Viscoelastic properties of semidilute poly(methyl methacrylate) solutions

Viscoelastic properties were examined for semidilute solutions of poly(methyl methacrylate) (PMMA) and polystyrene (PS) in chlorinated biphenyl. The number of entanglement per molecule, N, was evaluated from the plateau modulus, G _N . Two time constants, _s and ₁, respectively, characterizing the glass-to-rubber transition and terminal flow regions, were evaluated from the complex modulus and the relaxation modulus. A time constant _k supposedly characterizing the shrink of an extended chain, was evaluated from the relaxation modulus at finite strains. The ratios ₁/ _s and _k / _s were determined solely by N for each polymer species. The ratio ₁/ _s was proportional to N ^4.5 and N ^3.5 for PMMA and PS solutions, respectively. The ratio _k / _s was approximately proportional to N ^2.0 in accord with the prediction of the tube model theory, for either of the polymers. However, the values for PMMA were about four times as large as those for PS. The result is contrary to the expectation from the tube model theory that the viscoelasticity of a polymeric system, with given molecular weight and concentration, is determined if two material constants _s and G _N are known.     

Analysis of melt spinning transients in Lagrangian coordinates

Transients in melt spinning of isothermal power law and Newtonian fluids were found to be governed by an extremely simple partial differential equation ² ( ^1/n )/() = 0 in Lagrangian coordinates where is the cross-sectional area,n the power law exponent, the time and the the time at which a fluid molecule constituting the spinline left the spinneret. The general integral ^1/n =f() +g () of the above governing equation containing two arbitrary functions represents physically attainable spinline transients. Hitherto unknown analytical transient solutions of the above governing equation were obtained for the response of isothermal constant tension spinlines to a stepwise change in tension, spinneret hole area, extrusion speed or extrusion viscosity and for the starting transient in gravitational spinning. Linearized perturbation solutions and the stability limit of the spinline derived from the above new found nonlinear solutions were in agreement with previous findings and the above nonlinear response of the spinline to a step increase in the spinneret hole area was found to be equivalent to Orowan's tandem cylinder model of dent growth in filament stretching.     

Effect of suspended particles on the stability of plane poiseuille flow

The behavior of the neutral stability curves is investigated for various values of the particle relaxation time and mass concentration 0 100 and 0 f 0.1. It is shown that as increases from zero the flow is at first destabilized and then at >6 becomes stable, while at >40 the stabilizing effect of the dispersed phase grows weaker. It is found that there is a certain interval 10< <40 on which the flow is most stable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 46–53, January–February, 1986.     

The Preston tube in adiabatic compressible flow

A new procedure for the reduction of Preston tube data is introduced, based on the van Driest transformation. It appears to give results agreeing with the better calibration experiments, although a significant assumption in its derivation is violated.List of Symbols M _s Mach number sensed by Preston tube - M Friction Mach number (=u/wall sound speed) - R Gas constant - T _w Wall temperature - d Diameter of Preston tube - h Height of effective centre of Preston tube - p Preston tube pressure difference reading - p _i Equivalent incompressible Preston tube reading - p _w Wall pressure - r Recovery factor (=0.896) - u Friction velocity (=[_w/wall density]^1/2) - Empirical constant allowing for departure from Crocco temperature-velocity correlation (=0.975) - Specific heat ratio - Fluid kinematic viscosity - _w Wall shear-stress     

A method for the analysis of recirculating turbulent flows in a cavity

The complex fluid-dynamic aspects of a turbulent recirculating flow in a cavity with axial throughflow, and a rotating wall, were investigated by adopting a simple procedure for evaluating the turbulent stresses. The flow field was divided into two regions, a core and a wall region respectively. A wall function was adopted in the zones near to the solid boundaries, while a constant eddy diffusivity was assumed, in the core, following the indications of computed heat transfer coefficients in comparison with existing experimental data. The distributions of the stream function and of the tangential velocity are presented for a range of the rotational Reynolds number of the rotating wall and of the Reynolds number of the throughflow.

