Equivalent Linearization Technique Research Articles

Torsional vibration suppression for a high-damped rotor using dynamic vibration absorber based on weighted dual estimation of equivalent linearization techniques

In the rotating systems, the torsional vibration of damped rotors usually appears and induces the imbalance of systems. The dynamic vibration absorber (DVA) is used for reducing the torsional vibration of damped rotors. Recent researches have shown that traditional methods are suitable for reducing the torsional vibration of weak damped rotors. However, the traditional methods will cause significant errors for high-damped rotors. Therefore, a novel method is proposed to suppress the torsional vibration of high-damped rotors using a dynamic vibration absorber based on Weighted Dual Estimation (WDE) of equivalent linearization techniques. The high-damped rotor is replaced approximately by an equivalent undamped rotor based on the WDE of equivalent linearization techniques. Two closed-form expressions for a suitable system of springs and viscous dampers of the optimal DVA are determined to suppress the torsional vibration of high-damped rotors. The numerical analysis results show that the torsional vibration of the high-damped rotor has surprisingly reduced at the resonant frequency region by using the optimal DVA. The proposed expressions for a suitable system of springs and viscous dampers of the optimal DVA are reliable for suppressing the torsional vibration of a highly damped rotor.

Transient Response Analysis of Nonlinear Oscillators With Fractional Derivative Elements Under Gaussian White Noise Using Complex Fractional Moments

Abstract Complex fractional moment (CFM), which is defined as the Mellin transform of a probability density function (PDF), has been successfully employed to find the response PDF of a wide variety of integer-order nonlinear oscillators. In this paper, a CFM-based analysis is performed to determine the transient response PDF of nonlinear oscillators with fractional derivative elements under Gaussian white noise. First, an equivalent linear system is introduced for the purpose of deriving the Fokker–Planck (FP) equation for response amplitude. The equivalent natural frequency and equivalent damping coefficient of the system need to be determined, taking into account both the nonlinear and fractional derivative elements of the original oscillator. Moreover, to convert the FP equation into the governing equation of CFMs, these equivalent coefficients must be given in polynomial form of amplitude. This paper proposes formulas for appropriately determining the equivalent coefficients, based on an equivalent linearization technique. Then, applying stochastic averaging, the FP equation is derived from the equivalent linear system. Next, the Mellin transform converts the FP equation into coupled linear ordinary differential equations for amplitude CFMs, which are solved with a constraint corresponding to the normalization condition for a PDF. Finally, the inverse Mellin transform of the CFMs yields the amplitude PDF. The joint PDF of displacement and velocity is also obtained from the amplitude PDF. Three linear and nonlinear fractional oscillators are considered in numerical examples. For all cases, the analytical results are in good agreement with the pertinent Monte Carlo simulation results.

Futuristic study on seismic response analysis of North Indian soils– A sustainable approach

India's growing population and extensive unscientific structures put the country at risk of suffering major casualties during earthquakes. To make buildings more earthquake-resistant, and develop safe and durable structures for future, we need seismic sustainability. For this purpose, Cities should adopt ‘seismic microzonation’, which includes Site response analysis (SRA). In this paper, SRA for eight places in Varanasi (India) was performed using the equivalent linear technique subjected to four significantly different strong motions (Bhuj, Uttarkashi, Chamba, and Kobe motion). Geotechnical data from various government and private entities have been gathered for this purpose. Analysis has been conducted using equivalent linear approach in DEEPSOIL software. The results of the study have been formulated in terms of surface time histories, response spectra, Fourier spectra and PGA along the depth. These findings may be applied to dynamic analysis and structural design, as well as the preparation of advanced dynamic lab tests. Furthermore, sites in Varanasi have been found to considerably enhance ground motion, thus a site-specific design approach must be implemented for important constructions.

