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This paper introduces a novel numerical method for investigating the dynamic stability of a flutter panel exposed to a supersonic gas flow and a fluctuating axial excitation force. Initially, the system equation of motion was derived using Lagrange’s equation, where the first two modes are coupled Mathieu–Hill equations with damping, constituting a system of linear second-order differential equations with periodically variable coefficients. Subsequently, a new numerical method was proposed to analyze the dynamic stability of coupled Mathieu–Hill equations with damping. This method involves breaking down an arbitrary parametric load into discrete segments to approximate the variable excitation function using a step function. The system responses of each segment are then accumulated in matrix form. The proposed numerical method proves particularly effective for dynamic systems whose parameters cannot be treated as small. In practical application, the method allows the construction of instability regions corresponding to natural frequencies, subharmonics, and combination frequencies. Dynamic stability diagrams were generated based on dynamic pressure ratio, air/panel density ratio, Mach number, panel thickness—length ratio, and excitation frequency. The results demonstrated general agreement with those obtained through Hsu’s perturbation method, however, our numerical results have proven more accurate. The paper concludes by offering suggestions for suppressing panel flutter through appropriate parameter combinations. [ABSTRACT FROM AUTHOR]

Presented herein is an investigation for the nonlinear vibration and stability analysis of rotating functionally graded (FG) piezoelectric nanobeams based on the nonlocal strain gradient theory. The present model can be regarded as a simplified version for the rotating nanowire of biomechanical nanogenerators. The Hamilton principle is used to derive nonlinear equations of motion and their related boundary conditions, which are then discretized to form a set of algebraic equations. Accordingly, the nonlinear vibration frequencies and buckling loads of the nanobeams can be determined by an iterative method. A parametric study of rotational velocity, nonlocal parameter, material length parameter, power-law index, and electrostatic voltage on the dynamic stability behavior of such nanobeams is also presented. In the cantilever case, increasing the nonlocal parameter and material length parameter can result in a stiffness-hardening effect that is unaffected by rotational velocity and the material length parameter to nonlocal parameter ratio. Yet, this has not been reported previously. More importantly, incorporating the effect of geometric nonlinearity on the dynamic responses and stability results of the nanobeams is indispensable. In particular, new observations for the coupling effect of vibration amplitude and power-law index on the electric potential effect are useful for the design of rotating microelectromechanical devices. [ABSTRACT FROM AUTHOR]

This paper presents a dynamic analysis of trusses with an initial length imperfection of the elements, considering geometrical nonlinearity. In the nonlinear analysis of trusses, the hybrid finite-element formulation considers the initial length imperfection of the elements as a dependent boundary constraint in the master equation of stiffness. Moreover, it was incorporated into the establishment of a modified system of equations. To overcome the mathematical complexity of dealing with initial length imperfections, this study proposes a novel approach for solving nonlinear dynamic problems based on a hybrid finite-element formulation. In this study, the unknowns of the dynamic equilibrium equations were displacements and forces, which were obtained using virtual work. The hybrid matrix of elements of the truss is established based on the hybrid variation formulation with length imperfections of elements, considering large displacements. The authors applied Newmark integration and Newton–Raphson iteration methods to solve the dynamic equations with geometrical nonlinearity. An incremental iterative algorithm and calculation programming routine were developed to illustrate the dynamic responses of trusses with initial-length imperfections. The results verified the accuracy and effectiveness of the proposed approach. The uniqueness of the proposed method is that the length imperfection of the truss element is included in the stiffness matrix and is considered a parameter that affects the dynamic response of the system. This helps to solve the problem of the dynamic response of trusses with length imperfections becoming simpler. The numerical results show that the effect of length imperfection on the dynamic response of the trusses is significant, particularly on the dynamic limit load. In addition, to completely evaluate the behavior of the trusses, this study also developed formulas and analyses to consider the inelastic and local buckling of the truss structures, named 'Inelastic post-buckling analysis (IPB).' [ABSTRACT FROM AUTHOR]

This study aims at understanding the cause of vibrations observed during the operation of high speed centrifuges in a radioactive environment and materializing ways to mitigate these vibrations to ensure dynamic stability of centrifuges during operation. For this, a condition-based monitoring study was carried out for the centrifuge and its supporting structure using dynamic co-relation study in frequency domain during the operation of centrifuge at its operating speed of 19 700 rpm. Dynamic characteristics of the support structure were analyzed using standard finite element code. It was observed that high vibration amplitudes in the original support structure, before modification, owed to its low structural stiffness. Hence, structural modifications were carried out to reduce the dynamic crossover response of the support structure at the critical speeds and operating speeds of centrifuge. It was found that structural modification resulted in considerable reduction in the vibration amplitude of the support structure, compared to its initial configuration. [ABSTRACT FROM AUTHOR]

The focus of this paper is to examine the static and dynamic instabilities of functionally graded beam that contains multiple inclined cracks under the influence of an axial force comprising both static and time-varying harmonic components. The elasticity modulus and mass density of the functionally graded beam are assumed to vary exponentially along its thickness direction. Local stiffness matrix model-based finite element analysis (FEA) is conducted to determine the bending stiffness and tensile stiffness of the section with a crack, and the coupled effect of tensile and bending loadings. Two-node beam elements with three degrees-of-freedom per node are utilized. By combining the Euler–Bernoulli beam theory with Lagrange method, we derive the governing equations that describe the static and dynamic instabilities of a functionally graded beam with multiple inclined cracks. These equations can be solved as eigenvalue problems to obtain the natural frequency and static critical buckling load of the beam. Furthermore, to investigate the dynamic instability of the system, we use the Bolotin method to determine the boundary between the regions of instability and stability based on the same governing equations. By adopting this approach, the study comprehensively investigates the impacts of crack position, inclination angle, and length, as well as elasticity modulus ratio, static and dynamic load factors on both static and dynamic stabilities of a cracked functionally graded beam to gain valuable insights into the stability and performance of cracked functionally graded structures. [ABSTRACT FROM AUTHOR]

