Nonlinear Dynamic Equilibrium Equations Research Articles

Theoretical dynamic reduction for steady-state nonlinear vibration responses of complex jointed structures having hysteretic contact behavior

A novel nonlinear dynamic reduction method was developed to determine the steady-state vibration responses of complex jointed structures having hysteretic contact behavior. By using the harmonic balance method to reformulate nonlinear dynamic equilibrium equations into a set of nonlinear algebraic ones, a dynamic reduction technique based on nonlinearity transformation was theoretically developed to iterate nonlinear vibration solutions in the local coordinate that was related only to the degree-of-freedoms of nonlinear joints. Only odd-order harmonic components were truncated to approximate the hysteretic nonlinear contact forces of joint interfaces, which was conducive to further dimension reduction of the nonlinear algebraic equations and iteration matrix. Then, a nonlinear dynamic reduction solver was developed to associate the steady-state nonlinear vibration responses of overall structures with the dynamic characteristics of underlying linear substructures, nonlinear joint models and external excitations. Combined with finite element analysis, the steady-state nonlinear vibration responses of a complex assembled structure with four reduced-order nonlinear joint models were numerically simulated to validate the proposed nonlinear dynamic reduction method by comparing with the literature. The comparative results of nonlinear frequency response functions exhibited good agreement, and the proposed method indicated higher computational efficiency. The experimental investigations of a rubber isolator system were also performed to validate the proposed method, which demonstrated good performance.

Research on One-to-Two Internal Resonance of Sling and Beam of Suspension Sling–Beam System

An approach is presented to investigate the 1:2 internal resonance of the sling and beam of a suspension sling–beam system. The beam was taken as the geometrically linear Euler beam, and the sling was considered to be geometrically nonlinear. The dynamic equilibrium equation of the structures was derived using the modal superposition method, the D’Alembert principle and the Hamilton principle. The nonlinear dynamic equilibrium equations of free vibration and forced oscillation were solved by the multiple-scales method. We derived the first approximation solutions for the single-modal motion of the system. Numerical examples are provided to verify the correctness of formula derivation and obtain the amplitude–time response of free vibration, the primary resonance response, the amplitude–frequency response, and the amplitude–force response of forced oscillation. According to the analysis, it is evident that the combination system exhibits robust nonlinear coupling properties due to the presence of internal resonance, which are useful for engineering design.

Nonlinear flutter of 2D variable stiffness curvilinear fibers composite laminates by a higher-order shear flexible beam theory with Poisson’s effect

In this work, the nonlinear supersonic panel flutter characteristics of two-dimensional variable stiffness curvilinear fibres based laminated composite panels are studied using a higher-order shear flexible theory represented by sine function coupled with first-order approximation leading to quasi-aerodynamic theory. The structural formation takes care of geometric nonlinearity with von Karman’s assumptions. The beam constitutive equation is modified for the laminated beam with general lay-up by accounting for Poisson’s effect. The nonlinear dynamic equilibrium equations developed by Lagrangian equations of motion are solved using finite element approach in conjunction with the direct iterative solution procedure. For limit cycle oscillation, critical dynamic pressure is predicted iteratively through eigenvalue analysis, thereby identifying the first coalescence of vibrational modes. Also, the flutter behavior of two-dimensional panel under static differential pressure is investigated considering nonlinear static equilibrium position of panel obtained by Newton-Raphson’s iterative approach and then followed by modes coalescence approach. These solution procedures are tested against the results in literature. A thorough numerical investigation is done to show the effect of the curvilinear fiber path orientation, limited cycle amplitude, static differential pressure, panel thickness, panel end condition flexibilities and thermal environment on the nonlinear supersonic panel flutter of two-dimensional variable stiffness laminated panels.

