Spring stiffness

The air spring effect results from adiabatic expansion and compression of the air in the actuator casing, Niirnericallv, the small perturbation value for air spring stiffness in Newtons/rneter is given bv Eq, (8-107),... 

A simple springs-in-series model represents the representative volume element loaded in the 2-direction as in Figure 3-11. There, the matrix is the soft link in the chain of stiffnesses. Thus, the spring stiffness for the matrix is quite low. We would expect, on this basis, that the matrix deformation dominates the deformation of the composite material. 

The package has a mass of 2 kg or a weight of 4.415 lb. Assume a downward velocity of about 10 ft/s. The shock absorber develops a constant force (independent of the relative velocity) of 10 lb (44.48 N). The spring stiffness is 57.10 Ib/in. The potential energy due to gravity will be neglected. 

The assumption of independent oscillators allows us to study a simplified system containing only one atom, as illustrated in Fig. 14 where x and Xq denote, respectively, the coordinates of the atom and the support block (substrate). The dynamic analysis for the system in tangential sliding is similar to that of adhesion, as described in the previous section. For a given potential V and spring stiffness k, the total energy of the system is again written as... 

Stability of the atomic system depends on the spring stiffness and the potential corrugation, or more specifically, depends on the ratio of k/. The system would become more stable if the stiffness increases or the potential corrugation decreases, which means less energy loss and lower friction. 

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By definition, Hooke s law is valid when there is a linear relationship between the stress and the strain. Equation (3.97). For instance, if /q = 0.1 m then an extension (/ — /q) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system mass-spring are at the rest (equilibrium) ... 

To solve the Eqs. (20) and (21), we have to specify five parameters normal and tangential spring stiffness kn and kt, normal and tangential damping coefficient r n and t],.. and the friction coefficient nj. In order to get a better insight into how these parameters are related, it is useful to consider the equation of motion for the overlap in the normal direction <5n ... 

The contact force between two particles is now determined by only five parameters normal and tangential spring stiffness kn and kt, the coefficient of normal and tangential restitution e and et, and the friction coefficient /if. In principle, kn and k, are related to the Young modulus and Poisson ratio of the solid material however, in practice their value must be chosen much smaller, otherwise the time step of the integration needs to become unpractically small. The values for kn and k, are thus mainly determined by computational efficiency and not by the material properties. More on this point is given in the Section III.B.7 on efficiency issues. So, finally we are left with three collision parameters e, et, and which are typical for the type of particle to be modeled. 

If we further assume that e — et, then the relation between the normal and tangential spring stiffness is... 

Fig. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan-Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC configuration. Spring stiffness is K = 70,000, granular temperature is 9 = 1.0, and coefficient of normal restitution is e = 1.0. The system is driven by rescaling.

Fig. 21. The excess compressibility from soft-sphere simulations, with random initial particle positions, for different coefficients of normal restitution e (a) e = 1.0 (top-right) (b) e = 0.95 (top-left) (c) e = 0.90 (bottom-right) (d) e = 0.80 (bottom-left). The simulation results (symbols) are compared with Eq. (54) based on the Ma-Ahmadi correlation (solid line) or the Camahan-Starling correlation (dashed line). The spring stiffness is set to k = 70,000.

The classical picture that describes these anisotropic effects based on the Lorentz model is illustrated in Fig. 9.8, which is a generalization of the spring model of Fig. 9.1 note that the spring stiffness depends on direction. 

The diffusion bond strength between two identicai materiais can be determined from a singie normai incidence uitrasonic measurement based on the fact that imperfections in diffusion bonds resuit in reflection of some uitrasonic energy from the interface separating the two substrates. The imperfect diffusion bond is characterized by the interfaciai spring stiffness, which is determined from the reflected signai spectrum [25]. 

FIGURE 12.7 Simulated results for three values of spring stiffness ((a) 8Nm (b) 800Nm (c) 80 000 N m ). Soft sphere approach number of particles = 14 000, u = 3 Umf (from Kaneko, 2000). 

Kaneko, 2000) obtained for three values of spring stiffness constant (spread over four orders of magnitude) are shown in Fig. 12.7. It can be seen that if the objective is to understand the macroscopic behavior of the fluidized bed, low values of spring stiffness can be used for faster simulations. It must, however, be remembered that when such artificially low values of spring stiffness constant are used, the predicted values of contact time between solid particles are not realistic. When the objective is to understand local particle to particle heat or mass transfer, it is important to make accurate predictions of particle contact times. For such cases, it is necessary to use realistic values of spring stiffness constant at the expense of increased computational resources. 

Most real solids, and hence surfaces, are not perfectly periodic but contain a certain degree of disorder. This can change the tribological properties of a system qualitatively. There are many ways to introduce the effects of disorder into the FK model. One possibility is to assume that the substrate potential contains random elements, while another possibility is to assume that the bond lengths or the spring stiffnesses fluctuate around a mean value. 

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