\prime }}$ to ω₀, the steps from 2 to 4 will continue in circles if the absolute error of separation l compared to prior iteration is larger than the given constant until the condition to stop is satisfied. The output of separation l here is treated as the optimal parameter.

As demonstrated in Fig. 8, the separation l with respect to iterations in the optimization procedure indicates that the convergence of the proposed algorithm is so rapid that the optimization procedure can be completed with the loss of little time consumption. Meanwhile, the corresponding frequency-amplitude curves are also presented in Fig. 9, which manifests that the resonance of primary system is significantly reduced and the bandwidth of vibration absorption is simultaneously enlarged.

Fig. 8.

Separation l with respect to iterations.

Fig. 9.

Frequency-amplitude curves in the process of iteration.

5. Conclusion

A nonlinear absorber with MNSS is proposed in the paper. The geometry of the NDVA is depicted, and the characteristic of MNSS is also investigated in detail. Employing the NDVA to primary system, the dynamic governing equations are established, and averaging method is employed to obtain steady-state responses. With the purpose of excellent vibration absorption, an efficient algorithm is developed to obtain optimal parameters without numbers of iteration. The results manifest that the proposed method is feasible in term of efficiently performing parameter optimization, and the dramatic vibration reduction of primary system is achieved in a lager frequency range.

Footnotes

Acknowledgements

The authors of the paper gratefully acknowledge the financial support from National Science Foundation of China (Grant No. 11172225).

Appendix

The expressions of 𝛼₁, 𝛼₂, 𝛼₃, 𝛼₄ are (A.1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-1.0pc}{\alpha}_{1}=4{\lambda}^{2}\left({\zeta}{\omega}_{0}\left({\lambda}^{2}(1+{\mu})-1\right)-Z({\omega}_{0}^{2}-{\lambda}^{2})\right)^{2}+\left(\left({\lambda}^{2}(1+{\mu})-1\right)({\omega}_{0}^{2}-{\lambda}^{2})+4Z{\zeta}{\omega}_{0}{\lambda}^{2}+{\lambda}^{4}{\mu}\right)^{2}\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (A.2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{2}=\frac{{\kappa}}{{\mu}}\tilde{{\alpha}}_{2}=6\frac{{\kappa}}{{\mu}}Z{\lambda}^{2}\left({\zeta}{\omega}_{0}\left(1-{\lambda}^{2}(1+{\mu})\right)+Z({\omega}_{0}^{2}-{\lambda}^{2})\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad +∼\left(\frac{3}{2}\frac{{\kappa}}{{\mu}}\left({\lambda}^{2}(1+{\mu})-1\right)\right)\left(\left({\lambda}^{2}(1+{\mu})-1\right)({\omega}_{0}^{2}-{\lambda}^{2})+4Z{\zeta}{\omega}_{0}{\lambda}^{2}+{\lambda}^{4}{\mu}\right)\end{eqnarray}$ (A.3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{3}=\left(\frac{{\kappa}}{{\mu}}\right)^{2}\tilde{{\alpha}}_{3}=\left(\left(-\frac{3}{2}\frac{{\kappa}}{{\mu}}Z{\lambda}\right)^{2}+\left(\frac{3}{4}\frac{{\kappa}}{{\mu}}\left(1-{\lambda}^{2}(1+{\mu})\right)\right)^{2}\right)\end{eqnarray}$ (A.4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{4}=-{\lambda}^{4}.\end{eqnarray}$ And the relationship between 𝛼₂, 𝛼₃ and $\widetilde{{\alpha}_{2}}$ , $\widetilde{{\alpha}_{3}}$ are given by: (A.5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{2}=\frac{{\kappa}}{{\mu}}\widetilde{{\alpha}_{2}}\end{eqnarray}$ (A.6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{3}=\frac{{\kappa}^{2}}{{\mu}^{2}}\widetilde{{\alpha}_{3}}.\end{eqnarray}$

References

Lakes

R.S.

, Extreme damping in composite materials with a negative stiffness phase, Physical Review Letters 86(13) (2001), 2897.

Wang

Y.C.

and Lakes

, Negative stiffness-induced extreme viscoelastic mechanical properties: Stability and dynamics, Philosophical Magazine 84(35) (2004), 3785–3801.

Acar

M.A.

and Yilmaz

, Design of an adaptive–passive dynamic vibration absorber composed of a string–mass system equipped with negative stiffness tension adjusting mechanism, Journal of Sound and Vibration 332(2) (2013), 231–245.

Shen

Peng

, Analytically optimal parameters of dynamic vibration absorber with negative stiffness, Mechanical Systems and Signal Processing 85 (2017), 193–203.

Alabuzhev

, Vibration Protection and Measuring Systems with Quasi-Zero Stiffness, CRC Press, 1989.

Platus

D.L.

, Negative-stiffness-mechanism vibration isolation systems[C]//Vibration Control in Microelectronics, Optics, and Metrology, International Society for Optics and Photonics 1619 (1992), 44–55.

and Cui

, Novel design approach of a nonlinear tuned mass damper with duffing stiffness, Journal of Engineering Mechanics 143(4) (2017), 04017004.

Jordanov

I.N.

and Cheshankov

B.I.

, Optimal design of linear and non-linear dynamic vibration absorbers, Journal of Sound and Vibration 123(1) (1988), 157–170.

Fallahpasand

Dardel

Pashaei

M.H.

, Investigation and optimization of nonlinear pendulum vibration absorber for horizontal vibration suppression of damped system, The Structural Design of Tall and Special Buildings 24(14) (2015), 873–893.

10.

Dong

Zhang

Xie

, Simulated and experimental studies on a high-static-low-dynamic stiffness isolator using magnetic negative stiffness spring, Mechanical Systems and Signal Processing 86 (2017), 188–203.

11.

Asami

and Nishihara

, Closed-form exact solution to H∞ optimization of dynamic vibration absorbers (application to different transfer functions and damping systems), Journal of Vibration and Acoustics 125(3) (2003), 398–405.

12.

Tang

Brennan

M. J.

Gatti

, Experimental characterization of a nonlinear vibration absorber using free vibration, Journal of Sound and Vibration 367 (2016), 159–169.