Nonlinear Inertia Terms Research Articles

Nonlinear dynamics of modulated flow between a porous injector and an impermeable substrate

The stable, time-periodic flow between a porous injector disk and an impermeable substrate disk is explored for the case where fluid is injected into the gap region with a spatially uniform time-periodic velocity V(τ)=V0(1+α cos(στ)), where V0 is the mean injection velocity, α is the flow modulation amplitude, and σ is the flow modulation frequency. Fourier series expansions in time (τ) are combined with regular perturbation expansions of the Fourier coefficients in powers of α to describe the linear and nonlinear frequency dispersion in the system when 0<α<1. Generalized analytical expressions are obtained for the quasisteady response, and the method of matched asymptotic expansions is used to analyze the high frequency linear response of the system. Finite difference methods are used to calculate the frequency dispersion of the system for a wide range of modulation frequencies (σ) and Reynolds numbers. Oscillating harmonics are shown to interact destructively (via nonlinear inertial terms), resulting in the nullification of certain Fourier modes in the flow field. The Reynolds number and frequency dependence of harmonic nullification events are explored and their implications for creating multilayered alloys are briefly discussed.

NONLINEAR ANALYSIS OF THE PARAMETRIC RESONANCES OF A PLANAR FLUID-CONVEYING CANTILEVERED PIPE

This paper deals with the nonlinear dynamics and the stability of cantilevered pipes conveying fluid, where the fluid has a harmonic component of flow velocity, assumed to be small, superposed on a constant mean value. The mean flow velocity is near the critical value for which the pipe becomes unstable by flutter through a Hopf bifurcation. The partial differential equation is transformed into a set of ordinary differential equations (ODEs) using the Galerkin method. The equations of motion contain nonlinear inertial terms, and hence cannot be put into standard form for numerical integration. Various approaches are adopted to tackle the problem: (a) the centre manifold theory applied on the set of non-autonomousequations, followed by the normal form method, yielding both the principal and the fundamental resonances; (b) a perturbation method via which the nonlinear inertial terms are removed by finding an equivalent term using the linear equation; the system is then put into first-order form and integrated using a Runge-Kutta scheme; (c) a finite difference method based on Houbolt's scheme, which leads to a set of nonlinear algebraic equations that is solved with a Newton-Raphson approach; (d) periodic solutions and stability boundaries are obtained using an incremental harmonic balance method as proposed by S. L. Lau. Using the four methods, the dynamics of the pipe conveying fluid are investigated in detail. For example, the effects of (i) the forcing frequency, (ii) the perturbation amplitude, and (iii) the flow velocity are considered. Particular attention is paid to the effect of the nonlinear terms. These results are compared with experiments undertaken in our laboratory, utilizing elastomer pipes conveying water. The pulsating component of the flow is generated by a plunger pump, and the motions are monitored by a noncontacting optical follower system. It is shown analytically, numerically and experimentally, that periodic and quasiperiodic oscillations can exist, depending on the parameters.

NUMERICAL SOLUTIONS OF SECOND ORDER IMPLICIT NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

The existing literature usually assumes that second order ordinary differential equations can be put in first order form, and this assumption is the starting point of most treatments of ordinary differential equations. This paper examines numerical schemes for solving second order implicitnon-linear differential equations. Based on the literature review, three specific methods have been selected here. Two of them are based on series expansions: Picard iteration using Chebyshev series and Incremental Harmonic Balance (IHB), in which the non-linear differential equation is transformed into a set of algebraic ones that are solved iteratively. The third method is based on a fourth order (Houbolt's scheme) and an eighth order backward finite difference method (FDM). Each method is presented first, and then applied to specific examples. It is shown: (1) that the Picard method is not valid for solving implicit equations containing large non-linear inertial terms; (2) that the IHB method yields accurate periodic solutions, together with the frequency of oscillation, and the dynamical stability of the system may be assessed very easily; and (3) that both Houbolt's and the eighth order scheme can be used to compute time histories of initial value problems, if the time step is properly chosen. Finally, it is also illustrated how the combination of IHB and FDM can be a powerful tool for the analysis of non-linear vibration problems defined by implicit differential equations (including also explicit ones), since bifurcation diagrams of stable and unstable periodic solutions can be computed easily with IHB, while periodic and non-periodic stable oscillations may be obtained with FDM.

