Nonlinear coupled vibration of electrostatically actuated clamped–clamped microbeams under higher-order modes excitation

Nonlinear modal interactions have recently become the focus of intense research in micro-resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. Understanding and controlling nonlinear coupling between vibrational modes is critical for the development of advanced micromechanical devices. This article aims to theoretically investigate the influence of antisymmetry mode on nonlinear dynamic characteristics of electrically actuated microbeam via considering nonlinear modal interactions. Under higher-order modes excitation, two nonlinear coupled flexural modes to describe microbeam-based resonators are obtained by using Hamilton’s principle and Galerkin method. Then, the Method of Multiple Scales is applied to determine the response and stability of the system for small amplitude vibration. Through Hopf bifurcation analysis, the bifurcation sets for antisymmetry mode vibration are theoretically derived, and the mechanism of energy transfer between antisymmetry mode and symmetry mode is detailed studied. The pseudo-trajectory processing method is introduced to investigate the influence of external drive on amplitude and bifurcation behavior. Results show that nonlinear modal interactions can transit vibration energy from one mode to nearby mode. In what follows, an effective way is proposed to suppress midpoint displacement of the microbeam and to reduce the possibility of large deflection. The quantitative relationship between vibrational modes is also obtained. The displacement of one mode can be predicted by detecting another mode, which shows great potential of developing parameter design in MEMS. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.     

Geometrically exact planar beams with initial pre-stress and large curvature: Static configurations,natural frequencies,and mode shapes

Within this paper, an analytical formulation is provided and used to determine the natural frequencies and mode shapes of a planar beam with initial pre-stress and large variable curvature. The static configuration, mode shapes, and natural frequencies of the pre-stressed beam are obtained by using geometrically exact, Euler–Bernoulli beam theory. The beam is assumed to be not shear deformable and inextensible because of its slenderness and uniform, closed cross-section, as well as the boundary conditions under consideration. The static configuration and the modal information are validated with experimental data and compared to results obtained from nonlinear finite-element analysis software. In addition to the modal analysis about general static configurations, special consideration is given to an initially straight beam that is deformed into semi-circular and circular static configurations. For these special circular cases, the partial differential equation of motion is reduced to a sixth-order differential equation with constant coefficients, and solutions of this system are examined. This work can serve as a basis for studying slender structures with large curvatures.     

Nonlinear dynamics of doubly curved shallow microshells

The nonlinear dynamical characteristics of a doubly curved shallow microshell are investigated thoroughly. A consistent nonlinear model for the microshell is developed on the basis of the modified couple stress theory (MCST) in an orthogonal curvilinear coordinate system. In particular, based on Donnell’s nonlinear theory, the expressions for the strain and the symmetric rotation gradient tensors are obtained in the framework of MCST, which are then used to derive the potential energy of the microshell. The analytical geometrically nonlinear equations of motion of the doubly microshell are obtained for in-plane displacements as well as the out-of-plane one. These equations of partial differential type are reduced to a large set of ordinary differential equations making use of a two-dimensional Galerkin scheme. Extensive numerical simulations are conducted to obtain the nonlinear resonant response of the system for various principal radii of curvature and to examine the effect of modal interactions and the length-scale parameter.     

Analysis for the optimal location of cable damping systems on stayed bridges

Computational models are increasingly being used for the dynamic analysis of structures with nonlinear or uncertain behavior, such as cables in stayed bridges, which nowadays are progressively more used as an alternative for long span and slim structures. In this work, a 3D nonlinear model is described to evaluate the wind dynamic effects on cables for this type of bridges under different scenarios, but also for health monitoring and structural simulation to guarantee performance, evaluate load capacity and estimate life prediction. Fatigue is one of the most relevant and complex failure causes in highway bridges, particularly on the anchorage elements of the cables in stayed bridges; where dampers may be used to minimize the dynamic behavior of the structure and reduce fatigue damage. With this nonlinear simulation model, different damper locations and configurations are evaluated to find the optimal position. A feasibility function is used as a weighting function to take into account the damper’s size and design. Analysis is particularly focused for a real cable stayed bridge in the state of Veracruz in México. Although the geometry, the forces and the stresses on cable structures are a challenge, even for structural specialists, the results from this work using the proposed 3D nonlinear model showed to be accurate for the simulation of many different wind scenarios, and damper’s location and orientations. Finally, the feasibility weighting function enabled the geometrical limitations to estimate the best location of a damper system to minimize the risk for fatigue failure.     

