Parametric Resonance Region Research Articles

The onset of instability in a parametric resonance energy harvester under panchromatic excitations

This paper further explores the potential of parametric resonance to enhance wave energy conversion in a pendulum-based pitching floating energy harvester, considering power extraction and panchromatic excitation. Unlike traditional parametric systems that focus on inertial mechanical instability induced by floater motion, this concept, firstly proposed in a previous recent publication, leverages the parametric resonance induced by nonlinear hydrodynamic coupling between heave and pitch. By deliberately integrating parametric instability into the floater design with a 2:1 ratio of natural periods in two hydrodynamic degrees of freedom, the frequency bandwidth of the mechanical system is expanded, leading to two distinct regions of significant power production. The research advances current understanding in three key areas: (i) the use of Bezier curves to advance a computationally efficient nonlinear hydrodynamic model based on nonlinear Froude–Krylov force calculations tailored for prismatic floaters, (ii) investigation of the impact of power extraction on the severity of nonlinear parametric resonance, and (iii) consideration of more realistic panchromatic waves to quantify the onset of instability and the decrease in the severity of parametric resonance response, also addressing practical implications for numerical simulations and signal processing. Findings indicate that beneficial parametric resonance persists across various damping and wave conditions, although the relative importance of the contribution of the 2:1 parametric resonance region to power extraction decreases by 75%, on average.

3D Size-Dependent Dynamic Instability Analysis of FG Cylindrical Microshells Subjected to Combinations of Periodic Axial Compression and External Pressure Using a Hermitian C2 Finite Layer Method Based on the Consistent Couple Stress Theory.

This work develops a three-dimensional (3D) weak formulation, based on the consistent couple stress theory (CCST), for analyzing the size-dependent dynamic instability behavior of simply-supported, functionally graded (FG) cylindrical microshells that are subjected to combinations of periodic axial compression and external pressure. In our formulation, the microshells are artificially divided into nl layers. The displacement components of each individual layer are selected as the primary variables, which are expanded as a double Fourier series in the in-plane domain and are interpolated with Hermitian C2 polynomials in the thickness direction. Incorporating the layer-wise displacement models into our weak formulation, we develop a Hermitian C2 finite layer method (FLM) for addressing the current issue. The accuracy and the convergence rate of our Hermitian C2 FLM are validated by comparing the solutions it produces with the accurate two-dimensional solutions of critical loads and critical pressures of FG cylindrical macroshells and single-walled carbon nanotubes, which were reported in the literature. The numerical results show the effects of the material length-scale parameter, the inhomogeneity index, the radius-to-thickness and length-to-radius ratios, the load magnitude ratio, and the static and dynamic load factors on the first principal and first secondary instability regions of parametric resonance of simply-supported FG cylindrical microshells are significant.

Investigation of the Process of Separation of Fibers From Seeds During Roll Ginning

Abstract In the initial position in the process of roll ginning, the seed is drawn to the slot of the contact gap of the knife-drum due to the tension of the fibers. Subsequently, the seed is struck with a certain impulse. The separation of the fibers from the seed after the impact of the beater occurs sequentially, not simultaneously. As a result of theoretical and experimental studies, the optimum frequency of the shock pulse on the seed (f = 18-20) was found to be the most technologically successful in terms of providing the best quality indicators of seeds and fibers. Beating in the resonance mode allows to reduce the magnitude of the shock impulse on the seed, leading to a reduction in the degree of damage to the seeds. At this moment, the seeds at the end of the strand move only under the action of an elastic restoring force. The active part of the fiber bundle section in a dynamic process can be as little as 0.001 or less. The average value of the compliance coefficient for the raw cotton variety BA-440 was determined experimentally. This allowed to determine the stiffness coefficient of the volatiles indirectly. According to the parameters obtained, the dynamic state of the system practically falls into the region of parametric resonance, where the motion is unstable. This means that the damaged seeds at the end of the strand enlarge significantly after the impact of the beater, causing the seeds to separate from the fibers.