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Turbulente Rezirkulationsströmung in einem Hohlraum

Zusammenfassung Die komplizierten fluiddynamischen Aspekte einer turbulenten Rezirkulationsströmung in einem Hohlraum mit axialem Durchfluß und einer rotierenden Wand werden unter Verwendung einer vereinfachten Methode zur Berechnung der turbulenten Spannungen betrachtet. Das Strömungsfeld wird in einen Kern und einen Wandbereich aufgeteilt. Für die wandnahen Zonen wird eine Wandfunktion angenommen, während im Kern mit konstanter Wirbeldiffusivität gerechnet wird, was durch den Vergleich berechneter mit gemessenen Wärmeübergangskoeffizienten gerechtfertigt erscheint. Verteilungen der Stromfunktion und der tangentialen Geschwindigkeit sind für einen bestimmten Bereich der Reynoldszahlen für die Wandrotation und der für den Durchfluß angegeben.

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Nomenclature L axial length of enclosure - P dimensionless pressure, p^*2 - p static pressure - R dimensionless radial coordinate, r/r^* - r radial coordinate - r^* reference length, equal to r_O for enclosure - r_i radii of inlet and exit apertures - Re Reynolds number, v^*r^*/ - Re_i pipe Reynolds number, ¯v_zi(2r_i)/ - Re_t turbulent Reynolds number, Re(/) - Re rotational Reynolds number, r ₀ ² / - t dimensionless time,t/(r^*/v^*) - t time - V_r, V, Vz dimensionless velocity components, V_r/v^*, v, v_z/v^* - v_i turbulent fluctuation of the i-component of velocity - v_r, v, v_z velocity components - v^* reference velocity, equal to ¯v_zi for enclosure - X coordinate along a wall, x/r^* - Y coordinate normal to a wall, y/r^* - Z dimensionless axial coordinate, z/r^* - z axial coordinate - eddy diffusivity for momentum - dynamic viscosity - kinematic viscosity - density - shear stress - dimensionless shear stress, /v^*2 - dimensionless stream function, /r*²v*² - stream function - angular velocity - tangential vorticity component - ()_eff effective - ()_l laminar - ()_t turbulent - mean over the time     

Die swell of filled polymer melts

The Barus effect in polypropylene and polystyrene blended with a variety of fillers at various concentrations was investigated using a capillary extrusion rheometer. If the die swell is defined as the square of the ratio of the extrudate diameterd to the die diameterD, it is found to depend on the apparent shear stress _W . Below a certain value of _w the relation =B _B ^A applies. The die swell, _M , of a filled polymer depends on the type, size and volume fraction of the filler. In particular,A increases as the volume fraction increases and is largest for powders, smaller for flakes and smallest for fibres, whereasB shows the opposite trend but to a lesser extent.     

The eddy viscosity of turbulent equilibrium layers

Summary Similarity laws for the mean flow and scaling laws for the turbulent motion are used in an attempt to obtain a general expression for the eddy viscosity of equilibrium layers. It is found that =0.09 w ² /w*, in which w ² is a Reynolds stress representative for the region of overlap between the law of the wall and the velocity-defect law, while w* is the logarithmic slope of the mean velocity profile in that region. The distinction between w and w* is related to the strong inhomogeneity of the mean rate of strain in the inner layer. The results of the theory agree with experimental evidence obtained from transpired equilibrium layers.     

The onset of Taylor-like vortices in the flow induced by an impulsively started rotating cylinder

The onset of instability in the flow by an impulsively started rotating cylinder is analyzed under linear theory. It is well-known that at the critical Taylor number T_c=1695 the secondary flow in form of Taylor vortices sets in under the narrow-gap approximation. Here the dimensionless critical time _c to mark the onset of instability for TT_c is presented as a function of the Taylor number T. Available experimental data of water indicate that deviation of the velocity profiles from the primary flow occurs starting from a certain time 4_c. It seems evident that during _c4_c the secondary flow is very weak and the primary state of time-dependent annular Couette flow is maintained.     

