Nonlinear Vibration Analysis Research Articles

A Volterra-PEM approach for random vibration spectrum analysis of nonlinear systems

In this paper, the Volterra series and the pseudo-excitation method (PEM) are combined to establish a frequency domain method for the power spectral density (PSD) analysis of random vibration of nonlinear systems. The explicit expression of the multi-dimensional power spectral density (MPSD) of the random vibration response is derived analytically. Furthermore, a fast calculation strategy from MPSD to physical PSD is given. The PSD characteristics analysis of the random vibration response of nonlinear systems is effectively achieved. First, within the framework of Volterra series theory, an improved PEM is established for MPSD analysis of nonlinear systems. As a generalized PEM for nonlinear random vibration analysis, the Volterra-PEM is used to analyse the response MPSD, which also has a very concise expression. Second, in the case of computation difficulties with multi-dimensional integration from MPSD to PSD, the computational efficiency is improved by converting the multi-dimensional integral into a matrix operation. Finally, as numerical examples, the Volterra-PEM is used to estimate the response PSD for stationary random vibration of a nonlinear spring-damped oscillator and a non-ideal boundary beam with geometrical nonlinearity. Compared with Monte Carlo simulation, the results show that by constructing generalized pseudo-excitation and matrix operation methods, Volterra-PEM can be used for input PSD with arbitrary energy distribution, not only restricted to broadband white noise excitation, and accurately predict the secondary resonance phenomenon of the random vibration response of nonlinear systems in the frequency domain.

A Cartesian spatial discretization method for nonlinear dynamic modeling and vibration analysis of tensegrity structures

For vibration analysis of a tensegrity structure, the development of a dynamic model is a key step. A common issue in the traditional dynamic modeling methods for vibration analysis of tensegrity structures is that structural members are oversimplified. Member internal displacements, including those in longitudinal directions for bar and cable members and those in transverse directions for cable members, were neglected. This oversimplification would inevitably prevent the dynamic model of a tensegrity structure so developed from revealing accurate responses, especially for those in the high-frequency domain. To resolve this issue, a new method called the Cartesian spatial discretization method is developed for nonlinear dynamic modeling and vibration analysis of tensegrity structures. This method can successfully incorporate member internal displacements in dynamic modeling of a tensegrity structure by defining positions of structural members as a summation of internal terms and boundary-induced terms in a global Cartesian coordinate system. The proposed method is applied to vibration analysis of a planar Snelson’s X tensegrity structure, a three-dimensional tensegrity tower, and an irregular tensegrity grid in simulation, and compared with the Lagrangian method based on generalized coordinates, the commercial finite element analysis software ANSYS and the finite element analysis method in literatures, respectively. Results show that the proposed method is accurate in predicting dynamic responses of tensegrity structures, especially for vibration analysis in the high-frequency domain. It is also demonstrated that the proposed method is applicable to both simple and complex tensegrity structures, and computationally efficient as it converges in a super-linear rate by using only a small number of internal terms of member displacements.

Nonlinear forced vibration analysis of micro-rotating shaft–disk systems through a formulation based on the nonlocal strain gradient theory

Nonlinear forced vibration analysis of micro-rotating shaft–disk systems through a formulation based on the nonlocal strain gradient theory

Nonlinear vibration analysis of CFRR sandwich doubly-curved shallow shells with a porous microcapsule coating in hygrothermal environment

In this paper, the nonlinear vibration characteristics and response of laminated carbon fiber reinforced resin (CFRR) doubly-curved shallow shell with a porous microcapsule coating under hygrothermal environment is investigated. Based on the first-order shear deformation theory, the nonlinear equations of motion are obtained by using Hamilton’s principle, which are solved by utilizing the direct integration method and Runge–Kutta method. The effect of temperature and moisture, geometric parameters of microcapsules, geometric parameters of the shell, laying angle for carbon fiber and porosity of coating matrix are considered. Through numerical examples, it is found that temperature, moisture, length of edges, laying angle for carbon fiber and porosity of coating matrix have a positive effect on the dimensionless nonlinear frequency ratio, but negative effect on the dimensionless linear and nonlinear natural frequency of the shell. In addition, temperature, moisture, length of edges, laying angle for carbon fiber and porosity of coating matrix are proportional to the dimensionless nonlinear amplitude of the shell.