Nonlinear planar and non-planar vibrations of viscoelastic fluid-conveying pipes with external and internal resonances

In this study, nonlinear vibrations of a viscoelastic fluid-conveying pipe under in-plane and out-of-plane transverse excitations are investigated, with emphasis on external and internal resonances. For this purpose, a three-dimensional pipe model that couples the in-plane and out-of-plane vibrations is developed based on the Euler–Bernoulli beam theory together with the Kelvin–Voigt viscoelasticity. The Galerkin method is performed on the nonlinear governing equations, and the resultant discretized equations are solved via the pseudo-arclength continuation method and a direct integration method. The dynamical responses of the pipe are examined in both sub- and super-critical regions and presented by plotting the frequency–amplitude curves, force–amplitude curves, basins of the attraction, time histories, and phase portraits. A comparison between the simplified two-dimensional model and the present three-dimensional model is presented to showcase the possibility of the in-plane and out-of-plane vibrations in regions of primary and principal parametric resonances, highlighting the modal interaction between the in-plane and out-of-plane transverse modes. In the presence of a two-to-one internal resonance, in addition to typical jumping and hysteresis, the two-peak resonance behavior is observed. The numerical results not only show complex nonlinear dynamics of the fluid–structure interaction system, but also provide hints for safe design and novel application of such structures.

On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in One Case of Integer Nonequal Frequencies

This paper is concerned with the motions of a near-autonomous two-degree-of-freedom Hamiltonian system, $2\pi$-periodic in time, in a neighborhood of a trivial equilibrium. It is assumed that in the autonomous case, in the region where only necessary (which are not sufficient) conditions for the stability of this equilibrium are satisfied, for some parameter values of the system one of the frequencies of small linear oscillations is equal to two and the other is equal to one. An analysis is made of nonlinear oscillations of the system in a neighborhood of this equilibrium for the parameter values near a resonant point of parameter space. The boundaries of the parametric resonance regions are constructed which arise in the presence of secondary resonances in the transformed linear system (the cases of zero frequency and equal frequencies). The general case and both cases of secondary resonances are considered; in particular, the case of two zero frequencies is singled out. An analysis is made of resonant periodic motions of the system that are analytic in integer or fractional powers of the small parameter, and conditions for their linear stability are obtained. Using KAM theory, two- and three-frequency conditionally periodic motions (with frequencies of different orders in a small parameter) are described.

Research on Parametric Vibration of a Steel Truss Corridor under Pedestrians Excitation Considering the Time-Delay Effect

With the development of the Steel Truss Corridor (STC) toward long-span and gentle development, human-induced vibration often causes large lateral vibration problems and time-delay effects of the STC, which will have a non-negligible impact on the dynamic performance of the STC. In this paper, the parametric vibration model proposed by Piccardo is improved, and the nonlinear dynamic equation of the STC is established considering the longitudinal–lateral walking force coupled parametric vibration with the time-delay effect. Taking the Millennium STC as an example, the mechanism of lateral vibration under the time-delay effect is discussed by the numerical calculation method, and the influence of the time-delay effect on its dynamic response is analyzed. The results show that: considering the time-delay effect, when the frequency ratio θ/2Ω = 1, the value of the time-delay coefficient has no effect on the critical number of STCs in the parametric resonance region. As θ/2Ω moves away from 1, the more significant the effect. When the STC begins to excite the parametric resonance phenomenon, the existence of the time-delay effect will change the time for the STC to reach a stable amplitude and suppress the lateral vibration of the STC. When the STC generates parametric vibration, the value of the time-delay coefficient has no effect on the nonlinear dynamic response of the STC. For STCs in both the nonparametric resonance region and the critical region, there is a pair of staggered critical bifurcation time-delay coefficients, which increase or decrease the vibration response.

