Frequency-adaptive bilinear reduced-order model for structures with intermittent contacts

Nonlinear Dynamics - Computing the nonlinear forced response of structures with localized nonlinearity, such as intermittent contacts, is a time-intensive task. Temporal and spatial reduction...     

Nonlinear vibrations of a circular cylindrical shell with multiple internal resonances under multi-harmonic excitation

The nonlinear response of a water-filled, thin circular cylindrical shell, simply supported at the edges, to multi-harmonic excitation is studied. The shell has opportune dimensions so that the natural frequencies of the two modes (driven and companion) with three circumferential waves are practically double than the natural frequencies of the two modes (driven and companion) with two circumferential waves. This introduces a one-to-one-to-two-to-two internal resonance in the presence of harmonic excitation in the spectral neighbourhood of the natural frequency of the mode with two circumferential waves. Since the system is excited by a multi-harmonic point-load excitation composed by first and second harmonics, very complex nonlinear dynamics is obtained around the resonance of the fundamental mode. In fact, at this frequency, both modes with two and three circumferential waves are driven to resonance and each one is in a one-to-one internal resonance with its companion mode. The nonlinear dynamics is explored by using bifurcation diagrams of Poincaré maps and time responses.     

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.

     

A friction contact stiffness model of fractal geometry in forced response analysis of a shrouded blade

To investigate the nonlinear vibration behavior of a shrouded blade with friction dynamic contact interface, a friction contact stiffness model is proposed to describe the friction force at different rough interfaces and different normal loads. In the proposed model, the friction contact interface is discretized to a series of friction contact pairs and each of them can experience stick, slip, or separate states. Fractal geometry is used to simulate the topography of contact surfaces. The contact stiffness is calculated using the Hertz contact theory and fractal geometry, which is related to contact interfaces parameters including normal load, roughness, Young??s modulus, and Poisson??s ratio. The trajectory tracking method is used to predict the friction force and it is not necessary to judge the transition condition among stick, slip, and separate states. It is suitable for complicated periodic motion of the contact interfaces. The forced response of a real shrouded blade is predicted using the proposed model and the multi-harmonic balance method. The effect of surface roughness, initial normal load, and contact area on the forced response of a shrouded blade is studied. It is shown that contact stiffness increases with normal load and fractal dimension. The resonant amplitude is sensitive to the initial normal load and contact surface roughness. The response can be influenced by the contact area, which is an important parameter for blade designers.     

Crack parameter estimation in structures using finite element modeling

This paper addresses the problem of linear crack quantification, crack depth estimation and localization, in structures. An optimization technique based on a finite element model for cracked structural elements is employed in the estimation of crack parameters for beam, truss and two-dimensional frame structures. The modal data for the cracked structures are obtained by solving the corresponding eigenvalue problem. The error in the modal data is simulated by an additive noise that follows the normal distribution. The simulated reduced modal data is expanded using the eigenvector projection method. Numerical examples showed that this technique gives good results for cracks with high depth ratio. The accuracy of the estimated crack parameters depends on (1) the number of modes used, (2) the error level in the cracked structure modal data and (3) the number of measured degrees of freedom in the case of reduced modal data.     

Nonlinear normal modes in pendulum systems

Dynamics of the spring pendulum and of the system containing a pendulum absorber is considered by using the nonlinear normal modes?? theory and the asymptotic-numeric procedures. This makes it possible to investigate the pendulum dynamics for both the small and large vibration amplitudes. The vibration modes stability is analyzed by different methods. Regions of the nonlinear normal modes?? stability/instability are obtained. The nonlinear normal modes?? approach and the modified Rauscher method are used to construct forced vibration modes in the system with a pendulum absorber.     

