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Journal articles on the topic 'Nonlinear Restoring Force'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Xu, Bin, Bai-Chuan Deng, Jing Li, and Jia He. "Structural nonlinearity and mass identification with a nonparametric model using limited acceleration measurements." Advances in Structural Engineering 22, no. 4 (August 13, 2018): 1018–31. http://dx.doi.org/10.1177/1369433218792083.

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Structural nonlinearity identification is critical for post-event damage detection or condition evaluation of engineering structures after strong dynamic excitation such as earthquake where structural nonlinear behaviour should be considered. Structural nonlinear restoring force provides direct indicator describing structural damage initiation and development procedure. Considering the availability of structural dynamic response measurement and the difficulty in defining a parametric model for structural nonlinearity and in estimating structure mass accurately in practice, in this article, a time-domain structural nonlinear restoring force and mass identification approach for multi-degree-of-freedom structures under incomplete excitation using limited acceleration measurements but without using any parametric models of structural nonlinear restoring force is proposed. At first, a memory fading extended Kalman filter with a weighted globl iteration (MF-EKF-WGI) is used to identify the location of nonlinearities and then a Chebyshev polynomial nonparametric model is introduced to model the nonlinear restoring force. The unscented Kalman filter is used to identify the structural responses and the parameters of the Chebyshev polynomial to describe structural nonlinearity. Numerical and experimental studies with a four-storey frame model structure equipped with a magnetorheological damper, which is employed to mimic structural nonlinear behaviour, under impact excitations are carried out to validate the performance of the proposed approach using acceleration measurements at certain degrees of freedom. Numerical and experimental results show that the proposed approach is capable of identifying both structural nonlinear restoring force and mass with acceptable accuracy even with a very rough initial mass estimation. The proposed time-domain identification approach can be used to detect structural damage initiation and development process and to evaluate energy consumption quantitatively of engineering structures under dynamic loadings.

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12

Zhang, Xiao Long, Ya Bin Dong, Yu Min He, and Mei Juan Tong. "Forced Vibration of Rotor Suppressed by a Automatic Ball Balancer (Nonlinear Principal Resonances)." Advanced Materials Research 655-657 (January 2013): 551–57. http://dx.doi.org/10.4028/www.scientific.net/amr.655-657.551.

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The automatic ball balancer is equipment used to balance the rotor system online and control its forced vibration. Although the rotor system is a nonlinear system actually, especially the occurrence of the nonlinear elastic restoring force in the support of rolling elements bearings, many researches still focus on the linear rotor systems at present. The Jeffcott rotor acted by unsymmetrical nonlinear restoring force is studied in this paper. Through the resolving theoretically and simulating numerically, the principal resonance respond of the rotor system controlled by the balancer with two balls and its stability are studied, while the vibration characteristics and movement laws of the rotor and balls in every rotary speed region are analyzed. The results showed that the balancer can suppress the principal vibration of the nonlinear rotor system in high speed region very well.

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13

Chen, Yung-Wei. "Application of the Characteristic Time Expansion Method for Estimating Nonlinear Restoring Forces." Journal of Applied Mathematics 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/841690.

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This paper proposes a characteristic time expansion method (CTEM) for estimating nonlinear restoring forces. Because noisy data and numerical instability are the main causes of numerical developing problems in an inverse field, a polynomial to identify restoring forces is usually adopted to eliminate these problems. However, results of the way doing are undesirable for a high order of polynomial. To overcome this difficulty, the characteristic length (CL) is introduced into the power series, and a natural regularization technique is applied to ensure numerical stability and determine the existence of a solution. As compared to previous solutions presented in other researches, the proposed method is a desirable and accurate solver for the problem of restoring the force in the inverse vibration problems.

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14

Yang, Xiongjun, Han Su, Lijun Liu, and Ying Lei. "Identification of the nonlinear characteristics of rubber bearings in model-free base-isolated buildings using partial measurements of seismic responses." Journal of Low Frequency Noise, Vibration and Active Control 39, no. 3 (April 16, 2019): 690–703. http://dx.doi.org/10.1177/1461348419843385.

