Research Papers: Design Automation

Tuned Nonlinear Energy Sink With Conical Spring: Design Theory and Sensitivity Analysis

[+] Author and Article Information
Donghai Qiu

Changchun Institute of Optics,
Fine Mechanics and Physics,
Chinese Academy of Sciences,
Changchun 130033, China;
University of Chinese Academy of Sciences,
Beijing 100049, China;
Institut Clément Ader (ICA), CNRS,
Université de Toulouse,
3 rue Caroline Aigle,
Toulouse F-31400, France
e-mail: donghai.qiu@insa-toulouse.fr

Sébastien Seguy

Institut Clément Ader (ICA), CNRS,
Université de Toulouse,
3 rue Caroline Aigle,
Toulouse F-31400, France
e-mail: sebastien.seguy@insa-toulouse.fr

Manuel Paredes

Institut Clément Ader (ICA), CNRS,
Université de Toulouse,
3 rue Caroline Aigle,
Toulouse F-31400, France
e-mail: manuel.paredes@insa-toulouse.fr

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 24, 2017; final manuscript received October 17, 2017; published online November 9, 2017. Assoc. Editor: Ettore Pennestri.

J. Mech. Des 140(1), 011404 (Nov 09, 2017) (10 pages) Paper No: MD-17-1500; doi: 10.1115/1.4038304 History: Received July 24, 2017; Revised October 17, 2017

This paper is devoted to the study of a nonlinear energy sink (NES) intended to attenuate vibration induced in a harmonically forced linear oscillator (LO) and working under the principle of targeted energy transfer (TET). The purpose motivated by practical considerations is to establish a design criterion that first ensures that the NES absorber is activated and second provides the optimally tuned nonlinear stiffness for efficient TET under a given primary system specification. Then a novel NES design yielding cubic stiffness without a linear part is exploited. To this end, two conical springs are specially sized to provide the nonlinearity. To eliminate the linear stiffness, the concept of a negative stiffness mechanism is implemented by two cylindrical compression springs. A small-sized NES system is then developed. To validate the concept, a sensitivity analysis is performed with respect to the adjustment differences of the springs and an experiment on the whole system embedded on an electrodynamic shaker is studied. The results show that this type of NES can not only output the expected nonlinear characteristics, but can also be tuned to work robustly over a range of excitation, thus making it practical for the application of passive vibration control.