In the present exploration, the nonlocal stress and strain gradient microscale effects are adopted on the nonlinear dynamical instability feature of functionally graded (FG) piezoelectric microshells under a combination of axial compression, electric actuation, and temperature. To perform this objective, a unified unconventional shell model based on the nonlocal strain gradient continuum elasticity is established to capture the size effects as well as the influence of the geometrical nonlinearity together with the shear deformation along with the transverse direction on the dynamic stability curves. With the aid of an efficient numerical strategy incorporating the generalized differential quadrature strategy and pseudo arc-length continuation technique, the extracted unconventional nonlinear differential equations in conjunction with the associated edge supports are discretized and solved to trace the dynamic stability paths of FG piezoelectric microshells. It is revealed that the nonlocal stress and strain gradient effects result in, respectively, higher and lower values of the nonlinear frequency ratio in comparison with the conventional one due to the stiffening and softening characters associated with the nonlocality and strain gradient size dependency, respectively. In addition, it is observed that within the prebuckling territory, the softening character of nonlocality is somehow more than the stiffening character of strain gradient microsize dependency, while by switching to the postbuckling domain, this pattern becomes vice versa. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical simulation of both dynamic stability and responses of beams on elastic foundations under a pulsating axial parametric load in a single matrix method. First, the equation of motion of a beam on an elastic foundation with damping is derived and decoupled into a Mathieu–Hill equation. Three different elastic foundations are considered and compared: Winkler, Pasternak, and Hetenyi models. Then a novel numerical simulation algorithm is proposed to investigate both the dynamic stability and the responses of the beam simultaneously. Accurate instability diagrams are obtained by the numerical simulation and are substantiated by vibration response curves obtained from the same method. These numerically accurate diagrams are used to calibrate the approximate instability boundaries of various orders of Hill infinite determinants from the classical Bolotin method for the first time. A detailed discussion is presented on effects of various aspects including elastic foundation models, damping, and static and dynamic loads. The results provide insights into the efficient and safe application of beams on elastic foundations in engineering. The proposed numerical method can be extended to analyze dynamic stability and vibrations of systems under arbitrary parametric excitations where Mathieu–Hill equations are involved. [ABSTRACT FROM AUTHOR]

Cracks have a significant impact on the stability of rotating machinery. However, most studies focus on rotating machinery with only one crack. This paper discusses the influence of two cracks on the stability of a flywheel rotor-bearing system. In this study, the dynamic models of vertically placed flywheel rotor-bearing system with two open cracks and two breath cracks are established using the finite element method (FEM). Floquet theory is used to calculate the stability of the system and the influence of depth and positions of cracks are analyzed. The results show that breathing cracks cause more unstable regions than open cracks, and the range of rotating speeds over which the rotor is unstable increases with increasing crack depth. In addition, it is observed that when the crack locations are adjacent, the unstable region will become very large, covering most of the crack depth range and speed range, making the rotor extremely unstable. Besides, the bearing stiffness causes an offset in the unstable regions, whereas the damping influences the instability value of the rotor. [ABSTRACT FROM AUTHOR]

This paper analyzes the dynamic stability of an isotropic viscoelastic Euler–Bernoulli nano-beam using piezoelectric materials. For this purpose, the size-dependent theory was used in the framework of the modified couple stress theory (MCST) for piezoelectric materials. In order to capture the geometrical nonlinearity, the von Karman strain displacement relation was applied. Hamilton's principle was also employed to obtain the governing equations. Furthermore, the Galerkin method was used in order to convert the governing partial differential equations (PDEs) to a nonlinear second-order ordinary differential one. Dynamic stability analysis was performed and the effects of such parameters as viscoelastic coefficients, size effect, and piezoelectric coefficient were investigated. The results showed that in this system, saddle points, central points, Hopf bifurcation points, and fork bifurcation points could be created, and the phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, and homoclinic orbits. [ABSTRACT FROM AUTHOR]

The porous material is an emerging lightweight material, which is able to reduce structural weight and also keeps the superiority of the structure. Therefore, this work is devoted to the investigation of the functionally graded porous (FGP) annular and circular plates with general boundary conditions. The unified modeling method is proposed by combining the first-order shear deformation theory, the virtual spring technology, the multi-segment partition method, and the semi-analysis Rayleigh–Ritz approach. Afterwards, the convergency and correctness of the proposed method are verified, respectively. The frequency parameters, modal shapes, and forced vibration responses are uniformly calculated based on the proposed method. Finally, the dynamic analyses of the FGP annular and circular plates with different parameters, such as the porosity distribution types, porosity ratios, boundary condition types, geometry parameters, and load types, are conducted in detail. It is found that the reasonable porous design is able to keep the dynamic stability of the structure under different parameter variations. [ABSTRACT FROM AUTHOR]

This paper inspects the influence of a spring attachment provided on the top elastic layer on the stability of a pre-twisted, rotating sandwich beam having viscoelastic supports at the root under the impact of a periodically varying axial load. The spring is deployed on the beam to achieve more strength to weight ratio without compromising the stability. The beam is exponentially tapered, and a tip mass is at the free end to represent the rotating members in various types of machinery as closely as possible. The ruling equations and inter-related boundary conditions are attained by applying Hamilton's principle. To obtain the solution, a matrix equation was developed through the assumed-mode variational method. The resulting matrix equation was converted to a coupled Hill's equation of parametric vibration through the modal matrix corresponding to the free vibration problem. Finally, static and dynamic stability graphs were obtained for several system parameters such as position and length of the attached spring on the top elastic layer, the mass of the spring attachment, stiffness of the spring attachment, angle of pre-twist, tip mass, taper parameter, temperature gradient parameter, setting angle, viscoelastic spring stiffness, etc. to analyze their impact on the system's stability. Saito and Otomi conditions were used to obtain dynamic stability plots. Greater stability is achieved due to the spring attachment on the top of the top elastic layer. [ABSTRACT FROM AUTHOR]