Control of the homoclinic bifurcation in buckled beams: Infinite dimensional vs reduced order modeling

A method for controlling non-linear dynamics and chaos is applied to the infinite dimensional dynamics of a buckled beam subjected to a generic space varying time-periodic transversal excitation. The homoclinic bifurcation of the hilltop saddle is identified as the undesired dynamical event, because it triggers, e.g., cross-well scattered (possibly chaotic) dynamics. Its elimination is then pursued by a control strategy which consists in choosing the best spatial and temporal shape of the excitation permitting the maximum shift of the homoclinic bifurcation threshold in the excitation amplitude-frequency parameters space.The homoclinic bifurcation is detected by the Holmes and Marsden's theorem [A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Ration. Mech. Anal. 76 (1981) 135-165] constituting a generalization of the classical Melnikov's theory. Two classes of boundary conditions (b.c.) are identified: for the first, the Melnikov function is exactly the same as obtained with the reduced order models, while for the second, which is more general, this is no longer true, and the non-linear normal modes theory is used. Based on this distinction, the control method is then separately applied to the two cases, and the optimal spatial and temporal shapes of the excitation are determined.A detailed comparison of the infinite vs finite dimensional models is performed with respect to the control features, and it is shown that, depending on the b.c., the control based on the reduced order model provides either exact or engineering acceptable results, although more systematic investigations are required to generalize the last conclusion.     

静动载作用下牙冠的断裂失效模式

<正>常牙齿存在许多薄弱结构,如窝沟、楔状缺损和牙隐裂等.如何评估这些薄弱结构对牙齿裂纹产生和断裂的影响,一直缺乏有效的评价方法.利用实验和有限元分析两种方法,初步分析了牙冠在静动载荷作用下的断裂模式.研究结果表明,结构因素是决定牙冠断裂的最重要因素,牙冠断裂常常发生在结构薄弱的地方,且牙冠断裂符合第二强度准则.本研究为临床诊断和治疗提供一定的理论依据.     

High-efficiency nonlinear dynamic analysis for joint interfaces with Newton–Raphson iteration process

Nonlinear Dynamics - Nonlinear dynamic analysis of the assembled structures involves the complex nonlinearity of the joint interfaces. By combining the multi-harmonic balance method with the...     

Nonlinear vibration modes and instability of a conceptual model of a spar platform

Spar floating platforms have been largely used for deepwater drilling, oil and natural gas production, and storage. In extreme weather conditions, such structures may exhibit a highly nonlinear dynamical behavior due to heave-pitch coupling. In this paper, a 2-DOF model is used to study the heave and pitch dynamical response in free and forced vibration. Special attention is given to the determination of the nonlinear vibration modes (NNMs). Nonsimilar and similar NNMs are obtained analytically by direct application of asymptotic and Galerkin-based methods. The results show important NNM features such as instability and multiplicity of modes. The NNMs are used to generate reduced order models consisting of SDOF nonlinear oscillators. This allows analytic parametric studies and the derivation of important features of the system such as its frequency-amplitude relations and resonance curves. The stability is analyzed by the Floquet theory. The analytical results show a good agreement with the numerical solution obtained by direct integration of the equation of motion. Instability analyses using bifurcation diagrams and Mathieu charts are carried out to understand the fundamental mechanism for the occurrence of unstable coupled heave-pitch resonant motions of floating structures in waves and to study the dependencies of the growth rate of unstable motions on physical parameters.     

FREQUENCY DOMAIN CALCULATION METHOD FOR WIND-INDUCED RESPONSE OF MONOLAYER CABLE NET

以随机振动理论为基础,采用频域法对单层索网结构风振响应进行了研究,主要探讨不同计算方法和空间相干函数的选取等问题对索网结构风振响应的影响. 依据不同的计算方法和空间相干函数,建立了3 种分析模式进行了分析. 分析结果表明：对于频率密集的索网结构,需要考虑模态交叉项对计算结果的影响;空间相干函数的选取对计算结果影响较大,采用与风频率相干的空间相干函数更为合理.     

Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure

Nonlinear normal modes for elastic structures have been studied extensively in the literature. Most studies have been limited to small nonlinear motions and to structures with geometric nonlinearities. This work investigates the nonlinear normal modes in elastic structures that contain essential inertial nonlinearities. For such structures, based on the works of Crespo da Silva and Meirovitch, a general methodology is developed for obtaining multi-degree-of-freedom discretized models for structures in planar motion. The motion of each substructure is represented by a finite number of substructure admissible functions in a way that the geometric compatibility conditions are automatically assured. The multi degree-of-freedom reduced-order models capture the essential dynamics of the system and also retain explicit dependence on important physical parameters such that parametric studies can be conducted. The specific structure considered is a 3-beam elastic structure with a tip mass. Internal resonance conditions between different linear modes of the structure are identified. For the case of 1:2 internal resonance between two global modes of the structure, a two-mode nonlinear model is then developed and nonlinear normal modes for the structure are studied by the method of multiple time scales as well as by a numerical shooting technique. Bifurcations in the nonlinear normal modes are shown to arise as a function of the internal mistuning that represents variations in the tip mass in the structure. The results of the two techniques are also compared.     