Nonlinear time-history earthquake analysis for steel frames

This study presents a simple, efficient, accurate method for nonlinear time-history earthquake analysis of spatial steel frames. The proposed new fiber plastic hinge method simulating only one element for each member captures the time-history dynamic behavior of steel frames as accurately as sophisticated plastic zone methods. Stability functions and the geometric stiffness matrix are employed for predicting the second-order effects that aim to minimize computational time and computer resources. Residual stresses are considered through two fiber plastic hinges by assigning initial stress values. A numerical integration procedure using Newmark integration combined with the Newton-Raphson balanced iteration algorithm is developed to find a solution to nonlinear dynamic equilibrium equations of the structural system. Shear deformation effects are also considered in the dynamic analysis. The proposed software, named Direct Advanced Analysis and Design (DAAD), is proved to be accurate and reliable as compared with the analysis results generated by expensive commercial Finite Element Analysis (FEA) software packages. The proposed DAAD program can be improved for performance-based direct design and analysis.

Thermoelastic analysis of stiffened sandwich doubly curved plate with FGM core under low velocity impact

In this paper, dynamic analysis of a stiffened doubly curved sandwich composite plate with a functionally graded material (FGM) core and two isotropic layers in thermal environment is presented. Based on von Kármán non-linear strain–displacement relationships and classical plate theory, a list of non-linear dynamic equilibrium equations for stiffened FGM sandwich doubly curved plates under impact in thermal field are established by using the Hamilton’s variation principle, then combining with boundary and initial conditions, the whole problem is solved by adopting the finite difference method, Newmark method and iterative method synthetically. To verify the accuracy of the present work, comparisons are made with previously published results. The detailed studies about the influence of prime parameters like initial impact velocity, material property, temperature boundary condition, geometry parameters of doubly curved plate and non-uniform stiffeners on nonlinear dynamic response of the structure subject to impact with thermal effect are carried out.

Nonlinear vibration of a deploying laminated Rayleigh beam with a spinning motion in hygrothermal environment

The free vibration of a deploying laminated beam with spinning motion in hygrothermal environment is studied. The model of this system is given within the framework of the Rayleigh beam theory and the von Karman nonlinear strain theory. The nonlinear dynamic equilibrium equation of the cantilever boundary condition is established based on Hamilton's principle with considering the combined effects of the axial motion, transverse vibration, spinning motion, and hygrothermal environment. The frequencies of the system are numerically determined by using the Newmark method. The detailed parameter analyses are provided to investigate the effects of the deploying speed, the spinning speed, the temperature variation, the moisture concentration, and the fiber direction angle on frequency and the corresponding sensitivity.

Experimental and numerical studies of square concrete-filled double steel tubular short columns under eccentric loading

Square concrete-filled double steel tubular (CFDST) beam-columns consisting of an internal circular steel tube have increasingly been utilized in composite building structures because of their high structural performance. This paper describes experimental and numerical studies on the structural responses of square thin-walled CFDST columns loaded eccentrically. Tests on twenty short square CFDST columns were undertaken that included sixteen columns under eccentric loading and four columns under concentric loading. The parameters examined in the experiments included the cross-sectional dimensions, the width-to-thickness ratios of outer and internal tubes and loading eccentricity. The measured ultimate strengths, load-shortening responses, load-lateral displacement curves, stress-strain curves and observed failure modes are presented. A numerical model incorporating the fiber analysis is developed that predicts the moment-curvature responses and axial load-moment strength envelops of CFDST columns. The model explicitly accounts for the influences of the confinement exerted by the internal circular steel tube on the core concrete and the progressive post-local buckling of the external steel tube. Efficient computer algorithms implementing the inverse quadratic method is developed to produce converged solutions to the nonlinear dynamic equilibrium equations generated in the analysis. Measurements from the tests are employed to validate the proposed numerical model. It is shown that there is a good agreement between theory and experiment. The computer model is utilized to demonstrate the significance of various parameters on the behavior of thin-walled short CFDST beam-columns.