The dynamic stability of the foam-filled sandwich plates under arbitrary periodic loads is studied by using Lo's high-order shear deformation theory. The periodic load is simulated as a combination of axial stress and bending stress. The governing equations are established based on a perturbation method. Through the derived Mathieu-type equation and using the Bolotin method, the dynamic instability regions are determined by solving the eigenvalue problem. The effects of core thickness, static and dynamic load, and bending stress on the dynamic stability behavior of composite sandwich plates with stiff face sheets and soft foam core are studied. The results reveal that the presented higher-order plate theory adopted gives a better estimate of the dynamic stability of the foam-filled sandwich plate than the first-order plate theory. [ABSTRACT FROM AUTHOR]

The Homotopy Perturbation Method (HPM) is applied to investigate the nonlinear free and forced vibration behavior of the rotating cracked beam. Nonlinear governing equations considering the two-dimensional (2D) large flexible structure in a rotating reference frame considering centrifugal forces are obtained by the Lagrangian approach and the Assumed Mode Method (AMM). The crack is modeled as an elastic nonlinear massless rotational spring, which divides the beam into two parts. The Rayleigh–Ritz method is used to discretize the governing system of equations of the motion. Stability analysis along with bifurcation and phase portrait represents the different behavior of the system, depending on the variations of base angular velocity, crack location, and stiffness. Moreover, it is shown that as the rotational speed increases, a tensile force appears along the neutral axis, stiffening the cracked structure, which results in shifting the backbone to the right and highly affects the nonlinear features of the system. The results obtained through a comparative study of the HPM with first-order approximation and numerical simulations (Runge–Kutta algorithm) demonstrate an accurate and effective solution for structures with nonlinear dynamics. [ABSTRACT FROM AUTHOR]

Determination of the regions of dynamic instability has been an important issue for elastic structures. Under the extreme climate, the external load acting on structures is becoming more and more complicated, which can induce dynamic instability of elastic structures. In this study, we explore the dynamic instability and response characteristics of simply supported beams under multi-harmonic parametric excitation. A numerical approach for determining the instability regions under multi-harmonic parametric excitation is developed here by examining the eigenvalues of characteristic exponents of the monodromy matrix based on the Floquet theorem, and the fourth-order Runge–Kutta method is used to calculate the dynamic responses. The accuracy of the method is verified by the comparison with classical approximate boundary formulas of dynamic instability regions. The numerical results reveal that Bolotin's approximate formulas are only applicable to the low-order instability regions with a small value of the excitation parameter of simple parametric resonance. Multi-harmonic parametric excitation can significantly change the dynamic instability regions, it may cause parametric resonance on beams for longitudinal complex periodic loads. The influence of frequency and number of multiply harmonics on the parametrically excited vibration of the beam is explored. High-order harmonics with low-frequency have positive effects on the stable response characteristics for multi-harmonic parametric excitation. This paper provides a new perspective for the vibration suppression of parametric excitation. The developed procedure can be used for multi-degree-of-freedom (MDOF) systems under complex excitation (e.g. tsunami waves and strong winds). [ABSTRACT FROM AUTHOR]

This paper is concerned with the dynamic stability and response of an inclined Euler–Bernoulli beam under a moving mass and a moving follower force. The extended Hamilton's principle is used to derive the governing equation of motion and the boundary conditions for this general moving load/force problem. Considering a simply supported beam, one can solve the problem analytically by approximating the spatial part of the deflection with a Fourier sine series. Based on the formulation and method of solution, sample dynamic responses are determined for a beam that is inclined at 30 ∘ with respect to the horizontal. It is shown that the dynamic response of the beam under a moving mass is rather different from an equivalent moving follower force. Also investigated herein are the dynamic stability of inclined beams under moving load/follower force which are described by four key variables, viz. the speed of the moving mass/follower force, concentrated mass to the beam distributed mass, vibration frequency and the magnitude of the moving mass/follower force. The critical axial load and the critical follower force are different when they are located at different positions in the beam; except for the special case when they are at the end of the beam. [ABSTRACT FROM AUTHOR]

The paper presents a solution to the dynamic response of three-layered, annular plates that are thermo-mechanically loaded. Time-dependent forces are considered to act on the facings of the plate. The thermal loading is defined by a flat, axisymmetric temperature field whose profile is expressed by the stationary or increase in time temperature differences between the plate edges. The cases of the plates loaded only thermally or mechanically have been examined. The problem has been solved analytically and numerically using the approximation methods: orthogonalization and finite difference. The selected reactions of an exemplar plate have been compared with the results obtained for the plate analyzed by the finite element method (FEM). The critical temperature differences and critical loads and corresponding dynamic buckling modes have been analyzed in detail. The numerous results presented for the dynamic responses have an important and practical meaning in the design process of plate structures subjected to complex thermo-mechanical loads. [ABSTRACT FROM AUTHOR]

This paper deals with the stability and dynamic evolution of a sliding pipe conveying fluid in the three-dimensional sense. The pipe is assumed to slide out from a fixed channel so that its free end is moving at the same time, a problem often associated with instabilities in applications of aerial refueling operation. To tackle this problem, the nonlinear governing equations of motion are derived by using the Hamilton's principle and then reduced to a set of ordinary differential equations by the Galerkin's method. A parametric study is performed to explore the transient vibration responses of the pipe for different values of flow velocity and sliding rate. Various dynamic behaviors are detected for the pipe in sliding and conveying fluid. The results show that 3-D oscillations of the pipe occur when the flow velocity exceeds a certain value, which can be affected by the sliding rate. For various flow velocities, the evolution of the dynamic characteristics of the sliding pipe can be classified into three typical types of motion. When at low flow velocity, the pipe is mainly subjected to a single type of 3-D motion. When the flow speed increases to high values, multi-type of 3-D motion consisting of three typical types occurs on the pipe. In addition, the pipe can display planar motions, transferring from one plane to the other. The result presented herein is helpful to understand the stabilities and dynamic behaviors of sliding-pipe systems used in aerial refueling applications. [ABSTRACT FROM AUTHOR]

