Dynamic-mechanical properties of a bitumen-silica composite

Summary Dynamic-mechanical properties of bitumen-silica composite materials, measured at room temperature, do not vary with the volume ratio () in a simple manner as do usual bituminous concretes. However,E is a linear function of the interfacial area () between the filler and the binder per unit volume. ThusE = E ₀ +a(), wherea is a constant related to the storage modulus, in the absence of voids and with a void ratio factor. The loss moduli, plotted against () go through a maximum in a similar way as when plotted versus decreasing temperatures.

---

Zusammenfassung Die bei Zimmertemperatur gemessenen dynamischmechanischen Eigenschaften von Bitumen, welches mit Siliziumdioxid gefüllt worden ist, ändern sich nicht in einer so einfachen Weise mit dem Volumenanteil des Füllstoffs, wie es bei gewöhnlichen Asphaltbetonen der Fall ist. Dagegen ergibt sich der SpeichermodulE als eine lineare Funktion der Grenzfläche zwischen Bindemittel und Füllstoff pro Volumeneinheit:E = E ₀ +a(), wobeia eine Konstante bedeutet, die zum einen von dem Speichermodul bei Abwesenheit von Hohlräumen und zum andern von einem durch solche Hohlräume bedingten Faktor abhängt. Der Verlustmodul als Funktion von zeigt ein Maximum ähnlich wie bei der Auftragung gegen die Temperatur.

---

Résumé Les propriétés mécaniques dynamiques, mesurées à la température ambiante, d'un système composite bitume-silice, n'évoluent pas simplement avec la fraction volumique de charges comme dans le cas des enrobés usuels. En effet, le module de conservationE varie linéairement avec l'aire de l'interface liant-charge par unité de volume (). Ainsi la relation suivante a pu être mise en évidence:E = E ₀ +a(), la constantea étant fonction d'un module indépendant du taux de vide et d'un terme relié à ce dernier. Si augmente, les modules de perte passent par une valeur maximale. Ces variations sont semblables à celles que l'on aurait si l'on portait ces modules en fonction de la température.

---

---

With 9 figures and 2 tables     

An approximate theoretical analysis and experimental verification of turbulent entrance region flow of drag reducing fluids

Summary Entry lengths for pipe flows of moderately drag reducing fluids are determined using momentum integral technique. It is shown theoretically that the entry lengths for drag reducing fluids could be significantly larger than the Newtonian fluids flowing turbulently under otherwise identical conditions. The experimental data from the literature bear out the theoretical calculations.

---

Zusammenfassung Mit Hilfe der Impuls-Methode wird die Einlauflänge in einer Rohrströmung für Flüssigkeiten mit mäßig starker Widerstandsverminderung berechnet. Es wird vorausgesagt, daß die Einlauflänge für derartige Flüssigkeiten erheblich größer sein kann als für newtonsche Flüssigkeiten unter sonst identischen Bedingungen. Aus der Literatur entnommene experimentelle Daten bestätigen diese theoretischen Berechnungen.

---

Nomenclature A ₁ Coefficient in eq. [7] - A Slope of logarithmic velocity profile - a Exponent in eq. [10] - B Intercept function for logarithmic velocity profile - De Deborah number, - f Friction factor - F Function, eq. [30] - G Function given in eq. [11] - Static pressure, dynes/cm² - q Index of power law velocity profile - R Pipe radius, cm - r Radial distance, cm - R Core radius, cm - Re Reynolds number - Axial velocity, cm/s - u _c Core velocity, cm/s - u ⁺ Dimensionless velocity, eq. [5] - u ^* Friction velocity, , cm/s - Radial velocity, cm/s - V Average velocity, cm/s - x Axial distance, cm - x _e Entry length, cm - y Distance from the wall, cm - y ⁺ Dimensionless distance, eq. [5] - y _I ⁺ Dimensionless viscous sublayer thickness - coefficient in eq. [17] - exponent of Reynolds number in eq. [17] - shear rate, s^–1 - boundary layer thickness, cm - _fl fluid relaxation time, s - µ fluid viscosity, gm/cm s - v kinematic viscosity, cm²/s - _l laminar sublayer thickness, dimensionless - fluid density, gm/cm³ - shear stress, dynes/cm² - _w shear stress at the wall, dynes/cm² - ₁, ₂, ₃, ₄ functions in eq. [27] - ~ time averaged quantities - — dimensionless quantity With 3 figures and 1 table     

