Showing papers in "International Journal of Non-linear Mechanics in 2016"

Flow of a micropolar fluid due to a porous stretching sheet and heat transfer

Abstract: The present work investigates the micropolar fluid flow due to a permeable stretching sheet and the resulting heat transfer. Unlike the existing numerical works on the flow phenomenon in the literature, the prime interest here is to analytically work out shape of the solutions and identify whether they are unique. Indeed, unique solutions are detected and presented in the exact formulas for the associated boundary layer equations. Temperature field influenced by the microrotation is also mathematically resolved in the cases of constant wall temperature, constant heat flux and Newtonian heating. To discover the salient physical features of many mechanisms acting on the considered problem, it is adequate to have the analytical velocity and temperature fields and also closed-form skin friction/couple stress/heat transfer coefficients, all as given in the current paper. For instance, the practically significant rate of heat transfer is represented by a single formula valid for all three temperature cases.

Characterizing the creep of viscoelastic materials by fractal derivative models

Abstract: In this paper, we make the first attempt to apply the fractal derivative to modeling viscoelastic behavior. The methodology of scaling transformation is utilized to obtain the creep modulus and relaxation compliance for the proposed fractal Maxwell and Kelvin models. Comparing with the fractional derivatives reported in the literature, the fractal derivative as a local operator has lower calculation costs and memory storage requirements. Moreover, numerical results show that the proposed fractal models require fewer parameters, have simpler mathematical expression and result in higher accuracy than the classical integer-order derivative models. Results further confirm that the proposed fractal models can characterize the creep behavior of viscoelastic materials.

Damping for large-amplitude vibrations of plates and curved panels, Part 1: Modeling and experiments

Abstract: Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.

Pantographic 2D sheets: Discussion of some numerical investigations and potential applications

Abstract: The study of generalized continuum models through the numerical investigation of discrete systems considered as an approximation to a homogenized continuum limit is nowadays a well-known research approach in mechanics. In the present paper, a system constituted by a large number of beams interconnected via ideal hinges, called here a pantographic sheet, is considered, and some numerical simulations concerning the static and dynamic analysis of the system are presented and discussed. The observed behavior significantly differs from what one would expect from ordinary first gradient continuum models. Moreover, interesting application possibilities entailed by the specific characteristics of the structure, and in particular by the strong non-linear behavior of the mechanical variables, are discussed.

Damping for large-amplitude vibrations of plates and curved panels, part 2: Identification and comparisons

Abstract: A non-linear identification technique based on the harmonic balance method is presented to obtain the damping ratio and non-linear parameters of isotropic and laminated sandwich rectangular plates and curved panels, subjected to harmonic excitation orthogonal to the surface. The response of structures under consideration is approximated by a single-degree of freedom Duffing oscillator accounting for viscous damping, quadratic and cubic non-linear stiffness. The method uses experimental frequency-amplitude data and a least-squares technique to identify parameters and reconstruct frequency-response curves by spanning the excitation frequency in the neighborhood of the lowest natural frequencies. In particular, an iterative procedure is implemented to construct the mean displacement and identify the damping ratio. Close agreement is seen between the reconstructed non-linear frequency-amplitude curves, the experimental data and the results of the reduced-order model obtained in part 1 of the present study (Alijani et al., 2015 [1] ). The proposed identification technique confirms the very large increase of damping during large-amplitude vibrations, as observed in part 1 of the present study, and demonstrates a non-linear correlation between damping, vibration amplitude and excitation level.

Geometrically nonlinear large deformation analysis of triangular CNT-reinforced composite plates

Abstract: A first known investigation on the geometrically nonlinear large deformation behavior of triangular carbon nanotube (CNT) reinforced functionally graded composite plates under transversely distributed loads is investigated. The analysis is carried out using the element-free IMLS-Ritz method. In this study, the first-order shear deformation theory (FSDT) and von Karman assumption are employed to account for transverse shear strains, rotary inertia and moderate rotations. A convergence study is conducted by varying the supporting size and number of nodes. The effects of transverse shear deformation, CNT distribution and CNT volume fraction on the nonlinear bending characteristics under different boundary conditions are examined.

