Subharmonic solutions of the duffing equation with large non-linearity1

The subharmonic solutions of order $\text{1}\text{3}$ of the damped Duffing equation are determined in a suitable parametric form, following the procedure recently developed in [8, 9], and are compared with the results obtained by direct numerical integration of the same equation, carried out with respect to the time with the Runge-Kutta method. It can be deduced that the analytical solution gives satisfactory results in the approximation of the ‘predominantly’ subharmonic solutions of the above equation, even if the non-linearity of the system is very large.     

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.     

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).     

Two new methods for the approximate determination of the initial coefficient of the first normal stress difference

Two methods for determining the initial coefficient of the first normal stress difference are presented. They are based on the evaluation of the steady viscosity function η($\text{γ}\text{.}$) and the viscosity function η⁺($\text{γ}\text{.}$, t) at the start-up of a flow with a very small rate of deformation $\text{γ}\text{.}$ < $\text{γ}\text{.}$₀. For the functions η($\text{γ}\text{.}$) and η⁺($\text{γ}\text{.}$), equations are given which can be used for a simple evaluation of the integral relationships obtaiend for ψ₁₀. The values for ψ₁₀ calculated by the two methods are compared with values obtained by the well-known methods via measurement of the ψ₁($\text{γ}\text{.}$) or η″(ω)/ω functions and extrapolation to zero). Both methods give values which are in satisfactory agreement with the experimental values.     

Phenomenology of inertia effects in a dispersed solid-fluid mixture

Combining single particle results, average equations and thermodynamic considerations, we propose a way to build the equations describing a suspension of rigid spherical particles in a carrier fluid, with emphasis on inertia effects including virtual mass. The spatial fluctuations of the fluid velocity field are depicted by two phenomenological functions $\text{?(α}{}_{\mathrm{s}}\text{)}\text{and}\text{g(α}{}_{\mathrm{s}}\text{)}$ of the particle volume fraction, and a third function h(α_s) is necessary to describe the intensity of the particles internal stress. It is shown that all inertia effects occurring in the relative translational motion can be derived from the two functions $\text{?}\text{and}\text{g–h}$ only. The conditions under which the above system of equations is hyperbolic are determined and comparison is made with what is presently known about $\text{?, g}\text{and}\text{h}$ in the dilute limit.     

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.     

Arbitrary motion of an elliptic disc at a fluid interface

The problem of calculating the disturbance due to finite elliptic discs at the interface ($\text{x}{}_3\text{= 0}$) of two incompressible immiscible fluids of different coefficients of viscosity is solved, assuming that body and inertia forces are negligible. When the direction of motion is parallel to the interface, our solution, which is based on potential functions analogous to the Papkovitch-Neuber functions of linear elasticity, satisfies not only the interface conditions of continuity of fluid velocity and stresses but also that of zero normal velocity at the interface. It is also remarkable that this solution produces in each of the fluids a flow field that is totally independent of the properties of the other fluid. These results are not peculiar to elliptic discs, but also hold for finite discs of other shapes. The method of solution presented here can be readily applied to the more general cases where the two-phase fluid, in the absence of the disc, moves with an arbitrarily directed velocity which is a general polynomial function of the coordinates $\text{x}{}_1$ and $\text{x}{}_2$ at the interface. The procedure for carrying this out is demonstrated by treating the case of an elliptic disc in linear shear flow.     

Transient process of the vibrations of a cylindrical shell as a system with many degrees of freedom

A procedure is developed for investigating the transient process associated with the forced vibrations of a cylindrical shell as a system with many degrees of freedom. The amplitudes of the bending deflections of circular modes induced by a driving load of variable frequency and amplitude are compared with the same amplitudes for a uniform driving load and steady-state vibrations. The total maximum deflections of all the investigated cases are found to differ substantially. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 57–63, March, 1999.     

Subharmonic solution of a piecewise linear oscillator with tow degrees of freedom

In the present paper subharmonic resonance solution ofapiecewise linear oscillator with two degrees of freedom is studied. It is shown that in this system there exist a series of subharmonic resonance solutions, among them there are 1/2, 1/3, 1/4, 1/5, 1J6,……subharmonic resonance solutions. The calculated results by the analogy computer and the field experiments in the factory partly verify this theory. Under certain circumstances, the generation of chaotic states of the oscillation is observed in analogy computer solutions.     