Geometrically nonlinear dynamic analysis of FG-CNTRC plates subjected to blast loads using the weak form quadrature element method

Geometrically nonlinear dynamic analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) rectangular plates subjected to blast loads is conducted based on Reddy’s higher-order shear deformation theory using the weak form quadrature element method. The von-Kármán strain terms are introduced to consider the geometrically nonlinear effects. The polymer composite plate is reinforced by single-walled carbon nanotubes (SWCNTs) with the uniform and functionally graded distribution in the plate thickness direction. The effective material properties of the FG-CNTRC plates are estimated with the extended rule of mixture. The Newmark-β time integration scheme and the Newton–Raphson iteration technique are adopted to solve the nonlinear incremental dynamic equilibrium equation in temporal domain. Comparative and convergence studies are carried out to validate the accuracy, efficiency and numerical stability of the presented weak form quadrature element formulation. The effects of the carbon nanotube distribution and volume fraction, plate width-to-thickness ratio, plate aspect ratio, load type and boundary condition on the dynamic response of the FG-CNTRC plates under blast loads are systematically investigated through the parametric studies.

Vibration analysis of deploying laminated beams with generalized boundary conditions in hygrothermal environment

The free vibration of a deploying laminated beam in hygrothermal environment with a constant axial velocity is studied. The model of this system is given within the framework of the Euler-Bernoulli beam theory and von Karman nonlinear strain theory. The nonlinear dynamic equilibrium equation with generalized boundary conditions is established based on the Hamilton’s principle with considering the combined effects of the axial motion, transverse vibration and hygrothermal environment. Based on the Galerkin method, a set of ordinary differential equations is obtained. The numerical results of the discretization equation are performed adopting the eigenvalue method and Newmark method. In addition, the dynamic stability is discussed, and extensive numerical calculations are performed to illustrate the effects of varying extension velocities, temperature, humidity and ply angles on frequencies.

An incrementation-adaptive multi-transmitting boundary for seismic fracture analysis of concrete gravity dams

This study presents a numerical framework and methodology that features a nonlinear extended finite element formulation for earthquake fracture analysis of dam-reservoir-sediment-foundation coupled systems and an adaptive time incrementation multi-transmitting boundary for modeling seismic wave propagation in both infinite reservoir and foundation domains. To improve the convergence rate during the iterative solution of the highly nonlinear dynamic equilibrium equation, the original Multi-Transmitting Formula (MTF) is modified and coupled to the automatic incrementation time-stepping scheme through the introduction of a fictitious time scale in conjunction with an interpolation scheme that links the fictitious time scale to the real time scale. Meanwhile, the dynamic crack initiation and propagation in the body of the dam is represented by the extended finite element method including the effects of branching and intersecting cracks. As an application of the proposed formulation, the seismic cracking behavior of a representative concrete gravity dam system is analyzed.

Nonlinear dynamic responses of an axially moving laminated beam subjected to both blast and thermal loads

The nonlinear dynamic responses of an axially moving laminated beam subjected to a blast load in thermal environment is studied considering large-displacement. Firstly, the nonlinear dynamic equilibrium equation is established based on the large-displacement theory and the constitutive relation of the single layer material in thermal environment. Based on the Galerkin method, a set of ordinary differential equations is obtained. Secondly, the multiple scales method is adopted to get the nonlinear free vibration frequency. Then, the stability region of the axial velocity and temperature is derived and the truncation order is approximated by the convergence calculation of the natural frequencies. Finally, numerical calculations are performed to discuss the effects of different kinds of blast loads, axial velocity and temperature on the nonlinear dynamic responses adopting the Runge–Kutta technique.

Free vibration of summation and difference resonance of the vertical cable and other coupled structural members

Free vibration of summation and difference resonance of the vertical cable and other coupled structural members were investigated in this article. A model of a vertical cable and two mass–springs was built, with the sling considered to be geometrically nonlinear, and the upper and lower connecting structural members were taken as two mass–springs. Assuming the displacement of the sling, modal superposition method and D’Alembert principle were used to derive the dynamic equilibrium equations of the coupled structure. The nonlinear dynamic equilibrium equations were studied by means of multiple scales method, and the second-order approximation solutions of single-modal motion of the system were obtained. Numerical examples were presented to discuss the amplitude responses as functions of time of free vibration, with and without damping, respectively. Additionally, fourth-order Runge–Kutta method was directly used for the nonlinear dynamic equilibrium equations to complement and verify the analytical solutions. The results show that the coupled system performs strongly nonlinear and coupled characteristics, which is useful for engineering design.