Harmonic wavelets based response evolutionary power spectrum determination of linear and non-linear oscillators with fractional derivative elements

Abstract: A harmonic wavelets based approximate analytical technique for determining the response evolutionary power spectrum of linear and non-linear (time-variant) oscillators endowed with fractional derivative elements is developed. Specifically, time- and frequency-dependent harmonic wavelets based frequency response functions are defined based on the localization properties of harmonic wavelets. This leads to a closed form harmonic wavelets based excitation-response relationship which can be viewed as a natural generalization of the celebrated Wiener–Khinchin spectral relationship of the linear stationary random vibration theory to account for fully non-stationary in time and frequency stochastic processes. Further, relying on the orthogonality properties of harmonic wavelets an extension via statistical linearization of the excitation-response relationship for the case of non-linear systems is developed. This involves the novel concept of determining optimal equivalent linear elements which are both time- and frequency-dependent. Several linear and non-linear oscillators with fractional derivative elements are studied as numerical examples. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the technique.

A modified LuGre friction model for an accurate prediction of friction force in the pure sliding regime

Abstract: We propose a new friction model based on the well known LuGre friction model that can accurately describe the nature of friction force in the gross sliding regime. The modification is based on the responses observed from a single degree-of-freedom friction-induced vibration system. Numerical analysis shows that the friction curve in the gross sliding regime can only show counter clockwise hysteretic loops without violating other essential features. We then develop a new friction model by modifying the LuGre friction model that can describe both clockwise as well as counter clockwise hysteretic loops in the pure sliding domain.

Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation

Abstract: In this paper, the dynamics of a cantilever beam subjected to harmonic excitations and to the contact of an obstacle is studied with the help of experimental and numerical investigations. The steel flexible structure is excited close to the free end with a shaker and may come into contact with a deformable and dissipative obstacle. A technique for modeling contact phenomena using piece-wise linear dynamics is applied. A finite-dimensional modal model is developed through a Galerkin projection. Concentrated masses, dampers and forces are considered in the equations of motion in such a way that the boundary conditions are those of a cantilever beam. Numerical studies are conducted by assuming finite-time contact duration to investigate the frequency response of the impacted beam for different driving frequencies. Experimental results have been extrapolated through a displacement laser sensor and a load cell. The comparison between numerical and experimental results show many qualitative and quantitative similarities. The novelty of this paper can be synthetized in (a) the development of experimental results that are in good agreement with the numerical implementation of the introduced model; (b) the development of a comprehensive contact model of the beam with an unilateral, deformable and dissipative obstacle located close to the tip; (c) the possibility of accounting for higher modes for the cantilever beam problem, and hence of analyzing how the response varies when moving the excitation (and/or the obstacle) along the beam, and of investigating the effect of the linearly elastic deformability of the built‐in end of the beam; (d) an easy and intuitive solution to the problem of accounting for spatially singular masses, dampers, springs and forces in the motion equations; (e) the possibility of accounting for finite gap and duration of the contact between beam and obstacle.

Stochastic averaging of energy harvesting systems

Abstract: A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Ito equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.

Effect of uniaxial stress on the propagation of higher-order Lamb wave modes

Abstract: On the basis of the non-linear theory of elasticity and the invariant based formulation developed by Ogden, we analyse the effect of homogeneous stress on the propagation of Lamb waves. Using the theory of incremental deformations superimposed on large deformations, we derive the equations governing the propagation of small amplitude waves in a pre-stressed plate. By enforcing traction-free boundary conditions at the surfaces of the plate, we further obtain the characteristic equations for symmetric and anti-symmetric Lamb wave modes and investigate the effect of stress on the phase velocity, i.e. the acoustoelastic effect. A comparison with experimental data exhibits a better correlation than previously published results. The outcomes of this study can be utilised in the development of new techniques for the measurement of applied stresses based on the acoustoelastic effect. In particular, a strong sensitivity of the phase velocity to the applied stress near the cut-off frequencies of higher-order Lamb wave modes is a very promising option, which seems to have been overlooked in previous studies.

Overshoot of the non-equilibrium temperature in the shock wave structure of a rarefied polyatomic gas subject to the dynamic pressure

Abstract: The shock wave structure in a rarefied polyatomic gas is analyzed on the basis of non-linear extended thermodynamics with 6 independent fields (ET6); the mass density, the velocity, the temperature and the dynamic pressure, which permits us to study the shock profile also for large Mach numbers. The first result of this paper is that the shock wave structure is substantially the same as that obtained previously from the linear theory for small or moderately large Mach numbers. Only for very large Mach numbers there exist some differences in the relaxation part of the profile between the model with a non-linear production term and the one with a linear production term. The mathematical reason of this behavior is due to the fact that the non-linear differential system has the same principal part of the linear one. The classical Meixner theory of relaxation processes with one internal variable is fully compatible with the ET6 theory and this fact gives us the explicit expressions of the internal variable and the non-equilibrium temperature in the Meixner theory in terms of the 6 fields, especially, of the dynamic pressure. By using the correspondence relation, the shock wave structure described by the ET6 theory is converted into the variables described by the Meixner theory. It is shown that the non-equilibrium Meixner temperature overshoots in a shock wave in contrast to the kinetic temperature. This implies that the temperature overshoot is a matter of definition of the non-equilibrium temperature.

