Relevant bibliographies by topics / Nonlinear Restoring Force

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Nonlinear Restoring Force'

Author: Grafiati
Published: 4 June 2021
Last updated: 18 February 2022

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Journal articles on the topic "Nonlinear Restoring Force"

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Shimin, Wang, and Yang Lechang. "An Analytical Approximation Method for Strongly Nonlinear Oscillators." Journal of Applied Mathematics 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/958121.

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An analytical method is proposed to get the amplitude-frequency and the phase-frequency characteristics of free/forced oscillators with nonlinear restoring force. The nonlinear restoring force is expressed as a spring with varying stiffness that depends on the vibration amplitude. That is, for stationary vibration, the restoring force linearly depends on the displacement, but the stiffness of the spring varies with the vibration amplitude for nonstationary oscillations. The varied stiffness is constructed by means of the first and second averaged derivatives of the restoring force with respect to the displacement. Then, this stiffness gives the amplitude frequency and the phase frequency characteristics of the oscillator. Various examples show that this method can be applied extensively to oscillators with nonlinear restoring force, and that the solving process is extremely simple.
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Araki, Yoshikazu, Takehiko Asai, Kosuke Kimura, Kosei Maezawa, and Takeshi Masui. "Nonlinear vibration isolator with adjustable restoring force." Journal of Sound and Vibration 332, no. 23 (November 2013): 6063–77. http://dx.doi.org/10.1016/j.jsv.2013.06.030.

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Carboni, Biagio, Walter Lacarbonara, Patrick T. Brewick, and Sami F. Masri. "Dynamical response identification of a class of nonlinear hysteretic systems." Journal of Intelligent Material Systems and Structures 29, no. 13 (June 7, 2018): 2795–810. http://dx.doi.org/10.1177/1045389x18778792.

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The experimental dynamical response of three types of nonlinear hysteretic systems is identified employing phenomenological models togheter with the Differential Evolutionary algorithm. The mass–spring–damper system is characterized by hysteretic restoring forces provided by assemblies of shape memory and steel wire ropes subject to flexure or coupled states of tension and flexure. The energy dissipation due to phase transformations and inter-wire friction and the stretching-induced geometric nonlinearities give rise to different shapes of hysteresis cycles. The mechanical device subject to strong seismic excitations is investigated in its ultimate limit state whereby inelastic strains are induced in the steel wires together with a global nonsymmetric response of the system. The targeted dynamical characterization of the hysteretic oscillator up to its ultimate limit state has a special meaning when the device is employed in the field of vibration control. The dynamical response is identified exploiting the measurements of the oscillating mass relative displacement and inertia force that must be balanced, at each time instant, by the overall restoring forces provided by the mechanism. The restoring force is assumed to be the sum of different contributions such as a cubic nonsymmetric elastic force and a nonsymmetric hysteretic force modeled according to a modified version of the Bouc–Wen model. The parameters are identified minimizing the difference between the numerical and the experimental restoring force histories. High levels of accuracy are achieved in the identification with mean square errors lower than 2%.
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Ishida, Y., T. Ikeda, and T. Yamamoto. "Nonlinear Forced Oscillations Caused by Quartic Nonlinearity in a Rotating Shaft System." Journal of Vibration and Acoustics 112, no. 3 (July 1, 1990): 288–97. http://dx.doi.org/10.1115/1.2930507.

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This paper deals with nonlinear forced oscillations in a rotating shaft system which are caused by quartic nonlinearity in a restoring force. These oscillations are theoretically analyzed by paying attention to the nonlinear components represented by the polar coordinates. It is clarified which kind of nonlinear component has an influence on each oscillation. In experiments it was shown that, when the shaft was supported by double-row angular contact ball bearings, the restoring force had nonlinear spring characteristics involving quartic nonlinearity in addition to quadratic and cubic ones. Experimental results were compared with the theoretical results regarding the probability of occurrence and the shapes of the resonance curves.
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Zhang, Pin Le. "Restoring Force Models Investigation of Research Status and Problems Analysis of RC Structure." Applied Mechanics and Materials 275-277 (January 2013): 1045–48. http://dx.doi.org/10.4028/www.scientific.net/amm.275-277.1045.