Experimental Verification of the Importance of The Nonlinear Curvature in the Response of a Cantilever Beam

An experimental and theoretical investigation into the first- and second-mode responses of a parametrically excited slender cantilever beam is presented. Inclusion of quadratic damping in the analytical model significantly improves the agreement between the experimental and theoretical results. In addition, the experimental results verify that the often ignored nonlinear curvature terms play a dominant role in the response of the first mode and that the nonlinear inertia terms play a dominant role in the response of the second mode.

Intermittency Route to Chaos of a Cantilevered Pipe Conveying Fluid With a Mass Defect at the Free End

The nonlinear equations for planar motions of a vertical cantilevered pipe conveying fluid are modified to take into account a small lumped mass added at the free end. The resultant equations contain nonlinear inertial terms; by discretizing the system first and inverting the inertia matrix, these terms are transferred into other matrices. In this paper, the dynamics of the system is examined when the added mass is negative (a mass defect), by means of numerical computations and by the software package AUTO. The system loses stability by a Hopf bifurcation, and the resultant limit cycle undergoes pitchfork and period-doubling bifurcations. Subsequently, as shown by the computation of Floquet multipliers, a type I intermittency route to chaos is followed—as illustrated further by a Lorenz return map, revealing the well-known normal form for this type of bifurcation. The period between “turbulent bursts” of nonperiodic oscillations is computed numerically, as well as Lyapunov exponents. Remarkable qualitative agreement, in both cases, is obtained with analytical results.

Implementation of Sliding Mode Control for SCARA Robot

Implementation of a sliding mode control algorithm for the high speed/high precision SCARA robot has been demonstrated, Since nonlinear dynamics have undesirable effects on the high speed operations, the inaccurate modelling of the robot system degrades the performance. Instead of trying to obtain the exact dynamics of a manipulator and servo systems, the nonlinear dynamics, i.e., nonlinear inertia terms, Coriolis force term and centrifugal force term are treated as disturbance and a sliding mode control algorithm is derived to compensate for the modelling imprecision and disturbance. The algorithm is directly derived from the sliding condition, and it represents the control input explicitly by relating to the nonlinear terms. To evaluate performance of the proposed algorithm, a control system for SCARA robot has been designed using two TMS320C25 DSP’s and a host computer. Some preliminary results are shown.

Acoustic streaming at high Reynolds numbers in focused sound beams

In experiments reported recently on acoustic streaming in focused sound beams [Starritt et al., Ultrasound Med. Biol. 15, 363 (1989), and Matsuda et al., Advances in Nonlinear Acoustics: 13th ISNA, edited by H. Hobæk (World Scientific, Singapore, 1993), pp. 595–600], the Reynolds numbers for the streaming are of order 100. To model such experiments, nonlinearity in the momentum equation for the streaming velocity should be taken into account. Previous comparisons of theory with measurements of acoustic streaming in focused beams are based on linear equations for the streaming velocity, which are normally valid only for Stokes flows characterized by Reynolds numbers less than unity [Lighthill, J. Sound Vib. 61, 391 (1978)]. The numerical results presented here were obtained with nonlinear inertia terms for the streaming velocity retained in equations that describe the paraxial region of the beam. The primary wave is assumed to be a focused Gaussian beam, with parameters corresponding to the experiments cited above. Inclusion of nonlinearity yields predictions of streaming patterns that are more localized in the focal region, and in closer agreement with experiment, than predictions based on linearized equations for the flow. [Work supported by Schlumberger, NSF, and the Office of Naval Research.]

