Mechanical equilibrium

About: Mechanical equilibrium is a research topic. Over the lifetime, 1707 publications have been published within this topic receiving 29865 citations. The topic is also known as: equilibrium position.

The stress-dilatancy relation for static equilibrium of an assembly of particles in contact

Abstract: The dilatancy and strength of an assembly of individual particles in contact when subjected to a deviatoric stress system is found to depend on the angle of friction $\phi\_\mu$ between the particle surfaces, on the geometrical angle of packing, $\alpha$, and on the degree of energy loss during remoulding. The Mohr-Coulomb criterion of failure which is strictly applicable to a continuum is shown not to have general application to a discontinuous assembly of particles. A theoretical and experimental study of ideal assemblies of rods and uniform spheres establishes expressions for the relation between the rate of dilatancy and the maximum stress ratio for any ideal packing. The solution is extended to the case of a random assembly of irregular particles by investigating the conditions under which the mass dilates such that the rate of internal work absorbed in frictional heat is a minimum. Experiments on random masses of steel, glass, and quartz in which all the physical properties are measured independently show that the minimum energy criterion is closely obeyed by highly dilatant dense over-consolidated and reloaded assemblies throughout deformation to failure. An additional rate of energy has to be supplied to account for losses due to rearranging of loose packings, when the value of $\phi$ to satisfy the theory increases to $\phi\_f$ by an amount dependent on the degree of remoulding. The external stresses applied to an assembly are to be integrated over the $\alpha$-plane defined with reference to figure 14(a) as a plane of repetition of pattern over which the particles interlock, and the resulting forces are to be in equilibrium for sliding on particle interfaces at (45-$\frac{1}{2}\phi_f$) to the direction of the major principal stress. For the special case of no volume change these two planes are identical and the solution agrees then with that based on the Mohr-Coulomb theory. The well-known slip plane in drained discontinuous assemblies is proved to be the result of failure and nothing whatsoever to do with the peak strength. The findings are discussed in the light of previous contributions to the subject.

Stability of the lumbar spine. A study in mechanical engineering

Abstract: From the mechanical point of view the spinal system is highly complex, containing a multitude of components, passive and active. In fact, even if the active components (the muscles) were exchanged by passive springs, the total number of elements considerably exceeds the minimum needed to maintain static equilibrium. In other words, the system is statically highly indeterminate. The particular role of the active components at static equilibrium is to enable a virtually arbitrary choice of posture, independent of the distribution and magnitude of the outer load albeit within physiological limits. Simultaneously this implies that ordinary procedures known from the analysis of mechanical systems with passive components cannot be applied. Hence the distribution of the forces over the different elements is not uniquely determined. Consequently nervous control of the force distribution over the muscles is needed, but little is known about how this achieved. This lack of knowledge implies great difficulties at numerical simulation of equilibrium states of the spinal system. These difficulties remain even if considerable reductions are made, such as the assumption that the thoracic cage behaves like a rigid body. A particularly useful point of view about the main principles of the force distributions appears to be the distinction between a local and a global system of muscles engaged in the equilibrium of the lumbar spine. The local system consists of muscles with insertion or origin (or both) at lumbar vertebrae, whereas the global system consists of muscles with origin on the pelvis and insertions on the thoracic cage. Given the posture of the lumbar spine, the force distribution over the local system appears to be essentially independent of the outer load of the body (though the force magnitudes are, of course, dependent on the magnitude of this load). Instead different distributions of the outer load on the body are met by different distributions of the forces in the global system. Thus, roughly speaking, the global system appears to take care of different distributions of outer forces on the body, whereas the local system performs an action, which is essentially locally determined (i.e. by the posture of the lumbar spine). The present work focuses on the upright standing posture with different degree of lumbar lordosis. The outer load is assumed to consist of weights carried on the shoulders. By reduction of the number of unknown forces, which is done by using a few different principles, a unique determination of the total force distributions at static equilibrium is obtained.(ABSTRACT TRUNCATED AT 400 WORDS)

Neural, mechanical, and geometric factors subserving arm posture in humans

Abstract: When the hand is displaced from an equilibrium posture by an external disturbance, a force is generated to restore the original position. We developed a new experimental method to measure and represent the field of elastic forces associated with posture of the hand in the horizontal plane. While subjects maintained a given posture, small displacements of the hand along different directions were delivered by torque motors. The hand was held in the displaced positions and, at that time, we measured the corresponding restoring forces before the onset of any voluntary reaction. The stiffness in the vicinity of the hand equilibrium position was estimated by analyzing the force and displacement vectors. We chose to represent the stiffness both numerically, as a matrix, and graphically, as an ellipse characterized by three parameters: magnitude (the area), shape (the ratio of axis) and orientation (direction of the major axis). The latter representation captures the main geometrical features of the elastic force field associated with posture. We also evaluated the conservative and nonconservative components of this elastic force field. We found that the former were much larger than the latter and concluded that the behavior of the neuromuscular system of the multiarticular arm is predominantly spring-like. Our data indicated that the shape and orientation of the stiffness were invariant over subjects and over time. We also investigated the ability of our subjects to produce voluntary and adaptive changes in the stiffness. Our findings indicated that, when a disturbance acting along a fixed and predictable direction was imposed, the magnitude of the stiffness was increased but only minor changes in shape and orientation occurred. Taken together, all of these experiments represent a step toward the understanding of the interactions between geometrical and neural factors involved in maintaining hand posture and its interactions with the environment.

The control of hand equilibrium trajectories in multi-joint arm movements

Abstract: According to the equilibrium trajectory hypothesis, multi-joint arm movements are achieved by gradually shifting the hand equilibrium positions defined by the neuromuscular activity. The magnitude of the force exerted on the arm, at any time, depends on the difference between the actual and equilibrium hand positions and the stiffness and viscosity about the equilibrium position. The purpose of this paper is to test the validity and implications of this hypothesis in the context of reaching movements. A mathematical description of the behavior of an arm tracking the equilibrium trajectory was developed and implemented in computer simulations. The joint stiffness parameters used in these simulations were derived from experimentally measured static stiffness values. The kinematic features of hand equilibrium trajectories which were derived from measured planar horizontal movements gave rise to the suggestion that the generation of reaching movements involves explicit planning of spatially and temporally invariant hand equilibrium trajectories. This hypothesis was tested by simulating actual arm movements based on hypothetical equilibrium trajectories. The success of the predicted behavior in capturing both the qualitative features and the quantitative kinematic details of the measured movements supports the equilibrium trajectory hypothesis. The control strategy suggested here may allow the motor system to avoid some of the complicated computational problems associated with multi-joint arm movements.

Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders

Abstract: A general equation of mechanical equilibrium of fluid membranes subject to bending elasticity [reported in Phys. Rev. Lett. 59, 2486 (1987)] is derived in detail. The second variation of the shape energy, also obtained for arbitrary shapes, is used to analyze stability with respect to deformational modes for spherical and cylindrical vesicles. The former analysis is well known, while the latter is presented here for the first time. The theoretical results are shown to agree very well with previous numerical calculations. In addition, they provide the energies controlling the shape fluctuations and show that spontaneous curvature may transform cylinders into tapes or strings of beads. The study of the energy of infinitesimal deformations is finally extended to include the third variation. Applying the general result to the sphere, we obtain the critical value of spontaneous curvature below which oblate ellipsoids of a deformed sphere are more stable than prolate ones. It is shown to be the same regardless of whether volume or pressure is kept constant.