An effective model for bolted flange joints and its application in vibrations of bolted flange joint multiple-plate structures: Theory with experiment verification

Bolted flange joints are widely used in various engineering structures to combine two or more substructures together. However, the understanding of mechanical mechanism of bolted flange joints is still insufficient due to the lack of efficient mathematical models. This paper proposes an effective mathematical model for bolted flange joints, which is then employed to conduct vibration analysis on bolted flange joint multiple-plate structures. During the modeling process, the flange is modeled by the Kirchhoff plate theory, and the bolted joints are modeled by the two-dimensional face constraint in bolted joint affected region (BJAR). The artificial spring technique is adopted to simulate the face constraint. The friction contact model and equivalent linearization technique are adopted to analyze the stick and slip states of bolted joints. By adopting the Chebyshev orthogonal polynomials as admissible functions, governing equations are derived. Then, the Rayleigh-Ritz method is applied to study the free vibration characteristics, and the Newmark-beta method is employed to study the vibration response under base excitation. The experimental studies are performed on the bolted flange joint two-plate structure (BFJTPS) to illustrate the effectiveness of the proposed theoretical model. The results indicate that the proposed mathematical model can well predict the vibration characteristics of multiple-plate structures under both bolt looseness and no-looseness conditions. The soft nonlinearity phenomenon under bolt looseness condition can be successfully revealed by the model.

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

An Equivalent Linearization technique, termed an Equivalent Linearization Time and Ensemble Expectation (EL-TEE) approach, is used to develop an alternative method for estimating the response of a nonlinear oscillator to a combination of deterministic harmonic and random white noise excitation. The approach is based on applying equivalent linearization and averaging over the time period of one harmonic excitation cycle. This gives a set of coupled nonlinear equations that can be solved for the response averaged over time and across the ensemble. The primary advantages of the proposed method are its computational speed, ability to return physically meaningful linearization matrices and that it can be applied to a wide variety of nonlinearities. The method is applied to three example test systems: the well-known single degree of freedom Duffing oscillator; a single degree of freedom system with a displacement constraint imposing a discontinuous nonlinearity; and a multi degree of freedom oscillator with a localized polynomial nonlinearity that has also been examined experimentally. It is shown that the response predicted matches well with Monte Carlo results from direct time integration at a fraction of the computational cost, and the method is capable of reproducing key results observed experimentally.

Data-Driven Dynamic Event-Triggered Fault-Tolerant Platooning Control

This paper addresses the dynamic event-triggered (DET) fault-tolerant model-free adaptive platooning control (MFAPC) problem of vehicle platoon systems subject to sensor faults. First of all, a novel redefined output is introduced to assist in proving the synchronous tracking of the position and velocity. Based on which, the equivalent dynamic linearization technique is used for transforming the nonlinear vehicular platooning systems into an linear data model. Then, an observer is designed to introduce the estimation of the pseudo-partial derivative (PPD) parameter, which can also eliminate the symbol restriction on it. In addition, a DET mechanism is introduced to save network resources more effectively, and the neural network method is applied to deal with the sensor faults for the safe vehicle driving. Further, a novel DET-based fault-tolerant MFAPC strategy is developed to realize the vehicular position and velocity tracking only using the input-output data. At last, an example is provided to prove the effectiveness of designed MFAPC framework.

A unified dynamic analysis method for kilometre-scale ultra large spacecraft based on dynamic stiffness method

Ultra-kilometre and super-flexible spacecraft are the important development trends in the future. Classical multi-body dynamics and structural dynamics analysis theory have some limitations in dynamic modelling and solution of large deformation structures. In this paper, an improved dynamic stiffness method (DSM) is proposed to deal with the dynamical modelling and solution of ultra-kilometre spacecraft. The method retains the high accuracy and efficiency of the original DSM in dealing with continuous dynamical systems and breaks through the limitations of the original method in dealing with large deformation structures by introducing equivalent linearization techniques. Results show that the modal frequencies and mode shapes in this paper agree well with finite element solutions, and the maximum deviation is less than 1%; Besides, with the increase of spacecraft length, the structure has entered the nonlinear stage from the linear stage, the fundamental frequency of the structure will be lower than 0.001Hz, and the influence of structure length, density, and the mass of additional cabins on natural frequencies is less significant than that in linear stage. The proposed dynamic analysis framework can be further employed in the dynamic response and vibration control problems of super large spacecraft.