Double diffusive penetrative convection in porous media

A nonlinear energy stability analysis, for the model problem of convective motions due to temperature and salt fields in a porous medium, is carried out. Variations in viscosity and thermal expansion coefficients with temperature and the nonlinear inertia terms are also included in the analysis and the effect of their consideration is pointed out. A priori stability and continuous dependence results are also discussed.

Stability, double degeneracy and chaos in cantilevered pipes conveying fluid

The non-linear dynamics of planar motions of cantilevered pipes conveying fluid is studied via a two-mode discretization of the governing partial differential equation, after first replacing non-linear inertial terms by equivalent ones in the equations of motion through a perturbation procedure. For hanging cantilevers, as the flow velocity U is incremented to a critical value, the undeformed vertical configuration of the pipe becomes unstable and bifurcates into stable periodic orbits through a Hopf bifurcation. For a “standing” cantilever, in which the flow discharges from the free, upper end, the pipe is statically unstable for small U if the pipe is sufficiently long; it regains stability through a subcritical pitchfork bifurcation at higher U, and this is followed by a Hopf bifurcation and periodic motions at still higher U. For some parameter values these two bifurcations occur simultaneously (double degeneracy). By using centre manifold theory and normal forms, it is shown that heteroclinic cycles exist in the reduced subsystem, suggesting the possible existence of chaotic behaviour. Melnikov computations give guidance as to the likely location of chaotic regimes in the parameter space. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of phase portraits, bifurcation diagrams and Lyapunov exponents.

Non-Darcy flow and axial conduction effects on forced convection in porous material filled pipes

The effects of non-Darcy extensions and axial conduction on forced convective heat transfer in porous material filled pipes have been examined numerically employing successive accelerated replacement scheme. The non-Darcy extensions considered are classical convective terms, Forchheimer non-linear inertial terms and Brinkman viscous terms. Numerical results obtained indicate that inclusion of axial conduction leads to significant increase in local as well as average Nusselt number in the thermally developing region forPe<100. Among the aforementioned non-Darcy extensions, inclusion of Brinkman viscous terms to Darcy flow model lead to qualitatively significant changes in the flow and heat transfer. Inclusion of classical convective and/or Forchheimer non-linear inertial terms to Brinkman extended Darcy flow model lead to a slight increase in local Nusselt number. When all the non-Darcy extensions are included, the local Nusselt numbers coinside with the local Nusselt numbers obtained with Darcy-Forchheimer-Brinkman flow description.

Slip flow through porous media with permeable boundaries: implications for the dimensional scaling of packed beds

ABSTRACT: A theoretical description of single‐phase flow through porous media having permeable boundaries is presented. The volume averaged momentum transfer equation is solved numerically by a finite difference technique after neglecting the nonlinear inertia term. The influence of fluid slip on axial velocity profiles and wall skin friction is analyzed as a function of cross‐flow and the porous medium shape factor. Boundary effects on momentum transfer are quantified. The implications for the design of scale models for porous media such as filters, adsorber columns, and aeration towers are discussed. In accordance with experimental data obtained by other researchers, it is shown that in most cases, additional viscous dissipation effects due to the presence of boundaries are expected to be insignificant even in laboratory‐scale models only a few packing element diameters wide.

Nonlinear dynamics of a flexible beam in a central gravitational field—I. Equations of motion

The complete nonlinear differential equations governing the nonlinear motions of a beam able to undergo bending and pitching in space, are formulated in this paper. The formulation is based on a variational principle and accounts for all the nonlinearities due to deformation and gravity gradient effects. The nonlinearities due to deformation arise due to geometric effects, which consist of nonlinear curvature and nonlinear inertia terms. Expanded equations governing the nonlinear perturbed motion about an equilibrium are also developed for the case when the beam is in circular orbit. Such equations are suited for a perturbation analysis of the motion, and nonlinearities up to cubic order in a bookkeeping parameter are retained in them. Nonlinear motions involving interactions between bending and pitching of the beam are investigated in Part II of this work using the equations developed here.

