Nonlinear Structural Vibrations Research Articles

Comparative Study on PSD and HHT Techniques for Condition Assessment of Steel Truss Bridge Using Acceleration Data

Bridge structures deteriorate over time due to factors like corrosion, fatigue, aging of materials, and unexpected natural or accidental events. Consequently, the vibrations measured in these structures exhibit time-varying and non-stationary characteristics. Additionally, non-linearities are inherent in these structures due to material behavior, boundary condition changes, and joints or connections. Analysis of non-linear and non-stationary structural vibration data is necessary to extract information on the condition of the bridge. Most methods assessing bridge conditions from acceleration responses rely on the Power Spectral Density (PSD) which is based on the Fourier Transform. However, the accuracy of the Fourier transforms for non-linear and non-stationary signals limits its application potential. The Hilbert–Huang Transform (HHT) has emerged as an alternative for handling these non-stationary signals due to its time-frequency-energy representation. This study provides a comparative study of both techniques by utilizing the data obtained from continuously monitored Pamban Bridge. This steel truss bridge was fitted with accelerometers to record bidirectional acceleration time histories at twenty bottom node points on the bridge. This study examines the continuous variations in the frequency component by both methods for 136 days. A linear best-fit trendline is obtained for the frequency parameters at different sensor positions. The slope and intercept values of this linear best fit trendline is further studied. The HHT showcases various evolving dominant frequency components over time in contrast to the PSD representation. The frequency spectrum from the HHT displays less variation over time at each sensor position compared to the predominant peak value derived from the PSD of the acceleration signal. The spatial variability in the predominant frequency obtained from HHT is also less than the PSD. Therefore, the ability of HHT to discriminate damages needs further investigations.

Dynamic properties of large amplitude nonlinear oscillations using Hamiltonian-based frequency formulation

In the present work, we present a detailed investigation into the large amplitude vibration of nonlinear engineering structures by means of the Hamiltonian-based frequency formulation (HBFF), extracted from the Hamiltonian theory and residual equation method. Compared to the variational method, the frequency amplitude formulas have an excellent agreement. In addition, the compared curves between the HBFF, variational method and exact solution indicate that the HBFF is more accurate. The ideas in this research are expected to shed a new light to the exploration of the nonlinear oscillation problems arising in physics and engineering.

Linear vs nonlinear structural vibration behavior of steel-timber composite building elements

Composite structures including wood are increasingly coming into focus due to rising demand for sustainability and, at the same time, high-rise buildings, and long-span components. In this context, steel-timber hybrid building elements are gaining more attention since they offer sufficient load-carrying properties. However, such lightweight structures increase vibration levels and sound transmission. Accurate understanding of the structures is therefore indispensable. Consequently, herein, a novel steel-timber composite building element is studied concerning its vibrational behavior numerically and experimentally. Especially the influence of the joints between steel and timber components is investigated by considering nonlinear behavior due to frictional effects. Therefore, amplitude-dependency regarding eigenfrequencies and damping properties as well as multi-harmonic responses are studied. Hence, nonlinearities are revealed by slight amplitude-dependency and the occurrence of higher harmonics.

Reduced Order Model of Nonlinear Structures for Turbomachinery Aeroelasticity

Abstract This work concerns the numerical modeling of geometric nonlinear vibrations of slender structures in rotation using an original reduced order model based on the use of dual modes along with the implicit condensation method. This approach is an improvement of the classical ICE method in the sense that the membrane stretching effect is taken into account in the dynamic resolution. The dynamics equations are first presented and the construction of the reduced order model (ROM) is then proposed. The second part of the paper deals with numerical applications using the finite element method, first for a three-dimensional cantilever beam, then for an Ultra-High Bypass Ratio (UHBR) fan blade subject to aerodynamic loads. In the applications considered, the proposed method predicts more accurately the geometrically nonlinear behavior than the ICE method.

