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Rotary-impact nonlinear energy sink for shock mitigation: analytical and numerical investigations

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Abstract

The nonlinear energy sink (NES) is a lightweight, strongly nonlinear dynamical attachment coupled to a (typically linear) large-scale primary structure for passive vibration mitigation. There are two nonlinear mechanisms governing the dynamics of the coupled system: irreversible targeted energy transfer (TET) from the primary structure to the NES, where energy is confined and locally dissipated, and NES-induced nonlinear energy scattering between the structural modes of the primary structure. In the literature, different NES designs have been investigated to optimize their nonlinear effects on the primary structures. One such design is the rotary NES consisting of a small mass inertially coupled to the primary structure through a rigid arm; another is the vibro-impact NES with non-smooth nonlinearities and inelastic collisions with the primary structure. These types have been found to achieve strong and rapid TET and are less sensitive to energy fluctuations. In this work, a hybrid NES design is proposed based on the synergetic synthesis of the rotary and impact-based NESs in a single rotary-impact NES (RINES). The RINES incorporates a fixed rigid barrier attached (typically) to the top floor of the primary structure to inflict impacts between its rotating mass and the top floor. An analytical study to evaluate its capacity to engage in resonance capture with a primary structure is presented first, followed by numerical investigations of cases when the RINES is attached to the top floors of small- and large-scale linear primary structures under impulsive excitation. The non-smooth nonlinearities induced through the consecutive impacts resulted in effective broadband shock mitigation at highly energetic response regimes, whereas the nonlinear inertial coupling enables similar beneficial mitigation capacity at lower-energetic response regimes. Hence, the combined effect of non-smooth and inertial nonlinearities enables effective passive mitigation capacity for a broad range of applied impulsive energies.

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Authors and Affiliations

  1. Aerospace Engineering, Khalifa University of Science and Technology, 127788, Abu Dhabi, UAE

    Adnan S. Saeed, Mohammad A. AL-Shudeifat & Wesley J. Cantwell

  2. Mechanical Science and Engineering, University of Illinois at Urbana – Champaign, Urbana, IL, 61801, USA

    Alexander F. Vakakis

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  1. Adnan S. Saeed
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  2. Mohammad A. AL-Shudeifat
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Appendix: System parameters of the two-story (small-scale) and nine-story (large-scale) structures [37, 46, 61,62,63, 65, 68, 69]

Appendix: System parameters of the two-story (small-scale) and nine-story (large-scale) structures [37, 46, 61,62,63, 65, 68, 69]

The mass matrix of the linear two-story structure with locked NESs is given by:

$$\begin{aligned} M=\left[ {{\begin{array}{ll} {24.2}&{} 0 \\ 0&{} {24.3} \\ \end{array} }} \right] \hbox { (Kg)} \end{aligned}$$
(A1)

The stiffness matrix of the linear two-story structure is given by:

$$\begin{aligned} K=\left[ {{\begin{array}{ll} {8220}&{} {-8220} \\ {-8220}&{} {15040} \\ \end{array} }} \right] \hbox { (N/m)} \end{aligned}$$
(A2)

Assuming proportional viscous damping distribution, the damping matrix of the linear two-story structure is given by:

$$\begin{aligned} C=\left[ {{\begin{array}{ll} {0.9679}&{} {-0.2951} \\ {-0.2951}&{} {1.2155} \\ \end{array} }} \right] \hbox { (N/m)} \end{aligned}$$
(A3)

The derived modal damping values are:

$$\begin{aligned} \lambda _i =0.0343, 0.0604 \end{aligned}$$
(A4)

The mass matrix of the linear nine-story structure with locked NESs is given by:

$$\begin{aligned} M=\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {1037}&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ 0&{} {1074}&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} {1075}&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} {1075}&{} 0&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} {1075}&{} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} {1075}&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} {1075}&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} {1075}&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} {1098} \\ \end{array} }} \right] \hbox { (Kg)} \end{aligned}$$
(A5)

