The asymmetric flexible rotor turbocharger supported by floating ring bearings is studied. We use the model of flexible asymmetric rotor and the nonlinear bearing forces have been calculated by using the numerical solution of the Reynolds equation for both fluid films. It is shown that at this rotor speed range the rotor performs direct nonsynchronous regular precession, which corresponds to the conical shape of the rotor motion. The rotor speed at which the shape of the rotor precession abruptly changes from conical to cylindrical has been revealed. It is established that the cylindrical shape of the precession corresponds to the unacceptable increase of bearing loads. Thus, the maximum rotational speed above which the turbocharger rotor under study shuts has been found. The implications can be applied to the turbocharger rotors supported by two bearings with floating ring bearings and console location of the compressor and turbine wheels.
A variety of optimization problems in theoretical mechanics are related to the problem of finding the quickest operating mechanism among all possible mechanisms. Let us consider a mechanical system with one degree of freedom consisting of leading and lagging links with masses M1 and M2 respectively. Source of mechanical energy is attached to the leading link and its potential energy is known as a function of the position of the leading link. Positions of the leading - x and lagging links - y are related through the position function. Now the optimization problem can be formulated in the following way: find the position function such that the lagging link is transferred from initial to the final position in the shortest time. The analytic solution for this problem for the case when dissipative forces can be neglected was found using variational calculus method. In the case when dissipative forces can be described as viscous friction the problem can be solved using iterative methods. The differential equation that describes the stationary point of these iterations was obtained. The dependence of the optimal position function on the magnitude of friction is analyzed. In the case when dissipative forces are of the dry friction type the approach based on the variational calculus fails. We were able to find the optimal position function problem using Maximum Principle. New qualitative features of the solution arising due to dry friction are discussed. Approaches developed in this paper can be generalized for a variety of mechanisms where the operating time is critical.
A variety of optimization problems in theoretical mechanics are related to the problem of finding the quickest operating mechanism among all possible mechanisms. It can be formulated in the following way. Let us consider a mechanical system with one degree of freedom consisting of leading and lagging links with masses M1 and M2 respectively. Source of mechanical energy is attached to the leading link and its potential energy is known as a function of the position of the leading link. Positions of the leading - x and lagging links - y are related through the position function. Now the optimization problem can be formulated in the following way: find the position function such that the lagging link is transferred from initial position to the final position in the shortest time. We have obtained an analytic solution to this problem using variational calculus methods in the case then dissipative forces acting on the system can be neglected. We have shown that this problem is analogous to the classical brachistochtone problem. The qualitative feature of this result is that the optical mechanical system tends to accelerate the leading link first in order to maximize the power being extracted from the mechanical energy source. Analytical solution is applied for the optimization of the fast acting electric switch design. Leading link load and operating time reductions with the use of the optimal transfer function is demonstrated. This approach can be generalized for a variety of mechanisms where the operating time is critical.
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