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Restoring force model is a simplified mathematical model derived from restoring force-deformation curves in lots of tests. Selecting a reasonable restoring force model is the basis of conducting dynamic nonlinear analysis of structure. The work further investigates the advantage and disadvantage of the restoring force models presented in this paper. Classified and brief comments about the existing drawbacks of restoring force models and its application are conducted. Lastly, some useful suggestions are proposed for the further research.
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Lei, Y., SJ Luo, and MY He. "Identification of model-free structural nonlinear restoring forces using partial measurements of structural responses." Advances in Structural Engineering 20, no. 1 (July 28, 2016): 69–80. http://dx.doi.org/10.1177/1369433216646006.

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Identification of nonlinear structural system is an important but challenging task for structural health monitoring. Due to the complexities of structural nonlinearities, it is hard to establish proper mathematical models for some structural nonlinear behaviors. Moreover, only partial structural responses can be measured in practice; it is essential to conduct identification of nonlinear structural systems using only partial measurements of structural responses. To cope with these issues, an algorithm is proposed in this article for the identification of some model-free structural nonlinear restoring forces using only partial measurements of structural responses. First, an equivalent linear structural system is introduced for the identification of the locations of structural nonlinearities. Then, a model-free structural nonlinear restoring force is approximated by a power series polynomial. The unknown coefficients of the power series polynomials together with other structural parameters are identified by the extended Kalman filter so that the characteristics of the behaviors of the model-free of nonlinear restoring forces can be identified. Some numerical examples including the identification of two nonlinear multi-story shear frames and a planar nonlinear truss with different structural nonlinear restoring forces are used to validate the proposed algorithm.
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Bhattacharyya, K., and J. K. Dutt. "Unbalance Response and Stability Analysis of Horizontal Rotor Systems Mounted on Nonlinear Rolling Element Bearings With Viscoelastic Supports." Journal of Vibration and Acoustics 119, no. 4 (October 1, 1997): 539–44. http://dx.doi.org/10.1115/1.2889757.

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Unbalance response and stability analysis of a rotor shaft system, with the rotor mounted in the middle of a massless shaft, having linear elasticity and internal damping, with bearings having nonlinear restoring force characteristics at the ends mounted on viscoelastic support, has been carried out, taking the effect of gravity into account. The restoring force characteristics of the bearing has been linearized, by the method of effective linearization, thereby enabling an approximate stability analysis using simple techniques. It is found that, unlike the case with a bearing having linearly varying restoring force characteristics, gravity not only affects the unbalance response but also causes a decrease in the stability limits when the restoring force characteristics are nonlinear.
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Kovacic, Ivana. "Forced vibrations of oscillators with a purely nonlinear power-form restoring force." Journal of Sound and Vibration 330, no. 17 (August 2011): 4313–27. http://dx.doi.org/10.1016/j.jsv.2011.04.001.

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JO, Hoonhee, and Hiroshi YABUNO. "605 Paremetric resonance due to asymmetric nonlinear restoring force." Proceedings of the Dynamics & Design Conference 2007 (2007): _605–1_—_605–5_. http://dx.doi.org/10.1299/jsmedmc.2007._605-1_.

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TAUE, Katsuhira, and Tadayoshi KOIZUMI. "315 Characteristics of Chaotic Responses under nonlinear Restoring Force." Proceedings of Autumn Conference of Tohoku Branch 2010.46 (2010): 95–96. http://dx.doi.org/10.1299/jsmetohoku.2010.46.95.

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Dissertations / Theses on the topic "Nonlinear Restoring Force"

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Aykan, Murat. "Identification Of Localized Nonlinearity For Dynamic Analysis Of Structures." Phd thesis, METU, 2013. http://etd.lib.metu.edu.tr/upload/12615596/index.pdf.

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Most engineering structures include nonlinearity to some degree. Depending on the dynamic conditions and level of external forcing, sometimes a linear structure assumption may be justified. However, design requirements of sophisticated structures such as satellites, stabilized weapon systems and radars may require nonlinear behavior to be considered for better performance. Therefore, it is very important to successfully detect, localize and parametrically identify nonlinearity in such cases. In engineering applications, the location of nonlinearity and its type may not be always known in advance. Furthermore, as the structure will be excited from only a few coordinates, the frequency response function matrices will not be complete. In order to parametrically identify more than one type of nonlinearity which may co-exist at the same location with the above mentioned limitations, a method is proposed where restoring force surface plots are used which are evaluated by describing function inversion. Then, by reformulating this method, a second method is proposed which can directly evaluate the total describing function of more than one type of nonlinearity which may co-exist at the same location without using any linear frequency response function matrix. It is also aimed in this study to use the nonlinearity localization formulations for damage localization purposes. The validation of the methods developed in this study is demonstrated with case studies based on simulated experiments, as well as real experiments with nonlinear structures and it is concluded that the methods are very promising to be used in engineering structures.
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Paul, Bryan. "Analytical And Experimental Study Of Monitoring For Chain-Like Nonlinear Dynamic Systems." Master's thesis, University of Central Florida, 2013. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/5686.

