**Journal of Theoretical and Applied Mechanics**

Volume 32, Number 2, 2002

**Contents**

J. N. Boyd, R. G. Hudepohl, P. N. Raychowdhury

Department of Mathematical Sciences, Virginia Commonwealth University,

P.O. Box 842014, Richmond, VA 23248-2014, U.S.A.,

*e-mail: praychow@saturn.vcu.edu*

**Coupled Transverse Anharmonic Oscillations**

Abstract. In this note, we consider the motion of a system which consists of a linear array of two particles of the same mass and three identical, ideal Hooke’s Law springs. One spring connects the two particles to each other, and each of the other two springs is attached to a different particle and to a fixed point (as shown in Fig. 1). The particles vibrate in transverse modes perpendicular to the array. The system is, in effect, a coupled pair of simple anharmonic oscillators. The Lagrangian for the system is formulated in matrix notation. The original coordinates of the system are the displacements of the particles from their equilibrium positions. These coordinates are transformed to symmetry coordinates, and the Lagrangian is transformed accordingly by means of straightforward matrix algebra. Equations of motion in terms of the symmetry coordinates are obtained from the new Lagrangian. The natural frequencies of vibration associated with the symmetry coordinates are then obtained from these equations of motion.

Key words. anharmonic oscillator, Lagrangian, symmetry coordinates, natural frequency.

**V. Abadjiev, E. Abadjieva**

Institute of Mechanics, Bulgarian Academy of Sciences,

Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria,

*e-mail: abadjiev@imbm.bas.bg*

**ON THE CONIC CONVOLUTE HELICOIDS GENERATION FOR ACTIVE FLANKS OF THE
SPIROID PINION THREADS**

Abstract. Spiroid gear sets are used to transform rotations between shafts with crossed axes. The basic problem of analytical design of this type of gears is the tooth surface synthesis. The technology of the spiroid gear manufacture is based on the second Olivier's principle. It means that the spiroid gear teeth are generated by hob, which is counterpart of the spiroid pinion. For this reason we are interested exclusively in the synthesis of the spiroid thread pinion surfaces and spiroid thread hob surfaces, respectively. This study is dedicated of the synthesis of three types of conic convolute helicoids, which are suitable for these type surfaces.

Key words: spiroid gear set, spiroid pinion, spiroid gear, conic convolute helicoid.

F. Dong

Department of Theoretical and Applied Mechanics,

University of Illinois at Urbana-Champaign, Urbana, IL 61801,

e-mails: feidong@uiuc.edu

**A. T. Hsui**

Department of Geology,

University of Illinois at Urbana-Champaign, Urbana, IL 61801,

*e-mail*: *hsui@uiuc.edu *

D. N. Riahi

Department of Theoretical and Applied Mechanics,

University of Illinois at Urbana-Champaign, Urbana, IL 61801,

*e-mail: d-riahi@uiuc.edu*

**A STABILITY ANALYSIS AND SOME NUMERICAL COMPUTATIONS FOR THERMAL
CONVECTION WITH A VARIABLE BUOYANCY FACTOR**

Abstract. Linear and nonlinear analyses of thermal convection with a variable ``buoyancy factor'', which is defined as the product of thermal expansion coefficient and gravitational acceleration, are investigated for a fluid layer between two infinite horizontal plates. Isothermal boundary condition is applied for both boundaries, and the buoyancy factor throughout the fluid layer is chosen to be a function of depth. For several different profiles of variable buoyancy factor, the associated eigenvalue problem for the linear regime is solved numerically using a spectral method. Dependence of the critical conditions at the onset of motion on the buoyancy factor and its vertical rate of change are determined. A formula for the critical Rayleigh number as a function of the statistical features of the buoyancy factor is developed. For the nonlinear regime, computations based on a spectral Fourier-Chebyshev collocation method are carried out for six parabolic profiles of buoyancy factor. Flow patterns are found to be dominated by two--dimensional rolls for the Rayleigh numbers considered. Dependence of the Nusselt number on the buoyancy factor and its vertical rate of change is determined. When the vertical rate of change of the buoyancy factor is negative and sufficiently large to produce sign changes in the profile, our numerical simulations show that multiple layering in the vertical direction can be produced.