Simplified Approach to Nonlinear Vibration Analysis of Variable Stiffness Plates

A formulation for the analysis of the nonlinear vibrations of Variable Stiffness (VS) plates is presented. Third-order Shear Deformation Theory (TSDT) is employed in conjunction with a mixed variational formulation. The solution is sought via Ritz approximation for the spatial dependency, while time dependency is handled via Differential Quadrature (DQ) and Harmonic Balance (HB) methods. The main advantage of the framework is the reduced computational time, which is of particular interest to explore the large design space offered by variable stiffness configurations. The results are validated against reference solutions from the literature. Exemplary parametric studies are presented to demonstrate the potential of the approach as a powerful means for exploring the nonlinear vibration response of VS plates.

Nonlinear dynamic modeling and vibration analysis of whole machine tool

The machine tool structure vibration during machining operation is directly linked to machining quality, production efficiency as well as machine reliability. Hence, this research focuses on the dynamic modeling of the whole machine tool considering the milling process to predict vibration response and dynamic characteristics during machining. Applying the lumped parameter and finite element hybrid method, in order to improve virtual simulation efficiency, the analytical models of the workpiece clamping worktable system and the tool clamping column spindle system are theoretically developed considering the nonlinearity of moving joints and the position-dependent dynamics. The integrated model of the whole machine tool dynamics is established by coupling the models of two subsystems with the generated nonlinear dynamic load in the ball end milling process including the effects of vibrations and tool system errors. Subsequently, the frequency response functions and the instantaneous cutting forces and vibration responses under different cutting conditions are measured and compared with simulation results to evaluate the prediction ability of the proposed model. Finally, the experimentally confirmed model is used to carry out the parameter analysis of machine tool. The investigations reveal that the worktable system dynamic characteristics vary with the joint assembly preload, but the change becomes no longer significant when the preload increases to a certain value. Besides, cutting conditions have a significant influence on the vibration characteristics of machine tool during machining.

Nonlinear vibration analysis of sandwich plates with honeycomb core and graphene nanoplatelet-reinforced face-sheets

Nonlinear vibration analysis of sandwich plates with honeycomb core and graphene nanoplatelet-reinforced face-sheets

A novel nonlinear vibration feature-based approach for evaluating bolt-loosening faults in aerospace structures

Aerospace structures such as aircraft and satellites are frequently subjected to bolt-loosening faults in service, which will seriously affect structural integrity, reliability and safety. In this paper, the development of a novel approach based on nonlinear vibration features is presented to evaluate potential bolt-loosening faults in these structures. Firstly, the complex aerospace structure is decomposed into some simple T-type substructures based on structural configurations and features. Then, a general T-type multi-degree-of-freedom (MDOF) model simulating bolt-loosening faults and nonlinear boundaries as nonlinear damper-spring components is built for nonlinear vibration analysis. After that, two novel fault features, which are functions of structural properties, nonlinear output spectra and bolt-loosening fault-induced forces, are defined to form a novel bolt-loosening fault indicator. Finally, a novel approach with detailed operational procedures is proposed, and its effectiveness and superiority are verified through experimental studies on a simple lab structure with bolt-loosening faults and nonlinear boundaries.

Performance evaluation for the domain decomposition method in nonlinear vibration of the composite hard-coating cylindrical shell.

The applications of the modified domain decomposition method in nonlinear vibration analysis of the composite hard-coating cylindrical shells are still at a relatively superficial level, owing to the fact that its performance under different decomposition parameters has not been thoroughly investigated for achieving sufficient precision. A parametric domain decomposition method is developed to facilitate self-performance evaluation in nonlinear vibration analysis of the shell. Correspondingly, in order to avoid a mass of redundant computation of the segment stiffness and material damping matrices during iterations, a specialized preprocessing scheme is designed by pre-establishing the parametric analytical expressions and matrix databases. The resonant response is sensitive to the circumferential segment number, but weakly affected by the axial segment number. The optimum circumferential segment number in the present study is suggested to be Nθ = 70, which can achieve good calculation accuracy and efficiency. Highly consistency is shown for the distributions of axial equivalent strain under different axial segment numbers. Smaller circumferential segment numbers would result in larger equivalent strain and bad solution accuracy. The sufficient solution accuracy of nonlinear vibration of the composite hard-coating cylindrical shell can't be achieved by increasing the axial segment number with constant segment width, but only by enough circumferential segment number, which is fundamentally determined by its equivalent strain distributions and gradients, and is with close relation to the axial and circumferential wave numbers of the shell.

