Enhanced nonlinear performance of nonlinear energy sink under large harmonic excitation using acoustic black hole effect

Abstract

Although the frequency dependence of linear tuned-mass-damper (TMD) is overcome by the nonlinear energy sink (NES), it is sensitive to the input energy. When the structure with NES is subjected to harmonic excitation, a larger or smaller excitation amplitude may deteriorate the performance of the NES. In particular, the large excitation may make the coupled system of NES produce a high closed detached response (CDR). The appearance of CDR means that NES may be completely ineffective. To solve this problem, multiple acoustic black hole (ABH) beams with damping layers at their tip are attached to the mass block of NES to improve NES’s mitigation effect. This novel NES is called ABH-NES in this paper. Its theoretical model is first established using the Gaussian Expansion Element Method (GEEM) and modal approach. Then, the influence of ABH effects and the number of ABH beams on the ABH-NES’s vibration mode are analyzed based on this model. After that, the responses of ABH-NES, NES, and Uniform Beam-NES (UB-NES, using uniform beams to replace the ABH’s in ABH-NES) are compared using time-domain integration, energy method, and harmonic balance method (HBM). The results show that ABH-NES has a better and more robust damping effect, especially under large excitation. The introduction of the ABH beam has two main influences: First, it raises the CDR's excitation threshold by increasing the energy dissipation pathway. Second, it reduces the amplitude of the CDR utilizing the highly damped modes of ABH-NES. Parametric analysis is carried out to discuss the effects of the partial parameters, such as absorber mass, stiffness, damping, ABH-NES’s frequency, on the new absorber. Finally, an experiment is provided to verify the effectiveness of theoretical analysis.

Appendices

Appendix A: Harmonic balance method

Since the nonlinear stiffness of the coupled system is cubic-type, its periodic solution x_d and the nonlinear force F_dn are mainly composed of odd harmonics and can be assumed as [9, 42,43,44]

$$ {\varvec{x}}_{d} = \sum\limits_{k = 1}^{{N_{k} }} {{\varvec{a}}_{2k - 1} \cos \left( {2k - 1} \right)\omega_{d} \tau + {\varvec{b}}_{2k - 1} \sin \left( {2k - 1} \right)\omega_{d} \tau } $$

(A.1)

$$ {\varvec{F}}_{dn} = \left( {\sum\limits_{k = 1}^{{N_{k} }} {c_{2k - 1} \cos \left( {2k - 1} \right)\omega_{d} \tau + d_{2k - 1} \sin \left( {2k - 1} \right)\omega_{d} \tau } } \right){\varvec{f}}_{n} $$

(A.2)

where a_2k−1 and b_2k−1 are the harmonic coefficients of the periodic solution. And they also are n + 1 dimensional vectors. The N_k determines the order of the harmonics and is set as 3 in this paper. The c_2k−1 and d_2k−1 denote the harmonic coefficients of nonlinear force and are scalar.

Substituting Eqs. (A.1) and (A.2) into Eq. (26) and performing harmonic balance, a set of algebraic equations can be obtained

$$ \begin{array}{c} {\varvec{H}}\left( {\omega_{d} ,{\varvec{a}}_{1} ,{\varvec{b}}_{1} , \cdots ,c_{1} ,d_{1} , \cdots } \right) = {\mathbf{0}} \hfill \\ = \left[ {\begin{array}{*{20}c} {\left( {{\varvec{K}}_{d} - \omega_{d}^{2} {\varvec{M}}_{d} } \right){\varvec{a}}_{1} + \omega_{d} {\varvec{C}}_{d} {\varvec{b}}_{1} + c_{1} {\varvec{f}}_{n} - {\varvec{F}}_{d} } \\ {\left( {{\varvec{K}}_{d} - \omega_{d}^{2} {\varvec{M}}_{d} } \right){\varvec{b}}_{1} - \omega_{d} {\varvec{C}}_{d} {\varvec{a}}_{1} + d_{1} {\varvec{f}}_{n} } \\ \vdots \\ {\left( {{\varvec{K}}_{d} - \left( {2k - 1} \right)^{2} \omega_{d}^{2} {\varvec{M}}_{d} } \right){\varvec{a}}_{2k - 1} + \left( {2k - 1} \right)\omega_{d} {\varvec{C}}_{d} {\varvec{b}}_{2k - 1} + c_{2k - 1} {\varvec{f}}_{n} } \\ {\left( {{\varvec{K}}_{d} - \left( {2k - 1} \right)^{2} \omega_{d}^{2} {\varvec{M}}_{d} } \right){\varvec{b}}_{2k - 1} - \left( {2k - 1} \right)\omega_{d} {\varvec{C}}_{d} {\varvec{a}}_{2k - 1} + d_{2k - 1} {\varvec{f}}_{n} } \\ \vdots \\ \end{array} } \right]_{{2n \times N_{k} }} = {\mathbf{0}} \hfill \\ \end{array} $$