Key Words: thermal convection, buoyancy factor, convective flow, buoyancy driven flow, natural convection.

**A. K. Haghi**

Guilan University, P.O. Box 3756, Rasht, Iran,

*e-mail: Haghi@kadous.gu.ac.ir *

A Detailed Study on Moisture Sorption of

**Hygroscopic Fibers**

Abstract. A mathematical simulation is carried out to study the physical mechanisms of moisture diffusion into hygroscopic fabric during humidity transients. On the basis of a mathematical model developed to describe the coupled heat and moisture transfer in wool fabric, the moisture-sorption mechanisms are investigated for fabrics made from fibers with different degrees of hygroscopicity. Theoretical predictions on the moisture uptake and temperature changes under humidity transients are compared with those measured previously in a sorption-cell experiment for fabrics made from wool, cotton, acrylic fiber, and polypropylene fiber. It is concluded that the physical mechanism of moisture diffusion into highly hygroscopic fibers such as wool and cotton can be described by a two-stage moisture diffusion process: a fast Fickian diffusion with a concentration-dependent diffusion coefficient and a slow diffusion with a time-dependent diffusion coefficient. For weakly hygroscopic fibers such as polypropylene fiber, the moisture sorption process can be described by a single Fickian diffusion with a constant diffusion coefficient. Heat transfer in moist textiles takes place by conduction, infrared radiation, and distillation. Until the fiber is saturated, the evaporation process is determined mainly by the fiber sorption properties. Conductivity increases with water content and also depends on fiber sorption properties.

Key words: moisture diffusion, hygroscopic fabrics, humidity transients.

**V. Kaloyanova, M. Todorov **

University of Rousse,

8, Studentska Str., 7017 Rousse, Bulgaria,

*e-mail: *valja@mech.ru.acad.bg*, *mtodorov@ru.acad.bg*.*

**ON GENERALIZED PLANE STRAIN ELASTICITY**

Abstract**.** The most general state of plane strain in elasticity
(generalized plane strain) is considered. The displacement field has three-dimensional
spatial functional dependence, but the stress and strain components are functions of two
spatial dimensions. The displacement, strain and stress functions are derived and compared
to the classical plane strain case. Boundary conditions are discussed as well. Problems
are formulated-.to-.illustrate-.the-.range-.of-.use-.of-these-.ideas.

Key words: generalized plane strain, displacement field, stress, strain.

**R. P. Dobrev**

44, Bogdan Str., 1505 Sofia, Bulgaria,

*e-mail: dobrev@intech.bg .*

**D. A. Djerov, B. J. Vladimirov.**

University Hospital of Orthopaedics, Medical University of Sofia, Bulgaria,

56, Nikola Petkov Str., 1614 Sofia, Bulgaria,

*e-mails: dimdjerov@abv.bg,
vladimirov_borislav@hotmail.com*

**A MODEL OF A SUPPORTIVE LOAD-CARRYING STRATEGY MODULATING FORCES AND
STRESSES IN THE HUMAN ILIAC BONE AND FEMORAL NECK. PART I: THEORETICAL FORMULATION**

Abstract. A new supportive strategy of the human, which helps the iliac
bone and femoral neck to carry loads, is presented*.* It is based on the concept of
supportive bone-and-muscle interaction suggested by the authors. The functional necessity
of creating supportive mechanism as an instrument of this additional strategy is motivated
by a comparative anatomical and functional analysis. An additional isometric supportive
function is assumed to be fulfilled by the iliacus, obturator externus and vastus
lateralis muscles. The sensitivity of the bones to stress/strain distribution provokes the
supportive activity of the muscles. The intensity of this activity is controlled by the
central nervous system by means of osteoreceptor signals from the loaded bones. This is
one of the reasons for the supportive function to pass unnoticed and not to be described
by known classical biomechanical methods of studying the dynamic functions of muscles.

Key words: Biomechanics, hip joint, functions of iliacus, obturator externus and vastus lateralis muscles, load-carrying strategy.