Nonlinear Structural Vibration of Multi-Megawatt Vertical Axis Wind Turbine Blades-Part 1: Derivation of Motion Equations

Replacing fossil fuels with renewable energy is one of the important means to improve energy scarcity and environmental protection, and is of great significance for achieving sustainable development of human society. Vertical axis wind turbine (VAWT) is one of the key equipment for applying green energy — wind energy. During the operation of VAWT, its blades not only have to withstand periodic aerodynamic loads, but also suffer from centrifugal force effects and the Coriolis force effect caused by coupling with multi-dimensional deformation. The complex stress situation makes the blades susceptible to damage, resulting in structural resonance and instability of aeroelastic stability. This paper applies the Euler beam model and Hamilton principle to establish a nonlinear structural vibration motion equation for large-span vertical axis wind turbine blades with fully coupled four-dimensional deformation (i.e. lateral bending deformation, chord bending deformation, axial torsion deformation, and axial tensile deformation), and retains the nonlinear term of the motion equation to the third-order infinitesimal. Separating the motion equation into equilibrium equation and dynamic equation, it can be seen that the displacement of equilibrium state caused by the centrifugal force of the rigid body exists in the form of a coefficient in the dynamic equation, which will have an impact on the dynamic characteristics. This modeling work can provide a research foundation for exploring the dynamic characteristics and aeroelastic stability of VAWT blades in the future.

Nonlinear vibrations of earth structures

This article offers a detailed analysis of the current state of plane structure dynamics, addressing the complexities posed by non-homogeneous materials exhibiting nonlinear elastic and viscoelastic properties during real-life operations. The study proposes a comprehensive mathematical model and algorithm to investigate the dynamic behavior of such structures, employing the non-linear hereditary Boltzmann-Volterra theory to describe viscoelastic material properties accurately. Nonlinear oscillatory systems are analyzed using Lagrange equations based on the d’Alembert principle. The problem is approached through a multi-step process. Initially, the linear elastic problem of the structure’s natural oscillations is solved to determine its natural frequencies and modes of oscillations. Subsequently, these eigenmodes are employed as coordinate functions to address forced nonlinear oscillations in viscoelastic non-homogeneous systems. The complexity of the problem necessitates solving a Cauchy problem comprising a system of nonlinear integrodifferential equations. To illustrate the methodology, the study examines the Gissarak earth dam, considering real operational conditions and nonlinear, viscoelastic material properties near resonant modes of vibrations. Utilizing numerical methods, the dynamic behavior of the structure is analyzed, assessing displacements and stress components at different time points under non-stationary kinematic action. Stress concentration regions within the structure are identified for resonant vibrations, allowing the evaluation of its strength. Furthermore, the impact of nonlinear elasticity and viscoelasticity on the structural dynamics is quantified. This research provides valuable insights into the behavior of plane non-homogeneous structures, considering real-world scenarios and material complexities, ultimately contributing to an improved understanding of structural dynamics and facilitating the identification and mitigation of potential structural challenges.

The Extended Galerkin Method for Approximate Solutions of Nonlinear Vibration Equations

An extension has been made to the popular Galerkin method by integrating the weighted equation of motion over the time of one period of vibrations to eliminate the harmonics from thee deformation function. A set of successive equations of coupled higher-order vibration amplitudes is resulted, and a nonlinear eigenvalue problem is obtained for the frequency-amplitude dependence of nonlinear vibrations with successive displacements. The subsequent solutions of vibration frequencies and deformation are consistent with other successive approximate methods, such as the harmonics balance method. This is an extension of the Galerkin method which has broad applications for asymptotic solutions, particularly for problems in solid mechanics. This extended Galerkin method can also be utilized for the analysis of free and forced nonlinear vibrations of structures as a new technique with significant advantages in calculations.

Cross-flow-induced transverse–torsional vibrations of slender structures mitigation via coupled controllers

This study aims to propose a coupled controller of tuned mass damping (TMD) and nonlinear energy sink (NES) for enhancing the stability and suppressing nonlinear vibrations of slender structures, such as tall buildings subjected to cross airflows. The nonlinear governing equations are established to describe the coupled vibrations of the slender structure with the controller. Subsequently, a linear analysis is performed to indicate the positive effect of proposed controllers on increasing the onset speed of galloping for the slender structure. A nonlinear analysis is conducted to state the mechanism of controllers on reducing oscillations of the slender structure. The results show that the coupled controller has better effects on whether enhancing the onset speed or oppressing the structure’s vibration amplitudes. A parametric analysis is then carried out to design an optimal coupled controller. The distributed controllers are designed in the structure, indicating that there is an optimal number for the controller where the onset speed is peak depending on the mass ratio. The present study can provide beneficial guidance and design for controlling the cross-flow-induced vibrations of slender structures in engineering applications.

