Flutter of a viscoelastic strip

A new dimensionless parameter is chosen in the exact solution of the problem under study. Depending on this parameter, the critical flutter velocity can range from instantaneous modular to maximum modular.     

Flutter of a viscoelastic plate in a supersonic gas flow

The flutter of a viscoelastic plate in a supersonic gas flow is studied. A technique and algorithm for numerical solution of nonlinear integro-differential equations with weakly singular kernels are elaborated. The critical flutter speed of viscoelastic plates is determined     

Dynamic crack propagation in a viscoelastic strip

The dynamic propagation of a semi-infinite crack in a finite linear viscoelastic strip subjected to Mode I loading is investigated. Through the use of integral transforms the problem is reduced to solving a Wiener-Hopf equation. The asymptotic properties of the transforms are exploited to establish the stress intensity factor. Plane-stress and plane-strain stress intensity factors as a function of crack speed for both fully-clamped and shear-free lateral boundaries are presented for the standard linear viscoelastic solid. Comparisons are made with previously obtained asymptotic stress intensity factors and with stress intensity factors for the equivalent elastic strips.     

Flutter of an elastic strip in a longitudinal supersonic flow

An elastic strip in a longitudinal supersonic gas flow is considered. An excess pressure formula is obtained on the basis of the linearized theory of supersonic potential flow. It is shown that, in the case of high-speed supersonic flow, the critical flutter velocity is equal to the phase velocity of perturbation waves propagated along the strip. In the framework of the classical piston theory, the flutter problem is solved for the case of a rigidly fixed strip in a longitudinal flow.     

Experimental analysis of rolling contact stresses in a viscoelastic strip

An experimental approach to two-dimensional, viscoelastic, steadily moving rolling contact is described. The photoviscoelastic technique is employed for the analysis of rolling contact stresses between a viscoelastic plate and a rigid rolling cylinder in which the principal axes of stress, strain and birefringence are not coincident with each other. Using an elliptically polarized white light, the distribution of isochromatic fringe order and the principal axes of birefringence at an instant are determined from a single photoviscoelastic image. The time variations of the differences of the principal stresses and strains, as well as their directions, are obtained by use of the optical constitutive equations of photoviscoelasticity. The experimental results involving the time variation of the stresses around the contact surface and their distributions are analyzed.     

Constant velocity motion of a strip die on the boundary of a viscoelastic base

We consider a contact problem on the interaction of a rigid strip die with the boundary of a viscoelastic base. We assume that the die moves at a constant velocity on this boundary and is indented into it by a constant normal force. Friction in the die—surface contact region is neglected. The die base is corrugated in the direction perpendicular to the direction of motion. At the first stage, we determine the displacement of the base boundary due to the normal load applied to it. Then, at the second stage, we derive the integral equation of the contact problem for determining the contact pressure. At the third stage, we construct an approximate solution of this integral equation by using the modified Multhopp—Kalandiya method.     

Stability analysis of a fractional viscoelastic plate strip in supersonic flow under axial loading

The stability of a viscoelastic plate strip, subjected to an axial load with the Kelvin–Voigt fractional order constitutive relationship is studied. Based on the classical plate theory, the structural formulation of the plate is obtained by using the Newton’s second law and the aerodynamic force due to the fluid flow is evaluated by piston theory. The Galerkin method is employed to discretize the equation of motion into a set of ordinary differential equations. To determine the stability margin of plate the obtained set of ordinary differential equations are solved using the Laplace transform method. The effects of variation of the governing parameters such as axial force, retardation time, fractional order and boundary conditions on the stability margin of fractional viscoelastic panel are investigated and finally some conclusions are outlined.     

Flutter of a rectangular plate

We address theoretically the linear stability of a variable aspect ratio, rectangular plate in a uniform and incompressible axial flow. The flutter modes are assumed to be two-dimensional but the potential flow is calculated in three dimensions. For different values of aspect ratio, two boundary conditions are studied: a clamped-free plate and a pinned-free plate. We assume that the fluid viscosity and the plate viscoelastic damping are negligible. In this limit, the flutter instability arises from a competition between the destabilising fluid pressure and the stabilising flexural rigidity of the plate. Using a Galerkin method and Fourier transforms, we are able to predict the flutter modes, their frequencies and growth rates. The critical flow velocity is calculated as a function of the mass ratio and the aspect ratio of the plate. A new result is demonstrated: a plate of finite span is more stable than a plate of infinite span.     

Vibrational heating of a nonlinearly viscoelastic rod interacting with a viscoelastic element

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 5, pp. 95–102, May, 1991.     

Theoretical simulation of transient viscoelastic flows in a viscoelastic system

The flow of an Oldroyd fluid exposed to a sudden arbitrary time-dependent pressure is studied in a circular tube prolonged by a viscoelastic enclosure. the momentum equations are solved using a finite difference scheme. The results presented here reveal the influence of different parameters upon the transient flow.     