This study deals with the dynamic instability problem of spherical shells composed of carbon nanotubes/fiber/polymer composite (CNTFPC) with delamination around a central cutout. A multiscale analysis using the Hewitt and Malherbe equation was performed to determine the carbon nanotube (CNT) weight ratios, thickness–radius ratios, thickness–length ratios, and delamination area ratios around the cutout. A delamination around a central cutout was modeled in two dimensions by introducing the continuity conditions of displacements at the delamination boundaries. The proposed approach has been verified through a comparison of the results obtained with previous ones. The parametric study showed the significance of a proper CNT ratio and curvature for better structural performance on the dynamic stability of delaminated CNTFPC spherical shells. [ABSTRACT FROM AUTHOR]

The present paper concentrates on the active control of the static bending and dynamic response of a functionally graded piezoelectric material (FGPM) plate subjected to thermo-electro-mechanical loads. Using the first-order shear deformation theory and Hamilton's principle, the equation of motion for the FGPM plate is deduced. Based on the smart properties of piezoelectric ceramic, the active control of the static bending and vibration of the FGPM plate is studied by the mechanical load feedback, velocity feedback and LQR control methods in thermal environment. The effects of load parameter, temperature, volume fraction index and feedback control gain on the static bending and dynamic response of the FGPM plate are examined in detail. The simulation results show that the present control method can largely improve both the static and dynamic stability of the FGPM plate. [ABSTRACT FROM AUTHOR]

A direct eigenvalue analysis approach for solving the stability problem of periodically supported beams with multi-mode coupling vibration under general harmonic excitations is developed based on the Floquet theorem, Fourier series and matrix eigenvalue analysis. The transverse periodic supports are considered for improving the parametrically excited stability of beams under longitudinal periodic excitations. The dynamic stability of parametrically excited vibration of the beam with transverse spaced supports under longitudinal harmonic excitations is studied. The partial differential equation of motion of the beam with spaced supports under harmonic excitations is given and converted into ordinary differential equations with time-varying periodic parameters using the Galerkin method, which describe the parametrically excited vibration of the beam with coupled multiple modes. The fundamental solution to the equations is expressed as the product of periodic and exponential components based on the Floquet theorem. The periodic component and periodic parameters are expanded into Fourier series, and the matrix eigenvalue equation is obtained which is used for directly determining the parametrically excited stability. The dynamic stability of parametrically excited vibration of the beam with spaced supports under harmonic excitations is illustrated by numerical results on unstable regions. The influence of the periodic supports and excitation parameters on the parametrically excited stability is explored. The parametrically excited stability of the beam with multi-mode coupling vibration can be improved by the periodic supports. The developed analysis method is applicable to more general period-parametric beams with multi-mode coupling vibration under various harmonic excitations. [ABSTRACT FROM AUTHOR]

The static stability of slender concrete-filled steel tubular (CFST) columns has been explored thoroughly while few researches have been carried out on the dynamic stability of CFST columns even if all applied loadings are naturally time-dependent. This paper presents an analytical procedure for evaluating the dynamic stability of CFST columns of various composite cross-sections under general boundary conditions. This paper is featured by the following facts: (1) proportional damping is considered in derivation of the governing equations on the lateral parametric vibration of the CFST columns subject to axial excitation; (2) Bolotin's method is used to determine the boundaries of the regions of dynamic instability for the CFST columns with general supports; (3) the relationship of static and dynamic stability, and the effects of boundary conditions and cross-sectional forms are uncovered. New findings of this investigation are (1) larger amplitude or constant component of excitation make it easier for the dynamic instabilities of the CFST columns to occur, while increasing the constant component of excitation reduces the critical value of frequency ratio for the dynamic instability to occur; (2) the dynamic stability analysis can determine the critical loads for both the static and dynamic instability of CFST columns, and the critical instability load decreases with increasing disturbance on the static load; (3) under the same consumptions of steel and concrete, the square columns have better performance of dynamic stability than the circular columns, but there is no definite conclusion on the effect of hollow size on the dynamic stability of double-skin columns. [ABSTRACT FROM AUTHOR]

Subject to various pressure and voltage values, the deformation of a hyperelastic dielectric elastomer membrane may attain different stable and unstable equilibria. In this paper, the neo-Hookean material model is adopted to describe the hyperelastic behavior of a dielectric elastomer membrane. The effects of initial stretch ratio, pressure and voltage on the nonlinear free vibration of a spherical dielectric elastomer balloon are investigated qualitatively and quantitatively. Through a linear stability analysis of the equilibrium states, the safe regime of initial stretch ratio for the deformation of dielectric elastomer balloon is confined. Under specific static driving pressure and voltage, the system oscillates about the stable equilibrium and there is no oscillation in the neighborhood of the unstable equilibrium. Besides, the critical pressure and voltage values are determined. Beyond the critical values, there is no periodic oscillation. Along with the stability analysis, complex dynamical behavior such as drastic change of output regime, sporadic instability and sudden bifurcations can be predicted. By applying the Newton Harmonic Balance (NHB) method for quantitative analysis, the frequency response can be readily predicted. It is found that the nonlinear free vibration frequency decreases with increasing initial stretch ratio and control parameters (pressure and voltage). [ABSTRACT FROM AUTHOR]

The dynamic stability behavior of rotating functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells under combined static and periodic axial forces is investigated. The governing equations are derived based on the first-order shear deformation theory (FSDT) of shells. The initial mechanical stresses due to the steady state rotation of the shell are evaluated by solving the dynamic equilibrium equations. The equations of motion under different boundary conditions are discretized in the spatial domain and transformed into a system of Mathieu–Hill type equations using the differential quadrature method (DQM) together with the trigonometric series. The influences of both the initial mechanical stresses and Coriolis acceleration are considered. Then, the parametric resonance is analyzed and the dynamic instability regions are determined by employing the Bolotin's first approximation. After validating the approach, the effects of rotational speed, Coriolis acceleration, carbon nanotubes (CNTs) distribution in the thickness direction, CNTs volume fraction, length and thickness-to-mean radius ratios on the principal dynamic instability regions are examined in detail. [ABSTRACT FROM AUTHOR]