Plastic flow and dislocation structure at small strains in OFHC copper deformed in tension,torsion and combined tension-torsion

OFHC copper specimens of 39 μm grain size were deformed to small strains (up to 8%) in tension, torsion and combined tension-torsion at 300 K and the resulting dislocation structures, distributions and densities were determined using transmission electron microscopy. Employing the von Mises yield criterion and the plastic-work hypothesis good agreement was obtained for the three testing conditions for (i) equivalent stress $\text{}\text{?}$gs vs equivalent strain $\text{}\text{?}$g3^p curves, (ii) the dislocation structure, distribution and density ρ as a function of $\text{}\text{?}$g3^p, and (iii) $\text{}\text{?}$gs as a function of $\text{ρ}\text{1}\text{2}$. Furthermore, upon comparing the $\text{}\text{?}$gs vs $\text{ρ}\text{1}\text{2}$ curve for polycrystalline copper with the τ_RSS vs $\text{ρ}\text{1}\text{2}$ curve for single crystals, an average Taylor factor M= (σ/τ_RSS) of approximately 3.2 was obtained, which is in good accord with that predicted theoretically for FCC metals. Almost equally good correlations for the stressstrain curves and for the dislocation density were obtained on the basis of maximum shear stress τ_max and maximum shear strain γ^p_max as on the basis of $\text{}\text{?}$gs and $\text{}\text{?}$g3^P. Therefore, the present results do not permit a positive decision on the question whether the dislocation density correlates better with $\text{}\text{?}$gs and $\text{}\text{?}$g3^P or with τ_max and γ^P_max.A single test in which the direction of straining in torsion was reversed yielded a density and distribution of dislocations (and a corresponding value of $\text{}\text{?}$gs) equivalent to those that developed at a smaller strain in unidirectional straining.     

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.     

Investigation of droplet deposition from a turbulent gas stream

Turbulent deposition of particles from two-phase flow onto the smooth wall of a tube has been studied theoretically and experimentally. A model is proposed for the deposition motion of large particles based on turbulent diffusion in the core followed by a free flight towards the wall. The theory shows that within the Stokes regime, the dimensionless deposition velocity $\text{k-}{}_{\mathrm{<mtext>d</mtext>}}\text{/}\text{u}$* depends on Re and $\text{τ}{}^{\mathrm{+}}$ only, where u* is the friction velocity, Re is the tube Reynolds number and $\text{τ}{}^{\mathrm{+}}$ is the dimensionless particle relaxation time. Deposition data are obtained for air-water droplet flow through a 12.7-mm i.d. acrylic tubing at $\text{Re}\text{= 52,500}$ and 94,600. The proposed theory satisfactorily describes the existing deposition data as well as present measurements, covering a wide range of Re and $\text{τ}{}^{\mathrm{+}}$.     

An experiment to measure the pressure dependence of the zero-shear-rate viscosity

A simple experimental method, based on Stokes' law for falling spheres, has been devised and used to measure the pressure-dependence of the zero-shear-rate viscosity of a polypropylene melt. The experiment was performed by maintaining three thick-walled test cylinders containing the polymer melt and the falling sphere at the same elevated temperature but different pressures for periods of time ranging from 20 to 48 hours.When compared with experiments using high-pressure capillary or rotational viscometers, this experimental method has the advantages that viscous heating is non-existent and the apparatus and data analysis are relatively simple. The principal disadvantage encountered here, thermal degradation at high temperatures, could probably be reduced by molding specimens under vacuum and by shortening the exposure time. Since the falling-sphere experiment provides data at very low shear rates and the capillary and rotational viscometers generate data at high shear rates, the two experimental methods are complementary.The pressure coefficient b [=d(In η₀/dp] was determined for Hercules Pro-fax 6523 polypropylene in two series of experiments at different temperatures. For seven experiments at 218.3°C and pressures up to 97.9 $\text{MN}\text{m}{}^2$ (14,200 psi), the average value of b ± 95% confidence limits was found to be 14.8 ($\text{GN}\text{m}{}^2$)^?1 ± 2.9.The average b was 12.6 ($\text{GN}\text{m}{}^2$)^?1 ± 1.4 in a series of eight experiments at 232.2°C and pressures up to 123 $\text{MN}\text{m}{}^2$ (17,800 psi).     

Analytische Näherungen und physikalische Deutung nichtlinearer Schwingungen mit inneren Resonanzen

Übersicht In dieser Arbeit werden innere Resonanzen nichtlinearer Schwingungen analytisch genauer untersucht und physikalisch gedeutet. Die klassische kanonische Störungsmethode wird erweitert, so daß mit ihr auch angenäherte innere Resonanzen analytisch erfaßt werden können. Als Beispiel wird eine symmetrische Schwingerkette betrachtet, deren Instabilitäten durch innere Resonanzen erklärt werden.

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Analytical approximations and physical interpretation of nonlinear vibrations with internal resonances

Summary In this paper internal resonances of nonlinear vibrations are analytically investigated and physically interpreted. The classical canonical perturbation method is generalized in such a manner that nonlinear vibrations in near internal resonances can be analytically calculated too. As an example a symmetrical system with two degrees of freedom is considered and the instabilities of this system can be explained by internal resonances.