Curved flexoelectric and piezoelectric micro-beams for nonlinear vibration analysis of energy harvesting

Micro/nano-scale energy harvester is an enabling technology to provide sustained energy solutions for various micro/nano-electromechanical devices, this has attracted intensive research interests in both academia and industry in the past decades. Exploring various design possibilities for energy harvesters in terms of geometry, topology, materials as well as harnessing the interesting nonlinear responses have the potential to break the current limit of energy harvesters in terms of energy efficiency and power density. While most designs were limited to straight beams, in this work, we demonstrate that the nonlinear vibration of energy harvester is significantly affected by curved shapes and we show the possibility of improving its performance by tuning the structural curvature. By developing a couple stress-based piezo-flexoelectric curved beam model and using the multiple time scale method for the nonlinear frequency response analysis, the effect of structural curvature in the nanoscale arc-shaped beam is quantitatively evaluated.

Вимушені коливання тришарової оболонки подвійної кривизни з пружним стільниковим заповнювачем

This paper presents a mathematical model of vibrations of a three-layered double-curved shell under geometrically nonlinear deformation. The middle layer is a honeycomb manufactured using FDM additive technologies. The mechanical properties of the honeycomb were assessed by a homogenization procedure. The outer layers of the shell are thin, and they are made of carbon-filled plastic. The model is based on a higher-order shear theory and accounts for the orthotropy of the mechanical properties of all the shell layers. Each layer of the shell is described by five variables (three displacement projections and two rotation angles of the normal to the middle surface). The properties of linear vibrations were studied using discretization by the Rayleigh?Ritz method. Because the middle layer of the shell is far lighter and more compliant in comparison with the outer layers, the computational process has some features. The eigenferquencies and eigenmodes of the shell were found for a further analysis of nonlinear vibrations. The mathematical model of forced vibrations of the shell under geometrically nonlinear deformation is a system of nonlinear ordinary differential equations derived by the assumed-mode method. Nonlinear periodic vibrations and their bifurcations were studied using a numerical procedure, which is a combination of the continuation method and the shooting technique. The properties of the nonlinear periodic vibrations and their bifurcations in the regions of fundamental and subharmonic resonances were studied numerically. A spherical panel and a hyperbolic paraboloid panel were considered. It was shown that when a disturbing force is applied at a point out of the panel’s center of gravity, the panel’s eigenmodes interact, and the frequency response and the bifurcation diagram change qualitatively in comparison with the case where that force is applied at the panel’s center of gravity. An agreement between the results was studied as a function of the number of terms in the Rayleigh-Ritz and assumed-mode expansions.

Transverse forced nonlinear vibration analysis of a double-beam system with a supporting nonlinearity

To exploit the potential application of supporting nonlinearity in marine engineering, an attempt is made to establish the transverse forced vibration analysis model of a double-beam system supported by a spring-mass system that is nonlinear. This kind of vibration system consists of two beam sections, boundary supports, a coupling component, and a nonlinear spring-mass arrangement. The variational approach and the generalized Hamiltonian concept are used to develop the governing equations of such a double-beam system. The Galerkin truncation method (GTM) is a technique for obtaining the governing equations’ residual equations. By solving the associated residual equations numerically, the nonlinear responses of the double-beam system can be figured out. The GTM has good solidity and correctness in the prediction of the vibration system’s forced transverse vibration. The dynamic responses of the double-beam structure supported by a spring-mass system that is nonlinear are subtle to their initial calculation values. Appropriate parameters of the nonlinear support will subdue the level of vibration at the boundary of the double-beam system. In contrast, unsuitable parameters of the nonlinear support motivate complex dynamic responses of the double-beam system and harmfully influence the vibration repression at the boundary of the vibration system.