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Jiang, J. , and Iwai, Y. , 2009, “ Improving the B-Spline Method of Dynamically-Compensated Cam Design by Minimizing or Restricting Vibrations in High-Speed Cam-Follower Systems,” ASME J. Mech. Des., 131(4), p. 041003. [CrossRef]
Kim, S. , Dean, R. , Flowers, G. , and Chen, C. , 2009, “ Active Vibration Control and Isolation for Micromachined Devices,” ASME J. Mech. Des., 131(9), p. 091002. [CrossRef]
Trimble, A. Z. , Lang, J. H. , Pabon, J. , and Slocum, A. , 2010, “ A Device for Harvesting Energy From Rotational Vibrations,” ASME J. Mech. Des., 132(9), p. 091001. [CrossRef]
Okwudire, C. E. , 2012, “ Reduction of Torque-Induced Bending Vibrations in Ball Screw-Driven Machines Via Optimal Design of the Nut,” ASME J. Mech. Des., 134(11), p. 111008. [CrossRef]
Moeenfard, H. , and Awtar, S. , 2014, “ Modeling Geometric Nonlinearities in the Free Vibration of a Planar Beam Flexure With a Tip Mass,” ASME J. Mech. Des., 136(4), p. 044502. [CrossRef]
Vakakis, A. F. , Gendelman, O. V. , Bergman, L. A. , McFarland, D. M. , Kerschen, G. , and Lee, Y. S. , 2008, Targeted Energy Transfer in Mechanical and Structural Systems, Vol. 156, Springer Science & Business Media, Berlin.
Vakakis, A. F. , and Gendelman, O. V. , 2001, “ Energy Pumping in Nonlinear Mechanical Oscillators: Part II: Resonance Capture,” ASME J. Appl. Mech., 68(1), pp. 42–48. [CrossRef]
Lee, Y. , Vakakis, A. F. , Bergman, L. , McFarland, D. , Kerschen, G. , Nucera, F. , Tsakirtzis, S. , and Panagopoulos, P. , 2008, “ Passive Non-Linear Targeted Energy Transfer and Its Applications to Vibration Absorption: A Review,” Proc. Inst. Mech. Eng. Part K J. Multibody Dyn., 222(2), pp. 77–134.
Gourdon, E. , Alexander, N. , Taylor, C. , Lamarque, C. , and Pernot, S. , 2007, “ Nonlinear Energy Pumping Under Transient Forcing With Strongly Nonlinear Coupling: Theoretical and Experimental Results,” J. Sound. Vib, 300(35), pp. 522–551. [CrossRef]
Gourc, E. , Michon, G. , Seguy, S. , and Berlioz, A. , 2014, “ Experimental Investigation and Design Optimization of Targeted Energy Transfer Under Periodic Forcing,” ASME J. Vib. Acoust, 136(2), p. 021021. [CrossRef]
Gourc, E. , Michon, G. , Seguy, S. , and Berlioz, A. , 2015, “ Targeted Energy Transfer Under Harmonic Forcing With a Vibro-Impact Nonlinear Energy Sink: Analytical and Experimental Developments,” ASME J. Vib. Acoust, 137(3), p. 031008. [CrossRef]
Lamarque, C.-H. , Gendelman, O. V. , Savadkoohi, A. T. , and Etcheverria, E. , 2011, “ Targeted Energy transfer in mechanical Systems by Means of Non-Smooth Nonlinear Energy Sink,” Acta Mech., 221(1–2), p. 175. [CrossRef]
Sigalov, G. , Gendelman, O. , Al-Shudeifat, M. , Manevitch, L. , Vakakis, A. , and Bergman, L. , 2012, “ Resonance Captures and Targeted Energy Transfers in an Inertially-Coupled Rotational Nonlinear Energy Sink,” Nonlinear Dyn., 69(4), pp. 1693–1704. [CrossRef]
Jutte, C. V. , and Kota, S. , 2008, “ Design of Nonlinear Springs for Prescribed Load-Displacement Functions,” ASME J. Mech. Des., 130(8), p. 081403. [CrossRef]
Jutte, C. V. , and Kota, S. , 2010, “ Design of Single, Multiple, and Scaled Nonlinear Springs for Prescribed Nonlinear Responses,” ASME J. Mech. Des., 132(1), p. 011003. [CrossRef]
Wu, Y.-S. , and Lan, C.-C. , 2014, “ Linear Variable-Stiffness Mechanisms Based on Preloaded Curved Beams,” ASME J. Mech. Des., 136(12), p. 122302. [CrossRef]
Sönmez, Ü. , 2007, “ Introduction to Compliant Long Dwell Mechanism Designs Using Buckling Beams and Arcs,” ASME J. Mech. Des., 129(8), pp. 831–843. [CrossRef]
Sönmez, Ü. , and Tutum, C. C. , 2008, “ A Compliant Bistable Mechanism Design Incorporating Elastica Buckling Beam Theory and Pseudo-Rigid-Body Model,” ASME J. Mech. Des., 130(4), p. 042304. [CrossRef]
Chen, Y.-H. , and Lan, C.-C. , 2012, “ An Adjustable Constant-Force Mechanism for Adaptive End-Effector Operations,” ASME J. Mech. Des., 134(3), p. 031005. [CrossRef]
Al-Shudeifat, M. A. , 2017, “ Nonlinear Energy Sinks With Nontraditional Kinds of Nonlinear Restoring Forces,” ASME J. Vib. Acoust, 139(2), p. 024503. [CrossRef]
Gendelman, O. , Starosvetsky, Y. , and Feldman, M. , 2008, “ Attractors of Harmonically Forced Linear Oscillator With Attached Nonlinear Energy Sink I: Description of Response Regimes,” Nonlinear Dyn., 51(1–2), pp. 31–46.
Starosvetsky, Y. , and Gendelman, O. , 2007, “ Attractors of Harmonically Forced Linear Oscillator With Attached Nonlinear Energy Sink II: Optimization of a Nonlinear Vibration Absorber,” Nonlinear Dyn., 51(1), p. 47. [CrossRef]
Starosvetsky, Y. , and Gendelman, O. , 2008, “ Strongly Modulated Response in Forced 2DOF Oscillatory System With Essential Mass and Potential Asymmetry,” Phys. D: Nonlinear Phenom., 237(13), pp. 1719–1733. [CrossRef]
Rodriguez, E. , Paredes, M. , and Sartor, M. , 2006, “ Analytical Behavior Law for a Constant Pitch Conical Compression Spring,” ASME J. Mech. Des., 128(6), pp. 1352–1356. [CrossRef]
Paredes, M. , 2013, “ Analytical and Experimental Study of Conical Telescoping Springs With Nonconstant Pitch,” ASME J. Mech. Des., 135(9), p. 094502. [CrossRef]
Patil, R. V. , Reddy, P. R. , and Laxminarayana, P. , 2014, “ Comparison of Cylindrical and Conical Helical Springs for Their Buckling Load and Deflection,” Int. J. Adv. Sci. Technol., 73, pp. 33–50. [CrossRef]
Qiu, D. , Seguy, S. , and Paredes, M. , 2017, “ A Novel Design of Cubic Stiffness for a Nonlinear Energy Sink (NES) Based on Conical Spring,” Advances on Mechanics, Design Engineering and Manufacturing, Springer, Berlin, pp. 565–573. [CrossRef]
Harne, R. , Thota, M. , and Wang, K. , 2013, “ Concise and High-Fidelity Predictive Criteria for Maximizing Performance and Robustness of Bistable Energy Harvesters,” Appl. Phys. Lett., 102(5), p. 053903. [CrossRef]
Yamamoto, Y. , 1999, “ Spring's Effective Mass in Spring Mass System Free Vibration,” J. Sound. Vib, 220(3), pp. 564–570. [CrossRef]
Opgenoord, M. M. , Allaire, D. L. , and Willcox, K. E. , 2016, “ Variance-Based Sensitivity Analysis to Support Simulation-Based Design Under Uncertainty,” ASME J. Mech. Des., 138(11), p. 111410. [CrossRef]
Romeo, F. , Sigalov, G. , Bergman, L. A. , and Vakakis, A. F. , 2015, “ Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Numerical Study,” ASME J. Comput. Nonlinear Dyn., 10(1), p. 011007. [CrossRef]


