Direct Time Integration Research Articles

Nonlinear dynamics of cantilevered extensible pipes conveying fluid

In this paper the nonlinear planar dynamics of a fluid-conveying cantilevered pipe is investigated. The centreline of the pipe is considered to be extensible; i.e., coupled longitudinal and transverse displacements are considered. The extended version of the Lagrange equations for systems containing non-material volumes is employed to derive the equations of motion, resulting directly in a set of coupled nonlinear ordinary differential equations. The pseudo-arclength continuation technique along with direct time integration are used to solve these equations. Bifurcation diagrams of the system are constructed as the flow velocity is increased; these diagrams are supplemented by time traces, phase-plane portraits, and fast Fourier transforms for some sets of system parameters. As opposed to the case of an inextensible pipe, an extensible pipe elongates in the axial direction as the flow velocity is increased from zero; depending on the system parameters, this static elongation can be considerable. At the critical flow velocity, the system loses stability via a supercritical Hopf bifurcation, emerging from the trivial solution for the transverse displacement and leading to a flutter.

In-plane and out-of-plane nonlinear dynamics of an axially moving beam

In the present study, the nonlinear forced dynamics of an axially moving beam is investigated numerically taking into account the in-plane and out-of-plane motions. The nonlinear partial differential equations governing the motion of the system are derived via Hamilton’s principle. The Galerkin scheme is then introduced to these partial differential equations yielding a set of second-order nonlinear ordinary differential equations with coupled terms. This set is transformed into a new set of first-order nonlinear ordinary differential equations by means of a change of variables. A direct time integration technique is conducted upon the new set of equations resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are investigated for different system parameters and presented through use of time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms.

Global dynamics of an axially moving buckled beam

A parametric study for post-buckling analysis of an axially moving beam is conducted considering four different axial speeds in the supercritical regime. At critical speed, the trivial equilibrium configuration of this conservative system becomes unstable and the system diverges to a new non-trivial equilibrium configuration via a pitchfork bifurcation. Post-buckling analysis is conducted considering the system undergoing a transverse harmonic excitation. In order to obtain the equations of motion about the buckled state, first the equation of motion about the trivial equilibrium position is obtained and then transformed to the new coordinate, i.e. post-buckling configuration. The equations are then discretized using the Galerkin scheme, resulting in a set of nonlinear ordinary differential equations. Using direct time integration, the global dynamics of the system is obtained and shown by means of bifurcation diagrams of Poincaré maps. Other plots such as time traces, phase-plane diagrams, and Poincaré sections are also presented to analyze the dynamics more precisely.

Nonlinear stability and bifurcations of an axially accelerating beam with an intermediate spring-support

The present work aims at investigating the nonlinear dynamics, bifurcations, and stability of an axially accelerating beam with an intermediate spring-support. The problem of a parametrically excited system is addressed for the gyroscopic system. A geometric nonlinearity due to mid-plane stretching is considered and Hamilton's principle is employed to derive the nonlinear equation of motion. The equation is then reduced into a set of nonlinear ordinary differential equations with coupled terms via Galerkin's method. For the system in the sub-critical speed regime, the pseudo-arclength continuation technique is employed to plot the frequency-response curves. The results are presented for the system with and without a three-to-one internal resonance between the first two transverse modes. Also, the global dynamics of the system is investigated using direct time integration of the discretized equations. The mean axial speed and the amplitude of speed variations are varied as the bifurcation parameters and the bifurcation diagrams of Poincare maps are constructed.

Non-linear global dynamics of an axially moving plate

In the present study, the geometrically non-linear dynamics of an axially moving plate is examined by constructing the bifurcation diagrams of Poincaré maps for the system in the sub and supercritical regimes. The von Kármán plate theory is employed to model the system by retaining in-plane displacements and inertia. The governing equations of motion of this gyroscopic system are obtained based on an energy method by means of the Lagrange equations which yields a set of second-order non-linear ordinary differential equations with coupled terms. A change of variables is employed to transform this set into a set of first-order non-linear ordinary differential equations. The resulting equations are solved using direct time integration, yielding time-varying generalized coordinates for the in-plane and out-of-plane motions. From these time histories, the bifurcation diagrams of Poincaré maps, phase-plane portraits, and Poincaré sections are constructed at points of interest in the parameter space for both the axial speed regimes.