An analytical solution for the velocity and shear rate distribution of non-ideal Bingham fluids in concentric cylinder viscometers

An analytical solution is presented for the calculation of the flow field in a concentric cylinder viscometer of non-ideal Bingham-fluids, described by the Worrall-Tuliani rheological model. The obtained shear rate distribution is a function of the a priori unknown rheological parameters. It is shown that by applying an iterative procedure experimental data can be processed in order to obtain the proper shear rate correction and the four rheological parameters of the Worrall-Tuliani model as well as the yield surface radius. A comparison with Krieger's correction method is made. Rheometrical data for dense cohesive sediment suspensions have been reviewed in the light of this new method. For these suspensions velocity profiles over the gap are computed and the shear layer thicknesses were found to be comparable to visual observations. It can be concluded that at low rotation speeds the actually sheared layer is too narrow to fullfill the gap width requirement for granular suspensions and slip appears to be unavoidable, even when the material is sheared within itself. The only way to obtain meaningfull measurements in a concentric cylinder viscometer at low shear rates seems to be by increasing the radii of the viscometer. Some dimensioning criteria are presented.Notation A, B Integration constants - C Dimensionless rotation speed = µ/_y - c = 2µ - d = ₀ ²–2c_y - f() = (–₀)²+2c(–_y) - r Radius - r _b Bob radius - r _c Cup radius - r _y Yield radius - r ₀ Stationary surface radius - r Rotating Stationary radius - Y ₀ Shear rate parameter = /µ Greek letters Shear rate - = (r _y /r _b )²– 1 - µ Bingham viscosity - µ₀ Initial differential viscosity - µ µ₀-µ - Rotation speed - Angular velocity - Shear stress - _b Bob shear stress - _B Bingham stress - _y (True) yield stress - ₀ Stress parameter = _B +µ Y ₀ - _B - _y      

Stability of a radially bounded rotating fluid heated from below

The stability of a bounded rotating cylinder of fluid heated from below is treated mathematically under the assumptions of stationary onset and axisymmetry. Critical Rayleigh numbers are computed by Galerkin's method as a function of the Taylor number and cylinder aspect ratio for Taylor numbers,10⁶. The constraining effect of the side walls is shown to decrease with either increasing or increasing radius/height ratios. For>10⁶, most cylinders, excluding extremely tall ones, will appear infinite in horizontal extent as far as stability characteristics are concerned. The form of the motion at onset is discussed in relation to previous work.     

A direct formulation of correlations for wall-proximity corrections of hot-wire measurements

Correlations for corrections to hot-wire data for the effects of wall proximity within the viscous sublayer are usually presented in the form u/u = F (y u /). The application of such correlations requires a prior knowledge of the wall shear stress; alternatively, the correlation must be used in an iterative fashion. It is shown in the present note that any such correlation may be recast with no loss of generality in the explicit form u/u _m = f (y u _m/), which is more convenient for use.List of symbols u difference between measured and actual velocities, u _m – u - u _m measured velocity - u shear velocity, - u ⁺ on-dimensional velocity, u/u - y distance from wall - y ⁺ non-dimensional distance from wall, y u / - fluid density - fluid kinematic viscosity - _s wall shear stress     

Effect of surface layers on the constriction resistance of an isothermal spot. Part II: Analytical results for thick layers

Solution of a non-homogeneous Fredholm integral equation of the second kind [1], which forms the basis for the evaluation of the constriction resistance of an isothermal circular spot on a half-space covered with a surface layer of different material, is considered for the case when the ratio, , of layer thickness to spot radius is larger than unity. The kernel of the integral equation is expanded into an infinite series in ascending odd-powers of (1/) and an approximate kernel accurate to (^–(2M+1)) is derived therefrom by terminating the series after an arbitrary but finite number of terms, M. The approximate kernel is rearranged into a degenerate form and the integral equation with this approximate kernel is reduced to a system of M linear equations. An explicit analytical solution is obtained for a four-term approximation of the kernel and the resulting constriction resistance is shown to be accurate to (^–9). Solutions of lower orders of accuracy with respect to (1/) are deduced from the four-term solution. The analytical approximations are compared with very accurate numerical solutions and it is shown that the (^–9)-approximation predicts the constriction resistance exceedingly well for any 1 over a four orders of magnitude variation of layer-to-substrate conductivity ratio for both conducting and insulating layers. It is further shown that, for all practical purposes, an (^–3)-approximation gives results of adequate accuracy for > 2.     

