Showing papers in "International Journal of Non-linear Mechanics in 1992"

Magnetohydrodynamic flow of a power-law fluid over a stretching sheet

Abstract: Magnetohydrodynamic flow of an electrically conducting power-law fluid over a stretching sheet in the presence of a uniform transverse magnetic field is investigated by using an exact similarity transformation. The effect of magnetic field on the now characteristics is explored numerically, and it is concluded that the magnetic field tends to make the boundary layer thinner, thereby increasing the wall friction.

Non-linear vibration of a traveling tensioned beam

Abstract: Free non-linear vibration of an axially moving, elastic, tensioned beam is analyzed over the sub- and supercritical transport speed ranges. The pattern of equilibria is analogous to that of Euler column buckling and consists of the straight configuration and of non-trivial solutions that bifurcate with speed. The governing equations for finite local motion about the trivial equilibrium and for motion about each bifurcated solution are cast in the standard form of continuous gyroscopic systems. A perturbation theory for the near-modal free vibration of a general gyroscopic system with weakly non-linear stiffness and/or dissipation is derived through the asymptotic method of Krylov, Bogoliubov, and Mitropolsky. The method is subsequently specialized to non-linear vibration of a traveling beam, and of a traveling string in the limit of vanishing flexural rigidity. The contribution of non-linear stiffness to the response increases with subcritical speed, grows most rapidly near the critical speed, and can be several times greater for a translating beam than for one that is not translating. In the supercritical speed range, asymmetry of the non-linear stiffness distribution biases finite-amplitude vibration toward the straight configuration and lowers the effective modal stiffness. The linear vibration theory underestimates stability in the subcritical range, overestimates it for supercritical speeds, and is most limited in the near-critical regime.

Modal interactions in the non-linear response of elastic cables under parametric/external excitation

Abstract: A theoretical model is derived which describes the non-linear response of a suspended elastic cable to small tangential oscillations of one support. The support oscillations, in general, result in parametric excitation of out-of-plane motion and simultaneous parametric and external excitation of in-plane motion. Cubic non-linearities due to cable stretching and quadratic nonlinearities due to equilibrium cable curvature couple these motion components in producing full, three-dimensional cable response. In this study, a two-degree-of-freedom approximation of the model is employed to examine a class of in-plane/out-of-plane motions that are coupled through the quadratic non-linearities. A first-order perturbation analysis is utilized to determine the existence and stability of the planar and non-planar periodic motions that result from simultaneous parametric and external resonances. The analysis leads to a bifurcation condition governing planar stability and results highlight how planar stability is reduced and non-planar response is enhanced whenever a “two-to-one” internal resonance condition exists between a pair of in-plane and out-of-plane cable modes. This two-to-one resonant behavior is clearly observed in experimental measurements of cable response which are also in good qualitative agreement with theoretical predictions.

Hydromagnetic flow of a dusty fluid over a stretching sheet

Abstract: Analysis of hydromagnetic flow of a dusty fluid over a stretching sheet is carried out with a view to throw adequate light on the effects of fluid-particle interaction, particle loading, and suction on the flow characteristics. The equations of motion are reduced to coupled non-linear ordinary differential equations by similarity transformations. These coupled non-linear ordinary differential equations are solved numerically on an IBM 4381 with double precession, using a variable order, variable step-size finite-difference method. The numerical solutions are compared with their approximate solutions, obtained by a perturbation technique. For small values of β the exact (numerical) solution is in close agreement with that of the analytical (approximate) solution. It is observed that, even in the presence of a transverse magnetic field and suction, the transverse velocity of both the fluid and particle G phases decreases with an increase in the fluid-particle interaction parameter, β, or the particle-loading parameter, k . Moreover, the particle density is maximum at the surface of the stretching sheet, and the shearing stress increases with an increase in β or k .

Heat transfer in an electrically conducting fluid over a stretching surface

Abstract: An analysis is carried out to study the heat transfer characteristics in an electrically conducting fluid over a stretching sheet with variable wall temperature and internal heat generation or absorption. Two eases are studied, namely, (i) the sheet with prescribed surface temperature (PST case) and (ii) the sheet with prescribed wall heat flux (PHF case). The solutions for the temperature, the heat transfer characteristics and their asymptotic limits for large Prandtl number (σ) are obtained in terms of Kummer's and parabolic cylinder functions. It is shown that asymptotic limits are not possible for small Prandtl number; this is due to the solution changing by O(1) on length scale of 1/σ. For large Prandtl number, a boundary layer of width 1/σ at η = 0 and an internal layer of width ∝1/σ near the turning point are noticed.