A new direct time integration method for the equations of motion in structural dynamics

AbstractA direct time integration method is presented for the solution of the equations of motion describing the dynamic response of structural linear and nonlinear multi‐degree‐of‐freedom systems. It applies also to large systems of second order differential equations with fully populated, non symmetric coefficient matrices as well as to equations with variable coefficients. The proposed method is based on the concept of the analog equation, which converts the coupled N equations into a set of single term uncoupled second order ordinary quasi‐static differential equations under appropriate fictitious loads, unknown in the first instance. The fictitious loads are established from the integral representation of the solution of the substitute single term equations. The method is simple to implement. It is self starting, unconditionally stable and accurate and conserves energy. It performs well when large deformations and long time durations are considered and it can be used as a practical method for integration of the equations of motion in cases where widely used time integration procedures, e.g. Newmark's, become unstable. Several examples are presented, which demonstrate the efficiency of the method. The method can be straightforward extended to evolution equations of order higher than two.

Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support

In this paper, the nonlinear forced vibrations and stability of an axially moving Timoshenko beam with an intra-span spring-support are investigated numerically. Taking into account the shear deformation and rotary inertia, three coupled nonlinear partial differential equations of motion are obtained using Hamilton's principle along with stress–strain relations. These equations are discretized into a set of coupled nonlinear ordinary differential equations via the Galerkin method. The pseudo-arclength continuation technique is used to solve the governing set of equations. The frequency–response curves of the system in the subcritical regime are obtained and examined. In addition, direct time integration is employed to investigate the global dynamics of the system through constructing the bifurcation diagrams of Poincaré maps.

Parametric Stability and Bifurcations of Axially Moving Viscoelastic Beams with Time-Dependent Axial Speed#

The subject of this article is the investigation of the transverse vibration response, stability, and bifurcations of an axially moving viscoelastic beam with time-dependent axial speed. The force and moment balances as well as constitutive relations are employed to derive the equation of motion. Due to the presence of the time-dependent axial speed and steady dissipation terms, time-dependent coefficients and nonlinear dissipation terms are generated, respectively. The equation of motion is reduced into a set of coupled nonlinear ordinary differential equations with time-dependent coefficients. The subcritical resonant response of the system is obtained using the pseudo-arclength continuation technique, while the bifurcation diagrams of Poincaré maps are obtained via direct time integration of the discretized equations of motion.

Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).

Nonlinear dynamics of an axially moving Timoshenko beam with an internal resonance

This paper investigates the nonlinear forced dynamics of an axially moving Timoshenko beam. Taking into account rotary inertia and shear deformation, the equations of motion are obtained through use of constitutive relations and Hamilton’s principle. The two coupled nonlinear partial differential equations are discretized into a set of nonlinear ordinary differential equations via Galerkin’s scheme. The set is solved by means of the pseudo-arclength continuation technique and direct time integration. Specifically, the frequency-response curves of the system in the subcritical regime are obtained via the pseudo-arclength continuation technique; the bifurcation diagrams of Poincare maps are obtained by means of direct time integration of the discretized equations. The resonant response is examined, for the cases when the system possesses a three-to-one internal resonance and when not. Results are shown through time traces, phase-plane portraits, and fast Fourier transforms (FFTs). The results indicate that the system displays a wide variety of rich dynamics.

High-Dimensional Harmonic Balance Analysis for a Turning Process with State-Dependent Delay

The present study considers the state-dependent delay differential equations (SD-DDEs) for the turning process. In general, series expansion of the SD-DDEs turning system is essential in the nonlinear analysis such as the conventional methods of multiple scales and harmonic balance. Unfortunately, the mathematical theory of SD-DDEs, especially for those with an implicit function of delay, was just recently developed and any rigorous mathematical theory has not yet been proven. As one approach for the nonlinear analysis of the SD-DDEs, physically reasonable results could be obtained by extending the general theory of DDEs to the SD-DDEs through the use of the series expansion in conjunction with the implicit function, although there still remains an open issue of its mathematical rigorousness. The other approach may be treating the original SD-DDEs directly. To this end, the high-dimensional harmonic balance (HDHB) analysis is performed in this study in order to investigate the nonlinear behaviors of the turning system in the form of SD-DDEs without its series expansion. The results obtained by HDHB analysis are validated by comparing results with those of direct time integration. Using the resulting bifurcation diagrams, nonlinear chatter behaviors of the turning system are examined and discussed.

