Static equilibrium position

In this chapter we are seeking an equilibrium position of a multibody system. This is a position where the system does not move. The velocity and acceleration are zero. 

There are mainly two different ways to transfer a given multibody system into an equilibrium state  

We saw in Sec. 1.6 that the first task normally requires the solution of a system of linear equations. The second task yields a system of nonlinear equations of the form 

Solving nonlinear systems is not only interesting in this context, but it is also a subtask of many other problems, e.g. simulation, optimization. 

there is a free parameter s in the description of the problem and (3.1.1) takes the form 

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Displacement u is measured from the static equilibrium position under the weight mg. 

At the static equilibrium position the lubricant force on the runner is ... 

The characteristic motion of the journal center on stability threshold Is a stationary whirl on a closed elliptical crbit around the static equilibrium position of the journal center, with a definite whirl ratio expressing the ratio of whirl frequency to running frequency. In order to be able to predict the threshold speed, the eight dynamic coefficients of the oil film corresponding to such a characteristic motion should first be... 

Pig.1 shows the journal at its static equilibrium position relative to the bearing. 

For the case In which the small amplitude of oscillations Is Imposed around the static equilibrium position, the dimensionless bearing gap Is expressed by... 

A linearization of (A.2.9) around the static equilibrium position Xeqj yields... 

The atomic temperature factor, or B factor, measures the dynamic disorder caused by the temperature-dependent vibration of the atom, as well as the static disorder resulting from subtle structural differences in different unit cells throughout the crystal. For a B factor of 15 A2, displacement of an atom from its equilibrium position is approximately 0.44 A, and it is as much as 0.87 A for a B factor of 60 A2. It is very important to inspect the B factors during any structural analysis a B factor of less than 30 A2 for a particular atom usually indicates confidence in its atomic position, but a B factor of higher than 60 A2 likely indicates that the atom is disordered. 

When Wqi / Wq2 the magnetization recovery may appear close to singleexponential, but the time constant thereby obtained is misleading [50]. The measurement of 7) of quadrupolar nuclei under MAS conditions presents additional complications that have been discussed by comparison to static results in GaN [50]. The quadrupolar (two phonon Raman) relaxation mechanism is strongly temperature dependent, varying as T1 well below and T2 well above the Debye temperature [ 119]. It is also effective even in cases where the static NQCC is zero, as in an ideal ZB lattice, since displacements from equilibrium positions produce finite EFGs. 

In the previous sections, we have considered that the optical center is embedded in a static lattice. In our reference model center ABe (see Figure 5.1), this means that the A and B ions are fixed at equilibrium positions. However, in a real crystal, our center is part of a vibrating lattice and so the environment of A is not static but dynamic. Moreover, the A ion can participate in the possible collective modes of lattice vibrations. 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

Thus, within the context of the Newtonian force atom and the caloric theory of heat, solids, liqitids, and gases were all viewed as organized arrays of particles produced by a static equilibrium between the attractive interparticle forces, on the one hand, and the repulsive intercaloric forces, on the other. The sole difference was that the position of eqitilibriitm became greater as one passed from the solid to the liqitid to the gas, due to the increasing size of the caloric envelopes siuToittrding the component atoms (Figures 5 and 6). 

On the other hand, the application of a static or slowly varying electric field will be able to displace ions and electrons away from their equilibrium positions and, as a consequence, the polarizability of the electrons will be modified. In the description of H. A. Lorentz s electronic oscillators, the small shifts in the ionic positions modify the spring constants and restoring forces of the electronic oscillators. 

The presence of a near and far field in and around a non-equilibrium double layer leads to the distinction between (at least) two relaxation times. Relaxation to the static situation, after switching off the external field, can take place by conduction or by diffusion. Conduction means that ions relax to their equilibrium position by an electric field. Diffusion relaxation implies that a concentration gradient is the driving force. In double layers these two mechanisms cannot be separated because excess ion concentrations that give rise to diffusion, simultaneously produce an electric field, giving rise to conduction. For the same reason, if polarization has taken place under the Influence of an external field and this field is switched off, ions return to their equilibrium positions by a mixture of conduction and diffusion. 

Nevertheless, if one assumes a static, rather than a thermodynamic, equilibrium, one can attempt to estimate the dependence of the yield stress Oy and the modulus G on the shape and depth of the interparticle potential. Imagine that a gel is subjected to a shear strain Y that homogeneously displaces particles from their positions of static equilibrium. Pairs of particles are pulled apart by this strain, and separations between particle centers of mass should increase roughly by an amount yrQ, where ro = 2a + Dq is the separation between centers of mass in the absence of strain. Hence, the imposition of a strain y increases the gap between particle surfaces from Dq to... 

The source of elasticity in block copolymers containing well-ordered spherical domains is analogous to that for simple three-dimensional erystalline solids. When a mechanical deformation displaces spherical domains from their equilibrium lattice positions, the domains are pulled back by the thermodynamic forces that are responsible for the existence of the macrocrystalline order in the static sample. Similar forces exist when the domains are cylindrical or lamellar, but these latter morphologies, if well-ordered and oriented appro-priately, can sustain shearing deformations without displacement of the domains from their equilibrium positions, since they are not ordered in all dimensions as spherical domains are. 

Temperature factor An exponential expression by which the scattering of an atom is reduced as a consequence of vibration (or a simulated vibration resulting from static disorder). For isotropic motion the exponential factor is exp(—5iso sin 0/A ), where Biso is the isotropic temperature factor. It equals 87r (ti ), where (ti ) is the mean-square displacement of the atom from its equilibrium position. For anisotropic motion the exponential expression usually contains six parameters, the anisotropic vibration or displacement parameters, which describe ellipsoidal rather than isotropic (spherically symmetrical) motion or average static displacements. 

A paper by Prandtl [18] on the kinetic theory of solid bodies, which was published in 1928, one year prior to Tomlinson s paper [17], never achieved the recognition in the tribology community that it deserves. PrandtI s model is similar to the Tomlinson model and likewise focused on elastic hysteresis effects within the bulk. Nevertheless, Prandtl did emphasize the relevance of his work to dry friction between solid bodies. In particular, he formulated the condition that can be considered the Holy Grail of dry, elastic friction If the elastic coupling of the mass points is chosen such that at every instance of time a fraction of the mass points possesses several stable equilibrium positions, then the system shows hysteresis. In the context of friction, hysteresis translates to finite static friction or to a finite kinetic friction that does not vanish in the limit of small sliding velocities. Note that the dissipative term that is introduced ad hoc in Eq. (19) does vanish linearly with small velocities. 

Suppose for a moment that k > k. In this case there is a unique static solution for every x in Eq. (19), irrespective of the value of xq . When the upper sohd is moved at a constant (small) velocity vq relative to the substrate, each atom is always close to its unique equilibrium position. This equilibrium position moves with a velocity that is of the order of tiq- Hence the friction force is of the order of myvQ, and consequently Fk vanishes linearly with vq as vo tends to zero. The situation becomes different for k < k. Atoms with more than one stable equilibrium position will now pop from one stable position to another one when... 

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