Flutter and resonances of a flag near a free surface

We investigate the effects of a nearby free surface on the stability of a flexible plate in axial flow. Confinement by rigid boundaries is known to affect flag flutter thresholds and fluttering dynamics significantly, and this work considers the effects of a more general confinement involving a deformable free surface. To this end, a local linear stability is proposed for a flag in axial uniform flow and parallel to a free surface, using one-dimensional beam and potential flow models to revisit this classical fluid–structure interaction problem. The physical behaviour of the confining free surface is characterized by the Froude number, corresponding to the ratio of the incoming flow velocity to that of the gravity waves. After presenting the simplified limit of infinite span (i.e. two-dimensional problem), the results are generalized to include finite-span and lateral confinement effects. In both cases, three unstable regimes are identified for varying Froude number. Rigidly-confined flutter is observed for low Froude number, i.e. when the free surface behaves as a rigid wall, and is equivalent to the classical problem of the confined flag. When the flow and wave velocities are comparable, a new instability is observed before the onset of flutter (i.e. at lower reduced flow speed) and results from the resonance of a structural bending wave and one of the fundamental modes of surface gravity waves. Finally, for large Froude number (low effect of gravity), flutter is observed with significant but passive deformation of the free surface in response of the flag’s displacement.     

Numerical Investigation of the Flutter of a Rectangular Plate

The flutter of a rectangular plate with an arbitrary direction of the velocity vector relative to the plate side is studied. A numerical no-saturation algorithm is constructed to solve the eigenvalue problem. Calculation results for the critical flutter velocity and corresponding eigenmodes are given.     

Flutter suppression of a plate-like wing via parametric excitation

The present work is motivated by the well known stabilizing effect of parametric excitation of some dynamical systems such as the inverted pendulum. The possibility of suppressing wing flutter via parametric excitation along the plane of highest rigidity in the neighborhood of combination resonance is explored. The nonlinear equations of motion in the presence of incompressible fluid flow are derived using Hamilton's principle and Theodorsen's theory for modeling aerodynamic forces. In the presence of air flow, the bending and torsion modes possess nearly the same frequency. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the neighborhood of combination resonance, the nonlinear response is determined using the multiple scales method at the critical flutter speed and at slightly higher airflow speed. The domains of attraction and bifurcation diagrams are obtained to reveal the conditions under which the parametric excitation can provide stabilizing effect. The basins of attraction for different values of excitation amplitude reveal the stabilizing effect that takes place above a critical excitation level. Below that level, the response experiences limit cycle oscillations, cascade of period doubling, and chaos. For flow speed slightly higher than the critical flutter speed, the response experiences a train of spikes, known as ‘firing,’ a term that is borrowed from neuroscience, followed by ‘refractory’ or recovery effect, up to an excitation level above which the wing is stabilized. The results of the multiple scales method are verified using numerical simulation of the original nonlinear differential equations.     

Flutter of a Wide Strip Plate in a Supersonic Gas Flow

The stability of an elastic plate in the form of a wide strip in a supersonic inviscid gas flow is investigated in the linear approximation. An expression for the dependence of the pressure on the plate deflection, asymptotically exact for wide plates, is used. Two qualitatively different instability types are obtained: flutter with respect to a single oscillatory mode due to negative aerodynamic damping and flutter of a related type due to the interaction of oscillatory modes. For each type the stability criterion and the frequency at which the oscillation amplitude grows most intensely are found.     

Break-up of a viscoelastic liquid jet

Applying Green's continuum theory of a slender body, the process of liquid jet break-up is analysed for a viscoelastic upper-convected Jeffreys fluid. In contrast to a Newtonian liquid an enforced growth of the perturbation is received from a linear analysis. A non-linear numerical analysis shows the viscosity-dependent filament formation between growing droplets of the viscoelastic liquid. The radius of these filaments decreases in an uniaxial extensional flow.     

Stability of a viscoelastic inhomogeneous shell

Shells are considered for which the constants of viscoelasticity are random functions of curvilinear coordinates of the middle surface. Correlation functions are obtained for the first approximations of the deflection and the stress function, as well as the variance of the critical time.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 109–113, January–February, 1972.     

Rivulet flow of a viscoelastic liquid

A theoretical investigation is made of the rivulet flow of a viscoelastic liquid down an inclined plane. It is shown that elasticity causes a significant change in the shape of the rivulet, with height rise at the center. There is also a change in the relationship between the flow rate and the geometry of the rivulet. Elasticity is found to cause a flow in the cross-sectional plane in the form of counter-rotating vortices. In the Newtonian case the flow is purely axial.     

Attractors of a rotating viscoelastic beam

We investigate the non-linear oscillations of a rotating viscoelastic beam with variable pitch angle. The governing equations of motion are two coupled partial differential equations for the longitudinal and transversal displacements. Using a perturbation technique and Galerkin's projection, we reduce the equations of motion to a non-autonomous ordinary differential equation. Our regular perturbation technique is based on the expansion of longitudinal displacement and the amplitude of first transversal mode in terms of a small parameter. We numerically generate the Poincaré maps of the reduced equations and reveal that the system exhibits regular and chaotic attractors. The regular attractors are stable limit-cycles that are relevant to stable, short-period oscillations of the beam. A bifurcation analysis has also been performed when the pitch angle is constant.