A New Robust Method to Investigate Dynamic Instability of FTV for the Double Tripod Industrial Driveshafts in the Principal Parametric Resonance Region

The present work aims to design a robust method to detect and certify the deterministic chaos or ergodic process for the forced torsional vibrations (FTV) of a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR) which is considered by the researchers in the field as one of the most important resonance regions for the systems having parametric excitations. The DTID’s model for FTV considers the following effects: nonuniformities of inertial characteristics of the DTID’s elements, the harmonic torque excitation induced by the asynchronous electrical motor used for a heavy-duty grain mill, and the harmonic reaction torque generated by different granulation of the substance needed to be milled. Based on these aspects, a model of the FTV for the DTID was designed which was a modified, physically consistent model already used by the authors to investigate the FTV of automotive driveshafts (homokinetic transmission). For the DTID elements, the dynamic instability for nonstationary FTV in the PPRR using time–history analysis (THA) was analyzed—THA represents the phase portraits. Time–history analysis is a detection method for possible chaotic dynamic behavior for the nonstationary FTV (NFTV) in transition through PPRR. If this dynamic behavior was seen, a new robust method LEA–PM was created to certify and confirm the deterministic chaos for the NFTV of DTID. The new method, LEA–PM, is composed of the Lyapunov exponent’s approach (LEA) coupled with the Poincaré Map (PM) applied to the global system of differential equations that describe the FTV of DTID in the PPRR. This new robust method, which embeds LEA and PM, LEA–PM, establishes if the mechanical system has a deterministic chaotic dynamic behavior (strange attractor) or an ergodic dynamic process in this resonant region. LEA represents a new method that includes not only the maximal Lyapunov exponent method (MLEM) but also new mathematical criteria that is “the sum of all Lyapunov exponents has to be negative” which, coupled with MLEM, indicates the presence of deterministic chaos (strange attractors). THA–LEA–PM had been used for the NFTV of DTID computing the phase portraits, the Lyapunov exponents, and representing the Poincaré Maps of the NFTV for the DTID’s elements in transition through PPRR, founding deterministic chaos or ergodic dynamic behavior. Based on the obtained results, numerical simulations revealed the pitting manifestations of the DTID’s elements, typical for the geared systems transmission, mentioned recently in experimental data research for the homokinetic transmissions. Using the new robust method, THA–LEA–PM (time–history analysis coupled with LEA–PM) can be used in future research for chaotic dynamic analysis of DTID’s NFTV transition through superharmonic resonances, subharmonic resonances, combination resonances, and internal resonances. Time–history analysis as a detection method for chaos and LEA–PM as a certifying method for deterministic chaos can be integrated as a design tool for DTID’s FTV control of the homokinetic transmission.

Modeling and Analysis of FBV Movements for Automotive Driveshafts in the PPR Region

This research’s goal is to model and analyze the forced bending vibrating (FBV) movements for the elements of an automotive driveshaft using a perturbation technique, the asymptotic method approach (AMA), in the region of principal parametric resonance (PPR). The PPR region was chosen because the principal parametric resonance region is one of the essential resonance regions. The model of FBV movements for the automotive driveshaft (AD) considers the aspects of the following phenomena: geometric nonuniformity of the AD elements and shock excitation due to the road. To overcome the equations for the FBV movements of the AD elements, all inertia characteristics were reduced to the longitudinal ax of the midshaft using the variation of the geometric moments of inertia with the concurrent axis and Stener’s theorem. The midshaft of the AD was considered a Timoshenko simply supported beam with a concentrated mass at both ends and springs and dampers for linear and rotational movements at both ends. To determine the equations describing the FBV movements of the AD elements, Hamilton’s principle was used. After establishing the equations of motion for each AD element coupled with the specific boundary conditions, the amplitude and the phase angle were computed for stationary and nonstationary motion in the PPR region using the first order of the AMA, and the dynamic instability frontiers were determined based on the same equations. The dynamic behavior of the AD was investigated concerning the variation of the damping ratio and the variation of the parametric excitation coefficient. The AMA coupled with the model of FBV movements for the AD exhibits the future research directions for analyzing FBV movements for the AD in the regions of superharmonic resonances, subharmonic resonances, combination resonances, internal resonances, and simultaneous resonances. Additionally, the AMA can predict the endurance of the AD and design control of car damping systems.

