Subharmonic steady vibrations of a non-linear damper with two degrees of freedom and viscous damping

The steady-state. $\text{1}\text{3}$ subharmonic vibrations of a dynamic damper (or vibration absorber) with two degrees of freedom, sinusoidal forcing function and internal viscous damping. are presented. The study of these oscillations leads to the determination of suitable “form functions” of the solutions, by following a methodology recently introduced by Nocilla for studying the harmonic vibrations of non-linear systems with one and two degrees of freedom. The proposed theory. which is valid even if the non-linearity is large, gives satisfactory results in all the cases in which the subharmonic component is predominant in the steady-state oscillation of the system.     

Dynamic stability for a simply supported beam under periodic axial excitation

This paper studies the dynamic stability for a simply supported straight beam under periodic axial excitation by using the averaging method and the Routh-Hurwitz stability criteria. By considering the first two modes coupled, we discuss the effect of the stability-instability region and the amplitudes of vibration. Furthermore, by studying the principal parametric resonance i.e. subharmonic order $\text{1}\text{2}$, we investigate the effect of the amplitude of the main system by various kinds of non-linearities of the subsystem. Finally, by obtaining the transient results, we describe the beat phenomenon, and harmonic oscillation.     

On stability of non-linear differential systems

The object of this paper is to study the stability and asymptotic stability of solutions of the non-linear differential equation $\text{dx}\text{dt}\text{= A(t)x + f(t,x)}$ by using the method of equivalent inner products. This method enables one to determine a stability region without the ingenuity in constructing a Lyapunov function. It shows also that for an unstable linear system it is possible to choose a non-linear function so that the non-linear system is stable or asymptotically stable. Both global and regional stability are discussed.     

The MLP method for subharmonic and ultraharmonic resonance solutions of strongly nonlinear systems

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｙｏｆｔｈｅａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｏｆｓｔｒｏｎｇｌｙｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｓｉｓｓｔｉｌｌｔｈｅｍａｉｎｔａｓｋｉｎｔｈｅｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃａｌｒｅｇｉｏｎ .ＴｈｅｒｅａｒｅｍａｔｕｒｅｍｅｔｈｏｄｓｉｎｃｌｕｄｉｎｇｔｈｅｄｅｖｅｌｏｐｅｄＫＢＭｍｅｔｈｏｄ[1],ｔｈｅｍｏｄｉｆｉｅｄＬＰｍｅｔｈｏｄ[2 ]ａｎｄｔｈｅｓｔｒｏｂｏｓｃｏｐｉｃｍｅｔｈｏｄ[3]ｅｔａｌ.Ｈｏｗｅｖｅｒ,ｗｈｅｎｔｈｅｓｅｍｅｔｈｏｄｓａｒｅａ…     

Two-phase flow in a vertical pipe and the phenomenon of choking: Homogeneous diffusion model—I. Homogeneous flow models

The paper examines the topological structure of all possible solutions which can exist in flows through adiabatic constant-area ducts for which the homogeneous diffusion model has been assumed. The conservation equations are one-dimensional with the single space variable $\text{z.}$ but gravity effects are included. The conservation equations are coupled with three equations of state: a pure substance, a perfect gas with constant specific heats, and a homogeneous two-phase system in thermodynamic equilibrium. The preferred state variables are pressure $\text{P.}$ enthalpy $\text{h.}$ and mass flux $\text{G}{}^2$.The three conservation equations are first-order but nonlinear. They induce a family of solutions which are interpreted as curves in a four-dimensional phase space conceived as a union of three-dimensional spaces ($\text{P, h, G}{}^2\text{, z}$) with $\text{G}{}^2\text{=}\text{const}$ treated as a parameter. It is shown that all points in these spaces are regular, so that no singular solutions need to be considered. The existence and uniqueness theorem leads to the conclusion that through every point in phase space there passes one and only one solution-curve.The set of differential equations, treated as a system of algebraic equations of each point of the phase space, determines the components of a rate-of-change vector which are obtained explicitly by Cramer's rule. This vector is tangent to the solution curve. Each solution curve turns downward in $\text{z}$ at some specific elevation $\text{z}{}^1$, and this determines the condition for choking. Choking occurs always when the exit flow velocity at $\text{L = z}{}^1$ is equal to the local velocity of propagation of small plane disturbances of sufficiently large wavelength, that is when the flow rate $\text{G}$ becomes equal to a specified, critical flow rate, $\text{G}{}^1$. (The possible dependence of the sonic velocity on frequency in a real flow is ignored, because it has not been allowed for in the equations of the model under study.) A criterion, analogous to the Mach number, which indicates the presence or absence of choking in a cross section is the ratio $\text{K = G/G}{}^7$ of the mass-flow rate $\text{G}$ to the local critical mass flow rate. $\text{G}{}^7$, $\text{K = 1}$ denoting choking. The critical parameters depend only on the thermodynamic properties of the fluid and are independent of the gravitational acceleration and shearing stress at the wall.The topological characteristics of the solutions allow us to study all flow patterns which can, and which cannot, occur in a pipe of given length $\text{L}$ into which fluid is discharged through a rounded entrance from a stagnation reservoir and whose back-pressure is slowly lowered. The set of flow patterns is analogous to that which occurs with a perfect gas, except that the characteristic numerical values are different. They must be obtained by numerical integration and the influence of gravity must be allowed for.The preceding conclusions are valid for all assumptions concerning the shearing stress at the wall which make if dependent on the state parameters only, but not on their derivatives with respect to $\text{z}$. However, the study is limited to upward flows for which the shearing stress at the wall and the gravitational acceleration are codirectional.     

