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Establishment of the equal-peak principle for a multiple-DOF nonlinear system with multiple time-delayed vibration absorbers

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Abstract

In this study, a generalized equal-peak principle is established to suppress the multimodal vibration of a multiple-degree-of-freedom (M-DOF) nonlinear system. Based on the proposed generalized principle, the design procedure of the multiple time-delayed vibration absorbers (TDVAs) is carried out. By four conditions in the proposed generalized principle, the objective of suppressing all the resonance peaks around multiple modes to the equal minimum values is realized. For the existence of nonlinearity, the necessary and sufficient conditions in the design procedure can guarantee that the two resonance peaks around each mode are simultaneously equal. Moreover, the two equal resonance peaks are suppressed to minimum values with the minimum peak condition. Two case studies verify the efficiency of the TDVAs designed by the generalized equal-peak principle for multimodal vibration suppression. Due to the multimodal vibration suppression capacity of the proposed TDVAs designed by the generalized equal-peak principle, significant broad frequency band vibration suppression effects are achieved. Thus, TDVAs and the proposed equal-peak principle have potential applications in the fields of high-DOF vibration systems, such as civil engineering, precision machining and aerospace.

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Acknowledgements

The authors would like to gratefully acknowledge the support from the National Natural Science Foundation of China under Grants Nos. 11772229, 11932015 and 11972254.

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Appendices

Appendix A

The matrix and vectors in Eq. (2) are listed as follows:

$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {m_{p,1} } & 0 & 0 & 0 \\ 0 & {m_{p,2} } & 0 & 0 \\ 0 & 0 & {m_{a,2} } & 0 \\ 0 & 0 & 0 & {m_{a,1} } \\ \end{array} } \right], $$
(A1)
$$ {\mathbf{C}} = \left[ {\begin{array}{*{20}c} {c_{p,1} + c_{p,2} + c_{a,2} } & { - c_{p,2} } & { - c_{a,2} } & 0 \\ { - c_{p,2} } & {c_{p,2} + c_{a,1} } & 0 & { - c_{a,1} } \\ { - c_{a,2} } & 0 & {c_{a,2} } & 0 \\ 0 & { - c_{a,1} } & 0 & {c_{a,1} } \\ \end{array} } \right], $$
(A2)
$$ {\mathbf{K}} = \left[ {\begin{array}{*{20}c} {k_{p,1} + k_{p,2} + k_{a,2} } & { - k_{p,2} } & { - k_{a,2} } & 0 \\ { - k_{p,2} } & {k_{p,2} + k_{a,1} } & 0 & { - k_{a,1} } \\ { - k_{a,2} } & 0 & {k_{a,2} } & 0 \\ 0 & { - k_{a,1} } & 0 & {k_{a,1} } \\ \end{array} } \right], $$
(A3)
$$ {\mathbf{F}}_{n} = \left[ {\begin{array}{*{20}c} {k_{p,1,nl3} x_{p,1}^{3} + k_{p,2,nl3} \left( {x_{p,1} - x_{p,2} } \right)^{3} } & {k_{p,2,nl3} \left( {x_{p,2} - x_{p,1} } \right)^{3} } & 0 & 0 \\ \end{array} } \right]^{T} , $$
(A4)
$$ {\mathbf{G}}_{n} = \left[ {\begin{array}{*{20}c} {k_{a,2,nl3} \left( {x_{p,1} - x_{a,2} } \right)^{3} } \\ {k_{a,1,nl3} \left( {x_{p,2} - x_{a,1} } \right)^{3} } \\ {k_{a,2,nl3} \left( {x_{a,2} - x_{p,1} } \right)^{3} } \\ {k_{a,1,nl3} \left( {x_{a,1} - x_{p,2} } \right)^{3} } \\ \end{array} } \right], $$
(A5)
$$ {\mathbf{X}} = \left[ {\begin{array}{*{20}c} {x_{p,1} } & {x_{p,2} } & {x_{a,2} } & {x_{a,1} } \\ \end{array} } \right]^{T} , $$
(A6)
$$ {\mathbf{F}} = \left[ {\begin{array}{*{20}c} {f\cos \left( {\Omega t} \right)} & 0 & 0 & 0 \\ \end{array} } \right]^{T} , $$
(A7)