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Rubber-bearing isolation is one of the most successfully and widely used isolation technologies to provide lateral flexibility and energy dissipation capacity for reducing structural vibration and protecting the superstructure from damage. The seismic performance of the base-isolated structures partly depends on the nonlinear characteristics of the base isolation system. However, it is hard to establish proper mathematical models for the nonlinear hysteretic behaviors of base isolation due to the complexities of nonlinearities. Consequently, it is strongly desired to develop model-free methodologies for the nonlinear hysteretic performance identification with no assumption on the nonlinear hysteretic models of base isolation. In this paper, a novel method is proposed for this purpose. Firstly, the base isolation is in the linear state when the structure is under the weak earthquake, the restoring force is only provided by linear stiffness and viscous damping of base isolation, and the structural physical parameters can be estimated based on the extended Kalman filter approach. Then, the base isolation is in the nonlinear state when the structure is under the strong earthquake. The nonlinear hysteretic restoring forces from base isolation are treated as “unknown fictitious inputs” to the corresponding structural systems without base isolation. The generalized Kalman filter with unknown input algorithm is adopted for the simultaneous identification of the corresponding structural systems and the hysteretic restoring force of base isolation using only partial structural responses. No information about the structure is needed, and the responses at the location of the base isolation are not required, the proposed method is capable of identifying nonlinear characteristics of base isolation by the direct use of partial structural dynamic response. To validate the performances of the proposed method, some numerical simulation examples of identifying nonlinear hysteretic restoring forces of base isolation in different models are used.

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15

Hashimoto, Seiji, and Yusaku Fujii. "Material Tester Using a Controlled Oscillator and an Inertial Mass." Key Engineering Materials 381-382 (June 2008): 457–60. http://dx.doi.org/10.4028/www.scientific.net/kem.381-382.457.

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In this paper, a controlled oscillation using the levitation mass method (LMM) for material with nonlinear viscoelasticity is proposed. Since LMM is a very accurate and efficient method of measuring a varying force, it has a possibility to apply to the material tester. For LMM, system identification process is introduced to grasp the nonlinear components of the restoring force. The process consists of the experiment and frequency analysis for higher modes which originate for the nonlinear restoring force. The input signal is then derived based on the identification results. By adding the proposed identification process to LMM, the arbitrary waveform of the force can be investigated even in the open-loop excitation. The effectiveness of the proposed oscillation method is verified through the LMM-based experiments.

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16

Cifuentes, Arturo O., and Wilfred D. Iwan. "Nonlinear system identification based on modelling of restoring force behaviour." Soil Dynamics and Earthquake Engineering 8, no. 1 (January 1989): 2–8. http://dx.doi.org/10.1016/s0267-7261(89)80002-8.

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17

HOSHIYA, Masaru, and Osamu MARUYAMA. "Identification of restoring force characteristics of nonlinear systems during earthquakes." Doboku Gakkai Ronbunshu, no. 386 (1987): 397–405. http://dx.doi.org/10.2208/jscej.1987.386_397.

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18

KOIZUMI, Tadayoshi, and Ryousuke OOSONE. "Study on Vibration Characteristics and Chaos under Nonlinear Restoring Force." Transactions of the Japan Society of Mechanical Engineers Series C 69, no. 681 (2003): 1197–202. http://dx.doi.org/10.1299/kikaic.69.1197.

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19

Arnst, M., R. Ghanem, and S. Masri. "Maximum entropy approach to the identification of stochastic reduced-order models of nonlinear dynamical systems." Aeronautical Journal 114, no. 1160 (October 2010): 637–50. http://dx.doi.org/10.1017/s0001924000004115.

