Restoring force and dynamic loadings identification for a nonlinear chain-like structure with partially unknown excitations

Most of the currently employed vibration-based identification approaches for structural damage detection are based on eigenvalues and/or eigenvectors extracted from dynamic response measurements, and strictly speaking, are only suitable for linear system. However, the inception and growth of damage in engineering structures under severe dynamic loadings are typical nonlinear procedures. Consequently, it is crucial to develop general structural restoring force and excitation identification approaches for nonlinear dynamic systems because the restoring force rather than equivalent stiffness can act as a direct indicator of the extent of the nonlinearity and be used to quantitatively evaluate the absorbed energy during vibration, and the dynamic loading is an important factor for structural remaining life forecast. In this study, based on the instantaneous state vectors and partially unknown excitation, a power series polynomial model (PSPM) was utilized to model the nonlinear restoring force (NRF) of a chain-like nonlinear multi-degree-of-freedom (MDOF) structure. To improve the efficiency and accuracy of the proposed approach, an iterative approach, namely weighted adaptive iterative least-squares estimation with incomplete measured excitations (WAILSE-IME), where a weight coefficient and a learning coefficient were involved, was proposed to identify the restoring force of the structure as well as the unknown dynamic loadings simultaneously. The response measurements of the structure, i.e., the acceleration, velocity, and displacement, and partially known excitations were utilized for identification. The feasibility and robustness of the proposed approach was verified by numerical simulation with a 4 degree-of-freedom (DOF) numerical model incorporating a nonlinear structural member, and by experimental measurements with a four-story frame model equipped with two magneto-rheological (MR) dampers mimicking nonlinear behavior. The results show the proposed approach by combining the PSPM and WAILSE-IME algorithm is capable of effectively representing and identifying the NRF of the chain-like MDOF nonlinear system with partially unknown external excitations, and provide a potential way for damage prognosis and condition evaluation of engineering structures under dynamic loadings which should be regarded as a nonlinear system.     

Parameter identification of systems with multiple disproportional local nonlinearities

Identification of nonlinear systems, especially with multiple local nonlinearities exhibiting disproportional ratios of the degree of nonlinearity and present at a single or multiple spatial locations, is a highly challenging inverse problem. Identification of such complex nonlinear systems cannot be handled easily by the existing conventional restoring force or describing function methods. Further, noise-corrupted measured time history responses make the parameter identification process much more difficult. Keeping this in view, we propose a new meta support vector machine (meta-SVM) model to precisely identify the type, spatial location(s) and also the nonlinear parameters present in disproportionate levels using the noisy measurements. Apart from the conventional SVM model, we also explore the effectiveness of the non-batch processing models like incremental learning for lesser computational cost and increased efficiency. Both incremental and conventional support vector regression models are explored to precisely identify the nonlinear parameters. A numerically simulated multi-degree of freedom spring-mass system with limited multiple local nonlinearities at a few selected spatial locations is considered to illustrate the proposed meta-SVM model for nonlinear parametric identification. However, the extension of the proposed meta-SVM model is rather straightforward to include all types of nonlinearities and cases with the simultaneous existence of multiple numbers of same or different nonlinearities (i.e. combined nonlinearities) at single or multiple locations. It is also clearly established from the numerical simulation studies that the proposed incremental meta-SVM model paves way for online real-time identification of nonlinear parameters which is not yet been addressed in the existing literature.

     

Identification of the nonlinear vibration characteristics in hydropower house using transfer entropy

The nonlinear behavior is the primary attribute of the dynamic system stability. In this study, the time-delayed transfer entropy method is proposed to identify the nonlinear dynamic behavior of hydropower house and civil construction, including the transport directionality of information, cracked damage location, and degree of dynamic nonlinearity. Differing from the objects investigated in currently available literature, large-scale civil engineering structures such as hydropower house are more complex and larger size, which stimulate the demand for special identification techniques. Due to the fact that nonlinearity of hydropower house can be induced by many types of interactions between structure and mechanism, a simple similarity model of a multistory building, including damaged contact nonlinearity is studied first following by a discussion of the method for identifying information transmission directionality of the linear or nonlinear structure. The method for identifying the source and degrees of structural nonlinearity vibration is described. Furthermore, the procedure for identification of nonlinearity dynamic behavior in the hydropower house structure based on transfer entropy is studied based on a prototype field experiment under various load cases. Rather than the traditional linear signal processing tools and identification methods, the advantage of this proposed method is to identify the nonlinearity dynamic characteristics of hydropower house structure. This study provides a valuable reference for identifying the damage-induced nonlinearities in civil engineering structures as well as studying the nonlinear dynamic characteristics of hydropower house.     