A Physically Consistent Model for Forced Torsional Vibrations of Automotive Driveshafts

The aim of this research was to design a physically consistent model for the forced torsional vibrations of automotive driveshafts that considered aspects of the following phenomena: excitation due to the transmission of the combustion engine through the gearbox, excitation due to the road geometry, the quasi-isometry of the automotive driveshaft, the effect of nonuniformity of the inertial moment with respect to the longitudinal axis of the tulip–tripod joint and of the bowl–balls–inner race joint, the torsional rigidity, and the torsional damping of each joint. To resolve the equations of motion describing the forced torsional nonlinear parametric vibrations of automotive driveshafts, a variational approach that involves Hamilton’s principle was used, which considers the isometric nonuniformity, where it is known that the joints of automotive driveshafts are quasi-isometric in terms of the twist angle, even if, in general, they are considered CVJs (constant velocity joints). This effect realizes the link between the terms for the torsional vibrations between the elements of the driveshaft: tripode–tulip, midshaft, and bowl–balls–inner race joint elements. The induced torsional loads (as gearbox torsional moments that enter the driveshaft through the tulip axis) can be of harmonic type, while the reactive torsional loads (as reactive torsional moments that enter the driveshaft through the bowl axis) are impulsive. These effects induce the resulting nonlinear dynamic behavior. Also considered was the effect of nonuniformity on the axial moment of inertia of the tripod–tulip element as well as on the axial moment of inertia of the bowl–balls–inner race joint element, that vary with the twist angle of each element. This effect induces parametric dynamic behavior. Moreover, the torsional rigidity was taken into consideration, as was the torsional damping for each joint of the driveshaft: tripod–joint and bowl–balls–inner race joint. This approach was used to obtain a system of equations of nonlinear partial derivatives that describes the torsional vibrations of the driveshaft as nonlinear parametric dynamic behavior. This model was used to compute variation in the natural frequencies of torsion in the global tulip (a given imposed geometry) using the angle between the tulip–midshaft for an automotive driveshaft designed for heavy-duty SUVs as well as the characteristic amplitude frequency in the region of principal parametric resonance together the method of harmonic balance for the steady-state forced torsional nonlinear vibration of the driveshaft. This model of dynamic behavior for the driveshaft can be used during the early stages of design as well in predicting the durability of automotive driveshafts. In addition, it is important that this model be added in the design algorithm for predicting the comfort elements of the automotive environment to adequately account for this kind of dynamic behavior that induces excitations in the car structure.

On Nonlinear Oscillations of a Time-Periodic Hamiltonian System at a 2:1:1 Resonance

We consider the motions of a near-autonomous Hamiltonian system $2\pi$-periodic in time, with two degrees of freedom, in a neighborhood of a trivial equilibrium. A multiple parametric resonance is assumed to occur for a certain set of system parameters in the autonomous case, for which the frequencies of small linear oscillations are equal to two and one, and the resonant point of the parameter space belongs to the region of sufficient stability conditions. Under certain restrictions on the structure of the Hamiltonian of perturbed motion, nonlinear oscillations of the system in the vicinity of the equilibrium are studied for parameter values from a small neighborhood of the resonant point. Analytical boundaries of parametric resonance regions are obtained, which arise in the presence of secondary resonances in the transformed linear system (the cases of zero frequency and equal frequencies). The general case, for which the parameter values do not belong to the parametric resonance regions and their small neighborhoods, and both cases of secondary resonances are considered. The question of the existence of resonant periodic motions of the system is solved, and their linear stability is studied. Two- and three-frequency conditionally periodic motions are described. As an application, nonlinear resonant oscillations of a dynamically symmetric satellite (rigid body) relative to the center of mass in the vicinity of its cylindrical precession in a weakly elliptical orbit are investigated.