Free and forced vibration of repetitive structures

In this paper, the vibration problems of some repetitive structures, including symmetric, cyclic periodic, linear periodic, chain, and axi-symmetric structures is investigated. Eigen-value problems derived from the vibration equations of these structures are established based on their continuous models. The special properties of the structural modes of these structures are deduced. Applying these properties can provide effective reduction approach to solving the natural and forced vibration problems of these structures by either numerical or experimental methods. Furthermore, these properties can be applied in other aspects such as evaluating the reasonableness of the discrete models of these repetitive structures.     

Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods

Reduced order models for the dynamics of geometrically exact planar rods are derived by projecting the nonlinear equations of motion onto a subspace spanned by a set of proper orthogonal modes. These optimal modes are identified by a proper orthogonal decomposition processing of high-resolution finite element dynamics. A three-degree-of-freedom reduced system is derived to study distinct categories of motions dominated by a single POD mode. The modal analysis of the reduced system characterizes in a unique fashion for these motions, since its linear natural frequencies are near to the natural frequencies of the full-order system. For free motions characterized by a single POD mode, the eigen-vector matrix of the derived reduced system coincides with the principal POD-directions. This property reflects the existence of a normal mode of vibration, which appears to be close to a slow invariant manifold. Its shape is captured by that of the dominant POD mode. The modal analysis of the POD-based reduced order system provides a potentially valuable tool to characterize the spatio-temporal complexity of the dynamics in order to elucidate connections between proper orthogonal modes and nonlinear normal modes of vibration.     

A microslip friction model with normal load variation induced by normal motion

A two-dimensional microslip friction model with normal load variation induced by normal motion is presented in this paper. The model is a distributed parameter model, which characterizes the stick-slip-separation of the contact interface and determines the resulting friction force, including its time variance and spatial distribution, between two elastic structures. When the relative motion is simple harmonic motion, the stick-slip-separation transition angles associated with any point in the contact area can be analytically determined within a cycle of motion. In addition, if the relative motion is given, stick-slip-separation transition boundaries inside the contact area and their time variances can be determined. Along with an iterative multi-mode solution approach utilizing harmonic balance method (HBM), the developed model can be employed to determine the forced response of frictionally constrained structures. In the approach, the forced response is constructed in terms of the free mode shapes of the structure; consequently, it can be determined at any excitation frequency and for any type of normal load distribution. Two examples, a one-dimensional beam like damper and a more realistic blade to ground damper, are employed to illustrate the predictive abilities of the developed model. It is shown that while employing a single mode model, transition boundaries for the beam like damper agrees with the results given in the literature, the developed method identifies the phase difference along the slip to stick transition boundary when a multi-mode model is employed. Moreover, while partial slip is illustrated in the two examples, typical softening and hardening effects, due to separation of the contact surface, are also predicted for the blade to ground damper.     

Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures

Localization phenomena in one-dimensional imperfect continuous structures are analyzed, both in dynamics and buckling. By using simple models, fundamental concepts about localization are introduced and similarities between dynamics and buckling localization are highlighted. In particular, it is shown that strong localization of the normal modes is due to turning points in which purely imaginary characteristic exponents assume a non zero real part; in contrast, if turning points do not occur, only weak localization can exist. The possibility of a disturbance propagating along the structure is also discussed. A perturbation method is then illustrated, which generalizes the classical WKB method; this allows the differential problem to be transformed into a sequence of algebraic problems in which the spatial variable appears as a parameter. Applications of the method are worked out for beams and strings on elastic soil. All these structures are found to have nearly-defective system matrices, so their characteristic exponents are highly sensitive to imperfections.     

