Nonlinear Damped Systems Research Articles

Discussion on the accuracy of methods for determining interface force between numerical and physical substructures in shaking table-based real-time hybrid simulation

Real-time hybrid simulation (RTHS) is a developing experimental technology that combines the advantages of numerical simulation and physical test. In RTHS, the interface force is a key physical quantity coupling the dynamic response of physical substructure and numerical substructure. An inaccurate force determination method can distort RTHS results and may cause incorrect estimations of the dynamic performance of structures. This article aims to systematically discuss the impact of possible error in determining the interface force during shaking table-based RTHS. The errors caused by two typical force determination methods, (1) using force sensors and (2) using strain gauges, are comprehensively investigated. These two force determination methods are typically used for (1) performance evaluation of nonlinear damping systems and (2) investigation of soil–structure interaction effects. Virtual hybrid simulation numerical models (in which both physical and numerical substructures are simulated numerically) representing the interface force determination processes are established. The impacts of error on the RTHS results are evaluate quantitatively. The reported findings can assist in developing a deeper understanding of the causes of errors during RTHS and enhance the experimental designs. This, in turn, leads to more accurate and reliable results in RTHS testing.

Measurement and optimization of nonlinear damping systems for agricultural engineering vehicle cab

The issue of nonlinear dampness in the cab of agricultural engineering vehicles is examined by analyzing the vibration reduction system of a specific agricultural loader. Firstly, the specific loader was tested under different conditions. Then, the nonlinear vibration reduction system model of the cab–seat–human body is established by using the measured frame vibration signal as input. Finally, the multi–objective genetic algorithm is used to optimize the root mean square (RMS) value of vertical acceleration of the cab and seat. The test results show that the seat vibration is significantly greater than the acceleration of the cab floor under driving and working conditions, so the seat vibration is amplified and the seat parameter setting is unreasonable; the engine and the working device are also an important part of the cab vibration source, in addition to the uneven road surface. Comparing the RMS values of the vertical acceleration of the cab and seat, which were calculated by the model and obtained from the vehicle test, the error does not exceed 6%, indicating that the model’s accuracy meets the requirement. The vehicle experiment proves that the RMS value of the vertical acceleration of the cab and seat is reduced by 16% and 53%, respectively, after optimization. This study provides a theoretical basis for the design of the damping system for the cab of agricultural engineering vehicles.

Approximate solutions of the fractional damped nonlinear oscillator subject to Van der Pol system

This paper deals with fractal Van der Pol damped nonlinear oscillators equation having nonlinearity. By combining the techniques of the Laplace transform and the variational iteration method, we establish approximate periodic solutions for the fractal damped nonlinear systems. In this simple way, nonlinear differential equations can be easily converted into linear differential equations. Illustrative examples including the Van der Pol damped nonlinear oscillator reveal that this method is very effective and convenient for solving fractal nonlinear differential equations. Finally, comparison of the obtained results with those of the other achieved method, also reveals that this coupling method not only suggests an easier method due to the Lagrange multiplier but also can be easily extended to other nonlinear systems.

Interpolation‐based parametric model order reduction of automotive brake systems for frequency‐domain analyses

Brake squeal describes noise with different frequencies that can be emitted during the braking process. Typically, the frequencies are in the range of 1 to 16 kHz. Although the noise has virtually no effect on braking performance, strong attempts are made to identify and eliminate the noise as it can be very unpleasant and annoying. In the field of numerical simulation, the brake is typically modeled using the Finite Element method, and this results in a high-dimensional equation of motion. For the analysis of brake squeal, gyroscopic and circulatory effects, as well as damping and friction, must be considered correctly. For the subsequent analysis, the high-dimensional damped nonlinear equation system is linearized. This results in terms that are non-symmetric and dependent on the rotational frequency of the brake rotor. Many parameter points to be evaluated implies many evaluations to determine the relevant parameters of the unstable system. In order to increase the efficiency of the process, the system is typically reduced with a truncated modal transformation. However, with this method the damping and the velocity-dependent terms, which have a significant influence on the system, are neglected for the calculation of the eigenmodes, and this can lead to inaccurate reduced models. In this paper, we present results of other methods of model order reduction applied on an industrial high-dimensional brake model. Using moment matching methods combined with parametric model order reduction, both the damping and the various parameter-dependent terms of the brake model can be taken into account in the reduction step. Thus, better results in the frequency domain can be obtained. On the one hand, as usual in brake analysis, the complex eigenvalues are evaluated, but on the other hand also the transfer behavior in terms of the frequency response. In each case, the classical and the new reduction method are compared with each other.