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AbstractData-driven methodologies based on the restoring force method have been developed over the past few decades for building predictive reduced-order models (ROMs) of nonlinear dynamical systems. These methodologies involve fitting a polynomial expansion of the restoring force in the dominant state variables to observed states of the system. ROMs obtained in this way are usually prone to errors and uncertainties due to the approximate nature of the polynomial expansion and experimental limitations. We develop in this article a stochastic methodology that endows these errors and uncertainties with a probabilistic structure in order to obtain a quantitative description of the proximity between the ROM and the system that it purports to represent. Specifically, we propose an entropy maximization procedure for constructing a multi-variate probability distribution for the coefficients of power-series expansions of restoring forces. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed framework.

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20

He, Jia, Xiaoxiong Zhang, Mengchen Qi, and Bin Xu. "Model-Free Identification of Nonlinear Restoring Force with Modified Observation Equation." Applied Sciences 9, no. 2 (January 16, 2019): 306. http://dx.doi.org/10.3390/app9020306.

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Nonlinearity exists widely in civil engineering structures; for example, the initiation and growth of damage under dynamic loadings is a typical nonlinear process. To date, for the purpose of structural evaluation and a better understanding nonlinear characteristics of complicated structures, a number of parametric and nonparametric methods have been developed for the identification of nonlinear restoring force (NRF). However, due to the highly individualistic nature of nonlinear systems, it would be inefficient to attempt to express the structural NRF in a general parametric form. For many nonparametric techniques, their nonparametric models or approximations may result in undesirable results or oscillations around unsmooth points. In this paper, on the basis of extended Kalman filter (EKF), a model-free NRF identification approach is proposed to circumvent the limitations mentioned above. The NRF to be identified was treated as ‘unknown fictitious input’, and thus, no prior assumptions or approximations for the NRF model were required. With the aid of a projection matrix, a modified version of observation equation was obtained. Based on the principle of EKF, the recursive solution of the proposed approach was analytically derived. The NRFs provided by the nonlinear components were identified by means of least squares estimation (LSE) at each time step. Numerical examples, including building structures equipped with magnetorheological (MR) damper and shape memory alloy (SMA) damper, demonstrated that the proposed approach is capable of satisfactorily identifying NRF without knowledge or intuitive assumptions of any nonlinear model class in advance.

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21

Hong, Gan, Mei Li, and Yi Zhen Yang. "Nonlinear Dynamic Analysis of the Damping Frame Structure System." Advanced Materials Research 594-597 (November 2012): 886–90. http://dx.doi.org/10.4028/www.scientific.net/amr.594-597.886.

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Abstract. In the paper, take full account of energy dissipation operating characteristics. Interlayer shear-frame structure for the analysis of the Wilson-Θmethod ELASTOPLASTIC schedule, the design of a nonlinear dynamic time history analysis procedure. On this basis, taking into account the restoring force characteristics of the energy dissipation system, the inflection point in the restoring force model treatment, to avoid a result of the calculation results of distortion due to the iterative error. A frame structure seismic response time history analysis results show that: the framework of the energy dissipation significantly lower than the seismic response of the common framework, and its role in the earthquake when more significant.

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22

Zhang, Pin Le. "Restoring Force Model Investigation of Short Pier Shear Wall." Applied Mechanics and Materials 501-504 (January 2014): 700–703. http://dx.doi.org/10.4028/www.scientific.net/amm.501-504.700.

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To study the seismic behavior of short pier shear wall, 6 short pier shear wall specimens are tested under low cyclic loading with axial load ratio of 0.2. Hysteretic rules and stiffness degradation rate are determined base on test results and feature points are defined by theory, restoring force model that considers the effect of two loading direction to hysteretic characteristics is proposed based on test research, which could be applied to the nonlinear dynamic analysis of the structure.

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23

Chassiakos, A. G., S. F. Masri, A. W. Smyth, and T. K. Caughey. "On-Line Identification of Hysteretic Systems." Journal of Applied Mechanics 65, no. 1 (March 1, 1998): 194–203. http://dx.doi.org/10.1115/1.2789025.