Nonlinear dynamics, stability and control of the scan process in noncontacting atomic force microscopy

The nonlinear equations of motion for the scan process in noncontacting atomic force microscopy are consistently derived using the extended Hamilton’s principle. A modal dynamical system obtained from the continuum model reveals that scan control appears in the form of parametric excitation. The system is analyzed asymptotically and numerically to yield escape bounds limiting the noncontacting mode of operation. Approximate stability bounds are deduced from both a global Melnikov integral and a local Moon–Chirikov overlap criterion. The Melnikov–Holmes stability curve and the overlap criterion are found to be similar for small damping. However, for very small damping, typical of ultra-high vacuum conditions, where the Melnikov bound becomes trivial, the Moon–Chirikov criterion yields an improved stability threshold.     

大跨径斜拉—悬索协作体系桥动力分析

斜拉-悬索协作体系桥作为一种新的超大跨径桥型，综合了悬索桥和斜拉桥的特点，国内外对这种桥型的研究尚少。本文根据空间有限元计算模型，对伶仃东航道斜拉-悬索协作体系桥设计方案进行了模态分析，分析了斜拉-悬吊协作体系桥的固有特性，讨论了不同主梁纵向约束方式和辅助墩的设置情况的影响，为此类桥型结构的动力性能分析提供了有价值的参考。     

Primary pulse transmission in coupled steel granular chains embedded in PDMS matrix: Experiment and modeling

We present an experimental study of primary pulse transmission in coupled ordered steel granular chains embedded in poly-di-methyl-siloxane (PDMS) elastic matrix. Two granular one-dimensional chains are considered (an ‘excited’ and an ‘absorbing’ one), each composed of 11 identical steel beads of 9.5 mm diameter with the centerline of the chain spaced at fixed distances of 0.5, 1.5 or 2.5 mm apart. We directly force one of the chains (the excited one) by a transient pulse and measure, by means of laser vibrometry, the primary transmitted pulses at the end beads of both chains and at the first bead of the absorbing chain. It is well known that the dynamics of this type of ordered granular media is strongly nonlinear due, (i) to Hertzian interactions between adjacent beads, and (ii) to possible bead separations in the absence of compressive forces and ensuing collisions between neighboring beads. Accordingly, we develop a strongly nonlinear theoretical model that takes into account the coupling of the granular chains due to the PDMS matrix, with the aim to model primary pulse transmission in this system. After validating the model with experimental measurements, we employ it in a predictive fashion to estimate energy transfer between chains as a function of the interspatial distance between chains. Furthermore, based on this model we perform predictive matrix design to achieve maximum energy transfer from the excited to the absorbing chain, and provide a theoretical explanation of the nonlinear dynamics governing energy transfer (including energy equi-partition) in this system.     

Modal interactions in primary and subharmonic resonant dynamics of imperfect microplates with geometric nonlinearities

This paper analyses the modal interactions in the nonlinear, size-dependent dynamics of geometrically imperfect microplates. Based on the modified couple stress theory,the equations of motion for the in-plane and out-of-plane motions are obtained employing the von Kármán plate theory as well as Kirchhoff's hypotheses by means of the Lagrange equations. The equations of motions are solved using the pseudo-arclength continuation technique and direct timeintegration method. The system parameters are tuned to the values associated with modal interactions, and then nonlinear resonant responses and energy transfer are analysed.Nonlinear motion characteristics are shown in the form of frequency-response and force-response curves, time histories, phase-plane portraits, and fast Fourier transforms.     

