In the context of modern industry the molecular vacuum pumps (MVP) are widely used. The analyzed current market of vacuum technology enables drawing a conclusion that this equipment holds one of the leading positions among the high-vacuum facilities of pumping due to their advantages such as insensitivity to the atmosphere breakthrough, ability to pump out heavy gases quickly, rapid start-up time, and oil-free pumping.

The earlier developed authors’ mathematical model and calculation program are used to assess the influence efficiency of the key geometrical parameters of flowing part of the cylindrical molecular pump on its main characteristics. The obtained dependences allow us to solve a relevant, but not completely resolved as yet problem of optimizing the high-vacuum pumping facilities in case of their operation in a wide range of pressures on the suction side.

The paper presents graphs of the pumping speed and ratio of the cylindrical vacuum molecular pump pressures versus various parameters of the flowing part, such as the angle of inclination of the helix, the relative diameter and the number of helical starts. Conclusions are drawn.

There is a developed design of the external cylindrical nozzle with a vacuum camera. The paper studies the nozzle controllability of flow rate via regulated connection of the evacuated chamber to the atmosphere through an air throttle. Working capacity of the nozzle with inlet round or triangular orifice are researched. The gap is provided in the nozzle design between the external wall of the inlet orifice and the end face of the straight case in the nozzle case. The presented mathematical model of the nozzle with the evacuated chamber allows us to estimate the expected vacuum amount in the compressed section of a stream and maximum permissible absolute pressure at the inlet orifice. The paper gives experimental characteristics of the fluid flow process through the nozzle for different values of internal diameter of a straight case and an extent of its end face remoteness from an external wall of the inlet orifice. It estimates how geometry of nozzle constructive elements influences on the volume flow rate. It is established that the nozzle capacity significantly depends on the shape of inlet orifice. Triangular orifice nozzles steadily work in the mode of completely filled flow area of the straight case at much more amounts of the limit pressure of the flow. Vacuum depth in the evacuated chamber also depends on the shape of inlet orifice: the greatest vacuum is reached in a nozzle with the triangular orifice which 1.5 times exceeds the greatest vacuum with the round orifice. Possibility to control nozzle capacity through the regulated connection of the evacuated chamber to the atmosphere was experimentally estimated, thus depth of flow rate regulation of the nozzle with a triangular orifice was 45% in comparison with 10% regulation depth of the nozzle with a round orifice. Depth of regulation calculated by a mathematical model appeared to be much more. The paper presents experimental dependences of the flow coefficients of nozzle input orifice on the vacuum depth in the chamber. Research findings allowed us to express opinion that accepted in the works on "Fluid Mechanics " equality of pressure values in the center of cross-stream gravity and in its surrounding steam-gas medium is incorrect. The paper shows a possibility to create the nozzle design with updated device to connect a chamber to the atmosphere by the air throttle, which is flow pressure-controlled thus providing the nozzle operation as the flow rate stabilizer. The publication supplements information on nozzles provided in literature on " Fluid Mechanics". The developed design of the external cylindrical nozzle with controlled vacuum and of research results of its working capacity can be taken into consideration in designing hydraulic systems and devices of hydro-automatic equipment.

This work is devoted to creating a technique for calculating power characteristics of the screws with constant and variable step for the centrifugal pumps. The technique feature is that the reverse currents, which are observed in screws working at low flow, are numerically taken into account. The paper presents a diagram of the stream in the screw with flow to the network Q=0, and the static pressure of the screw in this mode is computed according to reverse current parameters. Maximum flow of screw is determined from the known formulas. When calculating the power characteristics and computing the overall efficiency of the screw, for the first time a volumetric efficiency of the screw is introduced. It is defined as a ratio between the flow into the network and the sum of the reverse current flows and a flow into the network. This approach allowed us to determine the efficiency of the screw over the entire range of flows.

A comparison of experimental characteristics of the constant step screw with those of calculated by the proposed technique shows their good agreement.

The technique is also used in calculating characteristics of the variable step screws. The variable step screw is considered as a screw consisting of two screws with a smooth transition of the blades from the inlet to the outlet. Screws in which the step at the inlet is less than that of at the outlet as well as screws with the step at the inlet being more than that of at the outlet were investigated. It is shown that a pressure of the screw with zero step and the value of the reverse currents depend only on the parameters of the input section of the screw, and the maximum flow, if the step at the inlet is more than the step at the outlet, is determined by the parameters of the output part of the screw. Otherwise, the maximum flow is determined a little bit differently.