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Using adaptive estimation approaches, a method is presented for the on-line identification of hysteretic systems under arbitrary dynamic environments. The availability of such an identification approach is crucial for the on-line control and monitoring of nonlinear structural systems to be actively controlled. In spite of the challenges encountered in the identification of the hereditary nature of the restoring force of such nonlinear systems, it is shown through the use of simulation studies and experimental measurements that the proposed approach can yield reliable estimates of the hysteretic restoring force under a very wide range of excitation levels and response ranges.

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24

NISHIDA, Shingo, Hiroshi MATSUHISA, Hideo UTSUNO, Keisuke YAMADA, and Katsutoshi SAWADA. "139 Torsional Vibration Suppression of Driveshaft with Magnetic Nonlinear Restoring Force." Proceedings of the Dynamics & Design Conference 2008 (2008): _139–1_—_139–6_. http://dx.doi.org/10.1299/jsmedmc.2008._139-1_.

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25

Zhang, Ying, Junyi Cao, Wei Wang, and Wei-Hsin Liao. "Enhanced modeling of nonlinear restoring force in multi-stable energy harvesters." Journal of Sound and Vibration 494 (March 2021): 115890. http://dx.doi.org/10.1016/j.jsv.2020.115890.

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26

Cazenave, T., A. Haraux, L. Vazquez, and F. B. Weissler. "Nonlinear effects in the wave equation with a cubic restoring force." Computational Mechanics 5, no. 1 (1989): 49–72. http://dx.doi.org/10.1007/bf01046879.

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27

Tsyfanskii, S. L., and A. B. Oks. "Stabilization of resonance vibrations of systems with a nonlinear restoring force." Soviet Applied Mechanics 22, no. 10 (October 1986): 1001–4. http://dx.doi.org/10.1007/bf01273683.

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28

Bruns, J. U., M. Lindner, and K. Popp. "Identification of the Nonlinear Restoring Force Characteristic of a Rubber Mounting." PAMM 2, no. 1 (March 2003): 270–71. http://dx.doi.org/10.1002/pamm.200310120.

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29

Shintani, Masanori, and Manabu Hamai. "Study on Analytical Model of Nonlinear Vibration for Elastic Plates With Gaps Under Random Waves." Journal of Pressure Vessel Technology 126, no. 4 (November 1, 2004): 504–9. http://dx.doi.org/10.1115/1.1689359.

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In this paper, an analytical model for the nonlinear elastic-plastic vibration for long plates with gaps subjected to random vibrations is considered. The nonlinear vibration is caused by the collision phenomena between a mass through a gap and plates with thickness of 0.5, 0.6, and 0.8 mm. An elastic perfectly plastic solid material is assumed in some cases, which adds another aspect to the nonlinear behavior of the system. The material characteristic of the steel is assumed to be an elasto-plasticity solid model. A restoring force characteristic is obtained as the nonlinear vibration of a cubic equation for 0.5, 0.6, and 0.8 mm, the thickness of the plates by experiments. Now the analytical model is proposed by the elasto-plasticity solid model. The relation between the displacement and the force is described by a complicated equation. The curve from the analytical model is called a deflection curve. The results by the analytical model are compared with the results by the experimental model. The restoring force characteristics by the analysis agree with those of the experiment. The restoring force characteristics of the analysis are described using cubic equations. The simple analysis model for evaluation of the vibration characteristic of the nonlinear vibration system, which performs collision vibration with gaps, is proposed by elasto-plasticity solid model in this paper. The results of this proposed analytical model agree with the experimental results better than the results of the minimum of error of square.

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30

Wu, Kenan, Hongbai Bai, Xin Xue, Tuo Li, and Min Li. "Energy Dissipation Characteristics and Dynamic Modeling of the Coated Damping Structure for Metal Rubber of Bellows." Metals 8, no. 7 (July 23, 2018): 562. http://dx.doi.org/10.3390/met8070562.