The vector \({\mathbf{G}}_{\tau }\) in Eq. (4) is:

$$ {\mathbf{G}}_{\tau } = \left[ {\begin{array}{*{20}c} {g_{2} x_{{a,2,\tau_{2} }} } & {g_{1} x_{{a,1,\tau_{1} }} } & { - g_{2} x_{{a,2,\tau_{2} }} } & { - g_{1} x_{{a,1,\tau_{1} }} } \\ \end{array} } \right]^{T} , $$
(A8)

Appendix B

The proposed incremental-iteration procedure for the generalized equal-peak principle is presented in Fig. 13.

Fig. 13
figure 13

Incremental-iteration procedure for the equal-peak principle. (The gray area represents the continuation process of time-delayed parameters. The cyan area represents the update process of structural parameters. The blue area indicates the proposed four conditions)

In Fig. 13, vectors p0 and \({\mathbf{R}}_{0}\) are the structural parameters of passive LTVAs and corresponding response solutions at all resonance frequencies; vector p is the structural parameters of TDVAs, which may be different from p0 due to some design requirements. The passive LTVAs with the optimal structural parameters p0 [22] can realize the equal-peak principle for M-DOF nonlinear primary systems under a small amplitude excitation. If the structural parameters p are different from p0 or the excitation amplitude increases, the resonance peaks are no longer equal and minimum around multiple modes. The detuned resonance peaks can be adjusted equally with proper time-delayed parameters. Thus, the incremental-iteration procedure consists of the update process of the structural parameters and the continuation process of the time-delayed parameters for the TDVAs. In the continuation process of time-delayed parameters targeted for the ith mode, four conditions are considered, including the stability condition, the necessary condition, the sufficient condition, and the minimum peak condition. The main steps of the proposed procedure are shown as follows.

Main steps of the incremental-iteration procedure

  • Step 0: The process is started.

  • Step 1: For arbitrary \({\mathbf{p}} \ne {\mathbf{p}}_{0}\), it is difficult to solve the nonlinear algebraic equations Eq. (13) because the Newton–Raphson algorithm is not self-starting and it needs to be provided with an initial solution. Since the nonlinearities of the system are not activated under a small force excitation, the LTVAs with the parameters shown in Table 1 can tune the peaks around the first and second modes to be approximately equal. In this case, the initial force is f = 0.001 N. Thus, the initial parameters for the proposed procedure are

    $$ {\mathbf{p}}_{0} = \left\{ {k_{a,1}^{0} ,k_{a,2}^{0} ,c_{a,1}^{0} ,c_{a,2}^{0} ,f^{0} } \right\} = \left\{ {0.031,0.019,0.0167,1.08 \times 10^{ - 3} ,0.001} \right\}. $$
    (B1)

By setting the initial control gains and time delays to \(g_{i}^{0} = 0\), \(\tau_{i}^{0} = 0\), \( \, i = 1,2\), and solving the corresponding linear system, the approximate responses at all resonance frequencies can be obtained as

$$ {\mathbf{R}}_{0} = \left\{ {{\mathbf{v}}_{1}^{0} ,\Omega_{1}^{0} ,{\mathbf{v}}_{2}^{0} ,\Omega_{2}^{0} ,{\mathbf{v}}_{3}^{0} ,\Omega_{3}^{0} ,{\mathbf{v}}_{4}^{0} ,\Omega_{4}^{0} ,g_{1}^{0} ,\tau_{1}^{0} ,g_{2}^{0} ,\tau_{2}^{0} } \right\}, $$
(B2)

where \({\mathbf{v}}_{i}^{0}\), \(\Omega_{i}^{0}\) for \(i = 1,2,3,4\) denote the harmonic coefficients and the corresponding ith resonance frequencies for the time-delayed system. The values are shown as Eqs. (B3B7).