Bursting phenomenon in a piecewise mechanical system with parameter perturbation in stiffness

Abstract: In this paper the existence condition and generation mechanism of the possible bursting phenomenon in a piecewise mechanical system with different time scales are studied. As an example of mechanical systems, a piecewise linear oscillator with parameter perturbation in stiffness and subject to external excitation is examined. The order gaps between the time scales are considered in the model, which are related to the periodic excitation and the changing rates of the variables. The focus-type periodic bursting oscillation with two time scales is presented, and the corresponding generation mechanism is revealed by using slow–fast analysis method. Furthermore, the analytical solution of piecewise linear subsystem as well as the stability condition of the fast subsystem are explored to explain the transition of bursting behaviors coming from the variation of intrinsic parameter and external excitation. The results about bursting phenomenon and its generation mechanism would provide important theoretical basis on the mechanical manufacturing and engineering practice.

Nonlocal continuum model for large deformation analysis of SLGSs using the kp-Ritz element-free method

Abstract: A geometrically nonlinear large deformation analysis of SLGSs is presented using the element-free kp-Ritz method. Classical plate theory (CLP) is applied to describe the geometrically nonlinear behavior of SLGSs. Nonlocal elasticity theory is incorporated into CLP to take the small-scale effect into consideration. The system nonlinear equations are derived from the Ritz procedure based on the total Lagrangian formulation. The modified Newton–Raphson method and arc-length continuation are employed to solve the nonlinear equations. The efficiency of the element-free kp-Ritz method is verified through comparison with results reported in previous research. Numerical cases are studied to examine the influence of boundary conditions, aspect ratio, side length and nonlocal parameters on the nonlinear large deformation behavior of SLGSs. An interesting phenomenon is observed in that the nonlocal parameter effect is related to the mathematical expression of the transverse load.

Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcott-rotor system

Abstract: This paper investigates the vibration control of a horizontally suspended Jeffcott-rotor system. A nonlinear restoring force and the rotor weight are considered in the system model. The system frequency (angular speed) -response curve is plotted at different values of the rotor eccentricity. The analysis illustrated that the system has a high oscillation amplitude and exhibits some nonlinear behaviors before control. A Proportional-Derivative (PD)-controller is integrated into the system via two pairs of electromagnetic magnetic poles. The nonlinearity due to the electromagnetic coupling is considered in the system model. A second-order approximate solution is obtained by utilizing multiple scales perturbation method. The bifurcation analyses of the controlled system are conducted. The results showed the high efficiency of the controller to mitigate the nonlinear vibrations of the considered system. Numerical simulations are carried out to validate the accuracy of the analytical results. The numerical results confirmed the excellent agreement with the analytical solutions. Then, the optimal working conditions of the system are concluded. Finally, a comparative study with previously published work is reported.

Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation

Abstract: This study analyses the nonlinear transverse vibration of an axially moving beam subject to two frequency excitation. Focus has been made on simultaneous resonant cases i.e. principal parametric resonance of first mode and combination parametric resonance of additive type involving first two modes in presence of internal resonance. By adopting the direct method of multiple scales, the governing nonlinear integro-partial differential equation for transverse motion is reduced to a set of nonlinear first order ordinary partial differential equations which are solved either by means of continuation algorithm or via direct time integration. Specifically, the frequency response plots and amplitude curves, their stability and bifurcation are obtained using continuation algorithm. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear dynamics of axially moving systems.

Steady state dynamics of a non-linear beam coupled to a non-linear energy sink

Abstract: A non-linear, simply supported beam under harmonic excitation coupled to a non-linear energy sink (NES) is considered here. The NES has a non-linear stiffness of order three. Steady state dynamic of the beam is investigated by two different theories of Euler–Bernoulli and Timoshenko. Complex averaging method combined with arc-length continuation is used to achieve an approximate solution for the steady state vibrations of the system based on 1:1 resonance condition. In order to design an optimized NES for the purpose of reducing the vibration amplitude of the beam, the effect of NES parameters on the amplitude of the primary system is investigated by varying the parameters, individually. The results demonstrated a significant reduction in the vibration amplitude of the original system. By illustrating the frequency spectrum, other harmonic components are detected and the steady state dynamic of the non-linear primary system is computed including the higher harmonics. Non-linear dynamic studies such as bifurcation analysis and Poincare׳ sections are also applied in order to study the effect of NES on the vibration behavior of the beam, in a more accurate manner. Numerical simulations confirm the accuracy of the approximate solutions. Robustness of the NES against changes in the amplitude of excitation is also investigated. Also the performance of NES is compared with linear vibration absorber.