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A novel coated damping structure for metal rubber (MR) of bellows is designed based on large-size metal rubber sheets. This structure is dynamically tested in the bending direction at normal temperature. According to the test results, a model of the nonlinear elastic restoring force is set up, which describes the dynamic characteristics of the coated damping structure for metal rubber of bellows, and identifies the parameters of the model. The results show that the coated damping structure for metal rubber of bellows has a strong damping energy dissipation ability, its dynamic vibration characteristics are related to the vibration amplitude and frequency, and it is a complex nonlinear hysteretic system with multiple damping components. After identification of the parameters, the model of nonlinear elastic restoring force shows highly accurate results.

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Zhang, Xiao Long, Ya Bin Dong, and Yu Min He. "Theoretical Analysis for Force Transmissibility and Jump Phenomena of Duffing Spring Type Vibration Isolator." Applied Mechanics and Materials 224 (November 2012): 11–17. http://dx.doi.org/10.4028/www.scientific.net/amm.224.11.

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In order to study the common nonlinear characteristics of the vibration isolator, a single degree of freedom system with cubic restoring force is introduced to describe the nonlinear vibration isolation system in the paper and the harmonic balance method was applied to investigate the primary resonance near the natural frequency of the system. Based on Routh–Hurwitz stability criterion, it was clarified theoretically that the region surrounded by the curve of the vertical tangential points in the curve cluster of the primary resonance amplitude frequency characteristics was instable. In addition, the equations of the jump frequency and force transmissibility were derived. The calculated results showed that the jump, hysteresis, stable and instable phenomena would take place for the force transmissibility of the isolator system and the effect of high frequency components of the transmitting force was limited; the damping, coefficient of nonlinear restoring force and the excitation amplitude had an influence on the force transmissibility whose frequencies were in the region of the resonance frequencies but not on those whose frequencies were lower than the resonance frequencies, and only the damping would affect the force transmissibility whose frequencies were higher than the resonance frequencies. Finally, the equation of the start frequency, on the condition that the force transmissibility was less than 1, was presented.

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32

Saeed, N. A., and H. A. El-Gohary. "On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force." Nonlinear Dynamics 88, no. 1 (December 16, 2016): 293–314. http://dx.doi.org/10.1007/s11071-016-3243-x.

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33

Appleby, John A. D. "On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations." International Journal of Differential Equations 2010 (2010): 1–25. http://dx.doi.org/10.1155/2010/508217.

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We consider the zero crossings and positive solutions of scalar nonlinear stochastic Volterra integrodifferential equations of Itô type. In the equations considered, the diffusion coefficient is linear and depends on the current state, and the drift term is a convolution integral which is in some sense mean reverting towards the zero equilibrium. The state dependent restoring force in the integral can be nonlinear. In broad terms, we show that when the restoring force is of linear or lower order in the neighbourhood of the equilibrium, or if the kernel decays more slowly than a critical noise-dependent rate, then there is a zero crossing almost surely. On the other hand, if the kernel decays more rapidly than this critical rate, and the restoring force is globally superlinear, then there is a positive probability that the solution remains of one sign for all time, given a sufficiently small initial condition. Moreover, the probability that the solution remains of one sign tends to unity as the initial condition tends to zero.

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34

Du, Hai-En, Guo-Kang Er, and Vai Pan Iu. "A Novel Method to Improve the Multiple-Scales Solution of the Forced Nonlinear Oscillators." International Journal of Computational Methods 16, no. 04 (May 13, 2019): 1843010. http://dx.doi.org/10.1142/s0219876218430107.