Finite element analysis of nonlinear deformation, stability and vibrations of elastic thin-walled structures

Thin-walled shell-type structures are widely used in various branches of technology and industry. Such structures under operating conditions are usually exposed to various loads, including thermomechanical ones. Real shell structures, as a rule, have a complex shapes. To increase reliability, reduce material consumption, for technological reasons, they are designed as inhomogeneous systems in thickness. This causes a great and constant interest of engineers and designers in the problems of investigating the behavior of elastic thin-walled shell structures.
 The work is devoted to the method of analysis of geometrically nonlinear deformation, stability, post-buckling behavior and natural vibrations of thin elastic shells of complex shape and structure under the action of static thermomechanical loads. The unified design model has been created on the basis of the developed universal spatial finite element with introduced additional variable parameters. The model takes into account the multilayer material structure and geometric features for structural elements of the thin shell. The shells can be reinforced with ribs and cover plates, weakened by cavities, channels and holes, have sharp bends in the mid-surface.
 Such a uniform formulation made it possible to create a unified finite element model of the shells with an inhomogeneous structure. It is shown on a number of problems that the method presented in this article is an effective tool for analyzing geometrically nonlinear deformation, stability, post-buckling behavior and natural vibrations of thin elastic shells of an inhomogeneous structure under the action of static thermomechanical loads.

Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow

This study deals with the development of an accurate, efficient and robust method for the numerical solution of the interaction of compressible flow and nonlinear dynamic elasticity. This problem requires the reliable solution of flow in time-dependent domains and the solution of deformations of elastic bodies formed by several materials with complicated geometry depending on time. In this paper, the fluid–structure interaction (FSI) problem is solved numerically by the space-time discontinuous Galerkin method (STDGM). In the case of compressible flow, we use the compressible Navier–Stokes equations formulated by the arbitrary Lagrangian–Eulerian (ALE) method. The elasticity problem uses the non-stationary formulation of the dynamic system using the St. Venant–Kirchhoff and neo-Hookean models. The STDGM for the nonlinear elasticity is tested on the Hron–Turek benchmark. The main novelty of the study is the numerical simulation of the nonlinear vocal fold vibrations excited by the compressible airflow coming from the trachea to the simplified model of the vocal tract. The computations show that the nonlinear elasticity model of the vocal folds is needed in order to obtain substantially higher accuracy of the computed vocal folds deformation than for the linear elasticity model. Moreover, the numerical simulations showed that the differences between the two considered nonlinear material models are very small.

The extended Rayleigh-Ritz method for an analysis of nonlinear vibrations

An extension has been made with the popular Rayleigh-Ritz method by integrating the energy functional over the time of one period of vibration to eliminate the harmonics from the deformation function. An eigenvalue problem is obtained for the frequency-amplitude dependence of nonlinear vibrations of small deformation. This is the extension of Rayleigh-Ritz method of the energy formulation and Galerkin method with the inclusion of time, which is equivalent to the harmonics matching in nonlinear vibrations. The extended Rayleigh-Ritz method can be utilized for the analysis of free and forced nonlinear vibrations of structures as a new technique with significant advantages.