Piezoelectric effect on transversal vibrations and buckling of a beam with varying cross section

In this study, the static and dynamic response of a system composed of an Euler-Bernoulli beam with axially restrained ends and a pair of piezo patches symmetrically bonded at a specified localization is investigated. The system is kinematically loaded as a result of the prescribed displacement of one or both supports. By applying an electric field to the piezo patches a residual in-plane stress is generated in the system. The residual force, depending on the direction of the electric field vector, may diminish or enhance the system buckling capacity as well as affecting its natural vibration frequency. In order to acquire approximate solutions to the non-linear dynamic equilibrium equation, a version of the Lindstedt-Poincare method is utilized. With this in mind, the transversal displacements, vibration frequency and axial dynamic force are expanded into exponential series with respect to the small amplitude parameter. The numerical results show the effect of the structural parameters and induced axial piezoelectric force on the stability of the system and its vibration frequency. The amplitude-frequency relationship of the actuated system is also investigated.

Thermoviscoelastic dynamic response for a rectangular steel plate under laser processing

The thermoviscoelastic dynamic response of a simply supported rectangular steel plate under laser processing is studied in this paper. Three-dimensional transient temperature field of the laser processing thermoviscoelastic rectangular steel plate is solved by the method of separation of variables. Meanwhile, based on von Karman nonlinear strain–displacement relationships and classical thin plate theory, a list of nonlinear dynamic equilibrium equations for a rectangular thermoviscoelastic steel plate are deduced. Then the entire problem is solved by utilizing the Newmark method and the iterative method. Finally, numerical results show that laser moving speed, laser power density and thickness-to-length ratio have a great influence on the thermoviscoelastic response of the rectangular DP980 steel plate.

Dynamic Analysis of Composite Laminated Circular Plate with Circular Delamination

AbstractIn this paper, the effect of delamination on free vibration and primary resonance behaviors of composite circular plate with circular delamination is investigated. Through Reissner Variational Principle, the nonlinear dynamic equilibrium equations, the generalized displacements continuity conditions and the generalized forces equilibrium conditions across delamination front and consistent boundary conditions of delaminated circular plate are obtained. In the work, by introducing Bessel Function and Modified Bessel Function and using Galerkin discretization method, the nonlinear dynamic partial differential equations of delaminated circular plate are transferred into a set of nonlinear ordinary differential equations. Then by using semi-analytic method and multiple scales method, the effects of delamination radius and delamination depth in the thickness-wise on the natural frequency and primary resonance behaviors of delaminated circular plate are presented. The Results show that delamination has considerable effects on the natural frequency and its primary resonance behaviors of delaminated plate.

Dynamic characteristics analysis of a misaligned rotor-bearing system with squeeze film dampers

In this paper, a dynamic model is established for a two-stage rotor system connected by a gear coupling and supported on ball bearings with squeeze film dampers (SFDs). The nonlinear dynamic behavior of the rotor system is studied under misalignment fault condition. The meshing force of the gear coupling is calculated considering the deformation of the tooth caused by torque transmission and dynamic vibration. The contact force between the ball and race is computed based on the Hertzian elastic contact deformation theory and the elastohydrodynamic lubrication theory. The supported force of SFD is simulated by integrating the pressure distribution derived from Reynolds’s equation. The equations of motion are rewritten in non-dimensional differential form, and the fourth-order Runge-Kutta method is employed to solve the nonlinear dynamic equilibrium equations iteratively. To verify the validity of the dynamic model and the correctness of the numerical solution method, the experimental power spectra of the rotor system under various misalignment degrees are compared with the analytical results. The effects of several important parameters, such as the lubrication of the ball bearing, the centralizing spring stiffness, the radial clearance of SFD, and the misalignment of gear coupling, on the dynamic characteristics of the rotor system are investigated and discussed mainly focusing on the system stability. The response spectra, bifurcation diagrams, and Pointcare maps are analyzed accordingly. These parametric analyses are very helpful in the development of a high-speed rotor system and provide a theoretical reference for the vibration control and optimal design of rotating machinery.