Nonlinear vibration analysis and stability analysis of rotor systems with multiple localized nonlinearities

This study proposes a methodology to analyze the nonlinear vibration characteristics of rotor systems with multiple localized nonlinearities adopting the Finite Element Method (FEM), free interface Component Mode Synthesis (CMS) method, and modified Incremental Harmonic Balance (IHB) method. The rotor system is supported by squeeze film dampers (SFDs) on both sides, and at the nodes of the SFD arrangement, strong local nonlinearities will appear due to fluid-film forces. The methodology to analyze the nonlinear vibration characteristics of the system by reducing the degree of freedom of the rotating system with multiple local nonlinear factors and combining with the IHB method is proposed for the first time in this paper. The FEM is used to write motion equations in components, and the CMS method is applied to reduce the degrees of freedom of linear components. The IHB method is used to solve the motion equations of the nonlinear system. The system has one linear component and two nonlinear components. For linear components, modal coordinates are used, and for nonlinear components, the original physical coordinate system is used. By synthesizing these three components, the motion equation of the whole system is created. In order to validate the effectiveness of the method, the results obtained by the proposed method are compared with the data in the published literature, and the system responses are considered when specific parameters are changed. The stability analysis of the calculated solutions is carried out using the Floquet theory.

Analysis of Nonlinear Vibration of Functionally Graded Simply Supported Fluid-Conveying Microtubes Subjected to Transverse Excitation Loads.

This paper presents a nonlinear vibration analysis of functionally graded simply supported fluid-conveying microtubes subjected to transverse excitation loads. The development of the nonlinear equation of motion is based on the Euler-Bernoulli theory, Hamilton principle and Strain gradient theory. The nonlinear equation of motion is reduced to a second-order nonlinear ordinary differential equation by the Galerkin method. The Runge-Kutta method is adapted to solve the equation, and the effects of the dimensionless microscale parameters, the amplitude and frequency of excitation loads on the stability of the microtubes system are analyzed. It is found that when the microtube diameter is equal to the material length scale parameter, the microtube movement pattern is quasi-periodic. With the increase of the dimensionless microscale parameter, the microtube movement changes from quasi-periodic to chaos. The smaller the power-law index of volume fraction, the smaller the vibration displacement of microtubes and the better the stability. The larger the amplitude of excitation loads is, the larger the vibration displacement of the microtubes will be. When the frequency of excitation loads is equal to the natural frequency of the microtubes, it will have resonance and the vibration displacement will increase significantly.

Nonlinear vibration analysis of carbon fiber-reinforced composites with frequency-dependence and strain-dependence: Experimental and theoretical studies

In this paper, with 3 kinds of carbon fiber unidirectional composite beams as test specimens, a test system based on the resonance method is established to test both the frequency-dependence and strain-dependence for carbon fiber-reinforced composites. The test captures the variations of relevant mechanical parameters for carbon fiber-reinforced composites with frequency and strain under vibration load. The expressions of frequency-dependence and strain-dependence for relevant mechanical parameters obtained by testing data processing and numerical fitting are introduced into the numerical models for 2 kinds of carbon fiber-reinforced composite plates with different fiber alternating laying modes. The comparison of the results from the numerical analysis and the corresponding experimental tests verifies the correctness and applicability for the expressions describing the two dependency characteristics. Obviously, clarifying the laws of the mechanical parameters of carbon fiber-reinforced composites varying with frequency and strain has guiding significance for the design of carbon fiber-reinforced composite structures subjected to vibration loads.