Slow flow past periodic arrays of cylinders with application to heat transfer

Solutions for the slow flow past a square and a hexagonal array of cylinders are determined using a somewhat non-conventional numerical method. The calculated values of the drag on a cylinder as a function of $\text{c}$, the volume fraction of the cylinders, are shown to be in excellent agreement with the corresponding asymptotic expressions for $\text{c ? 1}$ and for $\text{c → c}{}_{\mathrm{<mtext>max</mtext>}}$, the maximum volume fraction. These solutions are then used to calculate the average temperature difference between the bulk and the cylinders which are heated uniformly under conditions of small Reynolds and Péclet numbers.     

Nonuniqueness of solutions to differential equations for boundary-layer approximations in porous mediaNon unicité de solutions des équations différentielles pour un problème de couches limites en milieux poreux

The free convection, along a vertical flat plate embedded in a porous medium, can be described in terms of solutions to $\text{f?+}\text{α+1}\text{2}\text{ff″?αf′}{}^2\text{=0,}$ for all t∈(0,+∞). The purpose of this Note is to study the nonuniqueness of solutions to this problem, with the initial conditions, $\text{f(0)=a∈}\text{R}$ and f′(0)∈{0,1}, where $\text{α∈(?}\text{1}\text{3}\text{,0).}$ No assumption at infinity is imposed. We show that this problem has an infinite number of unbounded global solutions. Moreover, we prove that the first and the second derivative of solutions tend to 0 as t approaches infinity. To cite this article: M. Guedda, C. R. Mecanique 330 (2002) 279–283.     

Shear-layer acoustic radiation in an excited subsonic jet: experimental study

The subharmonic acoustic radiation of a tone excited subsonic jet shear-layer has been investigated experimentally. Two jet velocities $U_j=20\text{m}?{\text{s}}^{\mathrm{?1}}$ and $U_j=40\text{m}?{\text{s}}^{\mathrm{?1}}$ were studied. For $U_j=20\text{m}?{\text{s}}^{\mathrm{?1}}$, the natural boundary-layer at the nozzle exit is laminar. When the perturbation is applied, the fluctuations of the first and the second subharmonics of the excitation frequency are detected in the shear-layer. In addition, the first subharmonic near pressure field along the spreading jet is constituted of two strong maxima of sinusoidal shape. The far-field directivity pattern displays two lobes separated by an extinction angle ${\theta }^{?}$ at around 85° from the jet axis. These observations follow the results of Bridges about the vortex pairing noise. On the other hand, for $U_j=40\text{m}?{\text{s}}^{\mathrm{?1}}$, the initial boundary-layer is transitional and only the first subharmonic is observed in the presence of the excitation. The near pressure field is of Gaussian shape in the jet periphery and the acoustic far-field is superdirective as observed by Laufer and Yen. The state of the initial shear-layer seems to be the key feature to distinguish these two different radiation patterns. To cite this article: V. Fleury et al., C. R. Mecanique 333 (2005).     

Closed streamline flows past small rotating particles: Heat transfer at high péclet numbers

A heat transfer problem is solved, first for an infinitely long heated cylinder and then for a small heated sphere, each freely suspended in a general linear flow at Reynolds numbers $\text{Re ? 1}$. Asymptotic solutions to the convection problem are developed for very large values of the Péclet number $\text{Pe}$, and expressions are obtained for the asymptotic Nusselt number for two-dimensional flows ranging from solid body rotation to hyperbolic flow. Since the objects in these cases are surrounded by a region of effectively isothermal closed streamlines, the asymptotic Nusselt number becomes independent of the Péclet number in the limit $\text{Pe → ∞}$.     

Holdup in two phase flow

A wide range of experimental holdup data have been analysed on the basis of the general correlations of Chen & Spedding (1983). For upward inclined flow, holdup data in the range $\text{(}\text{R}{}_{\mathrm{G}}\text{/}\text{R}{}_{\mathrm{L}}\text{) = 4.0}$ to 275 were handled using a modification of the Chen & Spedding method, and for the case of $\text{(}\text{R}{}_{\mathrm{G}}\text{/}\text{R}{}_{\mathrm{L}}\text{) ? 4.0}$, the modified Armand equation was found to be suitable. Horizontal stratified flow was examined using the Bernoulli equation, and shown to be a limiting case of the free draining of a tube initially filled with liquid. For downward inclined stratified flow, the Manning equation predicted the holdup accurately for low liquid rates and small angles of inclination. In addition, for these two cases of horizontal and downward stratified flow, the holdup also was examined in terms of the critical depth of flow as determined using the total energy relation.     