Analytical parametric study of bi-linear hysteretic model of dry friction under harmonic,impulse and random excitations

Hysteretic behavior due to some nonlinear sources is a common phenomenon in many dynamical systems. One of the sources of this behavior in mechanical systems is dry friction. Dry friction in bolted or riveted joints of mechanical structures makes their dynamic behavior hysteretic. Bi-linear hysteresis is one of the models that can be used to study these systems which is used in this paper. A SDOF system containing a bi-linear hysteretic element called Jenkins element under harmonic, impulse and random excitations is considered. For all three types of excitations, the effects of system and excitation parameters on the defined equivalent system parameters and the response specifications are studied. Harmonic balance method is employed for harmonic excitation studies, and optimum friction threshold for minimizing response amplitude is obtained versus other system parameters and response amplitude. Energy balance method is used for impulse excitation through which the desired decaying ratio can be achieved by tuning the friction threshold, depending on stiffness ratio. System under random excitation is investigated by equivalent linearization technique in two steps. At the first step, equivalent properties are obtained versus instantaneous amplitude of response. In this step, the paper contains the parametric study of system in which the variations of equivalent parameters are described when physical parameters of system or input intensity vary. Overall variance of system response is determined in the second step, and optimum sliding threshold is obtained to have minimum overall variance of system response.     

The effect of underplatform dampers on the forced response of bladed disks by a coupled static/dynamic harmonic balance method

Friction contacts are often used in turbomachinery design as passive damping systems. In particular, underplatform dampers are mechanical devices used to decrease the vibration amplitudes of bladed disks.Numerical codes are used to optimize during designing the underplatform damper effectiveness in order to limit the resonant stress level of the blades. In such codes, the contact model plays the most relevant role in calculation of the dissipated energy at friction interfaces. One of the most important contact parameters to consider in order to calculate the forced response of blades assembly is the static normal load acting at the contact, since its value strongly affects the area of the hysteresis loop of the tangential force, and therefore the amount of dissipation.A common procedure to estimate the static normal loads acting on underplatform dampers consists in decoupling the static and the dynamic balance of the damper. A preliminary static analysis of the contact is performed in order to get the static contact/gap status to use in the calculation, assuming that it does not change when vibration occurs.In this paper, a novel approach is proposed. The static and the dynamic displacements of the system (bladed disk+underplatform dampers) are coupled together during the forced response calculation. Static loads acting at the contacts follow from static displacements and no preliminary static analysis of the system is necessary.The proposed method is applied to a numerical test case representing a simplified bladed disk with underplatform dampers. Results are compared with those obtained with the classical approach.     

Stress recovery algorithm for reduced order models of mechanical systems in nonlinear dynamic operative conditions

Nonlinear forced response analyses of mechanical systems in the presence of contact interfaces are usually performed in built-in numerical codes on reduced order models (ROM). Most of the cases these derive from complex finite element (FE) models, resulting from the high accuracy the designers require in modeling and meshing the components in commercial FE software. In the technical literature several numerical methods are proposed for the identification of the nonlinear forced response in terms of a kinematic quantity (i.e. displacement, velocity and acceleration) associated either to the master degrees-of-freedom retained in the ROM, or to the slave ones after having expanded the reduced response through the reduction matrix. In fact, the displacement is the quantity usually adopted to monitor the nonlinear response, and to evaluate the effectiveness of a partially loose friction interface in damping vibrations, with respect to a linear case where no friction interfaces exist and no energy dissipation can take place. However, when a ROM is used the engineering quantities directly involved in the mechanical design, i.e. the strains and stresses, cannot be retrieved without a further data processing. Moreover, in the case of a strong nonlinear behavior of the mechanical joints, the distributions of the nonlinear strains and stresses over the structure is likely different than the one obtained as a superposition of linear mode shapes whose definition require a-priori assumptions on the boundary conditions at the contact interface. This means that the mentioned approximation cannot be used to predict the safety margins of a structure working in real (nonlinear) operative conditions. This paper addresses this topic and presents a novel stress recovery algorithm for the identification of the strains and stresses resulting from a nonlinear forced response analysis on a ROM. The algorithm is applied to a bladed disk with friction contacts at the shroud joint, which make the behavior of the blades nonlinear and non-predictable by means of standard linear analyses in commercial FE software.