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Inverse analysis of nonlinear dynamic systems is an important area of research in the ?eld of structural health monitoring for civil engineering structures. Structural damage usually involves localized nonlinear behaviors of dynamic systems that evolve into different classes of nonlinearity as well as change system parameter values. Numerous parametric modal analysis techniques (e.g., eigensystem realization algorithm and subspace identification method) have been developed for system identification of multi-degree-of-freedom dynamic systems. However, those methods are usually limited to linear systems and known for poor sensitivity to localized damage. On the other hand, non-parametric identification methods (e.g., artificial neural networks) are advantageous to identify time-varying nonlinear systems due to unpredictable damage. However, physical interpretation of non-parametric identification results is not as straightforward as those of the parametric methods. In this study, the Multidegree-of-Freedom Restoring Force Method (MRFM) is employed as a semi-parametric nonlinear identification method to take the advantages of both the parametric and non-parametric identification methods. The MRFM is validated using two realistic experimental nonlinear dynamic tests: (i) large-scale shake table tests using building models with different foundation types, and (ii) impact test using wind blades. The large-scale shake table test was conducted at Tongji University using 1:10 scale 12-story reinforced concrete building models tested on three different foundations, including pile, box and fixed foundation. The nonlinear dynamic signatures of the building models collected from the shake table tests were processed using MRFM (i) to investigate the effects of foundation types on nonlinear behavior of the superstructure and (ii) to detect localized damage during the shake table tests. Secondly, the MRFM was applied to investigate the applicability of this method to wind turbine blades. Results are promising, showing a high level of nonlinearity of the system and how the MRFM can be applied to wind-turbine blades. Future studies were planned for the comparison of physical characteristic of this blade with blades created made of other material.
M.S.
Masters
Civil, Environmental, and Construction Engineering
Engineering and Computer Science
Civil Engineering; Structures and Geotechnical Engineering
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Hsieh, Fu-Hsuan, and 謝馥亘. "A Characteristic Time Expansion Method for Restoring-Force Identification in a Nonlinear Engineering Problem." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/48352544546464341240.

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碩士
國立臺灣海洋大學
系統工程暨造船學系
101
Since numerical instability phenomena always arise in the solving process for parameters identification of structural vibration, to resolve such a problem, the characteristic time expansion method in conjunction with the natural regularization method is adopted to overcome the higher order numerical oscillation problem when polynomial series expansion is necessary. Due to inclusion of the characteristic length in the scheme, the condition number of the constructed Vandermonde matrix will be reduced and will also increase the term number of polynomial series. Thus, the ill condition and numerical instability of numerical calculations can be resolved. Besides, to overcome the numerical instability problem of a noise disturbance, in contrast to the conventional Tikhonov regularization method, the natural regularization method is again adopted to resolve the problem. It is shown that single-scale and multi-scale characteristic time expansion methods with the natural regularization method can effectively those above mentioned problems through five benchmark examples.
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Chien, Shih-Ann, and 簡世安. "The Fictitious Time Integration Method and Characteristic Time Expansion Method for Estimating Nonlinear Restoring Force." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/38765095275543975855.

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碩士
國立臺灣海洋大學
機械與機電工程學系
97
For the inverse vibration problem, we propose the Fictitious Time Integration Method (FTIM) and the Characteristic Time Expansion Method (CTEM) to estimate the non-linear restoring force by using displacement data as input. In the Fictitious Time Integration Method, by introducing a fictitious time , we transform the Non-linear Algebraic Equations (NAEs) into the Ordinary Differential Equations (ODEs), and then we could obtain the numerical result by applying the Group Preserving Scheme (GPS). On the other hand, the Characteristic Time Expansion Method by introducing the characteristic time in the polynomial interpolation method, which may improve the ill-posedness of interpolation by high-order polynomials. From the numerical examples examined, both of the results in numerical methods for estimating non-linear restoring force have high stability and high accuracy.
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Book chapters on the topic "Nonlinear Restoring Force"

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Dimentberg, M. F., and A. A. Sokolov. "On the Cross-Correlation Method for Identification of Modal Restoring Force Nonlinearity from Random Vibration Data." In Nonlinear Stochastic Mechanics, 167–70. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/978-3-642-84789-9_14.