The modern theory of stochastic dynamic stability is founded on two main exponents: the largest Lyapunov exponent and moment Lyapunov exponent. Since any fractional viscoelastic system is indeed a system with memory, data normalization during iterations will disregard past values of the response and therefore the use of data normalization seems not appropriate in numerical simulation of such systems. A new numerical simulation method is proposed for determining the p th moment Lyapunov exponent, which governs the p th moment stability of the fractional stochastic systems. The largest Lyapunov exponent can also be obtained from moment Lyapunov exponents. Examples of the two-dimensional fractional systems under wideband noise and bounded noise excitations are presented to illustrate the simulation method. [ABSTRACT FROM AUTHOR]

The dynamic stability of vortex-induced vibration (VIV) of circular cylinders has been well investigated. However, there have been few studies on this topic for bridge decks. To fill this gap, this study focuses on the dynamic stability of a VIV system for bridge decks. Some recently developed techniques for nonlinear dynamics are adopted, for example, the state space reconstruction and Poincare mapping techniques. The dynamic stability of the VIV system is assessed by combining analytical and experimental approaches, and a typical bridge deck is analyzed as a case study. Results indicate that the experimentally observed hysteresis phenomenon corresponds to the occurrence of saddle-node bifurcation of the VIV system. Through the method proposed in this study, the evolution of dynamic stability of the VIV system versus wind velocity is established. The dynamic characteristics of the system are further clarified, which offers a useful clue to understanding the VIV system for bridge decks, while providing valuable information for mathematical modeling. [ABSTRACT FROM AUTHOR]

The dynamic stability of bidirectional woven fiber laminated glass/epoxy composite shallow shells subjected to harmonic in-plane loading in hygrothermal environment is considered. An eight-noded isoparametric shell element with five degrees of freedom is used in the analysis. In the present finite element formulation, a composite doubly curved shell model based on first-order shear deformation theory (FSDT) is used for the dynamic stability analysis of shell panels subjected to hygrothermal loading. A program is developed using MATLAB for the parametric study on the dynamic stability of shell panels under the hygrothermal field. The effects of various parameters like static load factor, curvature, shallowness, temperature, moisture, stacking sequence and boundary conditions on the dynamic instability regions of woven fiber glass/epoxy shell panels are investigated. The location of dynamic instability regions is shown to affect significantly due to presence of the hygrothermal field. [ABSTRACT FROM AUTHOR]

In this paper, the energy exponent is used to study the instability of parametrically excited systems governed by the damped Mathieu equation. Through the numerical tests, an energy-growth exponent (EGE) is adopted to evaluate the instability intensity and instability boundary of the system. The EGE can be expressed as a product of the modal frequency and a dimensionless coefficient, defined as the energy-growth coefficient (EGC). Based on the Floquet theory, the relationship between the EGE, Floquet exponent and Lyapunov exponent are derived. An energy criterion of parametric instability is proposed by using the EGE. Using a simple pendulum as an example, the geometric characteristics of the EGC are investigated, and approximate analytical formulae of the EGC/EGE for three different unstable patterns are developed. The EGC/EGE formulae are applicable to the parametrically excited systems governed by the damped Mathieu equation. The unstable behaviors and properties of parametric vibrations are analyzed and discussed in details with EGE/EGC for three systems including a triple pendulum, two-dimensional sloshing fluid, and a two-span continuous beam. The stability boundaries established by using EGE/EGC agree well with those by the conventional theory and experiment. As a practical tool, the EGE/EGC formulae can be easily applied to analyzing the unstable intensities and boundaries of the parametrically excited systems. [ABSTRACT FROM AUTHOR]

During the operation stage, parallel bridges may become nonparallel as a result of unequal load distribution between two parallel bridges and other special conditions. Aerodynamic performance could change significantly under nonparallel positions and become different from that under parallel positions. In this paper, the stationary aerodynamic performance of two parallel bridges under various nonparallel positions during operation stage is studied through a series of wind tunnel tests. This includes the investigation of two horizontal gap distances (HGDs), five relative vertical displacements (RVD) and five relative torsional displacements (RTD). First, sectional models of two closed box girders were tested in smooth flow for stationary aerodynamic force coefficients. An optimum iteration method was then used to calculate the structural displacements and torsional divergence critical wind velocities () of two assumed suspension bridges under stationary aerodynamic force. The research outcomes demonstrated that the changes of stationary aerodynamic force coefficients are dependent on the relative displacements of two girders and wind attack angles. In addition, it was revealed that interference effects are detrimental to stationary aerodynamic instability of two bridges with a larger gap-width ratio (i.e. D/B 1), which is related to the aerodynamic shape of girders and bridge structures. Further, the of the leeward bridge significantly decline when the vertical position of the leeward bridge become higher that of the windward bridge. Most importantly, it showed that the combination of RVD and RTD (e.g. RVD mm and RTD ) could potentially lead to the worst stationary aerodynamic performance by decreasing of the windward and leeward bridge with 12.03% and 7.89%, respectively. [ABSTRACT FROM AUTHOR]

The dynamic characteristics of hybrid composite plates under an arbitrary periodic load in hygrothermal environments are investigated. The material properties of the plate are assumed to be dependent on the temperature and moisture. The governing equations of motion of the Mathieu-type are established based on the Galerkin method with reduced eigenfunction transforms. The periodic stress is taken to be a combination of the pulsating axial and bending stress in the example problems. Based on Bolotin's method, the dynamic instability behaviors of hybrid composite plates are determined. The effects of layer thickness ratio, fiber volume fraction, temperature rise, moisture concentration and dynamic load on the instability regions of hybrid composite plates are studied, along with the dynamic instability index discussed. The results reveal that the layer thickness ratio and hygrothermal conditions have a significant impact on the dynamic instability of hybrid composite plates. [ABSTRACT FROM AUTHOR]