Modeling and analysis of a piezoelectric transducer embedded in a nonlinear damped dynamical system

This paper focuses on the dynamical analysis of the motion of a new three-degree-of-freedom (DOF) system consisting of two segments that are attached together. External harmonic forces energize this system. The equations of motion (EOM) are derived utilizing Lagrangian equations, and the approximate solutions up to the third order are investigated using the methodology of multiple scales. A comparison between these solutions and numerical ones is constructed to confirm the validity of the analytic solutions. The modulation equations (ME) are acquired from the investigation of the resonance cases and the solvability conditions. The bifurcation diagrams and spectrums of Lyapunov exponent are presented to reveal the different types of the system’s motion and to represent Poincaré maps. The piezoelectric transducer is connected to the dynamical system to convert the vibrational motion into electricity; it is one of the energy harvesting devices which have various applications in our practical life like environmental and structural monitoring, medical remote sensing, military applications, and aerospace. The influences of excitation amplitude, natural frequency, coupling coefficient, damping coefficient, capacitance, and load resistance on the output voltage and power are performed graphically. The steady-state solutions and stability analysis are discussed through the resonance curves.

Towards seismic performance quantification of viscous damped steel structure: Site-specific hazard analysis and response prediction

With an emphasis on predictable performance, the paramount importance of performance-based earthquake engineering (PBEE) in quantifying earthquake risks and facilitating the better-informed design of the built environment to achieve earthquake resilience has been widely acknowledged. And the uses of damping systems, i.e., structures incorporated with supplemental damping devices, have been recognized as an effective structural development to increase the resilience of structures to earthquakes. This paper presents a uniform hazard spectrum (UHS) based site-specific ground motion selection procedure, to implement the PBEE framework to relate earthquake hazard to structural performance for structures with variable dynamic properties. This ground motion selection procedure accounts for the effect of variable structural dynamic properties on hazard characterization for structures with damping systems. A paradigm building site, on which the site-specific hazard from the probabilistic seismic hazard analysis (PSHA) is consistent with the risk-targeted hazard from the design maps of ASCE/SEI 7-10, is selected for the implementation of the procedure. Three suites of 40 ground motions representing various hazard intensities at the building site are selected for time history dynamic analysis of a prototype structure with nonlinear viscous damping system. Evaluated is the probability distribution of major engineering demand parameters (EDPs) including story drift, residual story drift, floor velocity, and floor acceleration. The results demonstrated that the major EDPs of a building structure with supplemental nonlinear viscous dampers at a paradigm site can be estimated using the lognormal probability distribution, and the proposed UHS-based site-specific ground motion selection procedure is critical to the performance assessment of structures with variable dynamic properties in the PBEE framework.

Vibration control of double pendulum type crane system with uncertainty in parameters by elimination of the natural frequency component

Overhead cranes are widely used to transport heavy loads, and their efficient operation requires fast and accurate positioning. However, residual vibration which prevents positioning accuracy generally tends to be induced by fast transportation. In previous paper, we proposed a control method which is based on the characteristic that the residual vibration is completely suppressed in a linear undamped system when excited by an external force which does not contain the natural frequency component of the system. To apply this characteristic to nonlinear damped systems, we defined the apparent external force which includes the influence of system nonlinearity and damping. It was confirmed that the residual vibration of a double pendulum type system model of crane can be suppressed by eliminating two natural frequency components from each the apparent external forces. In this paper, in addition, to improve the robustness for a double pendulum type system against the estimation error, new conditions are added to reduce the component around the natural frequencies from each of the two apparent external forces. The effectiveness of these conditions for robustness is verified through numerical simulations. Furthermore, changes in the robustness are examined when the position of the mass point near the support point of the double pendulum is changed. The results showed that the robustness can be improved in many cases except when the control time is short and the mass point is close to the support point.

Determination of optimum design parameters of non-linear damped outrigger system for high-rise buildings

Determination of optimum design parameters of non-linear damped outrigger system for high-rise buildings

Vibration analysis of nonlinear damping systems by the discrete incremental harmonic balance method

Vibration analysis of nonlinear damping systems by the discrete incremental harmonic balance method

ON FOURTH ORDER MORE CRITICALLY DAMPED NON-LINEAR SYSTEMS UNDER SOME CONDITIONS

Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended for solving fourth order more critically damped non-linear systems. For different damping forces, the solutions obtained by the present method show good coincidence with numerical solutions. The method is illustrated by an example.

The design of nonlinear damped building isolation systems by using mobility analysis

Additional power law nonlinear damping has shown advantages in seismic isolation of buildings. The design of nonlinear damped building isolation systems can be conducted in the frequency domain based on the output frequency response function (OFRF) of the building’s frequency output responses. But this often requires many runs of simulations of nonlinear building systems, which will spend a long time when the finite element (FE) model or differential equation model of the building system is complex. In this study, the issue will be resolved by an integrated mobility analysis and equivalent linearization approach. In this approach, the complex linear building system without additional nonlinear damping is represented by several data-driven autoregressive with exogenous input (ARX) models. Mobilities of the building system can be evaluated from these ARX models, so that a mobility-based frequency domain representation of the linear building system can be achieved. After that, an equivalent linearization approach is applied to simulate the building system with additional nonlinear damping. Finally, the OFRF-based design can be conducted based on the proposed mobility analysis and equivalent linearization approach. A 4-degree of freedom (DoF) building system is discussed to demonstrate the advantages of the proposed method. The results indicate that the new approach can significantly increase the efficiency of the design and be effectively applied to the design of complex nonlinear building isolation systems.