Rippling effect on the structural response of electrostatically actuated single-walled carbon nanotube based NEMS actuators

Abstract: Several nonlinear phenomena have shown to have significant effect on the electromechanical performance of single-walled carbon nanotube (SWCNT) based nanoelectromechanical (NEMS) devices. To name few: the van der Waals forces, the Casimir forces, the tip charge concentration and the rippling phenomenon. Some of these effects have been take care of in previous investigation; however, some have been disregarded in the mechanical models suggested for simulation of the SWCNT based structures. In this paper, the influence of rippling deformation on the vibration characteristics of SWCNT based actuators is investigated using a nonlinear Euler-Bernoulli beam theory that incorporates the effect of rippling deformation using an improved function including some correcting terms for the SWCNT curvature (rippling deformation). The influence of the Casimir and the van der Waals attraction forces are considered in the proposed model as well as the size-dependent behavior assuming the so-called Eringen nonlocal elasticity theory. The dynamic response of CNT is investigated based on time history and phase portrait plots of the CNT based nano-actuator. It is shown that the rippling deformation can significantly decrease the static as well as the dynamic pull-in voltage of the SWCNT based actuator. The rippling deformation of SWCNT decreases the dynamic pull-in time as well. Effect of various factors such as the DC actuation load and the Casimir attractive forces on the dynamic stability and the pull-in characteristics of the nano-actuator are examined. Results of the present study are beneficial to accurate design and fabrication of electromechanical CNT based actuators. Comparison between the obtained results and those reported in the literature by experiments and molecular dynamics, verifies the integrity of the present numerical analysis.

Multi-modes approach to modelling of vortex-induced vibration

Abstract: In this work the fluid–structure interactions are considered by investigating a straight but slender pipe interacting with uniform water flow. Two configurations are studied, namely vertically and horizontally positioned pipes, which are modelled as an Euler–Bernoulli beam with flexural stiffness. Both pretension and length-wise mass distribution are considered. The structure is assumed to be moving only in the direction normal to flow (cross-flow motion) hence its in-line motion is neglected. The external fluid force acting on the structure is the result of the action of sectional vortex-induced drag and lift forces. Only mean drag force is considered, with time varying lift force modelled using a non-linear oscillator equation of the Van der Pol type. The obtained coupled system of non-linear partial differential equations is simplified employing Galerkin-type discretisation. The resulting ordinary differential equations are solved numerically providing multi-mode approximations of cross-flow displacement and non-dimensional lift coefficient. The comparison between the responses of vertical and horizontal structures shows that, as expected, due to a balancing between pretension and weight, in general a higher amplitude of vibration is observed for the vertical configuration than in the same location along the pipe for the horizontal configuration in the lower part of the structure. However, lower amplitudes are obtained in the upper part of the pipe. The horizontal configuration solutions are identical in symmetrical locations along the pipe due to constant pretension. The influence of the wake equation coefficients and the fluid force coefficients on the response amplitudes has been also considered together with the length of the pipe and pretension level, and the appropriate response curves are included. Finally, for the higher mode approximations it has been shown that the vibrations level at lower frequencies is predicted reasonably well by retaining only a small subset of modes.

A parametric study on nonlinear flow-induced dynamics of a fluid-conveying cantilevered pipe in post-flutter region from macro to micro scale

Abstract: A mathematical formulation is proposed to investigate the nonlinear flow-induced dynamic characteristics of a cantilevered pipe conveying fluid from macro to micro scale. The model is developed by using the extended Hamilton's principle in conjunction with the inextensibility condition and laminar and turbulent flow profiles as well as modified couple stress theory. The current model is capable of recovering the classical model of cantilevered pipe conveying fluid by neglecting the couple stress effect. The governing equation of motion is presented in dimensionless form in a convenient and usable manner. To solve the problem at hand, the integro-partial-differential equation of motion is discretized into a set of ordinary differential equations via Galerkin method. Afterward, a Runge–Kutta's finite difference scheme is employed to evaluate the nonlinear dynamic response of the cantilevered pipe conveying fluid. A parametric study is carried out to examine the influences of mass parameter and dimensionless mean flow velocity on the nonlinear dynamic characteristics of the cantilevered pipe conveying fluid in post-flutter region. The role of size-dependency in the nonlinear behavior of pipe is explored by converting the new set of dimensionless parameters into the conventional one. Eventually, some convergence studies are performed to indicate the reliability of present results.