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。     

Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments

We study targeted energy transfers (TETs) and nonlinear modal interactions attachments occurring in the dynamics of a thin cantilever plate on an elastic foundation with strongly nonlinear lightweight attachments of different configurations in a more complicated system towards industrial applications. We examine two types of shock excitations that excite a subset of plate modes, and systematically study, nonlinear modal interactions and passive broadband targeted energy transfer phenomena occurring between the plate and the attachments. The following attachment configurations are considered: (i) a single ungrounded, strongly (essentially) nonlinear single-degree-of-freedom (SDOF) attachment—termed nonlinear energy sink (NES); (ii) a set of two SDOF NESs attached at different points of the plate; and (iii) a single multi-degree-of-freedom (MDOF) NES with multiple essential stiffness nonlinearities. We perform parametric studies by varying the parameters and locations of the NESs, in order to optimize passive TETs from the plate modes to the attachments, and we showed that the optimal position for the NES attachments are at the antinodes of the linear modes of the plate. The parametric study of the damping coefficient of the SDOF NES showed that TETs decreasing with lower values of the coefficient and moreover we showed that the threshold of maximum energy level of the system with strong TETs occured in discrete models is by far beyond the limits of the engineering design of the continua. We examine in detail the underlying dynamical mechanisms influencing TETs by means of empirical mode decomposition (EMD) in combination with wavelet transforms. This integrated approach enables us to systematically study the strong modal interactions occurring between the essentially nonlinear NESs and different plate modes, and to detect the dominant resonance captures between the plate modes and the NESs that cause the observed TETs. Moreover, we perform comparative studies of the performance of different types of NESs and of the linear tuned mass dampers (TMDs) attached to the plate instead of the NESs. Finally, the efficacy of using this type of essentially nonlinear attachments as passive absorbers of broadband vibration energy is discussed.     

Buckling optimization of flexible columns

A novel approach to the optimization of flexible columns against buckling is presented. Previous published studies, considering either continuous or discrete finite element models, are always constrained to specific relations between stiffness and mass distributions of the column. These, besides yielding impractical configurations that do not conform to manufacturing and production requirements, result in designs that are certainly suboptimal. The present model formulation considers columns that can be practically made of uniform segments with the true design variables defined to be the cross-sectional area, radius of gyration and length of each segment. Exact structural analysis is performed, ensuring the attainment of the absolute maximum critical buckling load for any number of segments, type of cross section and type of boundary conditions. Detailed results are presented and discussed for clamped columns having either solid or tubular cross-sectional configurations, where useful design trends have been recommended for optimum patterns with two, three and more segments. It is shown that the developed optimization model, which is not restricted to specific properties of the cross section, can give higher values of the critical load than those obtained from constrained-continuous shape optimization. In fact, the model has succeeded in arriving at the global optimal column designs having the absolute maximum buckling load without violating the economic feasibility requirements.     

Steady-state dynamics of a 3D tensegrity structure: Simulations and experiments

This paper considers a modeling and analysis approach for the investigation of the linear and nonlinear steady-state dynamics of a base excited 3D tensegrity module carrying a top mass. The tensegrity module contains three compressive members, which may buckle and six cables (tendons). First, a dynamic model of the system is derived using Lagrange’s equation with constraints. The buckling modeling of the compressive members is based on the assumed-mode method with a single mode discretization. The tendons are modeled as piecewise linear springs, which can only take tensile forces. This research focusses on the dynamic stability of the tensegrity structure by defining the geometrical and material properties in such a way that the system is just below the static stability boundary. Static and linear dynamic analysis is performed. In the nonlinear steady-state analysis, frequency-amplitude plots, power spectral density plots, bifurcation point continuation diagrams, and Poincaré maps are presented. A tensegrity structure is designed and manufactured and an experimental set-up is realized in order to validate the model by comparing experimentally and numerically obtained responses. In the validation stage, the numerical results are based on an amplifier-shaker-tensegrity structure model. It can be concluded that the numerical results match partly quantitatively and partly qualitatively with the experimentally obtained responses.     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation

This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.     

Scaling Mode Shapes in Output-Only Systems by a Consecutive Mass Change Method

In operational modal analysis (OMA) mode shapes can be obtained only with arbitrary normalization. There are many applications where mass normalized mode shapes are required, such as response prediction and stress analysis. A method to scale the mode shapes in OMA is to modify the dynamic behaviour of the structure by adding masses and then to use the modal parameters of both the original and modified structure. Several mass change methods have been proposed in recent years for estimating the scaling factors, where a distributed array of added masses are needed to obtain good results. In this work a new mass change approach based on performing several individual mass changes is presented. This approach requires only a small number of masses that are located at different points in each individual experiment. The results of the individual tests are then combined to estimate the scaling factors. The approach is developed and validated by measurements carried out on a 15-tonne prestressed concrete slab strip and a steel cantilever beam. The results show that a good accuracy can be obtained by this method when a proper mass change strategy is used.     