Following the empirical-computational methodology, the contemporary investigations deal with inelastic stability and dynamics of concrete beam-columns. Even under service loads, the concrete structures exhibit physical nonlinearity due to presence of axio-flexural cracks. The objective of the present paper is to analyze the static and dynamic stability of conservative physically nonlinear fully cracked flanged concrete beam-columns. In this paper, using proper reference frames, analytical expressions are developed for the lateral displacement and stiffness of a flanged concrete cantilever under axial compressive and lateral forces. Two critical values of both the axial and lateral loads are identified. For constant lateral force smaller than its first critical value, the concrete beam-columns exhibit brittle buckling mode. Higher lateral forces lesser than the second critical value introduce alternate stable and unstable domains with increase in axial force. The lateral stiffness is predicted to vanish when the axial loads reach the critical values and when the limiting displacement is reached for axial load exceeding its second critical value. The load-space is partitioned into stable and unstable regions. Accessibility of these equilibrium states in the load space has been investigated. Such distinguishing aspects of the predicted behavior of elastic concrete beam-columns are discussed. Their dynamic stability is investigated in second part of the paper. [ABSTRACT FROM AUTHOR]

A nonclassical first-order shear deformation shell model is developed to analyze the axial buckling and dynamic stability of microshells made of functionally graded materials (FGMs). For this purpose, the modified couple stress elasticity theory is implemented into the first-order shear deformation shell theory. Unlike the classical shell theory, the newly developed shell model contains an internal material length scale parameter to capture efficiently the size effect. By using the Hamilton's principle, the higher-order governing equations and boundary conditions are derived. Afterward, the Navier solution is utilized to predict the critical axial buckling loads of simply-supported functionally graded (FG) microshells. Moreover, the governing equations are written in the form of Mathieu-Hill equations and then Bolotin's method is employed to determine the instability regions. A parametric study is conducted to investigate the influences of static load factor, axial wave number, dimensionless length scale parameter, material property gradient index, length-to-radius and length-to-thickness aspect ratios on the axial buckling and dynamic stability responses of FGM microshells. It is revealed that size effect plays an important role in the value of critical axial buckling load and instability region of FGM microshells especially corresponding to those with lower aspect ratios. [ABSTRACT FROM AUTHOR]

Noncontacting finger seals represent a new noncontacting and compliant seal in gas turbine sealing technology. The compliance and noncontacting nature make this kind of seals fully adaptive to rotor excursions in the radial direction without damaging the seal performance. A new semi-analytical method is developed for characterizing the linearized dynamic performance of noncontacting finger seals. The linearized dynamic characteristics of the gas film are numerically computed using the step jump method and then approximated analytically in the equations of motion using a Prony series. By combining the gas film analytical dynamic characterization with the constitutive model for the dynamic properties of noncontacting finger seals, the dynamic equation of motion for the system is derived in analytical form. The results are presented of the gas film properties, natural transient response to initial conditions, steady-state response to rotor excursion, transmissibility ratios and dynamic stability analysis using the analytical model. It is demonstrated that the steady-state responses from the closed-form solutions agree well with those by the numerical simulation, and the analytical solutions can be used as the benchmark for calibrating the results from numerical analysis. [ABSTRACT FROM AUTHOR]

In the present work, the vibration and dynamic stability of functionally graded ordinary (FGO) pre-twisted cantilever Timoshenko beam has been investigated. Finite element shape functions are established from differential equations of static equilibrium. Expressions for element stiffness and mass matrices are obtained from energy considerations. Floquet's theory is used to establish the stability boundary. The material properties along the thickness of the beam are assumed to vary according to the power law. The effects of power law index and pre-twist angle on the natural frequencies and dynamic stability of the beam have been investigated. Increase in pre-twist angle enhances the stability of the beam for first mode whereas it makes the beam more prone to parametric instability for the second mode. The increase in power law index is found to have a detrimental effect on the stability of the beam. The chance of parametric instability is enhanced with the increase in static load factor. [ABSTRACT FROM AUTHOR]

An energetic criterion for identifying the global dynamic instability of structures subjected to any kind of excitations is proposed in the paper. The concept of dynamic stability for structures is firstly interpreted and distinguished from the stability concept in the Lyapunov sense. It is demonstrated that the dynamic instability depends not only on the properties of the structure, but also relates to the change of external excitations on the structure, which distinguishes the essence of dynamic instability from the pseudo static stiffness criterion. This background leads to a novel energetic criterion with which the first passage of the variation of intrinsic energy over the input energy manifests the dynamic instability of structures. The proposed criterion has been extensively tested and verified in the numerical examples, with its advantages clearly illustrated. [ABSTRACT FROM AUTHOR]

The intrinsic relationship between deterministic system and stochastic system is profoundly revealed by the probability density evolution method (PDEM) with introduction of physical law into the stochastic system. On this basis, stochastic dynamic stability analysis of single-layer dome structures under stochastic seismic excitation is firstly studied via incorporating an energetic physical criterion for identification of dynamic instability of dome structures into PDEM, which yields to sample stability (stable reliability). However, dynamic instability is not identical to structural failure definitely, where strength failure can be experienced not only in the stable structure but also when the structure is out of dynamic stability. It is practically feasible to decouple the stochastic dynamic response of dome structures to be a stable one and an unstable one according to the generalized density evolution equation (GDEE). Consequently, the global failure probability can be investigated separately based on the corresponding independent stochastic response. For unstable failure probability assessment, the failure probability is the unstable probability if the dome's failure is attributed to instability, whereas inverse absorbing is firstly implemented to get rid of the stochastic response before instability and a complementary process is filled in the safe domain immediately to finally assess the probability of strength failure after dynamic instability. [ABSTRACT FROM AUTHOR]