An investigation of the static and dynamic behavior of electrically actuated rectangular microplates

Abstract: We present an investigation of the static and dynamic behavior of the nonlinear von-Karman plates when actuated by the nonlinear electrostatic forces. The investigation is based on a reduced order model developed using the Galerkin method, which rely on modeshapes and in-plane shape functions extracted using a finite element method. In this study, a fully clamped microplate is considered. We investigate the static behavior and the effect of different non-dimensional design parameters. The static results are validated by comparison with the results calculated by a finite element model. The forced-vibration response of the plate is then investigated when the plate is excited by a harmonic AC load superimposed to a DC load. The dynamic behavior is examined near the primary and secondary (superharmonic and subharmonic) resonances. The microplate shows a strong hardening behavior due to the cubic nonlinearity of mid-plane stretching. However, the behavior switches to softening as the DC load is increased. Finally, near-square plates are studied to understand the effect of geometric imperfections of microplates.

Referential description of the evolution of a 2D swarm of robots interacting with the closer neighbors: Perspectives of continuum modeling via higher gradient continua

Abstract: In the present paper a discrete robotic system model whose elements interact via a simple geometric law is presented and some numerical simulations are provided and discussed. The main idea of the work is to show the resemblance between the cases of first and second neighbors interaction with (respectively) first and second gradient continuous deformable bodies. Our numerical results showed indeed that the interaction and the evolution process described is suitable to closely reproduce some basic characteristics of the behavior of bodies whose deformation energy depends on first or on higher gradients of the displacement. Moreover, some specific qualitative characteristics of the continuous deformation are also reproduced. The model introduced here will need further investigation and generalization in both theoretical and numerical directions.

A simple nonlinear model to simulate the localized necking and neck propagation

Abstract: This paper deals with the equilibrium problem in nonlinear dissipative inelasticity of damaged bodies subject to uniaxial loading and its main purpose is to show the interesting potentialities offered by the damage theory in modeling the necking and neck propagation phenomena in polymeric materials. In detail, the proposed mechanical model is a two-phase system, with the same constitutive law but with different levels of damage for each phase. Despite its simplicity, it is shown that the model can straightforwardly reproduce the overall load–elongation curve provided by experimental tensile tests by involving only five parameters of clear physical meaning.

Equivalent nonlinear beam model for the 3-D analysis of shear-type buildings: Application to aeroelastic instability

Abstract: Tower buildings can be very sensitive to dynamic actions and their dynamic analysis is usually carried out numerically through sophisticated finite element models. In this paper, an equivalent nonlinear one-dimensional shear–shear–torsional beam model immersed in a three-dimensional space is introduced to reproduce, in an approximate way, the dynamic behavior of tower buildings. It represents an extension of a linear beam model recently introduced by the authors, accounting for nonlinearities generated by the stretching of the columns. The constitutive law of the beam is identified from a discrete model of a 3D-frame, via a homogenization process, which accounts for the rotation of the floors around the tower axis. The macroscopic shear strain in the equivalent beam is produced by the bending of columns, accompanied by negligible rotation of the floors. A coupled nonlinear shear–torsional mechanical model is thus obtained. The coupling between shear and torsion is related to a non-symmetric layout of the columns, while mechanical nonlinearities are proportional to the slenderness of the columns. The model can be used for the analysis of the response of tower buildings to any kind of dynamic and static excitation. A first application is here presented to investigate the effect of mechanical and aerodynamic coupling on the critical galloping conditions and on the postcritical behavior of tower buildings, based on a quasi-steady model of aerodynamic forces.