Hyperbolic phenomena in a strongly degenerate parabolic equation

We consider the equation u _t =((u) (u _x )) _x , where >0 and where is a strictly increasing function with lim _s = <. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u _t = ((u)) _x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.     

On a singular integral equation in the theory of radiative transfer

Summary A singular integral equation arising in the theory of radiative transfer with polynomial scattering indicatrices is studied in this paper. In the case of a semi-infinite atmosphere, brightness coefficients and auxiliary functions satisfy this equation when the free term is suitably specified. The general solution of the singular equation is given a closed form in terms of its moments _n and of the fundamental solution of a homogeneous Hilbert problem. The _n's (n 1) are seen to satisfy a set of linear equations, and ₀ turns out to be simply related to the subsequent _n's.

---

Sommario In questo articolo viene studiata un'equazione integrale singolare che si presenta nella teoria del trasporto radiativo con indicatrici di scattering polinomiali. I coefficienti di riflessione e trasmissione e le funzioni ausiliarie di un'atmosfera semiinfinita soddisfano questa equazione, con opportune determinazioni del termine noto. La soluzione generale dell'equazione viene espressa in forma chiusa mediante i suoi momenti _n e mediante la soluzione fondamentale di un problema omogeneo di Hilbert. Le _n (n 1) soddisfano a loro volta un sistema di equazioni lineari, e ₀ risulta legato alle successive _n da una semplice relazione.

---

---

This work was done under the auspices of the CNR Research Groups.     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - characteristic length for the-phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1,2,3 lattice vectors, m - m convolution product weighting function - m _V special convolution product weighting function associated with a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

Impulsive motion of a semi-infinite flat plate in a viscous fluid

The unsteady laminar boundary layer flow is investigated for a semi-infinite flat plate subjected to impulsive motion. An approximate solution is obtained by utilizing Meksyn's method. These results vary smoothly from Rayleigh's unsteady solution to the steady state solution of Blasius. Results are compared to those of Lam and Crocco.Nomenclature A expansion coefficient, see eq. (13) - a expansion coefficient, see eq. (10) - B expansion coefficient, see eq. (14) - b expansion coefficient, see eq. (12) - G function defined by eq. (6) - U free stream velocity - u velocity in x direction - v velocity in y direction - x coordinate along plate - y coordinate normal to plate Greek symbols (l, ) incomplete gamma function - function defined by eq. (15) - y(U/x) ^1/2 - kinematic viscosity - x/Ut - (Uvx)^1/2 f(, )     

Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

Stoffaustausch zwischen zwei fluiden Phasen,von denen sich eine als Strahl in der anderen bewegt

Zusammenfassung Der Übergang eines Stoffes zwischen zwei fluiden Phasen wird betrachtet, von denen sich einer als Strahl in der anderen bewegt. Die Geschwindigkeit der laminar strömenden Phase wird durch eine Gleichung ausgedrückt, die Geschwindigkeitsprofile zwischen der Kolben- und der Rohrströmung kontinuierlich beschreibt. Der Transport des Stoffes im Strahl durch Diffusion in radialer und durch Konvektion in axialer Richtung wird für den isothermen, stationären Fall untersucht. Die das Problem beschreibende Differentialgleichung wird anscheinend erstmals geschlossen gelöst. Die Lösungen beinhalten konfluente hypergeometrische Funktionen. Berechnet werden Eigenwerte, Koeffizienten, örtliche und mittlere Konzentrationsfelder sowie Stoffübergangszahlen.