Periodic motion of microscale cantilevered fluid-conveying pipes with symmetric breaking on the cross-section

A new theoretical model is developed for the three-dimensional (3D) nonlinear vibration analysis of cantilevered fluid-conveying micropipes with asymmetric cross-section. Fluid-conveying microscale pipes with ring-shaped cross-sections of O(2) symmetry are widely used in engineering. However, their cross-sections usually have a small symmetric breaking owing to manufacturing errors, which can be described by a small parameter. Hence, the dynamic behaviors of such geometrically imperfect micropipes are extremely required to be explored. This study mainly investigates the effects of the symmetric breaking parameter, flow velocity and dimensionless material length scale parameter on the dynamic behaviors of microscale cantilevered fluid-conveying micropipes. Based on the modified couple stress theory (MCST) and Hamilton's principle, the governing equation is derived, where the effects of the asymmetry of cross-section on the equation are reflected. Using the center manifold theory and normal form method, the original governing equation is rigorously reduced to a two-degree-of-freedom (2DOF) dynamical system, and the averaging equation of the reduced system is obtained by combining the averaging method. The existence and stability of the equilibrium point of the averaging equation are studied to obtain the results of the existence and stability of periodic motions of the original system. The results demonstrate that the types of periodic motions and stability when the pipe loses stability from different places on the curve of the critical flow velocity versus mass ratio (critical flow velocity curve) may differ. Meanwhile, the effects of various parameters on these periodic motions are presented. The increase in the dimensionless material length scale parameter increases the section corresponding to stable plane periodic motion and reduces the section corresponding to stable spatial periodic motion on the critical flow velocity curve. When dimensionless material length scale parameter is given, the existence and stability of various periodic motions of the system depend on the ratio of the increment of flow velocity around the critical value to the symmetric breaking parameter. In addition, some new phenomena are found in this paper. Stable 0 planar periodic motion of the macropipe can occur when pipe loses its stability from the hysteresis part of the critical flow velocity curve. Stable 0 planar periodic motion of microscale pipes can occur when pipe loses its stability from the non-hysteresis part of the critical flow velocity curve, and then stable 0 planar periodic motion and stable π planar periodic motion can coexist. Furthermore, stable planar and stable spatial periodic motions of pipes with symmetric breaking can coexist, and stable torus motion has also been observed; this is different from the dynamics of pipes with O(2) symmetry.

Nonlinear vibration analysis of meshing gear wheels considering the teeth elasticity

In this study, the nonlinear vibration of a meshing gear pair with the teeth elasticity is investigated. The elastic elements between the teeth and the body of each gear wheel are introduced to partially incorporate the teeth elasticity in the model. This model was originally considered by the second author as a rigid-elastic modeling approach of meshing gear wheels, which is well-suited for multibody systems. The contact analysis of the meshing teeth is performed using the Kelvin-Voigt method. By deriving the equations of motion for the mentioned model, the results of the numerical solution are presented. The method of multiple scales is utilized here for the first time to analytically analyze the nonlinear vibration of a gear system in which the teeth elasticity is partially incorporated by considering identical elastic elements between the teeth and the gear body. In this study, the effect of system parameters on the frequency response including the primary, sub-harmonic, and super-harmonic resonances is investigated. The results of the analytical solution show that the effect of external excitation amplitude and intensity on the system response amplitude is the same in the primary and sub-harmonic resonances. The results obtained numerically and analytically for the model have an acceptable agreement.

Nonlinear dynamic modeling and vibration analysis of bearing system considering shaft current damage

Based on the structure of damaged bearing, a nonlinear dynamic model is proposed. The nonlinear vibration response is obtained by solving the dynamic equation of the bearing system. Firstly, the influence of radial load and rotational speed on the nonlinear vibration response of the bearing system is analyzed. Then, the different vibration characteristics of the system when the inner and outer raceways of the bearing are damaged are discussed. Finally, the influence of the degree of shaft current damage on the nonlinear characteristics of the bearing system is studied. The results indicate that the bearing system presents bifurcation, jump, and chaos with the change of radial load. With the increase of the rotational speed, the path of the bearing system from chaos to quasi-periodic changes. With the increase of the degree of shaft current damage, the system gradually transitions from a stable state to a chaotic state.

Nonlinear torsional vibration analysis of shearer semi-direct drive cutting transmission system subjected to multi-frequency load excitation

Nonlinear torsional vibration analysis of shearer semi-direct drive cutting transmission system subjected to multi-frequency load excitation

Homotopy solution for nonlinear vibration analysis of multilayer graphene platelets-reinforced thin-walled pipes conveying fluid with rectangular cross-section

Homotopy solution for nonlinear vibration analysis of multilayer graphene platelets-reinforced thin-walled pipes conveying fluid with rectangular cross-section