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We propose a novel procedure to improve the solutions obtained by perturbation methods for analyzing the solutions of strongly nonlinear systems in this paper. The proposed procedure is presented and then combined with the multiple-scales method for the optimum solutions of a class of forced oscillators with strong nonlinearity. The solutions obtained from conventional multiple-scales method and the proposed method are examined by the results from numerical continuation method. The results show that the proposed method is effective for the oscillators with nonlinear restoring force as well as nonlinear inertial force even if the nonlinearities are strong. Numerical results and comparison show that the proposed method can improve the solution a lot in comparison to the solution obtained by conventional multiple-scales method.

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35

Zhao, Yan, Baofeng Zhang, Fangfang Han, Huan Tian, Xiao Yu, and Junchao Zhu. "Instantaneous Characteristics of Nonlinear Torsion Pendulum and Its Application in Parameter Estimation of Nonlinear System." Mathematical Problems in Engineering 2018 (2018): 1–10. http://dx.doi.org/10.1155/2018/7858403.

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The nonlinear model of torsion pendulum is presented by considering the nonlinear damping force and nonlinear restoring force. The analytic solution of the nonlinear model is calculated to analyze the relationship between the characteristics of torsion pendulum and the nonlinear factors. The instantaneous characteristics of nonlinear torsion pendulum are analyzed by instantaneous undamped natural frequency and instantaneous damping coefficient. The instantaneous characteristics can be used for the parameter estimation of nonlinear torsion pendulum system. The nonlinear characteristics of the torsion pendulum are validated by the torsion pendulum based on the air-hovered turntable. The parameter estimation method based on the instantaneous characteristics is validated by the moment of inertia measurement system based on the torsion pendulum.

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36

Lenaerts, V., G. Kerschen, J. C. Golinval, M. Ruzzene, and E. Giorcelli. "Validation of Two Nonlinear System Identification Techniques Using an Experimental Testbed." Shock and Vibration 11, no. 3-4 (2004): 365–75. http://dx.doi.org/10.1155/2004/203405.

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The identification of a nonlinear system is performed using experimental data and two different techniques, i.e. a method based on the Wavelet transform and the Restoring Force Surface method. Both techniques exploit the system free response and result in the estimation of linear and nonlinear physical parameters.

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37

Sarkar, A., and J. K. Bhattacharjee. "Renormalization Group for nonlinear oscillators in the absence of linear restoring force." EPL (Europhysics Letters) 91, no. 6 (September 1, 2010): 60004. http://dx.doi.org/10.1209/0295-5075/91/60004.

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38

Molla, M. H. U., M. S. Alam, and M. F. Alam. "Rapidly Convergent Solution of Nonlinear Oscillators with General Non-rational Restoring Force." Journal of Vibration Engineering & Technologies 7, no. 5 (June 18, 2019): 445–54. http://dx.doi.org/10.1007/s42417-019-00142-z.

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39

Allen, Matthew S., Hartono Sumali, and David S. Epp. "Piecewise-linear restoring force surfaces for semi-nonparametric identification of nonlinear systems." Nonlinear Dynamics 54, no. 1-2 (May 3, 2007): 123–35. http://dx.doi.org/10.1007/s11071-007-9254-x.

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Ghosh, Chandita Urmi, and M. Alhaz Uddin. "ANALYTICAL TECHNIQUE FOR DAMPED NONLINEAR OSCILLATORS HAVING GENERALIZED RATIONAL POWER RESTORING FORCE." Far East Journal of Mathematical Sciences (FJMS) 130, no. 1 (April 20, 2021): 25–41. http://dx.doi.org/10.17654/ms130010025.

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41

Sun, Kehui, A. Di-li Duo Li-kun, Yanqing Dong, Huihai Wang, and Ke Zhong. "Multiple Coexisting Attractors and Hysteresis in the Generalized Ueda Oscillator." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/256092.

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A periodically forced nonlinear oscillator called the generalized Ueda oscillator is proposed. The restoring force term of this equation consists of a nonlinear functionsgn(x)and an absolute function with a variant power. Dynamics is investigated by detailed numerical analysis as well as dynamic simulation, including the largest Lyapunov exponent, phase diagrams, and bifurcation diagrams. Multiple coexisting attractors and complex hysteresis phenomenon are observed. The results show that this system has rich dynamical behaviors, and it has a promising application in the fields of science and engineering.