---

Mass transfer between two fluids, one of the two fluids is moving as jet within the other

The mass transfer between two fluids is calculated, one of the two fluids is moving as a jet within the other. The velocity of the laminar flowing phase is expressed by an equation, which describes continously the velocity profiles from plug flow to tubular flow. For the isothermal, stationary state the transport of substance i by radial diffusion and by axial convection is investigated. It appears to be that the differential equations describing the problem are solved rigorously for the first time. The solutions contain confluent hypergeometrical functions. Results include eigenvalues, coefficients, local and mean concentration fields, mass transfer numbers.

---

Verwendete Zeichen und ihre Bedeutung a - A, A_n Koeffizienten - B, B_n Koeffizienten - c Konzentration, Konstante im Anhang - C_r=0 Mittenkonzentration - c₀ Konzentration in Phase I bis z=0 - c_II Konzentration in Phase II - ¯c mittlere Konzentration, definiert in Gl. (35) - C Koeffizient, definiert in Gl. (A 21) - D Diffusionskoeffizient - Da Damköhlerzahl - E Funktion, gegeben durch Gl. (A 12) - f, f(R) Funktion f von R - f_n, f_n (R) Funktionswerte - g, g(Z) Funktion g von Z - g_n, g_n (Z) Funktionswerte - h(Z) Funktion h von z - H_q Koeffizienten, gegeben durch Gl. (A 10) - j Massenstromdichte - J _k , J_q Besselfunktion der Ordnungk, q - k definiert durch Gl. (A 9) - n laufende Zahl - m laufende Zahl - p laufende Zahl - Pe=Re·Sc Pecletzahl - q laufende Zahl - Q_n Koeffizienten, definiert in Gl. (31) - r radiale Koordinate - r₀ Radius - R r/r₀ - Re=u₀r₀/ Reynoldszahl - S=2r₀z Zylinderfläche - Sc=/D Schmidtzahl - Sh=2r₀ /D Sherwoodzahl - Sherwoodzahl, definiert in Gl. (52) - Sh_u Sherwoodzahl, definiert in Gl. (54) - Sh_z Sherwoodzahl, definiert in Gl. (40) - Sherwoodzahl, definiert in Gl. (45) - t R² - u Geschwindigkeit - u₀ maximale Geschwindigkeit - v - Volumenstrom - w Variable - x Variable - y abhängige Variable - z axiale Koordinate, Lauflänge - Z z/r₀ - Z_Pe dimensionslose Lauflänge, definiert durch Gl. (34) - a_n Koeffizienten, definiert durch Gl. (A 19) - Stoffübergangskoeffizient - Stoffübergangskoeffizient, definiert in Gl. (48) - _u Stoffübergangskoeffizient, definiert in Gl. (49) - _z Stoffübergangskoeffizient, definiert in Gl. (38) - Stoffübergangskoeffizient, definiert in Gl. (44) - definiert in Gl. (A 21) - Gammafunktion - c Konzentrationsdifferenz - m Stoffmenge - Zahl zwischen Null und Eins - laufende Zahl - kinematische Zähigkeit - (v) (t) - konfluente hypergeometrische Funktion - (t) - konfluente hypergeometrische Funktion - , _n Eigenwerte Hochzeichen - * kennzeichnet asymptotische Lösungen     

Gaussian Closure of One-Dimensional Unsaturated Flow in Randomly Heterogeneous Soils

We propose a new method for the solution of stochastic unsaturated flow problems in randomly heterogeneous soils which avoids linearizing the governing flow equations or the soil constitutive relations, and places no theoretical limit on the variance of constitutive parameters. The proposed method applies to a broad class of soils with flow properties that scale according to a linearly separable model provided the dimensionless pressure head has a near-Gaussian distribution. Upon treating as a multivariate Gaussian function, we obtain a closed system of coupled nonlinear differential equations for the first and second moments of pressure head. We apply this Gaussian closure to steady-state unsaturated flow through a randomly stratified soil with hydraulic conductivity that varies exponentially with where =(1/) is dimensional pressure head and is a random field with given statistical properties. In one-dimensional media, we obtain good agreement between Gaussian closure and Monte Carlo results for the mean and variance of over a wide range of parameters provided that the spatial variability of is small. We then provide an outline of how the technique can be extended to two- and three-dimensional flow domains. Our solution provides considerable insight into the analytical behavior of the stochastic flow problem.     