A new look at the Equivalent Linearization Technique to predict LCO in aeroelastic systems with discrete nonlinearities

Control surfaces nonlinearities can lead to limit cycle oscillations (LCO). Several methods have been proposed to predict LCO, such as Harmonic Balance-based methods (HB). Describing function (DF) is the HB with a single harmonic motion assumed, and the approach can be combined with a classic eigenvalue stability analysis via the Equivalent Linearization Technique (ELT) to predict LCO. On the other hand, High-Order Harmonic Balance (HOHB) methods consider higher number of harmonics, but they lead to a more complex nonlinear algebraic system of equations. This paper introduces a new look at the ELT combining both eigenvalue analysis and describing functions. Two applications are considered. The first one is a new DF written in a matrix form for the ELT to consider both first and third harmonics in LCOs due to freeplay. The second application is an iterative procedure to combine the eigenvalue analysis with a describing function to predict LCOs in systems with both freeplay and friction nonlinearities. Numerical simulations are performed for the aeroelastic typical section airfoil. The results show that the new DF improves the classic DF by providing a more accurate prediction of LCOs that accounts for third harmonic. Also, the iterative-ELT is shown as an excellent predictor of LCO for systems with both freeplay and friction.

Seismic performance of a clutched inerter for structures with curved surface sliders

The performance of the clutched inerter in limiting the seismic response of structures isolated by curved surface slider (CSS) is investigated. The chosen structure is a flexible multi-floor building with an idealized linear shear-type behaviour. The equations governing the motion of an isolated structure with a clutched inerter under earthquake excitation have been derived. Initially, using the equivalent linearization technique, the isolated building's stationary floor absolute acceleration, isolator displacement, and device forces are evaluated. The stochastic earthquake excitation is considered to be a filtered white-noise Kanai-Tajimi model. The performance of the clutched inerter in mitigating the response of the isolated structure is studied by varying system parameters. The important parameters included are the inertance ratio of the clutched inerter, the period and friction coefficient of the CSS, and the flexibility of the superstructure. Further, there exists an optimum inertance of the clutched inerter for which the structure's highest floor absolute acceleration reaches its minimum level. The influence of important system parameters on the optimum inertance of the clutched inerter is also investigated. Finally, the performance of the clutched inerter as a supplemental device for the base-isolated structure is investigated when subjected to recorded earthquake motions consisting of near-fault as well as far-field types. The trends of results of isolated structures with clutched inerter under the recorded earthquake motions were in good agreement with that under stochastic excitation. It is demonstrated that the supplemental clutched inerter is effective in reducing the displacement response of base-isolated structures.

Application scope of equivalent linear technique for across-wind response analysis of tall flexible structures considering nonlinear aero-elastic effect

Large across-wind response of tall flexible structures becomes an increasing concern due to the significant nonlinear aero-elastic effect. The nonlinear aerodynamic damping, which is in phase of time-variant structural velocity, is usually employed to describe nonlinear aero-elastic effect. For the sake of evaluating the across-wind response of tall flexible structures within the framework of linear random vibration theory, equivalent linear aerodynamic damping model is required, which can be deduced from either harmonic balance (HB)- or statistical linearization (SL) technique. This study addressed clarifying application scope of various equivalent linear techniques for evaluating across-wind response of tall flexible structures. By conducting the Monte Carlo simulation and free-vibration wind tunnel test, it is clarified that the root-mean-squared (RMS) response calculated by the HB equivalent linear aerodynamic damping is acceptable, while is not satisfactory with respect to the SL technique especially when the structure is with lower damping level, and in the vicinity of vortex lock-in wind speed. Moreover, it is found that neither HB- nor SL technique can accurately predict the peak response. It is revealed in this study that accuracy of various equivalent linear techniques is essentially affected by probability density function of across-wind response.

Feasibility analyses of negative-stiffness dampers for seismic performance enhancement of a base-isolated liquid storage tank

Base-isolation systems are widely used to reduce the hydrodynamic base shear and overturning moment of liquid storage tanks with increased isolator displacements and liquid sloshing responses. Supplementing conventional damping devices in the isolation layer can effectively reduce isolator displacements whereas having limited influence on the liquid sloshing. In the present paper, simultaneous reduction on the isolator displacement and sloshing response of a base-isolated liquid storage tank is achieved by using negative-stiffness dampers (NSDs). NSDs consisting of a negative-stiffness component coupled in parallel with a fluid viscous damping component are considered, and an equivalent linearization technique is proposed for designing their properties on the basis of the principle of equal average energy dissipation. A four-lumped-mass model considering high sloshing modes of liquid storage tank is used as a numerical example. The feasibility of NSDs for improving the seismic performance of base-isolated storage tanks with varying aspect ratios is verified. Moreover, nonlinear response history analyses are conducted to illustrate the benefits of the proposed method for the seismic protection of base-isolated storage tanks. The seismic responses of base-isolated storage tanks controlled by using the proposed and traditional methods are comprehensively compared, and it is suggested that the proposed method can be advantageous over conventional ones by more effectively reducing base shear and sloshing responses without compromising isolator displacements.

