An equivalent linearization method for nonlinear Van der Pol oscillator subjected to random vibration using orthogonal functions

Younespour, Amir; Ghaffarzadeh, Hosein; Azar, Bahman Farahmand

doi:10.1007/s11768-018-7038-0

Cited by 5 publications

(2 citation statements)

References 28 publications

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“…Despite several studies on the analysis of nonlinear systems by the time‐domain EL methods, they only focused on specific kinds of excitations. Applying orthogonal functions into the EL introduces an alternate method in the time domain to accurately predict the response of nonlinear systems with low computational expenses under a general type of excitation . The block pulse (BP) functions, as the converted forms of orthogonal operators, are capable of easily leading to recursive computations with more accurate approximation and less mathematical costs in solving concrete problems such as differential and integral equations .…”

Section: Introductionmentioning

confidence: 99%

Orthogonal function‐based equivalent linearization for sliding mode control of nonlinear systems

Baghaei

Ghaffarzadeh

Younespour

2019

Struct Control Health Monit

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This paper proposes the equivalent linearization (EL) and sliding mode control (SMC) methods to address nonlinearity and enhance the performance of nonlinear systems subjected to nonstationary random excitations. The EL methods are commonly used to propose approximate solutions for nonlinear systems under random excitations due to the high computational demands in the exact analyses. Considering the application of orthogonal functions in improving the accuracy and efficiency of approximate solutions, an orthogonal block pulse (BP) function is applied to the EL method in this study to approximate the nonlinear system responses under nonstationary random excitations. The SMC, as a robust control method for systems with uncertainties and external disturbances, is capable of achieving reliable and accurate tracking control. This method is applied to effectively reduce the dynamic responses predicted by the proposed EL method under nonstationary random excitations. The proposed approach is tested on single-degreeof-freedom and two-degree-of-freedom Duffing systems subjected to a seismic type excitation. The results indicate that not only the orthogonal functionbased EL method can approximate the dynamic responses more accurately, at lower computational cost, and by a high agreement with the exact solution, but also the proposed SMC can improve the performance of nonlinear systems by effectively reducing the responses compared with the linear quadratic regulator control method.KEYWORDS block pulse functions, equivalent linearization, nonlinearity, sliding mode control Nomenclature: A, plant matrix of the system; B,B u , control location matrices; B r , excitation orientation vector; C,C eq , damping matrix and its equivalent value; E[·], mathematical expectation; g, nonlinear function of displacement and velocity; h, unit impulse-response function; J, cost function; J G ,J F , convolution operational matrix; K,K eq , stiffness matrix and its equivalent value; k, linear stiffness; M, mass matrix; m, positive integer value; P, sliding surface coefficient vector; Q,R, positive semidefinite and definite weighting matrices; q, width of block pulse

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Section: Introductionmentioning

confidence: 99%

Orthogonal function‐based equivalent linearization for sliding mode control of nonlinear systems

Baghaei

Ghaffarzadeh

Younespour

2019

Struct Control Health Monit

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“…In fact, the idea suggesting that structures collapse when the energy demand as a design parameter for an earthquake is greater than the amount provided by the structure, was introduced from the earliest days of earthquake engineering development [4][5][6][7][8][9]. In addition, in recent years, there has been a growing focus and studies on using energy dissipating devices and new control methods in structures [10][11][12][13]. Therefore, a reliable estimation of input-energy is of great importance in energybased seismic design methods.…”

Section: Introductionmentioning

confidence: 99%

A New Method for Estimating the Input-Energy of MDOF Structures

Davarnia

Azar

2019

NMCE

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Input-energy is the amount of energy that is imposed by an earthquake on a structure and its correlation with structural damage has been studied and demonstrated by many researchers. Since studies concerning seismic energy in multi-degree-of-freedom systems are relatively limited compared with single-degree-of-freedom, in this paper, firstly a theoretical exact method is discussed to calculate the input-energy of the multi-degree-of-freedom elastic oscillators. It is proved that unlike the general rule in mechanics, the superposition theorem is valid for inputenergy in conventional modal analysis. To estimate the input-energy, an approach based on PHSA to predict the Fourier amplitude spectrum, is proposed. The results indicate that the modal mass ratio is not the only decisive parameter in input-energy. Modal input-energy decomposition also confirms the possibility of greater input-energy of higher modes in comparison with fundamental ones or the ones with the higher mass participation ratio, especially for tall buildings located in the near-field seismic zones.

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Design of a tuned mass damper for damped structures using an orthogonal-function-based equivalent linearization method

2020

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An equivalent linearization method for nonlinear Van der Pol oscillator subjected to random vibration using orthogonal functions

Cited by 5 publications

References 28 publications

Orthogonal function‐based equivalent linearization for sliding mode control of nonlinear systems

Orthogonal function‐based equivalent linearization for sliding mode control of nonlinear systems

A New Method for Estimating the Input-Energy of MDOF Structures

Design of a tuned mass damper for damped structures using an orthogonal-function-based equivalent linearization method

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