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Noel, J. P., G. Kerschen, and A. Newerla. "Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure." In Topics in Nonlinear Dynamics, Volume 3, 1–19. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-2416-1_1.

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Moldenhauer, Benjamin, Daniel R. Roettgen, and Benjamin Pacini. "Implementing the Restoring Force Surface Method to Fit Experimentally Measured Modal Coupling Effects." In Nonlinear Structures & Systems, Volume 1, 79–82. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-47626-7_12.

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Aykan, Murat, and H. Nevzat Özgüven. "Identification of Restoring Force Surfaces in Nonlinear MDOF Systems from FRF Data Using Nonlinearity Matrix." In Topics in Nonlinear Dynamics, Volume 1, 65–76. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-6570-6_5.

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Villani, Luis G. G., Samuel da Silva, and Americo Cunha. "Application of a Stochastic Version of the Restoring Force Surface Method to Identify a Duffing Oscillator." In Nonlinear Dynamics of Structures, Systems and Devices, 299–307. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-34713-0_30.

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Belovodskiy, V. N., and M. Y. Sukhorukov. "Combination Resonances and Their Bifurcations in the Nonlinear Vibromachines with a Polynomial Characteristic of Restoring Force and Periodic Excitation." In Springer Proceedings in Physics, 235–40. Dordrecht: Springer Netherlands, 2011. http://dx.doi.org/10.1007/978-94-007-2069-5_32.

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Marthinsen, Tom. "The Statistics of Slow Drift Oscillations with Nonlinear Restoring Forces." In Nonlinear Water Waves, 459–66. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/978-3-642-83331-1_51.

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Léger, Alain, and Elaine Pratt. "The Case of the Nonlinear Restoring Force." In Qualitative Analysis of Nonsmooth Dynamics, 161–91. Elsevier, 2016. http://dx.doi.org/10.1016/b978-1-78548-094-2.50006-0.

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Husser, Nicholas, and Stefano Brizzolara. "A Linear Model Analysis of the Unsteady Force Response of a Planing Hull Through Forced Vertical Plane Motion Simulations." In Progress in Marine Science and Technology. IOS Press, 2020. http://dx.doi.org/10.3233/pmst200039.

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The prediction of planing hull motions and accelerations in a seaway is of paramount importance to the design of high-speed craft to ensure comfort and, in extreme cases, the survivability of passengers and crew. The traditional approaches to predicting the motions and accelerations of a displacement vessel generally are not applicable, because the non-linear effects are more significant on planing hulls than displacement ships. No standard practice for predicting motions or accelerations of planing hulls currently exists, nor does a nonlinear model of the hydrodynamic forces that can be derived by simulation. In this study, captive and virtual planar motion mechanism (VPMM) simulations, using an Unsteady RANSE finite volume solver with volume of fluid approach, are performed on the Generic Prismatic Planing Hull (GPPH) to calculate the linearized added mass, damping, and restoring coefficients in heave and pitch. The linearized added mass and damping coefficients are compared to a simplified theory developed by Faltinsen [6], which combines the method of Savitsky [12] and 2D+t strip theory. The non-linearities in all coefficients will be investigated with respect to both motion amplitude and frequency. Nonlinear contributions to the force response are discussed through comparison of the force response predicted by the linear model and force response measured during simulation. Components of the planing hull dynamics that contribute to nonlinearities in the force response are isolated and discussed.
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Zhou, Huawei, Fuhua Wang, Renchuan Zhu, and Kaiyuan Shi. "Numerical Study on Ship Parametric Roll in Head Waves." In Progress in Marine Science and Technology. IOS Press, 2020. http://dx.doi.org/10.3233/pmst200048.

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Ship parametric roll is one of the main reasons for marine accidents and is introduced into the second-generation intact stability criteria by the International Maritime Organization (IMO) recently. In this paper, a 6-DOF three-dimensional time-domain model based on the IRF (Impulse Response Function) method is constructed to predict large-amplitude ship motions and investigate the phenomenon of parametric roll in head waves as well as major factors. The F-K forces and the restoring forces are calculated on the instantaneous wet surface while the radiation and diffraction forces are kept linear and transformed from frequency-domain results calculated with the three-dimensional Havelock form translating-pulsating source green function method. The proposed weakly nonlinear time-domain model is used to simulate motions of the C11 containership, which predicts the occurrence of the parametric roll successfully and shows a good agreement with the experimental data in amplitude. The inner mechanism of parametric roll is revealed by investigating the time-history and resonance frequencies of restoring forces and coefficients numerically.
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Conference papers on the topic "Nonlinear Restoring Force"

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Alhadidi, Ali H., Amin Bibo, and Mohammed F. Daqaq. "Flow Energy Harvesters With a Nonlinear Restoring Force." In ASME 2014 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/smasis2014-7445.