Well-posedness and general decay for nonlinear damped porous thermoelastic system with second sound and distributed delay terms

Well-posedness and general decay for nonlinear damped porous thermoelastic system with second sound and distributed delay terms

Global existence and general decay of a weakly nonlinear damped Timoshenko system of thermoelasticity of type III with infinite memory

Global existence and general decay of a weakly nonlinear damped Timoshenko system of thermoelasticity of type III with infinite memory

Controllability analysis of multiple fractional order integro-differential damping systems with impulsive interpretation

We study the controllability criteria for a class of fractional integro-differential damped systems with impulsive perturbations. The solution representation is derived for both linear and non-linear damped systems, and the introduced formulations were constructed by employing Laplace transformation with Mittag-Leffler matrix function. We present necessary and sufficient conditions for the indicated systems to be controllable in finite dimensional spaces. Here, controllability conditions are proposed by using Grammian matrix as a powerful tool. Two numerical examples are also given at end to demonstrate the obtained theory.

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

Controllability of Higher Order Fractional Damped Delay Dynamical Systems with Time Varying Multiple Delays in Control

This paper is concerned with the controllability of higher order fractional damped delay dynamical systems with time varying multiple delays in control, which involved Caputo derivatives of any different orders. A necessary and sufficient condition for the controllability of linear fractional damped delay dynamical system is obtained by using the Grammian matrix. Sufficient conditions for controllability of the corresponding nonlinear damped delay dynamical system has established by the successive approximation technique. Examples have provided to verify the results.

Suppression of Residual Vibration in Nonlinear Systems With Temporal Variation and Uncertainty in Parameters by Elimination of the Natural Frequency Component

Abstract A systematic approach is developed for determining a control input for the point-to-point control of an overhead crane that exhibits temporal variation of rope length in addition to damping and nonlinearity, without inducing residual vibration. Complete suppression of the residual vibration is achieved by eliminating the natural frequency component of the cargo from the apparent external force, which is defined to include the effects of damping, nonlinearity, and parameter variation. Furthermore, an effective technique previously proposed by the authors for improving robustness to the modeling error of the natural frequency is extended. Numerical simulation results show that, even when cargo is hoisted up or down during operation, the proposed method realizes accurate positioning of the cargo without inducing residual vibration and sufficiently improves robustness. To the best of our knowledge, this is the first frequency-domain robust open-loop control strategy that ensures a theoretical zero amplitude for residual vibration in the absence of modeling error in nonlinear crane hoisting operation. The developed method is not only a contribution to the realization of low-cost and efficient crane hoisting operation, but is also applicable to the control of other nonlinear damped systems that include time-varying parameters.

Controllability of non-linear fractional-order systems with damping behaviour and multiple delays

Abstract A controllability analysis of both linear and non-linear fractional-order systems with damping behaviour and multiple delays is studied. We derived the controllability result for damped system with multi-term fractional order and multiple delays by introducing some lemmas with the help of Laplace transform properties and Mittag–Leffler function. Further, some new sufficient conditions ensuring controllability for a class of non-linear multi-term fractional-order damped systems with multiple delays are established by utilizing fractional Caputo derivatives and Schauder’s fixed point theorem. Moreover, as applications that demonstrate the proposed results, two examples are presented.

APPROXIMATE ANALYTICAL SOLUTIONS TO NONLINEAR DAMPED OSCILLATORY SYSTEMS USING A MODIFIED ALGEBRAIC METHOD

In the current paper, a modified algebraic method (MAGM) is proposed as an effective semi-analytical technique for solving nonlinear damped oscillatory systems. A polynomial is supposed as the trial solution, and its unknown coefficients are easily determined through the algebraic method (AGM). In order to improve the solution, the Laplace transformation is applied to the series solution, and then the Pade approximants of the resultant equation are constructed. Finally, the inverse Laplace transformation is adopted to obtain a periodic solution for the nonlinear problem under consideration. The proposed method is then applied for obtaining approximate analytical solutions of a damped rotatory oscillator as well as nonlinear vibrations of a flexible beam excited by an axial force. The results are compared with those obtained by the fourth-order Runge–Kutta method, and good agreement is observed.

Extended framework of Hamilton’s principle for single-degree-of-freedom nonlinear damping systems

The paper explores application of the variational formalism called extended framework of Hamilton’s principle to nonlinear damping systems. Single-degree-of-freedom systems with dominant source of nonlinearity from polynomial powers of the velocity are initially considered. Appropriate variational formulation is provided, and then the corresponding weak form is discretized to produce a novel computational method. The resulting low-order temporal finite element method utilizes non-iterative algorithm, and some examples are provided to verify its performance. The present temporal finite element method using small time step is equivalent to the adaptive Runge–Kutta–Fehlberg method with default error tolerances in MATLAB, and additional simulation shows its good convergence characteristics.