Mechanical restoring force

The force that can be developed by the comb-drive actuators is given by 

The gap between the teeth in the tangential direction of motion g which is determined by layout, is usually designed to be much larger than the spacing between the fingers, g, that is determined by the minimum design rules. To increase the total force, an array of N fingers is used. 

4 Graphical solution for balancing the mechanical and electrical The capacitor has an initial gap go equal to 2.1 pm. 

The slopes of the electrical and mechanical forces are also equal  

When one-third of the initial gap has been closed, the plates snap together or pull in. This pull-in instability limits the useful range of parallel plate electrostatic actuators with linear springs. For parallel plate electrostatic actuators formed in surface micromachining processes, the initial gap is defined by the sacrificial layer thickness, which is practically limited to a few microns, so that the useful actuation range is typically less than a micron. To find the pull-in voltage, the gap at pull-in, go/3, can be substituted into equation (3.22) and solved for the voltage  

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A spring is typically used to apply a mechanical restoring force for electrostatic actuators, as shown in Figure 3.3. The spring can be linear, following Hooke s Law ... 

The top plate of capacitor Ci has been released and the bottom plate is fixed in position. The spring provides a mechanical restoring force to the attractive electrostatic force when a voltage Vs is applied. Both the top and bottom plates of capacitor C2 are fixed. 

So far we have illustrated the classic and quantum mechanical treatment of the harmonic oscillator. The potential energy of a vibrator changes periodically as the distance between the masses fluctuates. In terms of qualitative considerations, however, this description of molecular vibration appears imperfect. For example, as two atoms approach one another, Coulombic repulsion between the two nuclei adds to the bond force thus, potential energy can be expected to increase more rapidly than predicted by harmonic approximation. At the other extreme of oscillation, a decrease in restoring force, and thus potential energy, occurs as interatomic distance approaches that at which the bonds dissociate. 

One of the classic problems of quantum mechanics that is very important for our study of nuclei is the harmonic oscillator. For a simple harmonic oscillator, the restoring force is proportional to the distance from the center, that is, F = — kx, so that V x) = kx2/2. The Schrodinger equation is... 

There is no simple, direct relationship between elasticity and emulsion or foam stability because additional factors, such as film thickness and adsorption behaviour, are also important [204]. Nevertheless, several researchers have found useful correlations between EM and emulsion or foam stability [131,201,203], The existence of surface elasticity explains why some substances that lower surface tension do not stabilize foams [25]. That is, they do not have the required rate of approach to equilibrium after a surface expansion or contraction as they do not have the necessary surface elasticity. Although greater surface elasticity tends to produce more stable bubbles, if the restoring force contributed by surface elasticity is not of sufficient magnitude, then persistent foams may not be formed due to the overwhelming effects of the gravitational and capillary forces. More stable foams may require additional stabilizing mechanisms. 

The viscoelastic fluids represent the 3rd material dass of non-Newtonian fluids. Many liquids also possess elastic properties in addition to viscous properties. This means that the distortion work resulting from a stress is not completely irreversibly converted into frictional heat, but is stored partly elastically and reversibly. In this sense, they are similar to solid bodies. The liquid strains give way to the mechanical shear stress as do elastic bonds by contracting. This is shown in shear experiments (Fig. 1.27) as a restoring force acting against the shear force which, at the sudden ending of the effect of force, moves back the plate to a certain extent. 

Statistical Mechanics of a System of Oscillators.—Dynamically, we have seen that a. crystal can be approximated by a set of ZN vibrations, if there are A afronas in the crystal. These vibrations have fre-quencies which we may label v. . . vzN varying through a, wide range of frequencies. To the approximation to which the restoring forces can be treated as linear, these oscillations are independent of each other, each one corresponding to a simple harmonic oscillation whose frequency is inde-... 

When an atom is displaced from its equilibrium position in a molecule, it is subject to a restoring force which increases with the displacement. A spring follows the same law (Hooke s law) a chemical bond is therefore formally similar to a spring that has weights (atoms) attached to its two ends. A mechanical system of this kind possesses a natural vibrational frequency which depends on the masses of the weights and the stiffness of the spring. 

From the repulsive term some important vibrational data, which allow the estimation of some mechanical properties of solids, can be determined. For example, in Chapter 10, it will be shown that the Young s modulus (an elastic constant) of a solid can be approximated by treating the interatomic bonds as springs. The restoring force for small... 

A number of different mechanisms have been proposed to account for the rubberlike properties of materials. In classical rubber theory these properties are attributed to a decrease in conformational entropy on deforming a network of kinetically free random polymer molecules. Stress orders the polymer chains and decreases their entropy by limiting their conformational freedom, thus providing the restoring force to the relaxed state. Such a theory was developed for elastin by Hoeve and Flory... 

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