Evaluation of Input Shaping Methods for the Nonlinear Vibration System Using a Furuta Pendulum

There are many nonlinear vibration problems of mechanical structures because of various reasons such as geometric parameters, impact loads, or property of materials. One simple solution for the suppression of nonlinear structural vibration is input shaping that generates a command signal to cancel its vibration. However, a motion platform to evaluate the performance of input shapers for nonlinear vibration is rare. This paper presents the evaluation of input shaping methods for the nonlinear vibration system using a Furuta pendulum. First, the mathematical model of the Furuta pendulum is introduced and its nonlinear vibration characteristic is analyzed. Then, commands for canceling the nonlinear vibration of the Furuta pendulum are generated with various input shapers such as ZV, ZVD, and ZVDD. Finally, we illustrate the effects of input shapers on the nonlinear Furuta pendulum by comparing the pendulum overshoot, settling time, and vibration-reduction ratio. The Furuta pendulum is shown to be a good motion platform to evaluate input shaping methods for nonlinear vibration systems.

Controlling uncertain nonlinear structural vibrations of moving continuum system by embedding a vibration monitoring unit to feedback algorithm

Controlling uncertain nonlinear structural vibrations of moving continuum system by embedding a vibration monitoring unit to feedback algorithm

Vibration and acoustic waves of multilayered cylindrical shells carrying internal components attached by nonlinear compliant mounts

The structural-acoustic interactions of a coupled multilayered composite cylindrical shell and internal equipment system immersed in an infinite fluid medium are analyzed. The internal equipment is installed on the cylindrical shell by a series of nonlinear compliant mounts, which are represented by nonlinear springs and linear viscous dampers. Both the translational and rotational vibrations of the internal equipment are considered, and they are strongly coupled together due to the nonlinear elastic mounts. The nonlinear dynamics model of the coupled cylindrical shell and internal equipment system is established using a modified variational method, and the fluid exterior to the shell is modeled by a time-domain boundary element method. A strongly coupled serially staggered procedure is adopted to solve the governing equations of the structural system and the acoustic fluid as a coupled system. The results of the present analysis are validated by comparing with those obtained from the finite element method. The nonlinear structural vibration and acoustic wave behaviors of the coupled system containing compliant mounts with linear or nonlinear stiffnesses (including quadratic, cubic, quartic, quadratic-quartic types of stiffness) are analyzed. The effects of the translational and rotational motions of the internal equipment as well as their coupling on the nonlinear vibration and acoustic wave responses of the coupled system are examined. Physical insights into the super-harmonics in the vibration and sound pressure responses of the coupled system are provided.

Static and dynamic structural analyses for a 750 kN class liquid rocket engine with TVC actuation

During the development stage of a liquid rocket engine structural analyses are needed for increasing structural reliability and identify necessary design changes. These analyses need enormous efforts for detail modeling and cause large computational cost. With increasing computational power, the modeling can be more detailed than in the past. In this study, non-linear static structural and free vibration analyses taking into account the static loading as preloading are performed for a 750 kN class liquid rocket engine system. The finite element (FE) structural analyses are performed using the general purpose commercial code, Abaqus 2016. The effects of the static loadings on stress distributions and the free vibration results are analyzed. Additionally a simple kinematic model using rigid body elements is made up to investigate the elasticity effects on the rotation angles due to engine gimbaling. The strains measured from a ground firing test are converted to equivalent stresses to be compared with the FE analysis results. And a certain leakage problem solved using the FE analysis results is also presented.

Nonlinear structural behavior and vibration control of a double curved cable net under the dynamic excitations

The paper is regarding to the structural dynamics of a saddle shaped cable-net structure with two high points and two low points at the corners. The net cables are pre-stressed by stretching the four edge cables that are supported at the corner points. A series of parametric nonlinear dynamic analysis is utilized to evaluate the dynamic response of the system under the different structural masses and geometries, pretensions and alignments of cables and the amplitudes of dynamic excitation. An equivalent single-degree-of-freedom model is presented as a simplified method to estimate the dynamic response of the full finite element model. The simulation results show that the equivalent model can accurately estimate the displacement response of multi-degree-of-freedom model under dynamic excitation. Furthermore, a nonlinear active control algorithm is applied to decrease the structural response under the transient wind excitation and the results indicate the effectiveness of the proposed control algorithm.