A boundary integral formulation for natural convection flows

A novel solver for natural convection flows based on the boundary integral equation method is presented here. The solver can be used to obtain flow solutions in arbitrary 2–D geometries with modest computational effort. Both the vorticity and thermal transport equations are treated as modified Helmholtz equations and solved using boundary integral methods. The non-linear inertial terms partially manifest themselves as volume sources which are computed iteratively using a method of characteristics approach. The buoyancy term is treated as a volume vorticity source. Rapid computation of the volume fields at each iteration level is achieved by precomputing the influence coefficient matrices. Results are presented for flow in a cavity driven by both shear and buoyancy.

A boundary integral equation method for Navier‐Stokes equations—application to flow in annulus of eccentric cylinders

AbstractA novel Navier‐Stokes solver based on the boundary integral equation method is presented. The solver can be used to obtain flow solutions in arbitrary 2D geometries with modest computational effort. The vorticity transport equation is modelled as a modified Helmholtz equation with the wave number dependent on the flow Reynolds number. The non‐linear inertial terms partly manifest themselves as volume vorticity sources which are computed iteratively by tracking flow trajectories. The integral equation representations of the Helmholtz equation for vorticity and Poisson equation for streamfunction are solved directly for the unknown vorticity boundary conditions. Rapid computation of the flow and vorticity field in the volume at each iteration level is achieved by precomputing the influence coefficient matrices. The pressure field can be extracted from the converged streamfunction and vorticity fields. The solver is validated by considering flow in a converging channel (Hamel flow). The solver is then applied to flow in the annulus of eccentric cylinders. Results are presented for various Reynolds numbers and compared with the literature.

A continuum model for low-pressure radio-frequency discharges

A computational model for low-pressure (&amp;lt;0.3–0.5 Torr) radio-frequency glow discharges is presented. The governing equations are the first three moments of the Boltzmann transport equations. In order to describe the weakly collisional sheaths, the nonlinear inertia terms in the momentum equations are retained. The model is demonstrated for a 13.56-MHz argon discharge. A parametric study involving variations in pressure, electrode spacing, and driving voltage is presented.

Effects of Shear Deformation and Rotary Inertia on the Nonlinear Dynamics of Rotating Curved Beams

In this investigation a method for the dynamic analysis of initially curved Timoshenko beams that undergo finite rotations is presented. The combined effect of rotary inertia, shear deformation, and initial curvature is examined. The kinetic energy is first developed for the curved beam and the beam mass matrix is identified. It is shown that the form of the mass matrix as well as the nonlinear inertia terms that represent the coupling between the rigid body motion and the elastic deformation can be expressed in terms of a set of invariants that depend on the assumed displacement field, rotary inertia, shear deformation, and the initial beam curvature. A nonlinear finite element formulation is then developed for Timoshenko beams that undergo finite rotations. The nonlinear formulation presented in this paper is applied to multibody dynamics where mechanical systems consist of an interconnected set of rigid and deformable bodies, each of which may undergo finite rotations. The equations of motion are developed using Lagrange’s equation and nonlinear algebraic constraint equations that mathematically describe mechanical joints and specified trajectories are adjoined to the system differential equations using the vector of Lagrange multipliers.

Non-linear modal coupling in planar and non-planar responses of inextensional beams

Non-linear inertia and curvature terms, as well as non-linear terms that couple flexural and torsional motions, can have a dominant role in the response of inextensional beams or beam-like structural members. In this paper the effect of modal coupling in the non-linear response of such members to a periodic excitation is analyzed and discussed. The analysis presented is based on a set of consistent differential equations of motion that take into account non-linear geometric effects. They include non-linear contributions from the beam's curvature and torsion expressions, and from inertia terms. Examples of amplitude-frequency response characteristics for cantilevers and for clamped-pinned/sliding beams are presented.

