The parallelogram of forces

Summary A brief historical outline and some epistemological suggestions are presented on the scientific debate about the parallelogram of forces.

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Sommario Si espone brevemente lo sviluppo storico del dibattito sulla regola del parallelogrammo delle forze e sui diversi tentativi di una sua dimostrazione razionale, alla luce di alcune considerazioni epistemologiche.

     

Analysis of thin parallelogram plates' bending by spline-finite-strip method

Spline finite strip has been successfully applied in solving right plates and shells by Cheung et al in 1982. In this paper, the method is extended to the analysis of parallelogram plate. This extension still retains the banded nature of the spline finite strip and only small amount of extra computing effort is required. Furthermore, the discretisation error of the above method is established theoretically as a general case for the spline finite strip method.     

Influence of elastic support on the bending of parallelogram plates

Summary The influence of elastic support on the centre deflections and maximum centre and edge moments in clamped parallelogram shaped plates is examined. A polynomial series is assumed for the deflection function, and by applying Galerkin's process, an approximate solution to the governing differential equation is obtained. Convergence of the results were verified. Results for various skew angles and aspect ratios are presented graphically.Nomenclature 2a, 2b lengths of the sides of the parallelogram - a _mn undetermined parameter in series representing the deflection W - c = cosine - D flexural rigidity of plate =Eh ³/[12(1– ²)] - E Young's modulus - h thickness of plate - k, l, m, n, M, N positive integers - p aspect ratio = b/a - q intensity of uniform load - s = sine - u, v oblique co-ordinates - W lateral deflection - x, y Cartesian co-ordinates - reaction of foundation per unit area for unit deflection - adjustable constant, zero or one - outwardly drawn normal - , dimensionless oblique co-ordinates equal to v/b and u/a respectively - skew angle - Poisson's ratio     

The center of a system of non-parallel forces

From a discrete system F of applied forces given by a collection of vectors F_k applied to corresponding points P_k, a new system QF can be obtained through a rotation by Q of all F_k without changing P_k. In this note we examine invariant properties of F under arbitrary rotations. We also examine invariant properties of the family QF when all rotations share a fixed axis, giving a coordinate-free approach to the results of Kolosov (1927) .     

The effects of fluctuating body forces on annular liquid jets

Summary The nonlinear dynamics of axisymmetric, inviscid, incompressible, thin, annular liquid jets subjected to fluctuating body forces is studied numerically by means of an adaptive finite difference method which maps the time-dependent, curvilinear geometry of the jet into a unit square. The fluctuating body forces may arise from fluctuations in the gravitational acceleration in inertial frames or from the acceleration of a non-inertial frame of reference which translates parallelly to an inertial one. It is shown that both the pressure coefficient and the axial location at which the annular jet becomes a solid one are periodic functions of time with a period equal to that of the imposed body force fluctuations, and that their magnitude increases as the amplitude of the body force fluctuations is increased. It has also been shown that, for both intermittent, sinusoidal or rectangular excitations, increases in the frequency of the excitation result in the creation of superharmonics, broad, albeit peaked, spectra, and closed phase planes with many loops.The research reported in this paper was supported by Project PB91-0767 from the D.G.I.C.Y.T. of Spain.     

The effect of long-range forces on the dynamics of a bar

The one-dimensional dynamic response of an infinite bar composed of a linear “microelastic material” is examined. The principal physical characteristic of this constitutive model is that it accounts for the effects of long-range forces. The general theory that describes our setting, including the accompanying equation of motion, was developed independently by Kunin (Elastic Media with Microstructure I, 1982), Rogula (Nonlocal Theory of Material Media, 1982) and Silling (J. Mech. Phys. Solids 48 (2000) 175), and is called the peridynamic theory. The general initial-value problem is solved and the motion is found to be dispersive as a consequence of the long-range forces. The result converges, in the limit of short-range forces, to the classical result for a linearly elastic medium. Explicit solutions in elementary form are given in a broad class of special cases. The most striking observations arise in the Riemann-like problem corresponding to a constant initial displacement field and a piecewise constant initial velocity field. Even though, initially, the displacement field is continuous, it involves a jump discontinuity for all later times, the Lagrangian location of which remains stationary. For some materials the magnitude of the discontinuity-jump oscillates about an average value, while for others it grows monotonically, presumably fracturing the material when it exceeds some critical level.     