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42

Messaoudi, Salim A., and Soh Edwin Mukiawa. "Existence and stability of fourth-order nonlinear plate problem." Nonautonomous Dynamical Systems 6, no. 1 (December 31, 2019): 81–98. http://dx.doi.org/10.1515/msds-2019-0006.

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AbstractIn this paper, we study a fourth-order plate problem as a model for a suspension bridge in the presence of a nonlinear frictional damping and a hanger restoring force. We establish the existence of a global weak solution and prove a stability result.

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43

Ma, Kai Ze. "Research on Restoring Force Model of High Strength Concrete-Filled Steel Tubular Columns." Applied Mechanics and Materials 166-169 (May 2012): 1746–51. http://dx.doi.org/10.4028/www.scientific.net/amm.166-169.1746.

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Based on the experimental results of twenty three high strength concrete-filled square steel tubular (HCFT) columns subjected to cyclic lateral load, the restoring force model of square HCFT columns is established. Through theoretical analysis and experimental results, the main parameters of the restoring force model which are axial load ratio, confinement effect coefficient as well as slenderness ratio are discussed. The theoretical method is put forward. The results show that the relation of elastic stiffness to degraded stiffness, the relation of ultimate bearing capacity to yielding loads are proportional with confinement effect coefficient, and inversely with axial load ratio as well as slenderness ratio. Many various influential factors are considered in the restoring force model of square HCFT columns. The model is close to the experimental results which can be conveniently applied for nonlinear dynamics analysis of composite structures.

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44

Gottlieb, O. "Bifurcations of a Nonlinear Small-Body Ocean-Mooring System Excited by Finite-Amplitude Waves." Journal of Offshore Mechanics and Arctic Engineering 119, no. 4 (November 1, 1997): 234–38. http://dx.doi.org/10.1115/1.2829101.

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We investigate the response of a nonlinear small-body ocean-mooring system excited by finite-amplitude waves. The system is characterized by a coupled geometrically nonlinear restoring force defined by a single elastic tether. The nonlinear hydrodynamic exciting force includes both dissipative and convective terms that are not negligible in a finite wave amplitude environment. Stability of periodic motion is determined numerically and the bifurcation structure includes ultrasubharmonic and quasi-periodic response. The dissipation mechanism is found to control stability thresholds, whereas the convective nonlinearity governs the evolution to chaotic system response.

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45

Kovacic, Ivana. "Nonlinear Oscillators with a Power-Form Restoring Force: Non-Isochronous and Isochronous Case." Applied Mechanics and Materials 430 (September 2013): 14–21. http://dx.doi.org/10.4028/www.scientific.net/amm.430.14.

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This work is concerned with single-degree-of-freedom conservative nonlinear oscillators that have a fixed restoring force, which comprises a linear term and an odd-powered nonlinear term with an arbitrary magnitude of the coefficient of nonlinearity. There are two cases of interest: i) non-isochronous, when the system has an amplitude-dependent frequency and ii) isochronous, when the frequency of oscillations is constant (amplitude-independent). The first case is associated with the constant coefficient of the kinetic energy, while the frequency-amplitude relationship and the solution for motion need to be found. To that end, the equation of motion is solved by introducing a new small expansion parameter and by adjusting the Lindstedt-Poincaré method. In the second case, the condition for the frequency of oscillations to be constant is derived in terms of the expression for the position-dependent coefficient of the kinetic energy. The corresponding solution for isochronous oscillations is obtained. Numerical verifications of the analytical results are also presented.

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Jiang, T., and T. E. Schellin. "Motion Prediction of a Single-Point Moored Tanker Subjected to Current, Wind and Waves." Journal of Offshore Mechanics and Arctic Engineering 112, no. 1 (February 1, 1990): 83–90. http://dx.doi.org/10.1115/1.2919840.