ТЕОРЕТИЧЕСКОЕ ИССЛЕДОВАНИЕ ЭФФЕКТОВ КОЛЕБАНИЙ ДВУХ НЕСМЕШИВАЮЩИХСЯ ЖИДКОСТЕЙ В ОГРАНИЧЕННОМ ОБЪЁМЕ

In the paper, the nonlinear oscillations of a two-layer fluid that completely fills a limited tank are theoretically studied. To determine any smooth function on the deflected interface, the Taylor series expansions are considered using the values of the function and its normal derivatives on the undisturbed interface of the fluids. Using two fundamental asymmetric harmonics, which are generated in two mutually perpendicular planes, the differential equations of nonlinear oscillations of the two-layer fluid interface are investigated. As a result, the frequency-response characteristics are presented and the instability regions of the forced oscillations of the two-layer fluid in the cylindrical tank are plotted, as well as the parametric resonance regions for different densities of the upper and lower fluids. The Bubnov-Galerkin method is used to plot instability regions for the approximate solution to nonlinear differential equations. At the final stage of the work, the nonlinear effects resulting from the interaction of fluids with a rigid tank that executes harmonic oscillations at the interface of the fluids are theoretically studied.

Study on Parametric Resonance of Support Parameters to Flow Pipeline

The influence of support parameters on the parametric resonance of fluid conveying pipeline under pulsating flow excitation is studied. Firstly, the mathematical model of fluid conveying pipeline and support system is established, Then the Galerkin method is used for discretization, and the incremental harmonic balance method is used to solve the problem. The effects of support stiffness, support damping, support position, support number and support spacing on the boundary of parametric resonance region and the parametric resonance response are discussed. The results show that under the fixed pulsating amplitude, the pulsating frequency range of system instability becomes smaller with the increase of support stiffness, support damping and the number of support, while the support position and support spacing have less influence on it. The change of support stiffness and support position has little influence on the pulsation frequency that the parametric resonance lasts to the end, but the increase of brace damping makes its value decrease obviously. The research results can provide a theoretical basis for the design and installation of pipeline support system.

Extremely large dynamics of axially excited cantilevers

Abstract The nonlinear parametric resonance of a cantilever under axial base excitation is examined while capturing extremely large oscillation amplitudes for the first time. A geometrically exact model is developed for the cantilever based on the Euler-Bernoulli beam theory and inextensibility condition. In order to be able to capture extremely large oscillation amplitudes accurately, the equation of motion is derived for centreline rotation while keeping trigonometric terms intact. The developed model is verified for the static case through comparison to a three-dimensional nonlinear finite element model. The internal energy dissipation model of Kelvin-Voigt is used to model the system damping in large amplitudes more accurately. The Galerkin modal decomposition scheme is utilised for discretisation procedure while keeping the trigonometric terms intact. It is shown that in parametric resonance region, the oscillation amplitudes grow extremely large even for smallest possible amplitudes of the base excitation, which highlights the significant importance of employing a geometrically exact model to examine the parametric resonance response of a cantilever.

Stability analysis of an axially moving nanocomposite circular cylindrical shell with time-dependent velocity in thermal environments

This article investigates the instability of an axially moving simply supported nanocomposite circular cylindrical shell reinforced by single-walled carbon nanotubes (SWCNTs) with time variant velocity in thermal environments considering the effect of viscous structure damping. The nanocomposite material properties are obtained from the extended mixture rule for uniformly distributed CNT, furthermore three different types of functionally graded distribution CNT in the matrix are represented. In addition, the Hamilton principle via the simplification assumption of the Donnell shallow-shell theory is employed to derive motion equations. At first, related displacements are obtained in terms of displacement in radius direction which is expressed in a combination of driven and companion modes from motion equations in axial and circumferential directions. The dimensionless motion equation in radius direction is discretized by the Galerkin method. The critical speed values of different types of CNT distribution reinforcement, various types of volume-fraction of CNT reinforcement and different temperatures are established using the multiple scale method. Moreover, the first order solution of the perturbation theory makes it possible to predict the unstable regions of principle parametric resonances and combination resonances. It occurs under the effect of temperature rising condition, changing the kind of SWCNTs as well as increasing the value of viscous structure damping, volume-fraction of CNT distribution, constant part of velocity function and fluctuation domain. Results show a system with the symmetric distribution of CNT reinforcement, while the outer and inner are CNT-rich, represents higher stability in axially moving nanocomposite cylindrical shells with constant and time-dependent velocity. Communicated by Eleonora Tubaldi