On the design of external excitations in order to make nonlinear oscillators respond as free oscillators of the same or different type

This study deals with nonlinear oscillators whose restoring force has a polynomial nonlinearity of the cubic or quadratic type. Conservative unforced oscillators with such a restoring force have closed-form exact solutions in terms of Jacobi elliptic functions. This fact can be used to design the form of the external elliptic-type excitation so that the resulting forced oscillators also have closed-form exact steady-state solutions in terms of these functions. It is shown how one can use the amplitude of such excitations to change the way in which oscillators behave, making them respond as free oscillators of the same or different type. Thus, in cubic oscillators, a supercritical or subcritical pitchfork bifurcation can appear, whilst in quadratic oscillators, a transcritical bifurcation can take place.     

随机最优控制方法识别动力学系统局部非线性

利用随机动态规划方法可以得到线性二次型高斯问题的最优控制解．基于这一结果与系统辨识问题最优控制解的概念,将动力学系统中局部非线性结构参数的辨识问题转化为求解对应线性系统的最优控制问题,利用线性系统随机最优控制的理论与方法,结合FSM（ForceStateMapping）方法,提出了识别动力学系统中局部非线性回复力类型及结构参数的新方法．所研究系统由大的线性子结构与一个或多个非线性子结构组成,其中线性结构的模型参数已知,待辨识量为局部非线性结构参数．     

Identification of nonlinear anti-vibration isolator properties

Vibrations are classified among the major problems for engineering structures. Anti-vibration isolators are used to absorb vibration energy and minimise transmitted force which can cause damage. The isolator is modelled as a parallel combination of stiffness and damping elements. The main purpose of the model is to enable designers to predict the dynamic response of systems under different structural excitations and boundary conditions. A nonlinear identification method, discussed in this paper, aims to provide a tool for engineers to extract information about the nonlinear dynamic behaviour using measured data from experiments. The proposed method is demonstrated and validated with numerical simulations. Thus, this technique is applied to determine the nonlinear parameters of a commercial metal mesh isolator. Nonlinear stiffness and nonlinear damping can decrease with the increase in the amplitude of the base excitation. The softening behaviour of the mesh isolator is clearly visible.     

Experimental identification of hardening and softening nonlinearity in circular laminated plates

This work investigates nonlinear characteristics of a circular laminated plate. A nonparametric identification method based on the Hilbert transform is applied to identify the nonlinear system. The results demonstrate that the force–displacement curve has a soft nonlinear characteristic under small displacements and a hard nonlinear characteristic under large displacements. The force–velocity curve also has a soft nonlinear characteristic. A circular isotropic plate is treated to test the method. The force-state method is adopted to confirm the identification results. The effects of the plate diameter are examined. A combination of a cubic polynomial and a hyperbolic tangent function is proposed to fit the experiment data. The fitting results are verified by time domain simulations under random excitations. The work illustrates some novel nonlinear characteristics in transverse vibration of a circular laminated plate via a nonlinear system identification process.     

建筑结构爆破地震反应弹塑性精细时程分析

针对爆破地震作用下建筑结构的安全评估问题,提出利用时程分析方法全面评估爆破地震波的安全度;建立了基于精细积分算法的结构弹塑性动力分析架构模式,编制了建筑结构爆破地震反应弹塑性精细时程分析程序;通过算例验证了该算法的准确性与高效性,弹性时程分析与不同恢复力模型弹塑性时程分析的结果曲线具有类似特征和数值差异;建议选择合理的恢复力模型,使用弹塑性时程分析方法模拟爆破地震作用下结构的动力响应,全面评估爆破地震波的安全性。     

预应力混凝土平面杆系结构的有限元方法

建立了基于有限元方法的考虑材料和几何非线性的任意截面预应力混凝土平面杆系结构的数值分析模型,可用于模拟预应力混凝土大跨度梁、单向偏压细长柱等的非线性全过程结构响应。引入修正的Rodriguez截面模型确定截面切线刚度,其中混凝土的贡献通过截面边界顶点定义的梯形单元来实现;在此基础上利用传统的平面非线性杆单元导出了标准有限元公式。通过两个算例验证了该模型的可靠性和适用性。     