Dynamics of spinup through resonance

Abstract: This work concerns the phenomenon of resonant capture, i.e. the failure of a rotating mechanical system to be spunup to a desired terminal state, due to its resonant interaction with another system or with itself. The phenomenon is important in the dynamics of dual-spin spacecraft. Starting from a simple mechanical system consisting of an unbalanced rotor attached to an elastic support and driven by a constant torque, we derive an abstract model of resonant capture. The model is investigated by using perturbation theory and elliptic functions. For a given system, the analysis predicts which initial conditions lead to capture. These predictions are shown to compare reasonably with the results of numerical integration.

Normal modes and global dynamics of a two-degree-of-freedom non-linear system—I. Low energies

Abstract: The high-energy global dynamics of an undamped, strongly non-linear, two-degree-of-freedom system are considered. As shown in an earlier work [A.F. Vakakis and R.H. Rand, Int. J. Non-Linear Mech. 27 , 861–874 (1992)], the oscillator under consideration contains “similar” non-linear normal modes and at certain values of its structural parameters a mode bifurcation is possible. For low energies, the mode bifurcation gives rise to a homoclinic orbit in the Poincare map of the system. For high energies, large- and low-scale chaotic motions are detected, resulting from transverse intersections of the stable and unstable manifolds of an unstable antisymmetric normal mode, and from the breakdown of invariant KAM-tori. The creation of additional free subharmonic motions is studied by a subharmonic Melnikov analysis, and the stability of the subharmonic motions is examined by an averaging methodology. The main conclusion of this work is that the bifurcation of similar normal modes results in a class of large-scale free chaotic motions, which do not exist in the system before the bifurcation.

A new approximate solution technique for randomly excited non-linear oscillators—II

Abstract: The method of weighted residuals is applied to the reduced Fokker-Planck equation associated with a non-linear oscillator, which is subjected to both additive and multiplicative Gaussian white noise excitations. A set of constraints are deduced for obtaining an approximate stationary probability density for the system response. One of the constraints coincides with the previously proposed criterion of dissipation energy balancing, and the others are useful for calculating the equivalent conservative force. It is shown that these constraints imply certain relationships among certain statistical moments; their imposition guarantees that such moments computed from the approximate probability density satisfy the corresponding exact equations derived from the original equation of motion. Moreover, the well-known procedure of stochastic linearization and its improved version of partial linearization are shown to be special cases of this scheme, and they are less accurate since the approximations are not chosen from the entire set of the solution pool of generalized stationary potential. Applications of the scheme are illustrated by examples, and its accuracy is substantiated by Monte Carlo simulation results.

The boundary layer equations of third-grade fluids

Abstract: Steady, two-dimensional, incompressible boundary layer equations for a fluid of grade three are derived using a special coordinate system. For the inviscid flow around an arbitrary object, the streamlines are the ψ-coordinates and velocity potential lines are the ψ-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, are then shown to be independent of the body shape immersed into the flow. In deriving the boundary layer equations, method of matched asymptotic expansion is used. Then, it is shown that the equations do not have similarity solutions. Finally, the shear stress on the boundary for the coordinate system is also calculated.

Response distribution of non-linear systems excited by non-Gaussian impulsive noise

Abstract: A perturbation scheme is devised to obtain an approximate stationary probability density for the response of a non-linear oscillator subjected to an impulsive-noise process which is statistically stationary but non-Gaussian. The excitation can be either additive or multiplicative. The effect of non-Gaussianity in the excitation process is found to be negligible if the product of the average arrival rate of the impulses and the relaxation time of the oscillator is of an order of 10 or greater. When this is not the case, the effect is not uniform; it may be more pronounced for one dynamic system than that of another. Numerical results are obtained for two examples, and they compare well with Monte Carlo simulations.