Post-buckling bifurcations and stability of high-speed axially moving beams

The nonlinear forced dynamics of an axially moving beam in the supercritical speed regime is numerically investigated in this paper, with special consideration to the case possessing an internal resonance between the first two modes. At critical speed, the initial equilibrium configuration of the beam becomes unstable and a new stable non-trivial equilibrium solution together with its symmetric counterpart arises. A closed-form solution for the post-buckling configuration is introduced and the system is assumed to be subject to a transverse harmonic excitation load at its buckled state. The equation of motion is cast into new coordinates which gives the equation governing the motion of the beam about the buckled state. This equation is discretized via the Galerkin method which yields a set of nonlinear ordinary differential equations (NODEs) with quadratic and cubic nonlinear terms. The set of NODEs are solved either via the pseudo-arclength continuation technique or by means of direct time integration so as to obtain frequency–response curves as well as bifurcation diagrams of Poincaré sections about the buckled state. Results are shown through time histories, phase-plane diagrams, and Poincaré sections.

Thermal effects on nonlinear vibrations of an axially moving beam with an intermediate spring-mass support

The thermo-mechanical nonlinear vibrations and stability of a hinged-hinged axially moving beam, additionally sup- ported by a nonlinear spring-mass support are examined via two numerical techniques. The system is subjected to a transverse harmonic excitation force as well as a thermal loading. Hamilton's principle is employed to derive the equations of motion; it is discretized into a multi-degree-freedom system by means of the Galerkin method. The steady state resonant response of the system for both cases with and without an internal resonance between the first two modes is examined via the pseudo-arclength continuation technique. In the second method, direct time integration is employed to construct bifurcation diagrams of Poincare maps of the system.

Thermal Effects on Nonlinear Vibrations of an Axially Moving Beam with an Intermediate Spring-Mass Support

The thermo-mechanical nonlinear vibrations and stability of a hinged-hinged axially moving beam, additionally supported by a nonlinear spring-mass support are examined via two numerical techniques. The system is subjected to a transverse harmonic excitation force as well as a thermal loading. Hamilton's principle is employed to derive the equations of motion; it is discretized into a multi-degree-freedom system by means of the Galerkin method. The steady state resonant response of the system for both cases with and without an internal resonance between the first two modes is examined via the pseudo-arclength continuation technique. In the second method, direct time integration is employed to construct bifurcation diagrams of Poincaré maps of the system.

Application of the Laplace transformation for the analysis of viscoelastic composite laminates based on equivalent single-layer theories

In this study, the linear viscoelastic response of a rectangular laminated plate is investigated. The viscoelastic properties, expressed by two basic spring-dashpot models, that is Kelvin and Maxwell models, is assumed in the range to investigate the influence of viscoelastic coefficients to mechanical behavior. In the present study, viscoelastic responses are performed for two popular equivalent single-layered theories, such as the first-order shear deformation theory (FSDT) and third-order shear deformation theory (TSDT). Compliance and relaxation modulus of time-dependent viscoelastic behavior are approximately determined by Prony series. The constitutive equation for linear viscoelastic material as the Boltzmann superposition integral equation is simplified by the convolution theorem of Laplace transformation to avoid direct time integration as well as to improve both accuracy and computational efficiency. The viscoelastic responses of composite laminates in the real time domain are obtained by applying the inverse Laplace transformation. The numerical results of viscoelastic phenomena such as creep, cyclic creep and recovery creep are presented.

Seismic performance assessment of highway bridges equipped with superelastic shape memory alloy-based laminated rubber isolation bearing

Seismic performance analysis is conducted for an isolated three-span continuous highway bridge, which is subjected to moderate to strong earthquake ground accelerations in the longitudinal direction. Two types of isolation bearings are used in the analysis: high damping rubber bearing (HDRB) and combined isolation bearing consisting of shape memory alloy (SMA) wires and natural rubber bearing (NRB) entitled as SMA-based rubber bearing (SRB). Two types of SMA wires, such as Ni–Ti and Cu–Al–Be, are used in fabricating the combined isolation bearings. In the first step of the work, analytical models for HDRB, NRB and SRB are introduced. Then, a three-span continuous highway bridge is modeled in a simplified form, using 2-DOF bridge pier-bearing system isolated by either HDRBs or SRBs. The hysteretic behavior of HDRB is evaluated using a strain-rate dependent constitutive model (i.e. visco-elasto-plastic model), while in the case of SRBs the hysteretic behavior is modeled by a nonlinear elasto-plastic model for NRB and a simplified visco-elastic model for SMAs. A standard bilinear force–displacement relationship is employed in the analytical model of the bridge pier to consider its nonlinear characteristic behavior. Nonlinear dynamic analysis of the bridge, based on the direct time integration approach, is conducted to evaluate the seismic responses of the bridge. This study shows that the seismic responses of the bridge are affected by the use of different types of isolation bearings; more specifically, residual displacement of the deck are noticeably reduced after earthquakes in the case of SRBs compared to HDRB for moderate and strong earthquakes; however, pier displacements are smaller in the cases of SRBs for moderate earthquakes and higher for strong earthquakes. Other response parameters of the system, such as deck displacement, bearing displacement and deck acceleration, are significantly larger in the cases of SRBs compared to those of HDRB. This study also depicts the effect of modeling of isolation bearings on the seismic responses of the system.