Nonlinear response of orthotropic thick shallow conical springs

Summary This study deals with the geometrically nonlinear axisymmetric deflection characteristics of orthotropic thick shallow truncated conical springs. Marguerre type governing equations including the effect of shear deformation have been formulated in terms of normal displacement , shear rotation and stress function . Polynmial approximations are used for these field variables and the discretised equations are obtained using the orthogonal point collocation method. The influence of thickness-to-radius ratio, orthotropic parameter and edge conditions has been investigated. Significant influence of shear deformation on the response is found for moderately thick high-modulus composite clamped caps. Its effect is less pronounced for simply-supported caps.

---

Nichtlineares Verhalten orthotroper, dicker, flacher Tellerfedern

Übersicht In dieser Arbeit werden die nichtlinearen Kennlinien von orthotropen, dicken, flachen Tellerfedern untersucht. Die dem Problem zugrunde liegenden Gleichungen vom Marguerre-Typ schließen den Effekt der Schubdeformation ein und werden mit Hilfe der Auslenkung , des Schubwinkels und einer Spannungsfunktion formuliert. Diese Größen werden in Potenzreihen entwickelt, und mit dem Kollokationsverfahren ergeben sich diskretisierte Gleichungen. Der Einfluß des Verhältnisses von Dicke zu Radius sowie der Einfluß von Orthotropieparametern und Randbedingungen wird untersucht.

     

Two-dimensional transonic vortex flows of an ideal gas

Equations are obtained for two-dimensional transonic adiabatic (nonisoenergetic and nonisoentropic) vortex flows of an ideal gas, using the natural coordinates (=const is the family of streamlines, and =const is the family of lines orthogonal to them). It is not required that the transonic gas flow be close to a uniform sonic flow (the derivation is given without estimates). Solutions are found for equations describing vortex flows inside a Laval nozzle and near the sonic boundary of a free stream.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–109, September–October, 1973.     

Nonlinear static behavior of a cantilever subjected to an inclined follower force

The nonlinear static behavior of a linearly elastic cantilever subjected to a nonconservative force of the follower type is formulated and examined. The formulation allows for finite rotations with small strains (the elastica). Exact solutions are found. The investigation is greatly facilitated by means of a phase plane analysis in which the phase plane variables are related to slope angle and bending moment. Some of the interesting and unusual effects occurring in this system are discussed and illustrated with a set of deflection curves for a typical case.Nomenclature x, y coordinates of a point on the deformed elastic axis - slope angle - s arc length - L length of beam (assumed constant) - x _L , y _L , _L values of x, y, at s=L - P applied force - constant angle between P and end tangent - angle between P and the horizontal - EI beam stiffness (assumed constant) - u, dimensionless variables defined by (7) and (8) - c ² load parameter defined by (10) - k, transformation parameters defined by (21) - F(, k), E(, k) elliptic integrals of the first and second kind - argument of elliptic integrals - ₀, ₁ values of at u=0 and u=1 - m, n positive integers - N mode number     

Theoretical investigation of the effect of an internal heat source on turbulent liquid metal heat transfer in a channel with uniform heat flux