空间网格结构动力分析的非线性模态方法

空间网格结构因自由度数多且无简化的力学模型,非线性动力分析通常要耗费大量时间．传统的非线性模态方法用于求解多高层结构的局部非线性问题已获得良好的效果,但对系统非线性问题的应用尚缺少研究．对比分析多高层结构和空间网格结构动力性能差异,指出网格结构动力非线性分析存在的问题．以主振型理论和切线刚度分离法为基础,将非线性模态方法用于几何非线性效应显著的空间网格结构动力分析．通过对运动方程的非线性恢复力进行拆分,形成线性表达形式,然后解耦到主振型所在的广义坐标系,以达到缩减自由度数量的目的．并通过实例验证非线性模态方法的高效性与适用性．     

Active vibration control of nonlinear benchmark buildings

The present nonlinear model reduction methods unfit the nonlinear benchmark buildings as their vibration equations belong to a non-affine system. Meanwhile, the controllers designed directly by the nonlinear control strategy have a high order, and they are difficult to be applied actually. Therefore, a new active vibration control way which fits the nonlinear buildings is proposed. The idea of the proposed way is based on the model identification and structural model linearization, and exerting the control force to the built model according to the force action principle. This proposed way has a better practicability as the built model can be reduced by the balance reduction method based on the empirical Grammian matrix. A three-story benchmark structure is presented and the simulation results illustrate that the proposed method is viable for the civil engineering structures.     

Wavelet basis expansion-based Volterra kernel function identification through multilevel excitations

Volterra series is a powerful mathematical tool for nonlinear system analysis, which extends the convolution integral for linear system to nonlinear system. There is a wide range of nonlinear engineering systems and structures which can be modeled as Volterra series. One question involved in modeling a functional relationship between the input and output of a system using Volterra series is to identify the Volterra kernel functions. In this article, a wavelet balance method-based approach is proposed to identify the Volterra kernel functions from observations of the in- and outgoing signals. The basic routine of the approach is that, from the system outputs under multilevel excitations, the Volterra series outputs of different orders are first estimated with the wavelet balance method, and then the Volterra kernel functions of different orders are separately estimated through their corresponding Volterra series outputs by expanding them with four-order B-spline wavelet on the interval. The simulation studies verify the effectiveness of the proposed Volterra kernel identification method.     

任意激励下结构动力响应的状态方程精细积分法

对只有弹性模态以及除此之外还有刚体模态的结构的瞬态响应给出了精细积分的通用公式，从而使得该方法不仅可以处理线性激励的情形，而且对激励是多项式形式或可以展开成多项式的激励也同样能够计算。对于非线性激励，只要可以用关于自变量的级数形式来近似表示，都可以用本文所给的方法进行计算，计算的精度可以通过变化级数的项数来调整。     

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.     

An intelligent parameter varying (IPV) approach for non-linear system identification of base excited structures

Health monitoring and damage detection strategies for base-excited structures typically rely on accurate models of the system dynamics. Restoring forces in these structures can exhibit highly non-linear characteristics, thus accurate non-linear system identification is critical. Parametric system identification approaches are commonly used, but require a priori knowledge of restoring force characteristics. Non-parametric approaches do not require this a priori information, but they typically lack direct associations between the model and the system dynamics, providing limited utility for health monitoring and damage detection. In this paper a novel system identification approach, the intelligent parameter varying (IPV) method, is used to identify constitutive non-linearities in structures subject to seismic excitations. IPV overcomes the limitations of traditional parametric and non-parametric approaches, while preserving the unique benefits of each. It uses embedded radial basis function networks to estimate the constitutive characteristics of inelastic and hysteretic restoring forces in a multi-degree-of-freedom structure. Simulation results are compared to those of a traditional parametric approach, the prediction error method. These results demonstrate the effectiveness of IPV in identifying highly non-linear restoring forces, without a priori information, while preserving a direct association with the structural dynamics.     