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic parameters

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic distribution parameters is proposed. The partial differential equation with spatial varying parameters for nonlinear vibration of beams with periodic parameters under harmonic excitations is derived. The procedure of the multimode perturbation method includes three main steps: first, the nonlinear partial differential equation is transformed into linear partial differential equations with varying parameters by applying perturbation method; second, the linear partial differential equations are transformed into ordinary differential equations with multimode coupling by applying Galerkin method, where multiple vibration modes of the beams are used and the equations are suitable to nonlinear vibration of periodic structures with high parameter-varying wave in wide frequency band; third, the ordinary differential equations are solved by applying harmonic balance method to obtain vibration response of the nonlinear periodic beam, which is used for characteristics analysis of frequency response and spatial mode. Furthermore, the stability problem of nonlinear harmonic vibration as multidegree-of-freedom system with periodic time-varying parameters is solved by applying direct eigenvalue analysis approach. The proposed method can incorporate multiple vibration modes into response analysis of nonlinear periodic structures and consider mode-coupling effects due to structural nonlinearity and parametric periodicity. Finally, a nonlinear beam with periodic supports under harmonic excitations is studied. Numerical results on frequency response of the beam are given to illustrate an application of the proposed method, new frequency response characteristics, and influences of periodic parameters on structural response. The results have potential application to nonlinear structural vibration control and support damage detection of nonlinear structures with periodic supports.

Analytical Formulation of Friction Contact Elements for Frequency-Domain Analysis of Nonlinear Vibrations of Structures With High-Energy Rubs

Abstract In gas-turbine engines and other rotating machinery structures, rubbing contact interactions can occur when the contacting components have large relative motion between components: such as in rotating bladed disk-casing rubbing contacts, and rubbing in rotor bearing and labyrinth seals. The analysis of vibrations of structures with rubbing contacts requires the development of a mathematical model and special friction contact elements that would allow for the prescribed relative motion of rubbing surfaces in addition to the motion due to vibrations of the contacting components. In the proposed paper, the formulation of the friction contact elements is developed, which includes the effects of the prescribed relative motion on the friction stick-slip transitions and, therefore, on the contact interaction forces. For a first time, the formulation is made for the frequency domain analysis of coupled rubbing and vibrational motion, using the multiharmonic representation of the vibration displacements. The formulation is made fully analytically to express the multiharmonic contact interaction forces and multiharmonic tangent stiffness matrix in an explicit analytical form allowing their calculation accurately and fast. The dependency of the friction and contact stiffness coefficients on the energy dissipated during high-energy rubbing contacts and, hence, on the corresponding increase of the contact interface temperature is included in the formulation. The efficiency of the developed friction elements is demonstrated on a set of test cases including simple models and a large-scale realistic blade.

The response of a random vibrations of nonlinear structure under white-noise excitations

Random vibration is the phenomenon wherein random excitation applied to a mechanical induces random response. Random vibration is the most practical of the areas discussed in this paper. In this paper, the sistem is excited by a force W which is a random process described. A method is to replace the governing set of non-linear defferential equations by an equivalent set of linear equations the defference betwen the sets being minimized in somme appropiate sense. In this way, the parameters of the linearised system are determined.

Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances

The study of both linear and nonlinear structural vibrations routinely circles the concise yet complex problem of choosing a set of coordinates which yield simple equations of motion. In both experimental and mathematical methods, that choice is a difficult one because of measurement, computational, and interpretation difficulties. Often times, researchers choose to solve their problems in terms of linear, undamped mode shapes because they are easy to obtain; however, this is known to give rise to complicated phenomena such as mode coupling and internal resonance. This work considers the nature of mode coupling and internal resonance in systems containing non-proportional damping, linear detuning, and cubic nonlinearities through the method of multiple scales as well as instantaneous measures of effective damping. The energy decay observed in the structural modes is well approximated by the slow-flow equations in terms of the modal amplitudes, and it is shown how mode coupling enhances the damping observed in the system. Moreover, in the presence of a 3:1 internal resonance between two modes, the nonlinearities not only enhance the dissipation, but can allow for the exchange and transfer of energy between the resonant modes. However, this exchange depends on the resonant phase between the modes and is proportional to the energy in the lowest mode. The results of the analysis tie together interpretations used by both experimentalists and theoreticians to study such systems and provide a more concrete way to interpret these phenomena.