Summary The effect of an internal heat source on the heat transfer characteristics for turbulent liquid metal flow between parallel plates is studied analytically. The analysis is carried out for the conditions of uniform internal heat generation, uniform wall heat flux, and fully established temperature and velocity profiles. Consideration is given both to the uniform or slug flow approximation and the power law approximation for the turbulent velocity profile. Allowance is made for turbulent eddying within the liquid metal through the use of an idealized eddy diffusivity function. It is found that the Nusselt number is unaffected by the heat source strength when the velocity profile is assumed to be uniform over the channel cross section. In the case of a 1/7-power velocity expression, the Nusselt numbers are lower than those in the absence of internal heat generation, and decrease with diminishing eddy conduction. Nusselt numbers, in the absence of an internal heat source, are compared with existing calculations, and indications are that the present results are adequate for preliminary design purposes.Nomenclature A hydrodynamic parameter - a half height of channel - a ₁ a constant, 1+0.01 Pr Re ^0.9 - a ₂ a constant, 0.01 Pr Re ^0.9 - C _p specific heat at constant pressure - D _h hydraulic diameter of channel, 4a - h heat transfer coefficient, q _w/(t _w–t _b) - I ₁ integral defined by (17) - I ₂ integral defined by (18) - k diffusivity parameter, (1+0.01 Pr Re ^0.9)^1/2 - m exponent in power velocity expression - Nu Nusselt number, hD _h/ - Nu ₀ Nusselt number in absence of internal heat generation - Pr Prandtl number, / - Q heat generation rate per volume - q _w wall heat flux - Re Reynolds number for channel, 2/ - s ratio of heat generation rate to wall heat flux, Qa/q _w - T dimensionless temperature, (t _w–t)/(t _w–t _b) - t fluid temperature, t _w wall temperature, t _b fluid bulk temperature - u fluid velocity in x direction, , fluid mean velocity - x longitudinal coordinate measured from channel entrance - x ⁺ dimensionless longitudinal coordinate, 2(x/a)/Pr Re - y transverse coordinate measured from channel centerline - z transverse coordinate measured from channel wall, a–y - molecular diffusivity of heat, /C _p - dummy variable of integration - dummy variable of integration - _H eddy diffusivity of heat - _M eddy diffusivity of momentum - dummy variable of integration - fluid thermal conductivity - _T dimensionless diffusivity, Pr ( _H/) - fluid kinematic viscosity - dummy variable of integration - fluid density - dummy variable of integration - ratio of eddy diffusivity for heat transfer to that for momentum transfer, _H/ _M - average value of - dimensionless velocity distribution, u/     

Existence of the vortex solutions of some semilinear elliptic equations

The purpose is to extend the existence result of vortex solutions to semilinear elliptic equations for a large class of nonlinearities. M. I. Weinstein used variational techniques to show the existence of nodal solutions for the specific nonlinear term f(¦¦)=(1–¦¦²). An ordinary differential equation phase space setting is used to show the unique transverse intersection of unstable and stable manifolds which contain the solutions satisfying the necessary boundary conditions under certain assumptions on the nonlinearity.     

An elementary proof of the polar factorization of vector-valued functions

We present an elementary proof of an important result of Y. Brenier [Br1, Br2], namely, that vector fields in ^d satisfying a nondegeneracy condition admit the polar factorization (*) u(x)=(s(x)), where is a convex function and s is a measure-preserving mapping. Brenier solves a minimization problem using Monge-Kantorovich theory; whereas we turn our attention to a dual problem, whose Euler-Lagrange equation turns out to be (*).     

Second order gravitational field theories with hilbert gauge

Summary We consider, in the field-theoretical approach, a class of gravitational theories deducible by a variational principle in the unrenormalized pseudo-Euclidean space-time. At first order in the coupling constant f we require the theories to coincide with the Einstein one. Moreover we assume the Hilbert gauge which assure the exclusion of the vector component of the gravitational potential . To get the higher order consistency we substitute the most general energy-momentum tensor for the particle tensorT ^(p) in the field equations. Requiring the latter to be deducible by a variational principle varying the potentials , we get a Lagrangian which, varying the particle coordinates, gives the equations of motion. So we get a class of theories depending on 5 arbitrary parameters. To have observable quantities we have to renormalize. So we realize that, to satisfy the equivalence principle, we have to put one of the arbitrary parameters equal to zero. With this choice the class of theories coincides at second order with general relativity.