Quantitative analysis of flow visualizations in an unsteady channel flow

Quantitative results concerning the modulation of the ejection and bursting frequency in an unsteady channel flow obtained by flow visualizations are presented and compared with probe measurements. The frequency of the imposed velocity oscillations f covers a large range going from the quasi steady limit to the time mean bursting frequency in the corresponding steady flow. The imposed amplitudes of the velocity oscillations are 13% and 20% of the centerline velocity. The bursting process is identified by the intermittent lift up of the dye injected at the wall. Qualitative analysis of the flow visualizations show that the ejection activity at a given phase of the oscillation cycle is repetitive from one cycle to the other. The modulation amplitude of the ejection frequency f _e is sensitive to the imposed frequency. At low imposed frequency f _e is modulated as the wall shear stress, but the inner scaling does not hold when f ⁺ is high. Here, (+) corresponds to the quantities normalized with the inner variables, i.e. the friction velocity u and the viscosity . The grouping of the ejections into bursts show the coexistence of two categories of events which react differently to the forcing. The groups of ejections (Multiple Ejection Bursts) are governed by the modulation of the wall shear stress in the whole imposed frequency range. The solitary ejections (or the Single Ejection Bursts) have modulation amplitudes and phases which differ significantly from those of in the intermediate and high imposed frequency range. There is a good agreement between the flow visualization data and the probe measurements.     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

A general model for the effective viscosity of pseudoplastic and dilatant fluids

Summary A new and very general expression is proposed for correlation of data for the effective viscosity of pseudoplastic and dilatant fluids as a function of the shear stress. Most of the models which have been proposed previously are shown to be special cases of this expression. A straightforward procedure is outlined for evaluation of the arbitrary constants.

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Zusammenfassung Eine neue und sehr allgemeine Formel wird für die Korrelation der Werte der effektiven Viskosität von strukturviskosen und dilatanten Flüssigkeiten in Abhängigkeit von der Schubspannung vorgeschlagen. Die meisten schon früher vorgeschlagenen Methoden werden hier als Spezialfälle dieser Gleichung gezeigt. Ein einfaches Verfahren für die Auswertung der willkürlichen Konstanten wird beschrieben.

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Nomenclature b arbitrary constant inSisko model (eq. [5]) - n arbitrary exponent in eq. [1] - x independent variable - y(x) dependent variable - y ₀(x) limiting behavior of dependent variable asx 0 - y(x) limiting behavior of dependent variable asx - z original dependent variable - arbitrary constant inSisko model (eq. [5]) andBird-Sisko model (eq. [6]) - arbitrary exponent in eqs. [2] and [8] - effective viscosity = shear stress/rate of shear - _A effective viscosity at = _A - _B empirical constant in eqs. [2] and [8] - ₀ limiting value of effective viscosity as 0 - ₀() limiting behavior of effective viscosity as 0 - limiting value of effective viscosity as - () limiting behavior of effective viscosity as - rate of shear - arbitrary constant inBird-Sisko model (eq.[6]) - shear stress - _A arbitrary constant in eqs. [2] and [8] - ₀ shear stress at inBingham model - _1/2 shear stress at = ( ₀ + )/2 With 8 figures     

Finite element solutions for laminar flow and combined convection around a square prism

The finite element solutions of the full Navier-Stokes and energy equations for steady laminar flow and combined convection around square prisms with attack angles of 0° and 45° are obtained for a gas having Pr=0.7. The variations of surface shear stress, local pressure and Nusselt number are obtained over the entire prism surface including the zone beyond the point of the separation. The predicted values of drag coefficients, the location of separation, average Nusselt number and the plots of velocity flow fields and isotherms are also presented. The trend of the present numerical results seems reasonable.

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Finite-Elemente-Verfahren für laminare Strömung und kombinierte Naturkonvektion um ein quadratisches Prisma

Zusammenfassung Es wird über Lösungen der Navier-Stokesund der Energiegleichungen mit Hilfe der Finite-Elemente-Methode für stationäre laminare Strömung, kombiniert mit Naturkonvektion, um ein quadratisches Prisma berichtet, wobei als Anströmwinkel 0° und 45° gewählt wurden und Gasströmung mitPr=0,7 angenommen wurde. Die Rechnung ergibt den Verlauf der Wandschubspannungen, des örtlichen Druckes und der Nusselt-Zahl über die gesamte Oberfläche des Prismas, einschließlich des Bereiches hinter dem Ablösepunkt. Weiterhin werden in dem Aufsatz Angaben gemacht über die Widerstandskoeffizienten, die Lage des Ablösepunktes, der mittleren Nusselt-Zahl sowie der Geschwindigkeits- und Temperaturfelder. Die numerischen Ergebnisse erscheinen im Trend vernünftig zu sein.