A hysteretic model of localized frictional contacts with instrumental stiffness

The steady-state dynamic response of a single-degree-of-freedom system comprising both a hysteretic element and a spring is considered. The Hertz–Cattaneo–Mindlin theoretical framework for modeling of local tangential contact with friction is applied in conjunction with the Masing model of hysteresis to describe the hysteretic behavior of the multiple localized frictional contact interface. The steady-state tangential displacement amplitude of a rigid body under harmonic tangential force excitation is approximately determined by means of the equivalent linearization technique, based on the harmonic balance principle. A special attention is paid to the evaluation of the frictional damping and the determination of the backbone curve of the Masing model from the dissipation-amplitude relation.

Dynamic performance of the energy harvester with a fractional-order derivative circuit

Vibration energy harvesters are expected to replace chemical batteries and achieve self-power for low-power electronic devices, and their design has attracted significant attention. In previous studies, circuits consistent with the constitutive relationship with integer derivatives were frequently used, but fractional-order derivative circuits have received less attention. Because of the dissipation effects, such as internal friction and thermal memory, the fractional-order derivative is an effective mathematical approach for determining such nonconservative behaviors. In this study, the dynamic performance of a vibration energy harvester with a fractional-order derivative circuit in a random environment was investigated. Both linear and nonlinear vibration energy harvesters were evaluated using the linear random vibration theory and the equivalent linearization technique. The effects of the system parameters on the mean output power were analyzed, and the optimal system parameters, such as the oscillation frequency of the circuit and the coupling coefficient, were determined. Furthermore, the relationships between the optimal parameters and the order of the fractional derivative were established. Our study will potentially guide future designs of vibration energy harvesters.

Random analysis of coupled vehicle–bridge systems with local nonlinearities based on explicit time-domain method

Based on the explicit time-domain method, an efficient analysis algorithm is developed for the random vibration analysis of the coupled vehicle–bridge system with local nonlinear components. In this work, the coupled vehicle–bridge system is divided into two subsystems, including the linear bridge subsystem and the nonlinear vehicle subsystem. Firstly, using the equivalent linearization technique, the equivalent linearized subsystem is constructed for the vehicle subsystem with the hysteretic suspension spring at a given time instant. Then, the explicit expressions of the responses of the linearized vehicle subsystem and the linear bridge subsystem are constructed corresponding to contact forces, respectively. Further, the explicit expression of the contact forces is derived in consideration of the compatibility condition. Lastly, the dimension-reduction vibration analysis for the equivalent linearized coupled vehicle–bridge system can be carried out based on the explicit time-domain method. The numerical example about a coupled vehicle–bridge system under the random irregular excitation is investigated, and the results indicate that the proposed approach is of feasibility.

Non-Linear Response of Cable-Mass-Spring System in High-Rise Buildings under Stochastic Seismic Excitation.

In high-rise buildings earthquake ground motions induce bending deformation of the host structure. Large dynamic displacements at the top of the building can be observed which in turn lead to the excitation of the cables/ropes within lift installations. In this paper, the stochastic dynamics of a cable with a spring-damper and a mass system deployed in a tall cantilever structure under earthquake excitation is considered. The non-linear system is developed to describe lateral displacements of a vertical cable with a concentrated mass attached at its lower end. The system is moving slowly in the vertical direction. The horizontal displacements of the main mass are constrained by a spring-viscous damping element. The earthquake ground motions are modelled as a filtered Gaussian white noise stochastic process. The equivalent linearization technique is then used to replace the original non-linear system with a linear one with the coefficients determined by utilising the minimization of the mean-square error between both systems. Mean values, variances and covariances of particular random state variables have been obtained by using the numerical calculation. The received results were compared with the deterministic response of the system to the harmonic process and were verified against results obtained by Monte Carlo simulation.