---

Sommario Si vuole ottenere una classe di teorie gravitazionali deducibili da un principio variazionale, nell'ambito della teoria dei campi e nello spazio-tempo pseudoeuclideo non-rinormalizzato. Si richiede che tali teorie coincidano, al primo ordine nella costante di accoppiamento f, con la teoria di Einstein. Si assume inoltre la gauge di Hilbert al fine di escludere la presenza della componente vettoriale del potenziale . Per ottenere la consistenza al secondo ordine delle equazioni di campo, si sostituisce, in queste ultime, al tensore della particellaT ^(p) il più generale tensore energia-quantità-di-moto . Imponendo alle equazioni di campo di essere deducibili mediante un principio variazionale ove si varino i potenziali , si ottiene una lagrangiana che, ove si varino le coordinate della particella di prova, dà le equazioni di moto. In tal modo si ottiene una classe di teorie dipendenti da 5 parametri arbitrari. Per un confronto con i dati sperimentali è necessario rinormalizzare, onde esprimere quantità osservabili. Si dimostra così che per soddisfare il principio di equivalenza al secondo ordine è necessario porre uno dei 5 parametri uguale a zero e che, con tale scelta, l'intera classe di teorie coincide, al secondo ordine, con la relatività generale.

---

---

Research sponsored by the CNR, Gruppi di ricerca Matematica     

Transcendentally small transversality in the rapidly forced pendulum

The rapidly forced pendulum equation with forcing sin((t/), where =<₀^p,p = 5, for ₀, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( ,t) plane satisfiesd(t) = sin(t/) sech(/2) +O( ₀ exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S _s = (/2) sech(/2) +O(( ₀ /) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.     

Nonstationary vibration of a rotating shaft with nonlinear spring characteristics during acceleration through a critical speed (A critical speed of a summed-and-differential harmonic oscillation)

Nonstationary vibration of a flexible rotating shaft with nonlinear spring characteristics during acceleration through a critical speed of a summed-and-differential harmonic oscillation was investigated. In numerical simulations, we investigated the influence of the angular acceleration , the initial angular position of the unbalance _n and the initial rotating speed on the maximum amplitude. We also performed experiments with various angular accelerations. The following results were obtained: (1) the maximum amplitude depends not only on but also on _n and : (2) when the initial angular position _n changes. the maximum amplitude varies between two values. The upper and lower bounds of the maximum amplitude do not change monotonously for the angular acceleration: (3) In order to always pass the critical speed with finite amplitude during acceleration. the value of must exceed a certain critical value.Nomenclature O-xyz rectangular coordinate system - , ₁, ₁ inclination angle of rotor and its projections to thexy- andyz-planes - I _r polar moment of inertia of rotor - I diametral moment of inertia of rotor - i _r ratio ofI _r toI - dynamic unbalance of rotor - directional angle of fromx-axis - c damping coefficient - spring constant of shaft - N _nt ,N _nt nonlinear terms in restoring forees in ₁ and ₁ directions - ₄ representative angle - a small quantity - V. V _u .V _N potential energy and its components corresponding to linear and nonlinear terms in the restoring forees - directional angle - _n coefficients of asymmetrical nonlinear terms - _n coefficients of symmetrical nonlinear terms - coefficients of asymmetrical nonlinear terms experessed in polar coordinates - coefficients of symmetrical nonlinear terms expressed in polar coordinates - rotating speed of shaft - t time - _n initial angular position of att=0 - p natural frequency - p ₁.p _t natural frequencies of forward and backward precessions - , ₁, ₁ total phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - , ₁, ₁ phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - P, R _t ,R _b amplitudes of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - difference between phases ( = _f – _u) - acceleration of rotor - initial rotating speed - t _t ,r _b amplitudes of nonstationary oscillation during acceleration - (r _t )_max, (r _b )_max maximum amplitudes of nonstationary oscillation during acceleration - (r ₁ ¹ )_max, (r _b ¹ )_max maximum value of angular acceleration of non-passable case - ₀ critical value over which the rotor can always pass the critical speed - p ₁,p ₂,p ₃,p ₄ natural frequencies of experimental apparatus     

The third-normal stress difference in entangled melts: Quantitative stress-optical measurements in oscillatory shear