Stability and nonlinear vibrations of a flexible pipe parametrically excited by an internal varying flow density

Pipes are often used to transport multiphase flows in many engineering applications. The total fluid flow density inside a pipe may vary with time and space. In this paper, a simply supported pipe conveying a variable density flow is modeled theoretically, and its stability and nonlinear vibrations are investigated in detail. The variation of the flow density is simulated using a mathematical function. The equation governing the vibration of the pipe is derived according to Euler–Bernoulli beam theory. When the internal flow density varies with time, the pipe is excited parametrically. The stability of the pipe is determined by Floquet theory. Some simple parametric and combination resonances are determined. For a higher mass ratio (mean flow mass/pipe structural mass), higher flow velocity, or smaller end axial tension, the pipe becomes unstable more easily due to wider parametric resonance regions. In the subcritical flow velocity regime, the vibrations of the pipe are periodic and quasiperiodic for simple and combination resonances, respectively. However, in the supercritical regime, the vibrations of the pipe exhibit much richer dynamics including periodic, multiperiodic, quasiperiodic, and chaotic behaviors.

On the Motions of One Near-Autonomous Hamiltonian System at a 1:1:1 Resonance

We consider the motion of a 2π-periodic in time two-degree-of-freedom Hamiltonian system in a neighborhood of the equilibrium position. It is assumed that the system depends on a small parameter e and other parameters and is autonomous at e = 0. It is also assumed that in the autonomous case there is a set of parameter values for which a 1:1 resonance occurs, and the matrix of the linearized equations of perturbed motion is reduced to a diagonal form. The study is carried out using an example of the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on an elliptical orbit with small eccentricity in the neighborhood of the cylindrical precession. The character of the motions of the reduced two-degree-of-freedom system in the vicinity of the resonance point in the three-dimensional parameter space is studied. Stability regions of the unperturbed motion (the cylindrical precession) and two types of parametric resonance regions corresponding to the case of zero frequency and the case of equal frequencies in the transformed approximate system of the linearized equations of perturbed motion are considered. The problem of the existence, number and stability of 2π-periodic motions of the satellite is solved, and conclusions on the existence of two- and three-frequency conditionally periodic motions are obtained.

PARAMETRIC RESONANCE OF A CANTILEVERED PIPE CONVEYING FLUID SUBJECTED TO DISTRIBUTED MOTION CONSTRAINTS

Pipes conveying fluid have been widely used in the fields of aerospace, mechanics, marine, hydraulic and nuclear engineering. The stability analysis, dynamic response and safety assessment of fluid-conveying pipes subjected to nonlinear constraints are particularly important for both engineering applications and scientific researches. Although the dynamical behaviors of fluid-conveying pipes subjected to single-point loose constraints have been discussed for four decades, the literature on the dynamics of fluid-conveying pipes subjected to distributed motion constraints is very limited. To obtain a better understanding of the dynamics of a cantilevered pipe conveying fluid subjected to distributed motion constraints, both cubic and modified trilinear spring models are employed in this study to describe the restraining force between the pipe and the motion constraints. This study is also concerned with the parametric resonance when the pipe is excited by an internal pulsating fluid. Firstly, the modified nonlinear equation of the pipe system was discretized via Galerkin's approach and solved using a fourth-order Runge-Kutta method. Via the Floquet theory, the nonlinear equation of motion was simplified to a linear one to calculate the parametric resonance regions versus the pulsating amplitude and frequency. Two representative values of mean flow velocity were employed to calculate the parametric resonance regions. Both the two values of mean flow velocity are assumed to be lower than the critical velocity for the cantilevered pipe system. Then, considering the geometric nonlinearity, the nonlinear dynamic responses focusing on the effect of external nonlinear restraining forces generated by the distributed motion constraints are discussed in detail. Results show that the stability regions of the nonlinear system agree well with that predicted by analyzing the linearized system. It is found that the distributed motion constraints would mainly affect the displacement amplitudes. Various oscillation types may arise when the pulsating frequency of the flow velocity is varied. Several bifurcation diagrams show that, however, a significant difference can be observed between the routes to chaos for the two constraint models, i.e., the pipe with a trilinear spring model can exhibit chaotic oscillations more easily than that with a cubic spring model.