Two-phase flow in bends

An equation is developed for use in predicting the two-phase multiplier for pressure drop in bends; the equation simplifies the use of an existing method. The method is also compared for the first time with data at high density ratios ($\text{?l/?g = 560}$).     

Method of characteristics solutions for non-equilibrium transient flow-boiling

A method of characteristics algorithm for the equal velocity, unequal temperature model of one-dimensional two-phase flow has been implemented. This algorithm uses a mesh of characteristics $\text{dz/dt}\text{= u ± a}$, and has greatly reduced numerical diffusion effects when compared with finite difference methods, making it useful for benchmark solutions and model testing. Results are presented from its applications to two standard problems.     

Espaces d'écoulements dits « universels »Spaces of universal flows

An isochoric motion can be performed both in perfect fluid, in Newtonian fluid, in Maxwell fluid (slow motions) and in Rivlin–Ericksen fluid of second grade whatever be viscosities and viscometric coefficients, iff the motion is universal. Every universal motion with steady vorticity is a generalised Belrami flow, and fulfils the Stokes equation. If the velocity $\text{u}$ of an universal motion complies with $\text{rot}\text{[(?}{}^{\mathrm{<mtext>t</mtext>}}\text{(}\text{Δ}\text{u}\text{))}\text{u}\text{]=}\text{0}$, the motion stands for feasible motion in every second order fluid. Brothers of the potential flows, all the sets of universal motions make up bundles of linear or cono??d spaces with various dimensions, finite or infinite, issued from the rest $\text{u}\text{≡}\text{0}$. The structures appear by scanning parallel to the potential flows. To cite this article: M. Bouthier, C. R. Mecanique 331 (2003).     

‘Pop’ safety valves: a compressible flow analysis

Pressure versus lift characteristics of Pop-type safety valves have been predicted from a model in which pressures on various parts of the valve disc can be deduced from compressible flow analysis of three passages in series. In particular, it has been shown that the characteristic snap-open and snap-shut phenomena are inherent in the solutions to the sets of equations, and are critically dependent on the setting of the adjusting ring, bearing out what is known in practice. The model has shown satisfactory agreement with experimental data from a $\text{1}\text{2}\text{″}$ (12.55 mm) pop valve operating in air.     

Exact solution of the non-linear differential equation RR̈ + 32R2 − AR−4 + B = 0

The non-linear equation $\text{R}\text{R}\text{?}\text{+}\text{3}\text{2}\text{R}{}^2\text{- AR}{}^{\mathrm{?4}}\text{+ B = 0}$ is shown to represent simply periodic motion with a minimum at R₁ and a maximum at R₁R₀ or a maximum at R₁ and a minimum at R₁R₀^?1. R₀ is a function of the ratio $\text{A}\text{B}$ and is greater than 1 for $\text{A}\text{B}$ > 1 and less than 1 for $\text{A}\text{B}$ > 1. The period of the motion satisfies the simple relation T(R₀^?1) = R₀^?1T(R₀). The exact solution to the above equation is represented in terms of elliptic integrals of the first and second kinds and a simple algebraic function.     

On the characteristic equations of elasto-plastic flow

Three dimensional characteristic surfaces (slip surfaces) of elasto-plastic Navier's equations and the criteria for their existence are discussed, and the solutions are also applied to two dimensional cases. By making use of isotropic yield function, the following results are proved. If, and only if the plastic/elastic moduli ratio is zero and Det$\text{(φ}{}_{\mathrm{ij}}\text{)=0,(φ}{}_{\mathrm{ij}}\text{=}\text{?φ}\text{?σ}{}_{\mathrm{ij}}$, φ: yield function,σ_ij: stress tensor), characteristic surfaces exist. There are two and only two characteristic surface elements at each point, and they are identical with the surfaces of maximum shearing stress.     

Hydromagnetic flow due to torsional oscillations of an infinite disk about a steady non-zero mean

The present investigation is the study of the laminar hydromagnetic flow due to torsional oscillations of an infinite disk about a steady non-zero mean in an electrically conducting fluid. Separate solutions have been obtained for the limiting cases of low and high frequency oscillations. The low frequency solution is obtained by expressing the flow functions in powers (ik) while for high frequencies, the flow functions are expressed in powers of $\text{(ik)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$, where $\text{k =}\text{ω}\text{Ω}$ being the ratio of the frequency of oscillations ω to the mean disk-angular velocity Ω and i² = ?1. It is found that the oscillating part of the transverse shearing stress has a phase-lead while that of the radial shearing stress has a phase lag behind the disk oscillations. The phase-lead in the former case and phase-lag in the latter case decrease with the increase in the strength of the applied magnetic field.