On Modal Interaction,Stability and Nonlinear Dynamics of a Model Two D.O.F. Mechanical System Performing Snap-Through Motion

Dynamics of a simple two degrees of freedom (d.o.f.) mechanical system is considered, to illustrate the phenomena of modal interaction. The system has a natural symmetry of shape and is subjected to symmetric loading. Two stable equilibrium configurations are separated by an unstable one, so that the model system can perform cross-well oscillations. Nonlinear statics and dynamics are considered, with the emphasis on detecting conditions for instability of symmetric configurations and analysis of bi-modal non-symmetric motions. Nonlinear local dynamics is analyzed by multiple scales method. Direct numerical integration of original equations of motions is carried out to validate analysis of modulation equations. In global dynamics (analysis of cross-well oscillations) Lyapunov exponents are used to estimate qualitatively a type of motion exhibited by the mechanical system. Modal interactions are demonstrated both in the local dynamics and for snap-through oscillations, including chaotic motions. This mechanical system may be looked upon as a lumped parameters model of continuous elastic structures (spherical segments, cylindrical panels, buckled plates, etc.). Analyses performed in the paper qualitatively describe complicated phenomena in local and global dynamics of original structures.     

Resonance,Parameter Estimation,and Modal Interactions in a Strongly Nonlinear Benchtop Oscillator

We study the vibrations of a strongly nonlinear, electromechanically forced, benchtop experimental oscillator. We consciously avoid first-principles derivations of the governing equations, with an eye towards more complex practical applications where such derivations are difficult. Instead, we spend our effort in using simple insights from the subject of nonlinear oscillations to develop a quantitatively accurate model for the single-mode resonant behavior of our oscillator. In particular, we assume an SDOF model for the oscillator; and develop a structure for, and estimate the parameters of, this model. We validate the model thus obtained against experimental free and forced vibration data. We find that, although the qualitative dynamics is simple, some effort in the modeling is needed to quantitatively capture the dynamic response well. We also briefly study the higher dimensional dynamics of the oscillator, and present some experimental results showing modal interactions through a 0:1 internal resonance, which has been studied elsewhere. The novelty here lies in the strong nonlinearity of the slow mode.     

Nonlinear analysis of structural elements under unilateral contact constraints by a Ritz type approach

A nonlinear modal solution methodology capable of solving equilibrium and stability problems of uni-dimensional structural elements (beams, columns and arches) with unilateral contact constraints is presented in this work. The contact constraints are imposed by an elastic foundation of the Winkler type, where special attention is given to the case in which the foundation reacts in compression only, characterizing the contact as unilateral. A Ritz type approach with moveable boundaries, where the coordinates defining the limits of the contact regions are considered as additional variables of the problem, is proposed to solve this class of unilateral contact problems. The methodology is illustrated by particular problems involving beams, beam-columns and arches, and the results are compared with available results obtained by finite element and mathematical programming techniques. It is concluded that the Ritz type approach proposed is particularly suited for the analysis of structural problems where the number, but not the length, of the contact regions between the bodies are known a priori. Therefore, it can substitute in these cases finite element applications and be used as a benchmark for more general and complex formulations as well.     

Study of milling stability with Hertz contact stiffness of ball bearings

This present work examines the stability and nonlinear responses of a spindle milling system supported by ball bearings. A shaft finite element based on Timoshenko beam theory is employed to model the spindle, and modal reduction method is therefore adopted for saving the numerical calculating time. The issues of evaluating the effects of the ball bearing Hertz contact stiffness are consequently addressed. It is found that suitable constant bearing stiffness can be adopted to replace the nonlinear nonsmooth Hertz stiffness in prediction of the critical cutting depth of the milling system in certain bearing configuration conditions. For the constant bearing stiffness can be obtained by experiment, this replacement will undoubtedly simplify the spindle-bearing milling system. But with the increase in the bearing clearance, the spindle milling system will present obvious nonlinear behaviors, and the nonlinear Hertz contact bearing stiffness will take over. Isolated islands of chatter vibration, which are induced by the nonlinear nonsmooth bearing Hertz stiffness, can be found exist in milling processes in large bearing clearance conditions.