Prediction of escape from a potential well under harmonic excitation

Abstract: Non-linear dynamical systems are considered which are conservative except for the presence of viscous damping. The systems are initially at rest in a potential well. Harmonic excitation is applied, and approximate conditions for escape from the potential well are sought. A method is described which uses the harmonic-balance technique to obtain an approximate motion, and then compares the maximum energy during that motion with the potential barrier. Numerical results are presented for three one-degree-of-freedom systems and a system with two-degrees-of-freedom. The approximate critical forcing amplitudes are plotted as a function of the forcing frequency, and the results are compared to those obtained by numerical integration of the equations of motion.

Thermoelasticity at cryogenic temperatures

Abstract: A theory of thermoelasticity is developed which is suitable for application at cryogenic temperatures. The thermodynamic functions are so chosen as to give correct predictions with experimental findings of second-sound wave speeds in NaF, in the temperature range 8–20 K and in Bi, in the temperature range 1–4 K. After the non-linear theory is presented, a linearized theory is developed. A uniqueness theorem is provided for the linearized theory on an unbounded domain. The question of a half-space heated on its boundary is addressed and, in particular, the question of transverse elastic wave propagation is studied. Finally, a simplified theory in which only unidirectional solutions are allowed is examined.

A generalization to stochastic averaging in random vibration

Abstract: Stochastic averaging is applied to a class of randomly excited single degree-of-freedom oscillators possessing linear damping and non-linear stiffness terms. The assumed excitation form involves an externally applied evolutionary Gaussian stochastic process. Special emphasis is placed on casting the problem in a more formal mathematical framework than that traditionally used in engineering applications. For the case under consideration, it is shown that a critical step involves the selection of an appropriate period of oscillation over which the temporal averaging can be performed. As an example, this averaging procedure is performed on a Duffing oscillator. The validity of the derived result is partially confirmed by reducing it to a special case, for which there is a known solution, and comparing both solutions.

On the paradox of the destabilizing effect of damping in non-conservative systems

Abstract: The nature of the critical states of a follower-type non-conservative elastic system with or without damping, is reexamined using a non-linear dynamic analysis. It is found that the destabilizing effect due to certain ratios of damping coefficients, regarded for a long time as a “paradox”, is associated with globally stable critical states. However, in the case of equal damping coefficients, the increase of damping increases the critical load (stabilizing effect). It is also established that this system is statically and dynamically stable for the entire region of variation of the non-conservativeness load parameter, contrary to existing analyses. Moreover, it is deduced that the critical states corresponding to both types of instability may become unstable if a slight material non-linearity is included; then, the mechanism of divergence and flutter instability change from stable to unstable and vice versa for a critical value of the material non-linearity which depends on the non-conservativeness loading parameter.

Non-linear optimal control of a Duffing system

Abstract: The Duffing oscillator is a useful current model for the behavior of structural systems including columns, gyroscopes, plates, pendulums and certain types of bridges. This paper studies a class of non-linear optimal controls for these oscillators, and determines the effects of higher-order feedback corrections based upon series expansions of the optimal cost function and the optimal control function in a Hamilton-Jacobi context. A novel representation of the solution is presented, in terms of the indicial formulation of tensor algebra. For the case in point, the indicial approach offers a conceptually attractive alternative to the general tensor solutions discussed by Buric in 1978 and by O'Sullivan and Sain in 1985. Numerical studies are provided, and an analysis of the effect of higher-order feedbacks upon the stability region of the controlled system is presented.

Self-similar solutions of the generalized Emden-Fowler equation

Abstract: When, in the generalized Emden-Fowler equation y” + f ( x ) y n = 0, the function f ( x ) takes certain forms, the equation possesses one or two Lie point symmetries and in some cases closed-form solutions can be obtained.

“Cubication” of non-linear oscillators using the principle of harmonic balance

Abstract: A new method is given of “cubication” of autonomous non-linear oscillators (NLO) of the class ẍ + c1x + c3x3+ ϵ[g(x)+ tf(x)ẍ] i.e.of constructing the cubic oscillator ẍ + λ∗ẍ + c1∗x + c3∗x3= 0 from the NLO. The solution, limit cycles, bifurcations, fixed points, and stability of this NLO are approached by studying its associated cubic oscillator which is equal at least in the largest harmonics (principle of harmonic balance) and by assuming as a first approximation a solution for the NLO problem in terms of Jacobian elliptic functions. When c3= 0, the elliptic functions become circular functions and the present method reduces to the well-studied harmonic-balance method of linearization. The present method is equivalent to a third-order Chebyshev expansion of the NLO force if this is conservative. For a dissipative NLO, it gives the position and features of limit cycles and bifurcations.