(A.3)

Equation (A.3) contains 2n × N_k algebraic equations obtained based on the Arc-length method. We first divided these variables in H into two parts. One is P = [ω_d, a₁, b₁, …, a_2k−1, b_2k−1, …]^T with 2n × N_k + 1 elements considered as unknown. The other is P_n = [c₁, d₁, …, c_2k−1, d_2k−1, …]^T seen as the functions of P. Its detailed form can be obtained using MATLAB or Maple but is not given in this paper for simplicity.

The procedure using the Arc-length method [9] is as follows:

(1) Step 1: Suppose a start P is known, written as P_i, and the next imprecise solution P_i+1⁰ can be calculated by

$$ {\varvec{p}}_{i + 1}^{0} = {\varvec{p}}_{i} + {\text{d}}s{\varvec{\nu}} $$

(A.4)

where ds is an arc length increment, set to 0.001 in this paper. The ${\varvec{\nu}} = {\varvec{\nu}}^{*} /\left| {{\varvec{\nu}}^{*} } \right|$ (ν^* = [v₁, v₂, …, v_2n×Nk+1]^T) is the tangent vector of the change in P and can be obtained by

$$ v_{i} = \left( { - 1} \right)^{i + 1} \det \left[ {\frac{{{\text{d}}{\varvec{H}}}}{{{\text{d}}p_{1} }} \cdots \frac{{{\text{d}}{\varvec{H}}}}{{{\text{d}}p_{i - 1} }}\frac{{{\text{d}}{\varvec{H}}}}{{{\text{d}}p_{i + 1} }} \cdots \frac{{{\text{d}}{\varvec{H}}}}{{{\text{d}}p_{2n \times Nk + 1} }}} \right] $$

(A.5)

where p_i−1 or p_i+1 is the i−1th or i + 1th element in P, and dH/dp_i−1 can be derived by the chain rule of derivation

$$ \frac{{{\text{d}}{\varvec{H}}}}{{{\text{d}}p_{i - 1} }} = \frac{{\partial {\varvec{H}}}}{{\partial p_{i - 1} }} + \frac{{\partial {\varvec{H}}}}{{\partial {\varvec{p}}_{{\text{n}}} }}\frac{{\partial {\varvec{p}}_{{\text{n}}}^{{\text{T}}} }}{{\partial p_{i - 1} }} $$

(A.6)

(2) Step 2: Use the Newton–Raphson method to handle P_i+1⁰ to get an exact P_i+1. Suppose P_i+1 ^k represents P_i+1⁰ after k Newton’s iterations. The P_i+1^k+1 can be calculated by

$$ {\varvec{p}}_{i + 1}^{k + 1} = {\varvec{p}}_{i + 1}^{k} - \left[ {\begin{array}{*{20}c} {\frac{{\partial {\varvec{H}}\left( {{\varvec{p}}_{i + 1}^{k} } \right)}}{{\partial {\varvec{p}}}} + \frac{{\partial {\varvec{H}}\left( {{\varvec{p}}_{i + 1}^{k} } \right)}}{{\partial {\varvec{p}}_{{\text{n}}} }}\frac{{{\text{d}}{\varvec{p}}_{n}^{{\text{T}}} }}{{{\text{d}}{\varvec{p}}}}} \\ {{\varvec{\nu}}^{{\text{T}}} \left( {{\varvec{p}}_{i + 1}^{k} } \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varvec{H}}\left( {{\varvec{p}}_{i + 1}^{k} } \right)} \\ 0 \\ \end{array} } \right] $$