The X-structure/mechanism approach to beneficial nonlinear design in engineering

Nonlinearity can take an important and critical role in engineering systems, and thus cannot be simply ignored in structural design, dynamic response analysis, and parameter selection. A key issue is how to analyze and design potential nonlinearities introduced to or inherent in a system under study. This is a must-do task in many practical applications involving vibration control, energy harvesting, sensor systems, robotic technology, etc. This paper presents an up-to-date review on a cutting-edge method for nonlinearity manipulation and employment developed in recent several years, named as the X-structure/mechanism approach. The method is inspired from animal leg/limb skeletons, and can provide passive low-cost high-efficiency adjustable and beneficial nonlinear stiffness (high static & ultra-low dynamic), nonlinear damping (dependent on resonant frequency and/or relative vibration displacement), and nonlinear inertia (low static & high dynamic) individually or simultaneously. The X-structure/mechanism is a generic and basic structure/mechanism, representing a class of structures/mechanisms which can achieve beneficial geometric nonlinearity during structural deflection or mechanism motion, can be flexibly realized through commonly-used mechanical components, and have many different forms (with a basic unit taking a shape like X/K/Z/S/V, quadrilateral, diamond, polygon, etc.). Importantly, all variant structures/mechanisms may share similar geometric nonlinearities and thus exhibit similar nonlinear stiffness/damping properties in vibration. Moreover, they are generally flexible in design and easy to implement. This paper systematically reviews the research background, motivation, essential bio-inspired ideas, advantages of this novel method, the beneficial nonlinear properties in stiffness, damping, and inertia, and the potential applications, and ends with some remarks and conclusions.

     

考虑几何与材料及静风荷载的非线性因素的大跨径桥梁静风稳定分析法

针对现有的桥梁静风稳定分析方法中存在的问题，提出了增量与内外两重迭代相结合的新方法，并且考虑了结构几何、材料和静风荷载非线性。在上述方法的基础上，编制了桥梁非线性空气静力稳定分析程序BNAP，并进行了相应的算例分析，所得结果表明该方法具有计算稳定和速度快的优点。最后，以一座主跨1000米的斜拉桥为例，分析了结构几何非线性、材料非线性和静风荷载非线性对大跨径桥梁空气静力稳定性的影响。     

Van der Pol type self-excited micro-cantilever probe of atomic force microscopy

A technique for dimensional reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. The DDEs considered here have a canonical form with at most cubic nonlinearities and periodic coefficients. The nonlinear terms are multiplied by a perturbation parameter. Perturbation expansion converts the nonlinear response problem into solutions of a series of nonhomogeneous linear ordinary differential equations (ODEs) with time-periodic coefficients. One set of linear nonhomogeneous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The linear part of the map is the monodromy operator whose eigenvalues characterize stability. Dimensional reduction on the map is then carried out. In the case of critical eigenvalues, this corresponds to center manifold reduction, while for the noncritical case resonance conditions are derived. The accuracy of the nonlinear Chebyshev collocation map is demonstrated by finding the solution of a nonlinear delayed Mathieu equation and then a milling model via the method of steps. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation. The method of this paper is shown by example to be applicable to systems with strong parametric excitations.     

Transition sets analysis based parametrical design of nonlinear metal rubber isolator

This research proposes the parametrical design of Metal Rubber (MR) isolation platform based on the investigation of nonlinear vibration properties under different types of excitation. Based on the mechanical model established by experiments, the restoring force of the isolation platform is proposed as a nonlinear function in consideration of the stiffness nonlinearity and Coulomb friction of metal wires. Then, the perturbation method is utilized to solve the steady states whose local stability is studied by singularity theory. The main results obtained by singularity theory show that there are five different types of vibration property, and the critical conditions for the transformation of different vibration properties are defined by transition sets. For impact excitation, the optimum structural parameters are obtained based on the vibration dissipation time via nonlinear backbone analysis; for periodic excitation, the optimum structural parameters are determined according to multiple standards including the bandwidth for effective isolation, bandwidth for multi-steady states, resonance peak and displacement transmissibility in high frequency band etc. The vibration performances for optimum structural parameters are verified by dynamical experiments. In conclusion, this paper carries out a novel sight of choosing optimum parameters, and therefore provides the guidance for the utilization of MR isolation platform for different types of excitation in engineering practices.     

Parametric characteristic of the random vibration response of nonlinear systems

Volterra series is a powerful mathematical tool for nonlinear system analysis,and there is a wide range of nonlinear engineering systems and structures that can be represented by a Volterra series model.In the present study,the random vibration of nonlinear systems is investigated using Volterra series.Analytical expressions were derived for the calculation of the output power spectral density(PSD) and input-output cross-PSD for nonlinear systems subjected to Gaussian excitation.Based on these expressions,it was revealed that both the output PSD and the input-output crossPSD can be expressed as polynomial functions of the nonlinear characteristic parameters or the input intensity.Numerical studies were carried out to verify the theoretical analysis result and to demonstrate the effectiveness of the derived relationship.The results reached in this study are of significance to the analysis and design of the nonlinear engineering systems and structures which can be represented by a Volterra series model.