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This ppppaper examines the performance of a galloping energy harvester possessing a nonlinear restoring force. To achieve this goal, a flow energy harvester consisting of a piezoelectric cantilever beam augmented with a square-sectioned bluff body at the free end is considered. Two magnets located near the tip of the bluff body are used to introduce the nonlinearity which strength and nature can be altered by changing the distance between the magnets. A lumped-parameter aero-electromechanical model adopting the quasi-steady assumption for aerodynamic loading is presented and utilized to numerically simulate the harvester’s response. Wind tunnel tests are also performed to validate the numerical simulations by conducting upward and downward wind velocity sweeps. Results comparing the relative performance of several harvesters with potential functions of different shapes demonstrate that a mono-stable potential function with a hardening restoring force can outperform all other configurations.
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Al-Shudeifat, Mohammad A. "Nonlinear Energy Sink With a Non-Traditional Kind of Nonlinear Restoring Force." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-47309.

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Enhanced nonlinear energy sink (NES) is addressed here by employing a non-traditional kind of a nonlinear restoring force. The usual nonlinear coupling element between the NES and the linear oscillator (LO) in the literature generates essentially nonlinear restoring force between the NES and the LO. Unlike Type I NES, here the nonlinear coupling force has varying components during the oscillation which appear in closed loops under the effect of damping terms. This NES attachment with the LO rapidly absorbs and immediately dissipates significant portion of the initial energy induced into the LO through a strong resonance capture between the NES and LO responses. The proposed design could also be promising for energy harvesting purposes. The obtained results by numerical simulation show that employing this type of nonlinear restoring force for passive targeted energy transfer (TET) is more promising than some other types of NESs in which purely cubic stiffness restoring forces have been incorporated.
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Lee, B., L. Jiang, and Y. Wong. "Flutter of an airfoil with a cubic nonlinear restoring force." In 39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1998. http://dx.doi.org/10.2514/6.1998-1725.

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Allen, Matthew S., Hartono (Anton) Sumali, and David S. Epp. "Restoring Force Surface Analysis of Nonlinear Vibration Data From Micro-Cantilever Beams." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14905.

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The responses of micro-cantilever beams, with lengths ranging from 100-1500 microns, have been found to exhibit nonlinear dynamic characteristics at very low vibration amplitudes and in near vacuum. This work seeks to find a functional form for the nonlinear forces acting on the beams in order to aide in identifying their cause. In this paper, the restoring force surface method is used to non-parametrically identify the nonlinear forces acting on a 200 micron long beam. The beam response to sinusoidal excitation contains as many as 19 significant harmonics within the measurement bandwidth. The nonlinear forces on the beam are found to be oscillatory and to depend on the beam velocity. A piecewise linear curve is fit to the response in order to more easily compare the restoring forces obtained at various amplitudes. The analysis illustrates the utility of the restoring force surface method on a system with complex and highly nonlinear forces.
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Carboni, Biagio, and Walter Lacarbonara. "Dynamic Response of Nonlinear Oscillators With Hysteresis." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46352.

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The nonlinear features of the steady-state periodic response of hysteretic oscillators are investigated. Frequency-response curves of base-excited single-degree-of-freedom (SDOF) systems possessing different hysteretic restoring forces are numerically obtained employing a continuation procedure based on the Jacobian of the Poincaré map. The memory-dependent restoring forces are expressed as a direct summation of linear and cubic elastic components and a hysteretic part described by a modified version of the Bouc-Wen law. The resulting force-displacement curves feature a pinching around the origin. Depending on the hysteresis material parameters (which regulate the shapes of the hysteresis loops), the oscillator exhibits hardening, softening and softening-hardening behaviors in which the switching from softening to hardening takes place above certain base excitation amplitudes. A comprehensive analysis in the parameters space is performed to identify the thresholds of these different behaviors. The restoring force features here considered have been experimentally obtained by means of an original rheological device comprising assemblies of steel and shape memory wire ropes. This study is carried out also with the aim of designing the restoring forces which give rise to dynamical behaviors useful for a variety of applications.
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Masuda, Arata, Yusuke Miyata, Sou Ushiki, and Zhao Feng. "Wideband operation of a nonlinear vibration energy harvester with asymmetric restoring force." In Active and Passive Smart Structures and Integrated Systems XIII, edited by Alper Erturk. SPIE, 2019. http://dx.doi.org/10.1117/12.2515423.