Parametric Resonance of Pipes with Soft and Hard Segments Conveying Pulsating Fluids

In this paper, the parametric resonance of pipes with soft and hard segments induced by pulsating fluids is investigated. The lowest six natural frequencies and mode shapes of the soft–hard combination pipe simply supported at both ends are obtained by the modified Galerkin's method. The Floquet method is used to numerically determine the parametric resonance regions, including subharmonic resonance regions and combination resonance regions. The parametric resonance results are verified by comparison with published ones, which confirm the validity of the present model establishment and numerical calculation. Compared with a uniform pipe conveying fluid simply supported at both ends, the soft–hard pipe conveying fluid is found to reveal different dynamical behaviors. Decreasing the length of the soft pipe, while increasing the stiffness ratio of the hard pipe compared to the soft one, can effectively improve the stability of the pipe system. The parametric resonance results show that the mean flow velocity and pulsation amplitude of the fluid have a great influence on the width of the parametric resonance regions. It is advisable that the ratio (the soft pipe/the whole pipe) of the length may be designed to be 0.4–0.5 for a flexural rigidity ratio (the hard pipe/the soft pipe) of 2. As the stiffness ratio (the hard pipe/the soft pipe) increases beyond 26, the hard pipe may be regarded as a rigid pipe. The probability of parametric resonance occurrence will be smallest if the soft–hard combination pipe is supported in a clamped–pinned way. For certain application cases, the safety design length of the two pipes with different materials can be determined through numerical calculation.

Dynamic response of a deepwater riser subjected to combined axial and transverse excitation by the nonlinear coupled model

In offshore engineering long slender risers are simultaneously subjected to both axial and transverse excitations. The axial load is the fluctuating top tension which is induced by the floater’s heave motion, while the transverse excitation comes from environmental loads such as waves. As the time-varying axial load may trigger classical parametric resonance, dynamic analysis of a deepwater riser with combined axial and transverse excitations becomes more complex. In this study, to fully capture the coupling effect between the planar axial and transverse vibrations, the nonlinear coupled equations of a riser’s dynamic motion are formulated and then solved by the central difference method in the time domain. For comparison, numerical simulations are carried out for both linear and nonlinear models. The results show that the transverse displacements predicted by both models are similar to each other when only the random transverse excitation is applied. However, when the combined axial dynamic tension and transverse wave forces are both considered, the linear model underestimates the response because it ignores the coupling effect. Thus the coupled model is more appropriate for deep water. It is also found that the axial excitation can significantly increase the riser’s transverse response and hence the bending stress, especially for cases when the time-varying tension is located at the classical parametric resonance region. Such time-varying effects should be taken into account in fatigue safety assessment.

Non-linear dynamic instability analysis of thin-walled stiffener beam subjected to uniform harmonic in-plane loading

Present analysis deals with nonlinear flexural-torsional vibration and dynamic instability of thin-walled stiffener beam with open section subjected to harmonic in-plane loading. The static and dynamic components of the applied harmonic in-plane loading are assumed to vary uniformly. A set of nonlinear partial differential equations (PDEs) describing the vibration of system is derived. Using Galerkin's method, these partial differential equations are reduced into coupled Mathieu equations. The steady state response of the system is determined by solving the condition for a non-trivial solution. The principal regions of parametric resonance are determined using the method suggested by Bolotin. The numerical results are presented to investigate the effect of aspect ratios, boundary conditions and static load factor on the frequency-amplitude responses and instability regions.