Moving Load Problems

We recall from Sect. 3 that once v x, t) is known over the contact region C(0, the extent of which has also to be determined, everything else can be calculated by elementary means. 

The general discussion in Sect. 2.5 on methods of solution of (2.5.1) applies here without alteration. 

We discuss the integral equation (3.4.2) in more detail for the problem of inden-tors in contact with a half-plane under the action of certain loads, and moving across it. In principle, the method outlined in the last section could handle any specified individual motion of the indentors. However, only the simplest will be considered, namely where the indentors are all moving in the same direction, taken to be along the negative x direction. 

Consider first the case of a single moving load. Let the lowest point of the indentor have position Xq(0 at time t. This specifies its overall motion. The displacement derivative in the contact region C(t) will then have the form 

In other words, it is the time at which the point x entered the contact interval. From (2.4.18) we have that the quantities n, are given by 

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We consider the moving load problem in this section to the extent of deriving an expression for the coefficient of hysteretic friction in the small viscoelasticity and small velocity approximations, respectively. 

Golden, J.M., Graham, G.A.C. (1987a) The transient quasi-static plane viscoelastic moving load problem. Int. J. Eng. Sci. 25, 65-84... 

Typically, rapid repair of main cracks in slabs is realized using bituminous masses covering only the damages. Unfortunately, this kind of protection is ineffective just after few months of exploitation, because the degraded mass (Fig. 2b, 2c) allows for infiltration of water under slabs [8]. The coming into being hydrodynamic pump effect (due to moving loads - Fig. 3a) destructs sealants in joints (Fig. 3b) and causes uneven settlement of concrete slabs (Fig. 3c). The solution of this problem is the use of special polymer flexible joints. 

Elastomers are frequendy subjected to dynamic loads where heat energy and motion control systems are required. One of the serious dynamic loading problems frequendy encountered in machines, vehicles, moving belts, and other products is vibration-induced deflection. Such effects can be highly destructive, particularly if a product resonates at one of the driving vibration frequencies. 

In Sect. 2.5, we discussed briefly the case where the ratio of viscoelastic (time-dependent terms in the functions G(t), J(t)) to purely elastic effects (constant terms in these functions) is small. The possibility of expressing solutions as power series in this ratio was noted. In the present section, we shall consider solutions to first order in this parameter. It turns out that the problem of the moving load greatly simplifies, to the extent that explicit expressions for all quantities of interest may be written down, in contrast to the highly implicit equations which emerged from the exact analysis in the previous section. 

Example Problem Dynamic Response of a Bridge (Beam) to a Random Train of Moving Loads... 

A fan blade is continuously vibrating millions of cycles up and down ia operatioa over a short period of time. Each time a blade tip moves past an obstmction it is loaded and then unloaded. If forced by virtue of tip speed and number of blades to vibrate at its natural frequency, the ampHtude is greatly iacreased and internal stresses result. It is very important when selecting or rating a fan to avoid operation near the natural frequency. The most common method of checking for a resonance problem is by usiag the relatioa ... 

Power to drive a belt conveyor is made up of five components power to drive the empty belt, to move the load against fric tion of the rotating parts, to raise or lower the load, to overcome inertia in putting material into motion, and to operate a belt-driven tripper if required. As with most other conveyor problems, if is advisable to work with formulas and constants from a specific manufacturer in making these calculations. For estimating purposes, typical data are given in Table 21-7. 

Moving now to two boilers, the heat load may comprise two elements. One may be a production process whose interruption would cause problems and the other, say, a heating load where any interruption would not be noticed immediately. Assuming the two elements were of equal duty, it would be reasonable to install two boilers each 50-50 per cent of the total load. One boiler would then be able to cover the process load. 

However, we are increasingly confronted with practical problems that involve material response that is inelastic, hysteretic, and rate dependent combined with loading which is transient in nature. These problems include, for instance, structural response to moving or impulsive loads, all the areas of ballistics (internal, external, and terminal), contact stresses under high speed operations, high... 

Owing to the high computational load, it is tempting to assume rotational symmetry to reduce to 2D simulations. However, the symmetrical axis is a wall in the simulations that allows slip but no transport across it. The flow in bubble columns or bubbling fluidized beds is never steady, but instead oscillates everywhere, including across the center of the reactor. Consequently, a 2D rotational symmetry representation is never accurate for these reactors. A second problem with axis symmetry is that the bubbles formed in a bubbling fluidized bed are simulated as toroids and the mass balance for the bubble will be problematic when the bubble moves in a radial direction. It is also problematic to calculate the void fraction with these models. 

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