The stiffness matrix of the linear nine-story structure is given by:

$$\begin{aligned} K= & {} \left[ {\begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r} 3.6962&{} -3.7544&{} 0.0375 &{} 0.0035 &{} 0.0032 &{} 0.0032 &{} 0.0032 &{} 0.0032 &{} 0.0030 \\ -3.7544&{} 7.7430 &{} -4.0534&{} 0.0646 &{} -0.0006&{} 0.0002 &{} 0.0002 &{} 0.0002 &{} 0.0002 \\ 0.0375&{} -4.0534&{} 8.2141 &{} -4.2657&{} 0.0678 &{} -0.0007&{} 0.0001 &{} 0.0001 &{} 0.0001 \\ 0.0035&{} 0.0646 &{} -4.2657&{} 8.3986 &{} -4.2680&{} 0.0677 &{} -0.0008&{} 0.0000 &{} 0.0000 \\ 0.0032&{} -0.0006&{} 0.0678 &{} -4.2680&{} 8.3986 &{} -4.2680&{} 0.0677 &{} -0.0008&{} 0.0000 \\ 0.0032&{} 0.0002 &{} -0.0007&{} 0.0677 &{} -4.2680&{} 8.3986 &{} -4.2680&{} 0.0677 &{} -0.0008 \\ 0.0032&{} 0.0002 &{} 0.0001 &{} -0.0008&{} 0.0677 &{} -4.2680&{} 8.3986 &{} -4.2680&{} 0.0677 \\ 0.0032&{} 0.0002 &{} 0.0001 &{} 0.0000 &{} -0.0008&{} 0.0677 &{} -4.2680&{} 8.3989 &{} -4.2658 \\ 0.0030&{} 0.0002 &{} 0.0001 &{} 0.0000 &{} 0.0000 &{} -0.0008&{} 0.0677 &{} -4.2658&{} 7.6583 \\ \end{array}} \right] \nonumber \\&(\times 10^{6}\hbox {N/m}) \end{aligned}$$
(A6)

Assuming proportional viscous damping distribution, the damping matrix of the linear nine-story structure is given by:

$$\begin{aligned} C= & {} \left[ {\begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r} 0.8749 &{} -0.5133&{} -0.1418&{} -0.0590&{} -0.0310&{} -0.0180&{} -0.0107&{} -0.0060&{} -0.0028 \\ -0.5133&{} 1.3401 &{} -0.5330&{} -0.1004&{} -0.0441&{} -0.0237&{} -0.0137&{} -0.0078&{} -0.0039 \\ -0.1418&{} -0.5330&{} 1.6460 &{} -0.5988&{} -0.1179&{} -0.0518&{} -0.0273&{} -0.0149&{} -0.0072 \\ -0.0590&{} -0.1004&{} -0.5988&{} 1.6963 &{} -0.5834&{} -0.1095&{} -0.0461&{} -0.0226&{} -0.0104 \\ -0.0310&{} -0.0441&{} -0.1179&{} -0.5834&{} 1.7047 &{} -0.5777&{} -0.1047&{} -0.0413&{} -0.0173 \\ -0.0180&{} -0.0237&{} -0.0518&{} -0.1095&{} -0.5777&{} 1.7095 &{} -0.5729&{} -0.0992&{} -0.0344 \\ -0.0107&{} -0.0137&{} -0.0273&{} -0.0461&{} -0.1047&{} -0.5729&{} 1.7151 &{} -0.5653&{} -0.0888 \\ -0.0060&{} -0.0078&{} -0.0149&{} -0.0226&{} -0.0413&{} -0.0992&{} -0.5653&{} 1.7273 &{} -0.5498 \\ -0.0028&{} -0.0039&{} -0.0072&{} -0.0104&{} -0.0173&{} -0.0344&{} -0.0888&{} -0.5498&{} 1.7450 \\ \end{array}} \right] \nonumber \\&(\times 10^{3}\hbox {N s /m}) \end{aligned}$$
(A7)

The derived modal damping values are:

$$\begin{aligned} \lambda _i =0.2033, 0.6084, 1.0131, 1.3995, 1.7459, 2.0181, 2.1911, 2.3257, 2.4698 \end{aligned}$$
(A8)

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Saeed, A.S., AL-Shudeifat, M.A., Vakakis, A.F. et al. Rotary-impact nonlinear energy sink for shock mitigation: analytical and numerical investigations. Arch Appl Mech 90, 495–521 (2020). https://doi.org/10.1007/s00419-019-01622-0

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