Thermoviscoelastic dynamic response for a composite material thin narrow strip

Based on von Karman nonlinear strain-displacement relationships and classical thin plate theory, a list of nonlinear dynamic equilibrium equations for a viscoelastic composite material thin narrow strip under thermal and mechanic loads are deduced. According to the material constitutive relationship and the relaxation modulus in the form of the Prony series, combing with the Newmark method and the Newton-cotes integration method, a new numerical algorithm for direct solving the whole problem in the time domain is established. By applying this numerical algorithm, the viscoelastic composite material thin narrow strip as the research subject is analyzed systematically, and its rich dynamical behaviors are revealed comprehensively. To verify the accuracy of the present work, a comparison is made with previously published results. Finally, the viscoelastic composite material thin narrow strip under harmonic excitation load and impact load are discussed in detail, and many valuable thermoviscoelastic dynamic characteristics are revealed.

Thermoviscoelastic Response for a Thin Narrow Composite Strip with Transverse Matrix Cracking

Based on von Kármán non-linear strain-displacement relationships and classical thin plate theory, a list of non-linear dynamic equilibrium equations for a thin narrow composite strip with transverse matrix cracking under thermal and mechanic loads are established and solved by the finite difference method, Newmark method, Newton-Cotes method and iterative method synthetically. Numerical examples show effect of the mechanical parameters, thermal environment, damage evolution and geometric parameters for thermoviscoelastic dynamic behavior of the thin narrow composite strip.

Dynamic damage criterion and damage mode for single layer lattice shell

When member buckling and joint fractures are considered in the numerical analyses of single layer lattice shells, the relationship curve between peak ground acceleration of the earthquake and the maximum nodal displacement, and the time–history curve of maximum nodal displacement, are erratic and unpredictable. Therefore, with such considerations, the current approach to determine dynamic damage of single layer lattice shells by comparing peak ground acceleration of the earthquake with critical ground acceleration derived from incremental dynamic analysis is inappropriate. An improved structure dynamic damage criterion is proposed for single layer lattice shells in this paper, which reviews the balance status of structure dynamic resistance against the earthquake action, and the structure damage time can be predicted by the occurrence of non-convergent solution to the structural nonlinear dynamic equilibrium equations in the iterative process from the mathematical point of view. Numerical examples are presented to illustrate the simplicity and practicality of the proposed criterion. This criterion serves clear physical meaning and is of considerable potential applicability in analyzing single layer lattice shell structures. Results of parametric analyses of single layer lattice shells under severe earthquake actions indicate that the structure dynamic damage is not determined by material strength failure. For single layer spherical lattice shell, it is determined by structural instability resulted from member buckling; and for single layer cylindrical lattice shell, it is determined by the combined effect of structural instability and the change of structural topology that resulted from member buckling and joint fractures respectively.

Nonlinear Time-History Analysis of 3-Dimensional Semi-rigid Steel Frames

This paper presents nonlinear elastic dynamic analysis of 3-D semi-rigid steel frames including geometric and connection nonlinearities. The geometric nonlinearity is considered by using stability functions and updating geometric stiffness matrix. The nonlinear behavior of the steel beam-to-column connection is considered by using a zero-length independent connection element comprising of six translational and rotational springs. The nonlinear dynamic equilibrium equations are solved by the Newmark numerical integration method. The nonlinear time-history analysis results are compared with those of previous studies and commercial SAP2000 software to verify the accuracy and efficiency of the proposed procedure. Keywords—Geometric nonlinearity, nonlinear time-history analysis, semi-rigid connection, stability functions.