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Xu, Bin, Ye Zhao, and Baichuan Deng. "NONPARAMETRIC NONLINEAR RESTORING FORCE AND EXCITATION IDENTIFICATION WITH LEGENDRE POLYNOMIAL AND DATA FUSION." In XI International Conference on Structural Dynamics. Athens: EASD, 2020. http://dx.doi.org/10.47964/1120.9062.18779.

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He, Qifan, and Mohammed F. Daqaq. "Load Optimization of a Nonlinear Mono-Stable Duffing-Type Harvester Operating in a White Noise Environment." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-13126.

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This paper investigates electric load optimization of nonlinear mono-stable Duffing energy harvesters subjected to white Gaussian excitations. Both symmetric and asymmetric nonlinear restoring forces are considered. Statistical linearization is utilized to obtain an approximate analytical expression for the optimal load as function of the other systems parameters. It is shown that the optimal load is dependent on the nonlinearity unless the ratio between the harvesting circuit time constant and the period of the mechanical oscillator is very large. Under optimal loading conditions, a harvester with a symmetric nonlinear restoring force can never produce more power than an equivalent linear harvester regardless of the magnitude or nature of the nonlinearity. On the other hand, asymmetries in the restoring force are shown to provide performance enhancement over an equivalent linear harvester.
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Masuda, Arata, and Feng Zhao. "Miniaturized Broadband Vibration Energy Harvester With Piecewise-Linear Asymmetric Restoring Force." In ASME 2019 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/smasis2019-5616.

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Abstract This paper presents a design study of a miniaturized broadband nonlinear vibration energy harvester (VEH) with piecewise-linear restoring force based on a mechanically-sprung resonator with stoppers. It is commonly recognized that a VEH based on a nonlinearly-sprung resonator can show broadband frequency characteristics while keeping its maximum power performance due to its bent resonance peak. The resonator to be investigated in this study consists of a magnet composite as a mass moving through an induction coil, two planar springs, and mechanical stoppers. The magnet composite is comprised of two repelling cylindrical magnets and a steel disk between them, all encapsulated in a thin stainless-steel cylinder. The planar springs with spiral-like shape are respectively connected to the both ends of the magnet composite so that they provide soft linear stiffness in a compact size. The mechanical stoppers installed to constrain the deformation of the spring give the resonator piecewise-linear hardening characteristics which effectively broaden the resonance band. In this study, the prototype VEH developed in the previous study is presented, and the gaps between the springs and stoppers are adjusted so that the resultant piecewise-linear restoring force shows symmetric or asymmetric property with respect to the equilibrium point. Experimental studies and analyses are carried out to examine the performance of the presented VEH in terms of the frequency response. The comparison of three different configurations of the stopper illustrates how the asymmetry in the bilinear restoring force affects the shape of the resonance peak. It is also suggested that the asymmetry may help the VEH operate in broader band by exploiting its ability of tailoring the resonance characteristics, which still needs further investigation.
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Shintani, Masanori, Hiroyuki Ikuta, and Hajime Takada. "Study on Nonlinear Vibration Characteristic Evaluation of Nonlinear Vibration System With Gaps by Transition Probability Density Function." In ASME/JSME 2004 Pressure Vessels and Piping Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/pvp2004-2949.

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In this paper, the transition probability density functions between response velocity and response displacement in nonlinear vibration systems which have the restoring force characteristic of a cubic equation are governed by the Fokker-Planck Equation. The experimental probability density functions are compared with analytical results. The analytical model of the cubic equation as Duffing Equation is proposed by the restoring force characteristic of the nonlinear vibration system with gaps in the experiments. However, a slight difference for the frequency range of the transfer function was shown by simulation results. Then, it is considered using transition probability density functions in the response characteristic. For stationary random input waves, the probability density function between the response displacement and the response velocity are easily estimated by the Fokker-Planck Equation and the Duffing Equation. The slight difference of the transfer function of the response acceleration is evaluated by the scattering of the restoring force characteristic estimated by the probability density function and self-natural frequency curve. The R.M.S. value and the transfer function of the experimental results are compared with the analytical results. It is thought that the estimation of the probability density function of the response has validity. It is thought that the evaluation of the nonlinear vibration characteristics by the probability density function is valid.
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