Non-linear non-planar oscillations of a cantilever beam under lateral base excitations

The non-planar responses of a cantilevered beam subject to lateral harmonic baseexcitation are investigated using two non-linear coupled integrodifferential equations of motion. The equations contain cubic non-linearities due to curvature and inertia. Two uniform beams with rectangular cross sections are considered: one has an aspect ratio near unity, and the other has an aspect ratio near 6.27. A combination of the Galerkin procedure and the method of multiple scales is used to construct a first-order uniform expansion for the case of a one-to-one internal resonance and a primary resonance. The results show that the non-linear geometric terms are important for the responses of low-frequency modes because they produce hardening spring effects. On the other hand, the non-linear inertia terms dominate the responses of high-frequency modes. We also obtain quantitative results for non-planar motions and investigate their dynamic behavior. For different range of parameters, the non-planar motions can be steady whirling motions, whirling motions of the beating type, or chaotic motions. Furthermore, we investigate the effects of damping.

Reduction of flexural vibrations by damping of axial motions

To realize a simple passive method for the reduction of the amplitude of flexural vibrations in structures, a method of damping the axial displacements of a structural component is introduced. The non-linear deformation of slender beam is considered, and the axial force acting at one end of it is assumed to be linearly proportional to the beams' axial velocity there. This, in turn, is expressed in terms of an integral over the length of the beam of the time derivative of the quantity ν s 2, where ν s = ∂ν ∂s, s denoting the arc length along the axis of the beam. This is a second order non-linear effect. The derivation of the equation of motion of the beam includes non-linear effects to the third order. The beam is then assumed to be driven by a transverse force that varies sinusoidally in time. For a simply supported beam, a one-term Galerkin approximation is applied to remove the spatial dependence of the partial differential equation. The novel feature of this differential equation is the presence of a non-linear damping term that consists of the product of the transverse velocity and the square of the transverse displacement. Non-linear stiffness and inertia terms also appear in this reduced equation of motion. A system of non-linear first order differential equations is then obtained. For the case of the steady state response, the dependence of the steady state amplitude upon the frequency ratio is determined and a plot of its variation is reported for several values of the non-linear damping parameter. The Runge-Kutta method is used to determine the variation of the amplitude of the motion as a function of time when the driving frequency is equal to the natural frequency of the system. Numerical calculations reveal that significant reductions in the transverse amplitudes of the motion can be realized through an increase in the degree of damping that is exerted on the axial motion of the beam. Poincaré maps are plotted for undamped and non-linearly damped beam motions. A transition into and then out of a state of chaos is observed as the value of the non-linear damping parameter is increased.

Boundary-Layer Flows in Rotating Cavities

A rotating cylindrical cavity with a radial outflow of fluid provides a simple model of the flow between two corotating air-cooled gas-turbine disks. The flow structure comprises a source region near the axis of rotation, boundary layers on each disk, a sink layer on the peripheral shroud, and an interior core of rotating inviscid fluid between the boundary layers. In the source region, the boundary layers entrain fluid; outside this region, nonentraining Ekman-type layers are formed on the disks. In this paper, the differential boundary-layer equations are solved to predict the velocity distribution inside the entraining and nonentraining boundary layers and in the inviscid core. The equations are discretized using the Keller-box scheme, and the Cebeci–Smith eddy-viscosity model is used for the turbulent-flow case. Special problems associated with reverse flow in the nonentraining Ekman-type layers are successfully overcome. Solutions are obtained, for both laminar and turbulent flow, for the “linear equations” (where nonlinear inertial terms are neglected) and for the full nonlinear equations. These solutions are compared with earlier LDA measurements of the radial and tangential components of velocity made inside a rotating cavity with a radial outflow of air. Good agreement between the computations and the experimental data is achieved for a wide range of flow rates and rotational speeds.