$$ \Omega_{1}^{0} = 0.5413, \Omega_{2}^{0} = 0.6477, \Omega_{3}^{0} = 1.5733, \Omega_{4}^{0} = 1.6648, $$
(B3)
$$ \begin{aligned} {\mathbf{v}}_{1}^{0} & = \left\{ {0.0029489933899181607,0.002683551652457273,} \right. \\ \, & \quad 0.004031090863229077,0.004564122204181016, \\ \, & \quad - 0.0032764823051420824,0.01978517889595882, \\ \, & \left. {\quad 0.003310645619155455,0.003036213586086393} \right\}, \\ \end{aligned} $$
(B4)
$$ \begin{aligned} {\mathbf{v}}_{2}^{0} & = \left\{ { - 0.0004578245240697892,0.003958878333272922,} \right. \\ \, & \quad - 0.0017109492194471339,0.006237515727242439, \\ \, & \quad - 0.013394206491315563, - 0.008481300504015903, \\ \, & \quad \left. { - 0.0005786852704034095,0.004713530256144929} \right\}, \\ \end{aligned} $$
(B5)
$$ \begin{aligned} {\mathbf{v}}_{3}^{0} & = \left\{ {0.0017997131951469386,0.003525165900110686,} \right. \\ \, & \quad - 0.001367189446220872, - 0.0023740574788185666, \\ \, & \quad 0.0005426967041943076,0.00011812323759052738, \\ \, & \quad \left. { - 0.018730726002204462,0.034181068875830695} \right\}, \\ \end{aligned} $$
(B6)
$$ \begin{aligned} {\mathbf{v}}_{4}^{0} & = \left\{ { - 0.001752514371509075,0.0035497620318474692,} \right. \\ \, & \quad 0.0009106766795588835, - 0.0020833660662084945, \\ \, & \quad 0.00017721148312708785,0.00037572734400900505, \\ \, & \quad \left. { - 0.019968049353075664, - 0.028625375777991315} \right\}. \\ \end{aligned} $$
(B7)

In the update process of structural parameters, the range between p and p0 can be divided into N intervals, and each augmentation is denoted by Δp = (p-p0)/N. From Step 2 to Step 6, the parameters \(k_{a,1}\), \(k_{a,2}\), \(c_{a,1}\), \(c_{a,2}\), \(f\) are sequentially updated from the initial values p0 to the values p.

  • Step 2: The update process for the structural parameter \(k_{a,1}\) and the continuation process of the time-delayed parameters for suppressing the peaks around the first and second modes are carried out.

  • Step 2.1: The update process for the structural parameter \(k_{a,1}\) is carried out.

  • \(k_{a,1}\) is updated from the value p0 to p. N1 = 100, \({\mathbf{\Delta}} \mathbf{{p}}_{1} = \left\{ {{{\left( {k_{a,1} - k_{a,1}^{0} } \right)} \mathord{\left/ {\vphantom {{\left( {k_{a,1} - k_{a,1}^{0} } \right)} {N_{1} }}} \right. \kern-\nulldelimiterspace} {N_{1} }},0,0,0,0} \right\}\), \({\mathbf{p}}_{k} = {\mathbf{p}}_{k - 1} + {\mathbf{ \Delta}} {\mathbf{p}}_{1}\), \({1} \le k \le {100}\).

  • Step 2.2: The continuation process of time-delayed parameters for the first and second modes is carried out.

  • For k from 1 to 100, \(\tau_{2}\) is fixed, and \(g_{1}\), \(g_{2}\), \(\tau_{1}\) are calculated. Then for the first mode, the equal-peak case is determined according to the sufficient condition Eq. (16), as described in Sect. 3.3, and the peak amplitudes are suppressed to the minimum values according to the minimum peak condition Eq. (17), as described in Sect. 3.4. Next, for the second mode, the optimal \(g_{1}\), \(g_{2}\) and \(\tau_{1}\) are selected as the initial values, \(\tau_{1}\) is fixed, and \(g_{1}\), \(g_{2}\), \(\tau_{2}\) are calculated. The same process is repeated, and the final optimal time-delayed parameters are obtained for all modes.