The paper compares experimental characteristics with characteristics calculated by the technique for variable step screws, thereby demonstrating good compliance.

Results of the research allowed us to draw the following conclusions.

1. Taking into account the reverse currents enabled us to refine the calculation of power characteristics of the constant and variable step screws over the entire range of flows.

2. A program developed on the basis of the proposed technique enables us to calculate screw characteristics from their geometrical parameters and frequency of rotation. This is very convenient when performing the design calculations of the flow part of the screwcentrifugal pump.

3. If the screw-set pump is intended for use in the continuous flow mode, it is advisable to set either constant step screws S1=S2 or variable step screw S1<S2.

4. If the pump is supposed to be used in a wide range of flows, it is advisable to install the screw with variable step S1>S2.

Extent of increasing pressure in a vortex stage is limited to value of gas temperature on forcing. Because of the rather low efficiency (E) values of a vortex stage, gas temperature in the course of compression in a stage significantly increases. Therefore, to receive a higher extent of compression in the compressor the multistage scheme with intermediate cooling between the stages is applied. However, linkage of stages in the multistage scheme requires taking into consideration a need to match all stages, and first of all, by a value of the coefficient of the discharge defining a zone of the maximum efficiency. It is especially important for last stages in which as a result of compression and intermediate cooling the volume discharge significantly decreases.

The article considers various options of linkages of stages, namely: with identical rotor frequency; with identical diameters of driving wheels on stages, but with different frequencies of rotation. The stages were assumed to run at a constant value of the coefficient of discharge which size was defined when calculating the first stage. Except the above-named parameters of stages, the paper also considers a possible pattern of loading distribution by stages either a uniform distribution, or that of with increasing load at the first or last stages.

All options of linkages are illustrated using the calculation of the three-stage vortex compressor with intermediate cooling as an example. The calculation uses characteristics of a vortex stage in dimensionless parameters. Calculation results allowed us to analyze changes of key parameters of vortex stages and choose the most rational option of the three-stage compressor linkage.

It is shown that matching of stages can be achieved, in particular, through the use of driving wheels with the identical outer diameter and geometry of flowing part, but with various rotation frequencies of stages. For this purpose the paper offers to make the vortex compressor according to the known scheme of the cross-compound integrally geared compressor in which with raising the number of stage the rotor frequency decreases.

The article deals with the problem of stability of the truck while unloading. The main symptom is a loss of stability of a vehicle rollover. The most common vehicle rollover occurs when unloading on sloped sites. The unloading area can be both longitudinal and cross slope. In this regard, the main task of the calculation is to determine the stability of the car the maximum allowable angle of discharge site and loads acting on the bearing system of the car.

The paper presents a developed mathematical model of the truck when unloading on the longitudinal slope. It also offers the procedure of the step change in the parameters for the calculation of the critical angle of the site. The formulas for calculating the force interaction platform and the support system in the truck unloading at the site with a longitudinal slope are given. On the basis of the developed technique, a program for calculating longitudinal stability and loads acting on the bearing system dump trucks is created.

Modern methods to assess stability while unloading the truck on a cross slope are analyzed. It is noted that the solution of this problem is developing in two directions. The first approach involves the testing truck samples on the special stands. Examples of bench tests on the stability of trucks are given. Another approach uses computational methods for the study. However, this requires the creation of a complex mathematical model of the truck, which should take into account the structural features of the suspension, the nonlinear elastic suspension components and tires, as well as the possibility of plastic deformation of the support system in torsion. In this formulation, the problem can only be solved using numerical methods and may be the subject of further research.

The proposed approach allows multiple calculations for truck strength and stability already at an early designing stage in order to find the optimal design solution.

Elastic elements are widely used in instrumentation. They are used to create a particular interference between the parts, for accumulating mechanical energy, as the motion transmission elements, elastic supports, and sensing elements of measuring devices. Device reliability and quality depend on the calculation accuracy of the elastic elements. A corrugated membrane is rather common embodiment of the elastic element.

The corrugated membrane properties depend largely on its profile i.e. a generatrix of the meridian surface.

Unlike other types of pressure elastic members (bellows, tube spring), the elastic characteristics of which are close to linear, an elastic characteristic of the corrugated membrane (typical movement versus external load) is nonlinear. Therefore, the corrugated membranes can be used to measure quantities, nonlinearly related to the pressure (e.g., aircraft air speed, its altitude, pipeline fluid or gas flow rate). Another feature of the corrugated membrane is that significant movements are possible within the elastic material state. However, a significant non-linearity of membrane characteristics leads to severe complicated calculation.