The dynamic instability of functionally graded material (FGM) plates under an arbitrary periodic load is studied. The properties of the functionally graded plates (FGPs) are assumed to vary continuously across the plate thickness according to a simple power law. With the derived Mathieu equations, the dynamic instability regions of the FGPs are determined by using the Bolotin's method. The in-plane periodic load is taken to be a combination of periodic axial and bending stress in the example problems. The influences of the volume fraction index, layer thickness ratio, static and dynamic load on the dynamic instability of ceramic-FGM-metal plates are discussed. The results reveal that the excitation frequency, instability region and dynamic instability index of these plates are significantly affected by the static load, dynamic load, volume fraction index and layer thickness. [ABSTRACT FROM AUTHOR]

This study examines how the vibration absorbers influence the stability of nonlinear flow-solid interaction systems. A novel approach is proposed for the analysis of dynamic stability of the two-dimensional nonlinear system using the internal resonance contour plot (IRCP) and flutter speed contour plot (FSCP). The system considered is a planar rigid-body with plunge and pitch vibrations. The two ends of the body are supported by cubic nonlinear springs with quadratic damping. The vibration absorber attached beneath is also considered as a rigid body, with the mass and position of the absorber adjusted for optimization of vibration reduction. The method of multiple scales (MOMS) is employed to obtain a fixed point solution. The P-method is also adopted to obtain the eigen plot and Poincaré Map for comparison. Finally, both the IRCP with FSCP are cross-referenced to provide guide-lines for identifying the optimal location for the absorber with regard to stability and vibration reduction without the need to alter the framework of the main body. [ABSTRACT FROM AUTHOR]

In this paper, dynamic stability analysis and vibration control for a rotating elastic beam connected with an end mass driven by a direct current (DC) motor is considered. A complete strategy including mathematical modeling, dynamic analysis, vibration controller design and simulation for linear and nonlinear systems are presented. Once the rotating flexible physical system has been described by a set of governing partial differential equations, they are manipulated to achieve an appropriate mathematical format for vibration control system design and computer simulation, respectively. Hamilton principle, Lagrange's equation, assumed-modes and the fourth-order Runge-Kutta methods are applied in the system modeling derivation, descretization, and numerical analysis. The correctness of the numerical results and the characteristic property between mathematics and dynamics are demonstrated as well. Also, a realizable vibration control scheme is developed which not only can stabilize all the vibration modes but also make this rotating elastic beam system efficient for good transient response. [ABSTRACT FROM AUTHOR]

This study examines the dynamic stability regions of damped columns on a Winkler foundation that are subjected to sub-tangentially distributed follower forces. A nondimensionalized equation of motion for the column subjected to linearly distributed follower forces is firstly derived based on the extended Hamilton's principle. A finite element procedure, using Hermitian interpolation functions, is employed to develop the mass matrix, Rayleigh damping matrix, Winkler foundation matrix, elastic and geometric stiffness matrices due to distributed axial forces, and a load correction stiffness matrix to account for sub-tangential follower forces. Subsequently, a time history analysis using the Newmark-β method and an evaluation method for the flutter and divergence loads of the nonconservative system are presented. Finally, the dynamic stability characteristics of the nonconservative system that display the jumping phenomenon in the second flutter load are explored through a parametric study. In particular, how the stable and unstable regions of the undamped and damped Leipholz columns translate with changes in the Winkler foundation stiffness is demonstrated and discussed. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear instability of dished shallow shells under a uniformly distributed load is investigated. The dimensionless governing differential equations for the problem are derived and the equations solved by using the Free-Parameter Perturbation Method with the Spline Function Method. By analyzing the instability modes of dished shallow shells, we obtain the variation rules of the maximum deflection area of initial instability of the uniformly loaded dished shallow shell, and discuss the relationship between the initial instability area and the maximum deflection area of initial instability. These results provide some theoretical basis for engineering design and instability prediction and control of shallow shell structures. [ABSTRACT FROM AUTHOR]

This paper presents the dynamic stability analysis of a spinning, thick, annular disk loaded by a circumferentially distributed frictional traction. Considering the automotive/aircraft disc brake as a representative system for this study, the brake rotor is modeled as a spinning annular disk in the context of the Mindlin's thick plate theory to ensure a more accurate estimation of the eigenvalues when the disk involves high circumferential vibration modes that are often observed in unstable disc brake rotors. The frictional traction is decomposed into in-plane and transverse components. The in-plane component is equilibrated by membrane stresses while the transverse component is a nonconservative follower-type force that is the source of the dynamic instability of the disk. The corresponding nonconservative eigenvalue problem is formulated based on the modal expansion theory for traveling waves and direct discretization technique. The brake pad or stator is modeled as a second order viscoelastic subgrade that reacts to both transverse and shearing motion of the disc brake rotor. The instability behavior of the disk is investigated by examining the resulting complex eigenvalues under various combinations of the system para-meters such as frictional traction, geometry of the disk, and viscoelastic properties of the pad lining. In order to assess the effects of shear deformations and rotary inertia of the disk model under such loading conditions, results are compared with those from the classical Kirch-hoff--Love's thin disk model. Based on the phase angle information of the eigenvalues, it is found that there exists a critical value of friction, slenderness ratio of the disk, and the size of pad lining at which the transverse amplitude growth rates of high circumferential vibration modes drastically increase. Unless damping is present in the disc brake rotor or pad lining, vibration modes with repeated eigenvalues are found to be always unstable either by means of divergence or flutter even for a very small nonconservative traction load. [ABSTRACT FROM AUTHOR]