The aerodynamic forces and pressure distribution of a revolving pigeon wing

The aerodynamic forces acting on a revolving dried pigeon wing and a flat card replica were measured with a propeller rig, effectively simulating a wing in continual downstroke. Two methods were adopted: direct measurement of the reaction vertical force and torque via a forceplate, and a map of the pressures along and across the wing measured with differential pressure sensors. Wings were tested at Reynolds numbers up to 108,000, typical for slow-flying pigeons, and considerably above previous similar measurements applied to insect and hummingbird wing and wing models. The pigeon wing out-performed the flat card replica, reaching lift coefficients of 1.64 compared with 1.44. Both real and model wings achieved much higher maximum lift coefficients, and at much higher geometric angles of attack (43°), than would be expected from wings tested in a windtunnel simulating translating flight. It therefore appears that some high-lift mechanisms, possibly analogous to those of slow-flying insects, may be available for birds flapping with wings at high angles of attack. The net magnitude and orientation of aerodynamic forces acting on a revolving pigeon wing can be determined from the differential pressure maps with a moderate degree of precision. With increasing angle of attack, variability in the pressure signals suddenly increases at an angle of attack between 33° and 38°, close to the angle of highest vertical force coefficient or lift coefficient; stall appears to be delayed compared with measurements from wings in windtunnels.     

The characteristics of soil reaction forces on a single movable lug

In order to understand clearly the characteristics of the soil reaction forces on a single movable lug, the resultant of measured soil reaction forces was determined and presented along with its position on the lug plate. The resultant of soil reaction forces acting on the movable lug increased gradually and reached the maximum value when the lug was on about its lowest position in the soil, then it decreased without offering any downward resistance to the lug till the lug left the soil. The maximum resultant force of the movable lug was higher than that of a fixed lug. The point of action of the resultant force on the movable lug shifted in a similar way in all test cases, that is, it moves to the center of the lug from the outer tip until it reaches the position where it becomes the maximum, then it moves to the outer tip till the lug leaves the soil. The inclination angle of the resultant force increased with the decrease of lug inclination angle. The bigger lug sinkage of the movable lug produced bigger soil reaction forces and shifted the point of action of the resultant force from the tip part to the central part of the lug. However, there was no significant effect of the lug sinkage on the direction of the resultant force. The increase in lug slippage from 25% to 50% brought bigger soil reaction forces on the movable lug, but did not influence the direction and point of action of the resultant force.     

On the decomposition of forces

We show that any continuously differentiable force is decomposed into the sum of a Rayleigh force and a gyroscopic force. We also extend this result to piecewise continuously differentiable forces. Our result improves the result on the decomposition of forces in a book by David Merkin and further extends it to piecewise continuously differentiable forces.     

Apparent forces of analytical mechanics

Summary Some extensions of the apparent forces of classical mechanics are expounded, taking the following cases of motion: point referred to general coordinates involving time, point tied to a moving surface and a holonomic system with moving constraints.For a holonomic system with moving constraints and with stationary kinetic energy the apparent forces formally coincide (allowing, obviously, for a conversion ratio) with the ponderomotive forces of a stationary electromagnetic field.Lastly, the apparent forces introduced by the method of ignoration of coordinates are considered and the analogy of this case with the previous one is elucidated.

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Sommario Si mettono in evidenza alcune estensioni delle forze apparenti della Meccanica classica, prendendo in considerazione i seguenti casi di movimento: punto riferito a coordinate generali involgenti anche il tempo, punto vincolato a superficie mobile, sistema olonomo con vincoli mobili.Si rileva in particolare che per un sistema olonomo a vincoli mobili con energia cinetica stazionaria le forze apparenti coincidono formalmente (salvo naturalmente un coefficiente di ragguaglio) con le forze ponderomotrici di un campo elettromagnetico stazionario.Si considerano infine le forze apparenti introdotte dal metodo di ignorazione delle coordinate e si mostra il perchè dell'analogia di questo caso col precedente.

     

The stability of the motion of a liquid,due to thermocapillary forces

A study is made of the stability of the plane-parallel flow of a viscous liquid in a layer with a free boundary, under weightless conditions. The motion of the liquid is due to the dependence of the surface tension on the temperature. An exact solution for an unperturbed boundary is obtained by the same method used in [1], but with a more general boundary condition for the temperature. A study of the stability was carried out by the method of small vibrations, taking account of the perturbation of the free boundary. The article discusses the asymptotic behavior of long waves at small Reynolds numbers, and the conditions for instability are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 94–98, November–December, 1971.     

The effect of unsteady and history forces in the gravity convection of suspensions

The effect of unsteady and history forces on the motion of a particle is studied in the case of gravitational sedimentation (rise) of a spherical particle in the harmonic velocity field of a viscous carrier phase. An algorithm is proposed to calculate the history Basset-Boussinesq force. A parameter range is found for the case when the consideration of unsteady and history forces is required to correctly describe the mesoscale motions in a settling (rising) suspension.