This article is aimed at calculating the corrugated membrane to obtain the elastic characteristics and the deformed shape of the membrane meridian, as well as at investigating the processes of buckling. As the calculation model, a thin-walled axisymmetric shell rotation is assumed. The material properties are linearly elastic. We consider a corrugated membrane of sinusoidal profile. The membrane load is a uniform pressure.

The algorithm for calculating the mathematical model of an axisymmetric corrugated membrane of constant thickness, based on the Reissner’s theory of elastic thin shells, was realized as the author's program in C language. To solve the nonlinear problem were used a method of changing the subspace of control parameters, developed by S.S., Gavriushin, and a parameter marching method developed by N.V. Valishvili. The principle of the method of changing the subspace of control parameters is piecewise smooth parameter marching process. In each smooth section a numerical analysis is reduced to the one-parameter problem.

The problem is solved by two-stage predictor-corrector scheme. The predictor stage uses extrapolation to predict initial values of unknown on the basis of historical data. At the corrector stage a modified method of Newton - Raphson is used to specify initial approximation solutions.

As a result of the programme, the following results were obtained: elastic characteristic of the corrugated membrane and deformed shapes of a corrugated shell meridian in appropriate points of the elastic characteristic.

The paper has considered a phenomenon of local buckling. It has shown the elastic characteristic obtained and a deformed shape of the corrugated membrane meridian. The method to have an isolated solution by changing a subspace of control parameters has been proposed. The proposed algorithm enables efficient investigation of membrane behaviour during nonlinear deformation.

Control actions are provided on the basis of inverse kinematic problem. Now there is a set of methods to solve this task.

This article considers an example of the author’s approach application to the inverse kinematic problem.

The main idea of approach is as follows:

1. The limited set of the joints necessary to implement the chosen gait is selected from all joints of the robot. For these joints a strict sequence of the movement within each step and restriction of changing generalized coordinates are specified. 2. The joints non-involved in implementing the chosen gait are disabled, with no calculations performed for them.

Thus, the sources of basic data for the inverse kinematic problem are the kinematic scheme of the executive mechanism of the walking robot and the chosen gait.

To use the offered approach it is necessary:

1. To number the legs and their joints.

2. To choose joints to be involved in realization of the chosen gait.

3. To appoint a sequence of the change of supporting legs when moving by the chosen gait.

4. To specify a motion sequence of the chosen joints within a step for each leg.

5. To specify restrictions of changes of the generalized coordinates in the chosen joints.

The inverse kinematic problem process consists in gradual approach to the solution by change (increase or decrease) of the generalized coordinates in the same order in which the joints of a leg corresponding to these coordinates move within a step by the chosen gait when walking.

Criterion of completing calculations is the limits reached or the fact that a leg is fixed on a supporting plane by a contact sensor (or a condition in the modeling program). Changes of generalized coordinates are within a cycle; each generalized coordinate changes by a certain value at each of iterations of a cycle. The total time of a cycle corresponds to the estimated time of a step to be done.

Advantages of the approach are following: unambiguity of the received solution, possibility to consider degenerate configurations of the executive mechanism of the walking robot, possibility to calculate in the conditions of kinematic redundancy, and high-speed calculations.

The possibility to consider degenerate configurations is owing to the fact that, when calculating, the device of matrix transformations is not used (in particular, calculation of the return matrix of Jacobi).

The possibility to calculate in the conditions of kinematic redundancy, unambiguity of the received solution and high speed are caused by fact that the sequence of the joints movement initially defines a certain desirable leg configuration in space and, in fact, specifies a solution direction.

There is no need to verify the received solution as kinematic restrictions has been are already taken into consideration in the course of calculations.

One more important advantage of the approach is the fact that robots can have any number of legs, and joints in legs.

A lack of possibility to create universal algorithms for the walking robots with essentially different kinematic schemes (for example, anthropomorphous and arachnoid) because of constructive differences in executive mechanisms can be the shortcoming of the approach. But other heuristic methods, such as FABRIK, FTL or CCD also have the same shortcoming. The approach is akin to these methods by the fact that there is no certain formula which use allows us to receive the solution with these methods at once. There are only algorithms of search checking a set of the specified conditions, which are criteria of correction or end of the solution when calculating.