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Nomenclature a side length of square - C _f friction drag coefficient - C _p pressure drag coefficient - C _D total drag coefficient - F _f total friction drag force - F _P total pressure drag force - Gr Groshoff number,g (T _w -T )a ³/v ² - g gravitational acceleration - h local heat transfer coefficient - K thermal conductivity - L dimensionless location of surface from the front stagna tion point,L ^*/a - L ^* dimension location of prism surface - Lc location of separation - N _j shape function - Nu, local and average Nusselt numbers - M _l shape function - P dimensionless pressure,p ^*/u ² - P ^* pressure - p _x ^* x-componentP ^* - Pe Peclet number,Re Pr - Pr Prandtl number, c/K - Ra Rayleigh number,Gr Pr - Re Reynolds number,a u /v - s direction along the sides of prism - u dimensionlessX-direction component of velocity,u ^*/u - u ^* X-direction component of velocity - u free stream velocity - dimensionlessY-direction component of velocity,v*/u - ^* Y-direction component of velocity - X X-direction axis - x dimensionlessX-direction coordinate,x ^*/a - x ^* X- direction coordinate - Y Y-direction axis - y ^* dimensionless 7-direction coordinate,y ^*/a - y ^* Y-direction coordinate Greek symbols coefficient of volumetric thermal expansion - attack angle - dynamic viscosity - kinematic viscosity,/ - density of fluid - _w dimensionless surface shear stress, _w ^* /u ² - _w ^* surface shear stress - _wx ^* x-component of _w ^/* - dimensionless temperature,      

Turbulent boundary layer over a smooth wall with widely separated transverse square cavities

Laser-Doppler velocimetry (LDV) measurements and flow visualizations are used to study a turbulent boundary layer over a smooth wall with transverse square cavities at two values of the momentum thickness Reynolds number (R =400 and 1300). The cavities are spaced 20 element widths apart in the streamwise direction. Flow visualizations reveal a significant communication between the cavities and the overlying shear layer, with frequent inflows and ejections of fluid to and from cavities. There is evidence to suggest that quasi-streamwise near-wall vortices are responsible for the ejections of fluid out of the cavities. The wall shear stress, which is measured accurately, increases sharply immediately downstream of the cavity. This increase is followed by a sudden decrease and a slower return to the smooth wall value. Integration of the wall shear stress in the streamwise direction indicates that there is an increase in drag of 3.4% at bothR .Nomenclature C _f skin friction coefficient - C _fsw friction coefficient for a continuous smooth wall - k height of the cavity - k ⁺ ku / - R Reynolds number based on momentum thickness (U ₁ /v) - Rx Reynolds number based on streamwise distance (U ₁ x/) - s streamwise distance between two cavities - t time - t ⁺ tu ₂ / - U ₁ freestream velocity - mean velocity inx direction - u,v,w rms turbulent intensities inx,y andz directions - u local friction velocity - u _sw friction velocity for a continuous smooth wall - w width of the cavity - x streamwise co-ordinate measured from the downstream edge of the cavity - y co-ordinate normal to the wall - z spanwise co-ordinate - y ⁺ yu / - boundary layer thickness - ₀ boundary layer thickness near the upstream edge of the cavity - _i thickness of internal layer - kinematic viscosity of water - ⁺ zu / - momentum thickness     

Momentum balance in highly distorted turbulent pipe flows

An in depth study into the development and decay of distorted turbulent pipe flows in incompressible flow has yielded a vast quantity of experimental data covering a wide range of initial conditions. Sufficient detail on the development of both mean flow and turbulence structure in these flows has been obtained to allow an implied radial static pressure distribution to be calculated. The static pressure distributions determined compare well both qualitatively and quantitatively with earlier experimental work. A strong correlation between static pressure coefficient Cp and axial turbulence intensity is demonstrated.List of symbols C _p static pressure coefficient = (pw-p)/1/2 - D pipe diameter - K turbulent kinetic energy - (r, , z) cylindrical polar co-ordinates. / 0 - R, y pipe radius, distance measured from the pipe wall - U, V axial and radial time mean velocity components - mean value of u - u, u/ , / - u, , w fluctuating velocity components - axial, radial turbulence intensity - turbulent shear stress - u friction velocity, (u ² = ₀/p) - ₀ wall shear stress - ^* boundary layer thickness A version of this paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

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