The symmetry group of second-grade materials

Abstract: The mathematically correct form of the material symmetry group in the second-grade elasticity is obtained. Its possible ramifications for the theory of continuous distributions of disclinations are explored.

A Recurrence Formula for Viscoelastic Constitutive Equations

Abstract: The viscoelastic constitutive equations are generally represented by integral equations with kernels. These kernels are functions of current time, an integration limit of the hereditary integral. Therefore, the values of these kernels change as the time increases and the integral must be evaluated from time equals zero to the current time for every increment of time. Thus, as time increases, the required computing time becomes longer and longer. Furthermore, all physical values from time equals zero to the current time must be stored for later evaluations of these integrals. Additionally, for finite deformation viscoelastic problems, the constitutive equation is an integral part of the equilibrium equations that result in a set of nonlinear differential-integral equations. These equations usually can only be solved numerically and iteratively. Hence, computing time and data storage are the main concerns in solving finite deformation viscoelastic problems. The main object of this paper is to develop a method that saves both computing time and data storage in evaluating these integral equations.

Equivalent lagrangians and the solution of some classes of non-linear equations ?

Abstract: The second-order ordinary differential equation q + p(t)q + r(t)q = μq2q−1 + f(t)qn, where μ ≠ 1 is linearizable(sl(3, R) algebra) via a point transformation if and only if n = μ or n = 1. We construct a quadratic Lagrangian L = 12A(t,q)q2 + B(t,q), which determines the point transformation Q = F(t,q) and = G(t,q) that maps the Lagrangian to the simple completely integrable Lagrangian L = case 12Q2. For n = 4μ − 3 the equation admits the sl(2, R) algebra. In this case we again construct a quadratic Lagrangian and then obtain the corresponding point transformation that reduces the original Lagrangian to the representative Lagrangian L = QT + 1/(2TQ). For both cases, sl(2,R) and sl(3,R), we obtain complete solutions (cf. [1,2]).

Similarity solutions to some unsteady flows of non-Newtonian fluids of power law behavior

Abstract: The question of the non-linear effects in some unsteady flows of non-Newtonian fluids is addressed. Exact similarity solutions in closed form are presented for Stokes' first problem, for Couette rotating flow, for the spreading of a thin fluid layer and for unsteady flow through a porous medium. From these solutions the conditions for the existence of traveling-wave characteristics are found for the velocity, shear stress and pressure distributions.

Stochastic linearization of MDOF systems under parametric excitations

Abstract: The stochastic linearization approach is examined for non-linear systems subjected to parametric type excitations. It is shown that, for these systems too, stochastic linearization and Gaussian closure are two equivalent approaches if the former is applied to the coefficients of the Ito differential rule. A critical review of other stochastic linearization approaches is also presented and discussed by means of simple examples.

Non-linear multi-degree-of-freedom system random vibration by equivalent statistical quadratization

Abstract: The statistical linearization method is often inadequate for estimating spectral properties of random responses of non-linear systems. This is sometimes due to the fact that the power spectra of responses of linear systems span only the frequency range of the excitation spectrum, whereas significant responses outside this range are possible for non-linear systems. Recently, the concept of the statistical “quadratization” method was introduced to address this shortcoming of the linearization methods. The effectiveness of statistical quadratization was demonstrated on several single-degree-of-freedom systems. In this paper the method is generalized to multi-degree-of-freedom systems. The statistical quadratization solution procedure involves replacing the non-linear system by an “equivalent” system with polynomial non-linearities up to quadratic order. The non-linear equivalent system has a form whose solutions can be approximated by using the Volterra series method. The non-Gaussian joint response probability distribution is approximated by a third-order Gram-Charlicr expansion. The method is formulated for systems with general non-linearities and with non-linearities of a special form. To demonstrate the usefulness of the method, solutions are obtained for a specific system. The corresponding results compare well with Monte Carlo simulation data. Further, it is shown that the quadratization method is notably superior to the linearization method for the considered system.