Non-linear interactions in the flexible multi-body dynamics of cable-supported bridge cross-sections

Abstract: An elastic section model is proposed to analyze some characteristic issues of the cable-supported bridge dynamics through an equivalent planar multi-body system. The quadratic non-linearities of the four-degree-of-freedom model essentially describe the geometric coupling which may strongly characterize the dynamic interactions of the bridge deck and a pair of identical suspension cables (hangers or stays). The linear modal solution shows that the flexural and torsional modes of the deck (global modes) typically co-exist with symmetric or anti-symmetric modes of the cables (local modes). The combinations of parameters which realize remarkable 2:1:1 internal resonance conditions among one of the global modes (with higher natural frequency) and two local modes (with lower and close natural frequencies) are obtained by virtue of a multiparameter perturbation method. The non-linear response of the resonant systems shows that the global deck motion – directly forced at primary resonance by an external harmonic load – can parametrically excite the local cable motion, when the deck vibration amplitude overcomes the critical value at which a period-doubling bifurcation occurs. The relevant effects of both viscous damping and internal detuning on the instability boundaries are parametrically investigated. All the internal resonance conditions as well as the critical vibration amplitudes are expressed as an explicit, though asymptotically approximate, function of the structural parameters.

Axisymmetric rotational stagnation point flow impinging on a radially stretching sheet

Abstract: The normal impingement of the rotational stagnation-point flow of Agrawal (1957) [8] on a sheet radially stretching at non-dimensional stretch rate β is studied. A similarity reduction of the Navier–Stokes equations yields an ordinary differential equation which is solved numerically over a range of β. A unique solution exists at the turning point β = β t and dual solutions are found in the region β > β t where β t = − 0.657 is the turning point in the parametric shear stress curve separating upper from lower branch solutions. An analysis of solutions near the Agrawal point β = 0 is provided, and the large-β asymptotic behavior of solutions is determined. Sample velocity profiles along both solution branches are presented. A linear temporal stability analysis reveals that solutions along the upper branch are stable while those on the lower branch are unstable.

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

Abstract: The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.

Stochastic damage hysteretic model for concrete based on micromechanical approach

Abstract: In the present paper, a kind of stochastic damage hysteretic model is proposed to describe the damage and hysteretic behaviors of concrete material. According to the model, a parallel system made up of micro-elements, which are developed based on the micro-tensile and shear damage mechanism respectively, is adopted to obtain the overall responses of concrete. The influence of plastic strain and hysteretic energy dissipation of the material are also considered in the model. To reflect the stochastic properties of concrete, the fracture strain of the micro-element is set as a random variable. Then the monotonic, loading and unloading curves of the parallel system are derived analytically by averaging the stochastic micro-elements and two hysteretic rules are combined to the proposed model to account for complicated loading conditions. Furthermore, a nonlocal process is introduced to the model to overcome the mesh dependence issues of softening materials. Finally, several numerical examples are conducted, demonstrating that the proposed model can provide reliable results reflecting the damage, plasticity and hysteretic behaviors of concrete material.

Usefulness of passive non-linear energy sinks in controlling galloping vibrations

Abstract: The suppression of vibration amplitudes of an elastically-mounted square prism subjected to galloping oscillations by using a non-linear energy sink is investigated. The non-linear energy sink consists of a secondary system with linear damping and non-linear stiffness. A representative model that couples the transverse displacement of the square prism and the non-linear energy sink is constructed. A linear analysis is performed to determine the impacts of the non-linear energy sink parameters (mass, damping, and stiffness) on the coupled frequency and onset speed of galloping. It is demonstrated that increasing the damping of the non-linear energy sink can result in a significant increase in the onset speed of galloping. Then, the normal form of the Hopf bifurcation is derived to identify the type of instability and to determine the effects of the non-linear energy sink stiffness on the performance of the aeroelastic system near the bifurcation. The results show that the non-linear energy sink can be efficiently implemented to significantly reduce the galloping amplitude of the square prism. It is also shown that the multiple stable responses of the coupled aeroelastic system are obtained as well as the periodic responses, which are dependent on the considered non-linear energy sink parameters.

The applications of the partial Hamiltonian approach to mechanics and other areas

Abstract: The partial Hamiltonian systems of the form q i = ∂ H ∂ p i , p i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) arise widely in different fields of the applied mathematics. The partial Hamiltonian systems appear for a mechanical system with non-holonomic nonlinear constraints and non-potential generalized forces. In dynamic optimization problems of economic growth theory involving a non-zero discount factor the partial Hamiltonian systems arise and are known as a current value Hamiltonian systems. It is shown that the partial Hamiltonian approach proposed earlier for the current value Hamiltonian systems arising in economic growth theory Naz et al. (2014) [1] is applicable to mechanics and other areas as well. The partial Hamiltonian approach is utilized to construct first integrals and closed form solutions of optimal growth model with environmental asset, equations of motion for a mechanical system with non-potential forces, the force-free Duffing Van der Pol Oscillator and Lotka–Volterra models.