  • Step 3–6: Step 2 is repeated for the update process of \(k_{a,2}\), \(c_{a,1}\), \(c_{a,2}\), and \(f\) sequentially from the value p0 to p.

  • According to the stability condition in Eq. (7), the calculation of the eigenvalues for the time-delayed system is embedded in the procedure to ensure the stability of the system. After the solution Rk corresponding to pk is obtained, the k + 1 iteration starts with pk+1 = pk + Δp and \({\mathbf{R}}_{{k{\mathbf{ + }}1}}^{{\mathbf{0}}} = {\mathbf{R}}_{k}\), where the subscript k indicates the evaluation at iteration k and the superscript 0 denotes the initial value at each iteration. The time-delayed parameters can be calculated to any desired accuracy by setting the appropriate residual error.

Appendix C

Figures 14 and 15 show the comparison between analytical and numerical results for the multiple steady-state response in Fig. 6 at Ω = 0.85 rad/s and Ω = 2.3 rad/s, respectively.

Fig. 14
figure 14

Comparison of the analytical (black lines) and numerical (red dots) results for the multiple steady-state responses in Fig. 6 at Ω = 0.85 rad/s, f = 0.08 N. (ac) are time history, phase trajectory, and frequency spectrum for the response attracted to MRC, (df) are time history, phase trajectory, and frequency spectrum for the response attracted to DRC

Fig. 15
figure 15

Comparison of the analytical (black lines) and numerical (red dots) results for the multiple steady-state responses in Fig. 6 at Ω = 2.3 rad/s, f = 0.08 N. (ac) are time history, phase trajectory, and frequency spectrum for the response attracted to MRC, (df) are time history, phase trajectory, and frequency spectrum for the response attracted to DRC

In Figs. 14 and 15, time history, phase trajectory, and frequency spectrum are given for the response attracted to MRC and DRC with different initial conditions. The phase trajectories show that the numerical and analytical results of MRC match better than that of DRC. The frequency spectrum indicates that the error is caused by the third-order harmonic response. The third-order harmonic response exists in numerical results and becomes larger for the response of DRC. While for the analytical results, only the first-order harmonic is considered and higher-order terms are overlooked.

Figures 16 and 17 show the comparison between analytical and numerical results for FRCs in Fig. 9 at f = 0.03 N, 0.05 N and FRCs in Fig. 12 at f = 0.03 N and 0.06 N, respectively.

In Figs. 16 and 17, time history, phase trajectory, and frequency spectrum are given at Ω = 1.7 rad/s and Ω = 1.9 rad/s, respectively. It indicates that the numerical results obtained by the Runge–Kutta method are in good agreement with the analytical results obtained by the Averaging Method.

Fig. 16
figure 16

Comparison of the analytical and numerical results for the TDVAs in Fig. 9. (a) FRC obtained by analytical and numerical methods. (bd) are time history, phase trajectory, and frequency spectrum at Ω = 1.7 rad/s, f = 0.03 N, (eg) are time history, phase trajectory, and frequency spectrum at Ω = 1.7 rad/s, f = 0.05 N

Fig. 17
figure 17

Comparison of the analytical and numerical results for the TDVAs in Fig. 12. (a) FRC obtained by analytical and numerical methods. (bd) are time history, phase trajectory, and frequency spectrum at Ω = 1.9 rad/s, f = 0.03 N, (eg) are time history, phase trajectory, and frequency spectrum at Ω = 1.9 rad/s, f = 0.06 N

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Meng, H., Sun, X., Xu, J. et al. Establishment of the equal-peak principle for a multiple-DOF nonlinear system with multiple time-delayed vibration absorbers. Nonlinear Dyn 104, 241–266 (2021). https://doi.org/10.1007/s11071-021-06301-w

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