This article presents the evaluation of static and dynamic behavior of functionally graded ordinary (FGO) beam and functionally graded sandwich (FGSW) beam for pined-pined end condition. The variation of material properties along the thickness is assumed to follow exponential and power law. A finite element method is used assuming first order shear deformation theory for the analysis. The element chosen is different from the conventional elements as the shape functions of the element are obtained from the exact solution of the static part of governing differential equation derived according to Hamilton's principle. Moreover, the shape functions depend on length, cross-section and material properties which ensure better accuracy of the solution. The effect of power law index on critical buckling load and natural frequencies of FGO beam is investigated. The critical buckling load of FGO beam with steel-rich bottom increases with power law index whereas the trend reverses for beam with Al-rich bottom. The first three natural frequencies of FGO beam are found to decrease to a minimum value and then increase as the power law index increases from one. The dynamic stability of FGO beam with steel-rich bottom is found to be more than that of beam with Al-rich bottom. An FGSW beam with alumina as bottom skin, steel top skin, and mixture of alumina and steel as core is chosen for analysis. The critical buckling load of FGSW beam increases with the increase of core thickness for variation of material properties in core as per power law with index more than one, whereas it decreases with the increase of core thickness for variation of material properties in core as per exponential law. The first three natural frequencies of the FGSW beam increase with the increase of FGM core for both the types of property distributions. The dynamic stability of the FGSW beam is enhanced as the thickness of the FGM core is increased. [ABSTRACT FROM AUTHOR]

A mathematical model suitable for static and dynamic analyses of curved-in-plane cable-stayed bridges is proposed. By expressing the tensile forces of the cables in relation to the deck and pylon deformations, the problem is reduced to the solution of a beam curved-in-plane that is subjected to the usual permanent and external loads and to the tensile forces of the cables, the latter being functions of the deformation of the beam. The theoretical formulation presented is based on a continuum approach, which is suitable for the three-dimensional (3D) analysis of long span cable-supported bridges. Numerical examples will be analyzed to illustrate the applicability of the proposed approach. [ABSTRACT FROM AUTHOR]

Presented herein is a modified Galerkin discretization procedure for determining the qualitative dynamic behavior of elastic cantilevers with internal damping under partial follower step loading at their tips. For this strong nonlinear nonconservative system, the scheme proposed makes use of basic functions that are a product of nonlinear corrections of approximate linear shape functions. These corrected modes are computed in a way that all the nonlinear nonhomogeneous boundary conditions of the actual problem are satisfied throughout the motion. Numerical results obtained using a two-mode approach are found to be in very good qualitative agreement with the finite element results presented in the literature, not only in the vicinity of the critical states, but also in remote unstable domains. The effect of variation of initial conditions is also investigated and the advantages of the proposed procedure compared with conventional ones are discussed. Further research is required for establishing its capabilities and the range of its applicability for a broader class of nonconservative dynamic problems. [ABSTRACT FROM AUTHOR]

It is well-known that the domains of static stability and dynamic stability (even for a linear approach) do not match each other when the system is no more conservative and the dynamic approach is usually privileged, meaning that the dynamic stability domain is included in the static one. Following previous works proposing a new criterion of static stability of nonconservative systems and prolonging a paper of Gallina devoted to linear dynamic instability (flutter), we show in this paper some remarkable relations between the two approaches: contrary to the common thought, the new static stability criterion implies partially the dynamic one. [ABSTRACT FROM AUTHOR]

This paper deals with the stability of the pylons of a cable-stayed bridge under the action of time-dependent loads, due to the vibration of the bridge deck. The stability of such problems of cable-stayed bridges is solved by a technique developed in the Laboratory of Metal Structures and Steel Bridges, of National Technical University of Athens (NTUA), as well as Bolotin's technique for the solution of nonlinear problems of dynamic stability. Three cases are studied: pylons with damping, pylons under forced vibration, and pylons subjected to an arbitrary external dynamic load. Useful relations are established by the aforementioned solution method, examples for a variety of pylons are presented, and interesting results regarding the stability of each case are given in diagrams. [ABSTRACT FROM AUTHOR]

The present study deals with the dynamic stability of laminated composite pre-twisted cantilever panels. The effects of various parameters on the principal instability regions are studied using Bolotin's approach and finite element method. The first-order shear deformation theory is used to model the twisted curved panels, considering the effects of transverse shear deformation and rotary inertia. The results on the dynamic stability studies of the laminated composite pre-twisted panels suggest that the onset of instability occurs earlier and the width of dynamic instability regions increase with introduction of twist in the panel. The instability occurs later for square than rectangular twisted panels. The onset of instability occurs later for pre-twisted cylindrical panels than the flat panels due to addition of curvature. However, the spherical pre-twisted panels show small increase of nondimensional excitation frequency. [ABSTRACT FROM AUTHOR]

In this paper, static and dynamic stabilities of a cantilever laminated composite beam, having a linear translation spring as elastic support whose position is changeable from the free end to midspan of the beam, subjected to periodic vertical end loading, are examined. The beam is assumed to be an Euler beam and the finite element model used can accommodate symmetric and antisymmetric lay-ups. Solutions referred to as combination resonance are investigated for the dynamic stability analysis. The effects of length-to-thickness ratio, the variation of cross-section in one direction, orientation angle, static and dynamic load parameters, stiffness and position of the elastic support on stability are examined. [ABSTRACT FROM AUTHOR]

Using Fourier series as shape functions practically eliminates the ill-conditioning problem associated with high-order polynomials in structural analyses as confirmed by the newly constructed condition number diagram. This paper extends the Fourier p-elements to study the dynamic stability of frame structures. Both conservative and follower forces are considered. The results compare very well to the exact method of dynamic stiffness. It is found that the Fourier p-elements outperform the dynamic stiffness method in terms of versatility in applications and numerical stability at the very low and high ends of the frequency spectrum. New results of follower tension are given. Follower tension buckling under uniformly distributed follower tension is originally reported. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the vibration, buckling and dynamic instability behavior of laminated composite, cross-ply, doubly-curved panels with a central circular hole subjected to in-plane static and periodic compressive loads. A generalized shear deformable Sanders' theory is used to model the curved panels, considering the effects of transverse shear deformation and rotary inertia. Bolotin's approach is used for studying the dynamic instability regions of doubly-curved panels. The effects of non-uniform edge loads, curvature with different cutout ratios, static and dynamic load factors, and lamination parameters on curved panels are investigated with the results discussed. [ABSTRACT FROM AUTHOR]