Coupled global dynamics of an axially moving viscoelastic beam

The nonlinear global forced dynamics of an axially moving viscoelastic beam, while both longitudinal and transverse displacements are taken into account, is examined employing a numerical technique. The equations of motion are derived using Newton′s second law of motion, resulting in two partial differential equations for the longitudinal and transverse motions. A two-parameter rheological Kelvin–Voigt energy dissipation mechanism is employed for the viscoelastic structural model, in which the material, not partial, time derivative is used in the viscoelastic constitutive relations; this gives additional terms due to the simultaneous presence of the material damping and the axial speed. The equations of motion for both longitudinal and transverse motions are then discretized via Galerkin’s method, in which the eigenfunctions for the transverse motion of a hinged-hinged linear stationary beam are chosen as the basis functions. The subsequent set of nonlinear ordinary equations is solved numerically by means of the direct time integration via modified Rosenbrock method, resulting in the bifurcation diagrams of Poincaré maps. The results are also presented in the form of time histories, phase-plane portraits, and fast Fourier transform (FFTs) for specific sets of parameters.

Non-linear dynamic instability of functionally graded plates in thermal environments

Geometrically non-linear parametric instability of functionally graded rectangular plates in thermal environments is investigated via a multi-degree-of-freedom energy approach. Non-linear higher-order shear deformation theory is used and the non-linear response to in-plane static and harmonic excitation in the frequency neighborhood of twice the fundamental mode is investigated. The boundary conditions are assumed to be simply supported movable. Numerical analyses are conducted by means of pseudo arc-length continuation and collocation technique to obtain force–amplitude relations in the presence of temperature variation in the thickness direction. The effect of volume fraction exponent and temperature variation on the onset of instability for both static and periodic in-plane excitation are fully discussed and the post-critical non-linear responses are obtained. Moreover, direct time integration of equations of motion is carried out and bifurcation diagrams, phase-space plots, Poincaré maps and time histories are obtained showing complex non-linear dynamics through period-doubling and Neimark–Sacker bifurcations.

Active vibration control of smart composite structures in hygrothermal environment

The composite materials may be exposed to environmental (thermal or hygral or both) condition during their service life. The effect of environmental condition is usually adverse from the point of view of design of composite structures. In the present research study the effect of hygrothermal condition on the design of laminated composite structures is investigated. The active fiber composite (AFC) which may be utilized as actuator or sensor is considered in the present analysis. The sensor layer is used to sense the level of response of the composite structures. The sensed voltage is fed back to the actuator through the controller. In this study both displacement and velocity feedback controllers are employed to reduce the response of the composite laminate within acceptable limit. The Newmark direct time integration scheme is employed along with modal superposition method to improve the computational efficiency. It is observed from the numerical study that the laminated composite structures become weak in the presence of hygrothermal load. The response of the structure can be brought to the acceptable level once the AFC layer is activated through the feedback loop.

Approach for Selection of Rayleigh Damping Parameters Used for Time History Analysis

Nonlinearities, whether geometric or material, need to be addressed in seismic analysis. One good analysis method that can address these nonlinearities is direct time integration with Rayleigh damping. Modal damping is the damping typically specified in seismic analysis Codes and Standards (ASCE 4-98, 1998, “Seismic Analysis of Safety-Related Nuclear Structures and Commentary,” American Society of Civil Engineers, Reston, Virginia and ASCE/SEI 43-05, 2005, “Seismic Design Criteria for Structures, Systems, and Components in Nuclear Facilities,” American Society of Civil Engineers, Reston, Virginia.). Modal damping is constant for all frequencies where Rayleigh damping varies with frequency. An approach is proposed here for selection of Rayleigh damping coefficients to be used in seismic analyses that is consistent with given modal damping. The approach uses the difference between the modal damping response and the Rayleigh damping response along with effective mass properties of the model being evaluated to match overall system response levels. This paper provides a simple example problem to demonstrate the approach. It also provides results for a finite element model representing an existing piping system. Displacement, acceleration, and stress results are compared from model runs using modal damping and model runs using Rayleigh damping with coefficients selected using the proposed method.