Geometrically Nonlinear Random Response of Stiffened Laminated Plates by Proper-Orthogonal-Decomposition-Based Reduced-Order Modeling

The geometrically nonlinear random vibration response of stiffened laminated plates under acoustic excitation is carried out by using the equivalent linearization (EL) technique combined with the reduced-order model and the finite element method. The nonlinear equations of motion of stiffened laminated plates are obtained based on large deflection finite element formulation. The proper orthogonal decomposition technique is used to select the modal basis, and a reduced-order model of the structures is established. Based on the force error minimization approach, the EL formulation for solving nonlinear random vibration response is derived. Then the EL technique is incorporated into finite element software NASTRAN by the direct matrix abstraction program, and the direct matrix abstraction program also realizes the iterative process of determining the modal equivalent linear stiffness matrix. Then, the statistical dynamic response results obtained by EL and linear method are compared and analyzed. It is verified that the proposed method has high accuracy and efficiency, and it can be applied to the geometric nonlinear random vibration response analysis of stiffened laminated plates. The results show that geometric nonlinearity plays an important role in the vibration response of stiffened laminated plates, especially under high sound pressure loads.

Damping-Ductility Relationships for Flag-Shaped Hysteresis

AbstractDamping-ductility relationships, as applied in direct displacement-based design (DDBD), are products of equivalent linearization techniques used to estimate the nonlinear response of struct...

Stochastic analysis of locally resonant linear and hysteretic metamaterials for seismic isolation of process equipment

Periodic metafoundations have proven to inherit valuable properties from wave propagating in phononic periodic structures in the very low-frequency regime. Therefore, finite locally resonant metafoundations (LRMs) represent a novel type of seismic isolation for ultralow-frequency applications. In this context, it is still unknown the impact that massive resonators with varying frequencies or devices with hysteretic behavior can entail on the whole system performance. For this purpose, we develop and optimize two finite locally resonant multiple degrees of freedom (MDoF) metafoundations in this paper: i) a foundation endowed with resonators, linear springs and linear viscous dampers tuned to multiple frequencies; and ii) a foundation equipped with fully nonlinear hysteretic dampers. Both are optimized considering the stochastic nature of ground motion, modelled with a modified Kanai-Tajimi filter in the stationary frequency domain, and a massive MDoF superstructure, chosen to be a fuel storage tank. In order to take all of the above-mentioned effects into account, we establish a procedure based on nonlinear programming that is able to optimize any number of parameters. More precisely, to optimize the nonlinear behavior of damper devices we employ a Bouc-Wen hysteretic model. Therefore, we reduce the nonlinear differential equations of Bouc-Wen models to a system of linear equations through the stochastic (equivalent) linearization technique. Moreover, we test the optimized systems against natural seismic records both with linear and nonlinear time history analyses. To investigate the role of hysteresis on the nonlinear band structure, we derive linearized and nonlinear dispersion relationships for the uncoupled periodic metafoundation. Finally, we obtain further detailed information on the nonlinear wave propagation by means of a spectro-spatial analysis.

Prescribed performance model-free adaptive sliding mode control of a shape memory alloy actuated system

This paper concentrates on a simple and robust control method for the discrete time nonlinear systems to fulfill the requirement of predefined accuracy. A sliding mode control method is designed by introducing equivalent dynamic linearization technique according to the input/output (I/O) information merely. A square-root type error transformation method is presented for the tracking error to be restricted within a preassigned zone. The performance of presented control method is demonstrated through experiments on a nonlinear system. Experiment results show that the presented control method has a superior tracking accuracy compared with PID controller and model-free adaptive control (MFAC).

Vibration mitigation of an MDoF system subjected to stochastic loading by means of hysteretic nonlinear locally resonant metamaterials

In this paper, we intend to mitigate absolute accelerations and displacements in the low-frequency regime of multiple-degrees-of-freedom fuel storage tanks subjected to stochastic seismic excitations. Therefore, we propose to optimize a finite locally resonant metafoundation equipped with massive resonators and fully nonlinear hysteretic devices. The optimization process takes into account the stochastic nature of seismic records in the stationary frequency domain; the records are modelled with the power spectral density S0 and modified with a Kanai–Tajimi filter. Moreover, the massive superstructure of a fuel storage tank is also considered in the optimization procedure. To optimize the nonlinear behaviour of dampers, we use a Bouc–Wen hysteretic model; the relevant nonlinear differential equations are reduced to a system of linear equations through the stochastic equivalent linearization technique. The optimized system is successively verified against natural seismic records by means of nonlinear transient time history analyses. Finally, we determine the dispersion relations for the relevant periodic metafoundation.