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Horizontal motions of a tanker attached to a single-point mooring (SPM) terminal were predicted using digital simulation in the time domain. Excitations from steady current, gusting wind, and irregular seaway were included. Hydrodynamic forces generated by the ship’s motion and the action of its propeller and rudder were calculated in accordance with a previously validated, nonlinear quasi-steady four-quadrant maneuvering model, extended to include linear memory effects due to waves generated by the moving ship. Memory effects were approximated by a vectorial recursive state space model corresponding to a set of higher order differential equations. A nonlinear relationship of the force in the mooring hawser was assumed to represent restoring force characteristics of the SPM system. Wave excitation forces comprised first-order forces at wave frequencies and second-order drift forces at low frequencies. First-order wave forces were obtained by superposition of force components corresponding to regular wave components comprising the wave spectrum. Based on the low-frequency wave envelope, drift forces were calculated using mean drift force coefficients in regular waves. Selected sample simulations are presented to illustrate the use of this digital simulation method.

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Patrikalakis, N. M., G. A. Kriezis, H. N. Gursoy, and T. Maekawa. "Dynamics of Compliant Risers." Journal of Ship Research 34, no. 02 (June 1, 1990): 123–35. http://dx.doi.org/10.5957/jsr.1990.34.2.123.

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The objective of this paper is first to formulate the three-dimensional dynamic equations of a compliant riser, idealized as a rotationally nonuniform rod, around a nonlinear static configuration with linearized restoring force and inertial components in the presence of general current and monochromatic wave excitation. Next, to harmonically linearize nonlinear forces such as quadratic drag for the general three-dimensional problem by minimizing the mean square error between the linear approximation and the nonlinear force. Finally, to present an efficient numerical solution method appropriate for nonlinear boundary-value problems with sharp boundary layers such as the problem at hand. Numerical examples and comparisons with time-domain solutions for a catenary riser with a three-dimensional configuration and a steep-wave riser are included. Comparisons of our theoretical predictions with experimental results obtained from a small-scale riser model are also summarized to evaluate our theoretical ability to predict the response of compliant riser systems.

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Tang, Bin, MJ Brennan, V. Lopes, S. da Silva, and R. Ramlan. "Using nonlinear jumps to estimate cubic stiffness nonlinearity: An experimental study." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 230, no. 19 (August 9, 2016): 3575–81. http://dx.doi.org/10.1177/0954406215606746.

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Attempts are being made to improve mechanical design by using nonlinearity rather than eliminating it, especially in the area of vibration control and in energy harvesting. In such systems, there is a need to both predict the dynamic behavior and to estimate the system properties from measurements. This paper concerns an experimental investigation of a simple identification method, which is specific to systems in which the behavior is known to be similar to that of a Duffing-type system. It involves the measurement of jump-down frequencies and the amplitudes of displacement at these frequencies. The theoretical basis for the method is briefly described as, is an experimental investigation on a beam-shaker system. The results are comparable with those determined by the restoring force surface method. The method described in this article has the advantage that the data can be collected and processed more easily than the restoring force surface method and can be potentially more suitable for the engineering community than existing identification measures.

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Bibo, Amin, Ali H. Alhadidi, and Mohammed F. Daqaq. "Exploiting a nonlinear restoring force to improve the performance of flow energy harvesters." Journal of Applied Physics 117, no. 4 (January 28, 2015): 045103. http://dx.doi.org/10.1063/1.4906463.

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Tian, Rui-Lan, Xin-Wei Yang, Qing-Jie Cao, and Qi-Liang Wu. "Bifurcations and chaotic threshold for a nonlinear system with an irrational restoring force." Chinese Physics B 21, no. 2 (February 2012): 020503. http://dx.doi.org/10.1088/1674-1056/21/2/020503.

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