(A.7)

When $\left| {{\varvec{p}}_{i + 1}^{k + 1} - {\varvec{p}}_{i + 1}^{k} } \right|/\left| {{\varvec{p}}_{i + 1}^{k} } \right| < {\text{tolerable error}}$, we consider P_i+1^k+1 to be sufficiently accurate and treat it as P_i+1. Here, the tolerable error is set as 0.0001.

(3) Step 3: Repeat the above steps with P_i+1 as a new initial condition to get the next solution P_i+2 until ω_d reaches the excitation frequency boundary.

Appendix B: Floquet theory

Suppose x_d^* = [x_dm^*, x_d1^*, …, x_dn^*]^T is the solution of Eq. (26), and its period is T_d. After it is subjected to a perturbation ∆x_d, the new solution is x_d = x_d^* + ∆x_d. Substituting it into Eq. (26) and preserving the linear part of ∆x_d, we get

$$ \Delta {\varvec{X}}_{{\text{d}}} ^{\prime} = {\varvec{A}}_{{\text{d}}} \left( \tau \right)\Delta {\varvec{X}}_{{\text{d}}} $$

(B.1)

where

$$ \Delta {\varvec{X}}_{{\text{d}}} = \left[ {\begin{array}{*{20}c} {\Delta {\varvec{x}}_{{\text{d}}} } \\ {\Delta {\varvec{x}}_{{\text{d}}} ^{\prime}} \\ \end{array} } \right],{\varvec{A}}_{{\text{d}}} \left( \tau \right) = \left[ {\begin{array}{*{20}c} {\varvec{I}} & {} \\ {} & {{\varvec{M}}_{{\text{d}}} } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {} & {\varvec{I}} \\ { - \left( {{\varvec{K}}_{{\text{d}}} + \frac{{{\text{d}}{\varvec{F}}_{{{\text{d}}n}} }}{{{\text{d}}{\varvec{x}}_{{\text{d}}} }}} \right)} & { - {\varvec{C}}_{{\text{d}}} } \\ \end{array} } \right] $$

$$ \frac{{{\text{d}}{\varvec{F}}_{{{\text{dn}}}} }}{{{\text{d}}{\varvec{x}}_{{\text{d}}} }} = 3\varepsilon_{1} \gamma \left( {\sum\limits_{i = 1}^{n} {x_{{{\text{d}}i}}^{*} } - x_{{{\text{dm}}}}^{*} } \right)^{2} {\varvec{f}}_{{\text{n}}} \left[ {\begin{array}{*{20}l} { - 1} \hfill & 1 \hfill & 1 \hfill & \cdots \hfill & 1 \hfill \\ \end{array} } \right] $$

where I denotes the unit matrix, the A_d(τ) is a time-varying matrix with period T_d. Its monodromy matrix is ${\varvec{B}} = \exp \left( {\int_{0}^{{{\text{T}}_{{\text{d}}} }} {{\varvec{A}}_{{\text{d}}} \left( \tau \right){\text{d}}\tau } } \right)$. By calculating the eigenvalues of B, we can get the stability of x_d^* (The absolute values of all eigenvalues are smaller than 1, and the x_d^* is stable, otherwise unstable) [9].

It is worth noting that the exact monodromy matrix B is usually difficult to obtain. To address this problem, Hsu et al. [46] divided T_d into N_t segments and considered A_d to be constant in each subinterval, so that an approximate formula for B can be derived

$$ {\varvec{B}} = \prod\limits_{k = 1}^{{N_{{\text{t}}} }} {\exp \left( {{\varvec{A}}_{{\text{d}}} \left( {\frac{2k - 1}{{2N_{t} }}{\text{T}}_{{\text{d}}} } \right)} \right)} $$

(B.2)

When N_t is large enough, Eq. (35) can accurately predict the stability of x_d^*. And it is set as 1000 in this paper.