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Fig. 2

Cubic NES under periodic forcing with parameters K = 3000, λ1 = 0.6, λ2 = 0.3, ε = 0.01, G = 0.2 mm and initial conditions x0 = 0, x˙0=0, y0=0 and y˙0=0. (a) time-displacement of LO and NES. (b) SIM structure and trace between LO and cubic NES: the thin curve represents the transient projected motion.

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Fig. 1

Schematic of a harmonically excited LO coupled with a NES

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Fig. 9

Polynomial components of the two optimized conical springs

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Fig. 3

Frequency response function of LO with cubic NES (points) and without NES (thin line) in different types of excitation: (a) G = 0.08 mm, (b) G = 0.13 mm, (c) G = 0.18 mm, and (d) G = 0.23 mm. The points and the crosses represent stable and unstable fixed points, respectively.

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Fig. 4

Maximum amplitude of LO with the variation of (a) nonlinear stiffness and excitation amplitude, (b) nonlinear stiffness, and (c) excitation amplitude. Each point is extracted from the maximum amplitude of FRF.

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Fig. 5

Evolution of the multiplicity of periodic solutions for the system with parameters K = 3000, λ1 = 0.6, λ2 = 0.3, and ε = 0.01

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Fig. 6

Critical excitation amplitude as a function of the nonlinear stiffness, λ1 = 0.6, λ2 = 0.3, and ε = 0.01

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Fig. 7

Force characteristic of conical spring, T represents the transition point between linear and nonlinear phase

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Fig. 8

State and corresponding force of two conical springs (the dashed line): (a) and (c) at original length; (b) and (d) precompressed at the transition point. The solid line represents the composed force.

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Fig. 10

Schematic of NES system: (a) negative stiffness mechanism, (b) conical spring, and (c) the composed system

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Fig. 20

Experimental results: (a) frequency response curve of LO with (thick line) and without (thin line) the designed NES, and (b) SMR of LO (solid line) and NES (dashed line)

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Fig. 18

The nonlinear NES force versus the response of LO and NES with G = 0.205 mm, σ = −0.2: (a) and (d) positive linear stiffness, γ1 = γ2 = 1, η1 = η2 =−1; (b) and (e) negative linear stiffness, γ1 = γ2 = −1, η1 = η2 = 1; (c) and (f) unsymmetrical stiffness, γ1 = 1, γ2 = −1, η1 = η2 = 0

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Fig. 11

Assembly of NES system. x and y correspond to the displacement of LO and NES, respectively.

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Fig. 12

Conical spring manufactured and measuring equipment

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Fig. 13

Experimental force–displacement relation of conical springs

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Fig. 14

Nonlinear energy sink system with linear springs of large mean diameter

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Fig. 15

Details of the experimental setup and measuring system

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Fig. 16

Force–displacement relation of the designed NES

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Fig. 19

Experimental setup: (a) global view of the system and (b) detailed view of LO and NES

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Fig. 17

Differences of the adjustment length for conical springs (γi) and linear springs (ηi)



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