Stress-optical measurements are used to quantitatively determine the third-normal stress difference (N ₃ = N ₁ + N ₂) in three entangled polymer melts during small amplitude (<15%) oscillatory shear over a wide dynamic range. The results are presented in terms of the three material functions that describe N ₃ in oscillatory shear: the real and imaginary parts of its complex amplitude ₃ ^* = ₃ - i ₃ , and its displacement ₃ ^d . The results confirm that these functions are related to the dynamic modulus by ² ₃ ^* ()=(1-)[G ^*())– G ^*(2)] and ² ₃ ^d ()=(1- )G() as predicted by many constitutive equations, where = –N ₂/N ₁. The value of (1-) is found to be 0.69±0.07 for poly(ethylene-propylene) and 0.76±0.07 for polyisoprene. This corresponds to –N ₂/N ₁ = 0.31 and 0.24±0.07, close to the prediction of the reptation model when the independent alignment approximation is used, i.e., –N ₂/N ₁ = 2/7 – 0.28.     

Beitrag zur Auswertung von Anlauf-Vorgängen in viskoelastischen Flüssigkeiten

Zusammenfassung In diesem Bericht wird die Kraftübertragung zwischen zwei unbegrenzt ausgedehnten und durch eineMaxwell-Flüssigkeit gekoppelten Platten bei beliebigem Zeitgesetz der Bewegung der oberen auf die elastisch aufgehängte untere Platte unter Einbeziehung der Trägheit von unterer Platte undMaxwell-Flüssigkeit berechnet.Diese Fragestellung führt mathematisch auf ein Randwertproblem, dessen eine Randbedingung die Form ³ u/x ² y + ₁ u + ₁ u/y – ₂ ² u/xy = 0 fürx = 0 annimmt, also gemischte Ableitungen nach 2 unabhängigen Veränderlichen enthält.Es wird gezeigt, daß sich die Lösung als unendliche Reihe darstellen läßt, deren 1. Glied aus abklingenden Exponentialfunktionen, alle weiteren hingegen aus gedämpften Schwingungen bestehen.Als spezielle Zeitgesetze für die obere Platte werden die impulsförmige Erregung, die zeitproportionale und die periodische Bewegung untersucht.Es wird abgeleitet, wie sich die Kenngrößen undG (Viskosität und Schubmodul) aus Eigenfrequenz und Dämpfung bzw. aus den beiden Exponenten dere-Funktionen in den beiden ersten Fällen berechnen lassen. Bei periodischer Beanspruchung werden undG aus der Resonanzfrequenz und der Halbwertbreite bzw. dem Phasenwinkel ermittelt.Abschließend wird gezeigt, daß und wie ein Rotationsviskosimeter auf den behandelten Fall ebener Platten zurückgeführt werden kann.

---

Summary In this report the force transmission between two infinitely extended plates coupled by aMaxwell-Fluid is calculated. In this calculation the inertia of the fluid and the influence of the elastically suspended lower plate is included, while the upper plate can perform any time-dependent movement.Mathematically this question leads to a boundery value problem; one boundery condition has the form ³ u/ ² y + ₁ u + ₁ u/y – ₂ ² u/xy = 0 (x = 0) ( ₁, ₂, = constant), that means it contains mixed differential-quotients of two independent variables.The solution is demonstrated as an infinite series, the form of the first term is of two negative exponential functions, all the other terms are damped oscillations.The excitation from a single pulse, the uniform timeproportional and the periodic movement as timedependent laws for the upper plate are analysed.The values of andG (viscosity and shear modulus) are derived from eigenfrequency and damping-constant respectively from the values of the two exponentials. In the periodic case andG are calculated from the resonance frequency and halfwidth or phase-angle respectively.Finally the conditions of the rotation-viscometer are reduced to the case of parallel-plate-viscometer.

---

---

Vorgetragen auf der Jahrestagung der Deutschen Rheologen vom 7.–9. Mai 1973 in Berlin.

---

Mit 6 Abbildungen und 1 Tabelle