Buckling and post-buckling behaviour of laminated shallow spherical shells subjected to external pressure

Abstract: The present paper deals with the static buckling and post-buckling behaviour of clamped elastic laminated shallow spherical shells subjected to a uniform external pressure Applying the Rayleigh-Ritz procedure to Marguerre's equations combined with precise pre-buckling numerical analysis, reasonably accurate solutions are obtained for buckling upper and lower pressures The effects of fibre orientations on pre- and post-buckling behaviour, buckling loads and modes are considered The results for composite shells are compared with those calculated for quasi-isotropic ones

Gyroscopic stability and its loss in systems with two essential coordinates

Abstract: Cyclic systems with two essential coordinates are studied. The particular case is investigated, when a steady motion of the system is neutrally stable in linear approximation due to gyroscopic effects on an originally unstable equilibrium. Conditions are given to ensure the preservation of stability for the non-linear conservative system. In addition, a universal loss of stability is proved as a result of non-total dissipation or acceleration acting on the system.

Effect of non-linearities in elastomeric material dampers on torsional vibration control

Abstract: The effect of non-linearities in clastomeric-material dampers used to quiet torsional oscillations of internal-combustion engines shafts is investigated. The method of multiple scales is used to solve the equations for the case of primary resonance. Steady-state solutions are obtained and their stability examined. A relationship that relates the elastomer properdes (for stable solutions) and other system parameters is found. The steady-state response shows a softening behavior of the system although the actual material non-linearity is of the hardening type.

Dynamic buckling of a simple elastic-plastic model under pulse loading

Abstract: The dynamic buckling of an elastic-plastic imperfection-sensitive model subjected to rectangular- and triangular-shaped loading pulses is examined to provide some insight into the dynamic buckling behaviour of structures. The loading pulse is expressed as a function of the horizontal displacement, which allows an analytical method to be used for determining the stability domains for both pulses. The estimates obtained are compared with some previously published results on the dynamic elastic-plastic buckling of the same model under a step loading. It transpires that for the pulse loading of models with large imperfections dynamic instability occurs either elastically or plastically depending on the pulse duration, while for a step loading only an elastic instability is possible for the parameters examined.

Sedimentation in a bounded column

Abstract: A non-linear model of a sedimentation process in a bounded column is analysed by means of a reciprocal transformation. Reduction is thereby made to an associated linear moving boundary problem and thence to a standard rigid boundary-value problem by an appropriate in variance transformation of the heat equation. An exact solution, which serves also as an appropriate small-time solution for the original non-linear boundary-value problem, is presented.

Large deformations under axial force and moment loads of initially flat annular membranes

Abstract: Finite deformations of homogeneous, isotropic, incompressible initially flat, unstressed annular membranes under axisymmetric axial force and moment loads are studied. The general equilibrium equations and boundary conditions were established through the principle of stationary potential energy and the coupled, non-linear differential equations solved numerically, to obtain the angle of twist, the deflection and the radial deformation at every point of the membrane. Two strain energy densities are considered: neo-Hookean, for which it is possible to obtain a closed approximated solution, and Mooney-Rivlin. The results are prese'nted in non-dimensional form.

Dynamic buckling of composite cylindrical panels with high-order transverse shears subjected to a transverse concentrated load

Abstract: The characteristics of dynamic buckling of a geometrically non-linear cylindrical laminated composite panel subjected to a transverse concentrated step load applied at the center is studied. Attention is focused on the dynamic stability of a finite element discrete structural system. The sufficient condition for dynamic buckling, from the energy transfer consideration, is defined as the smallest load for which an unbounded motion is initiated at one generalized displacement. In other words, the dynamic buckling load associated with that generalized coordinate can be predicted by the intersecting point on the static equilibrium curve and the zero potential energy curve. This dynamic buckling criterion can also be observed by the existence of an inflection point on the generalized displacement response curve. Considering the multi-degree-of-freedom for the entire structure without damping, the dynamic buckling criterion used in a single-degree-of-freedom model gives the lower-bound dynamic buckling estimate, which will be shown in the results of the dynamic buckling analysis for a laminated composite arch. The possibility of parametric resonance due to the transverse concentrated step load on a geometrically non-linear system is discussed. Furthermore, the dynamic effects for different loading rates of the applied concentrated load are examined. Finally, the study of the damping effect, which raises the dynamic buckling load, is included.