CONTRIBUTION TO THE CINEMATIC AND DYNAMIC STUDIES OF HYDRAULIC RADIAL PISTON MOTORS.

Transcription

1 Ing. MIRCEA-TRAIAN CHIMA CONTRIBUTION TO THE CINEMATIC AND DYNAMIC STUDIES OF HYDRAULIC RADIAL PISTON MOTORS. PhD Thesis Abstract Advisor, Prof. dr. ing. matem. Nicolae URSU-FISCHER D.H.C. Cluj-Napoca

2 Chapter 1. Current achievements and trends in construction and use of hydraulic motors In this chapter the general framework that fits the theme of this thesis is presented, namely hydraulic drives. As the cam is a component of a (radial piston) hydraulic motor, it is important to know the different types of hydraulic motors, as well as how they are divided. Briefly, the most common hydraulic motors are presented from the main three categories, namely: - high speed hydraulic motors - intermediate speed hydraulic motors, and - slow speed hydraulic motors The radial piston hydraulic motors are presented in more detail, as also the operating principle and its main components. Some examples, as well as the industries and applications where the hydraulic motors are used are briefly presented. From this multitude of applications and industries where the radial pistons motors are used, results the crucial role these are nowadays playing in the construction of various types of machines. Consequently we considered a theoretical study of all aspects of the geometry, kinematics and force distribution as particularly useful. In the subsequent chapters several contributions to solving theses problems, are presented. Chapter 2. Contributions to determine the profile geometry of a multilob cam of a radial piston hydraulic motor. There are several possibilities to choose the type of the closed curves which are used as inner cam profile. The study of two potential curves is intended in this chapter and as a result of a few calculations regarding their geometry, the kinematic aspects and the mode of interaction between pistons and the cam in order to choose an optimal variant. One of the possible alternatives is a shortened epicycloids curve, defined by the radii of two circles (one fixed and one mobile) and the distance from the center of the mobile circle to the point generating the curve. It is shown that for some imposed numerical values for elements of epitrochoidal (peritrochoidal) movement one can obtain the corresponding values for the elements of peritrochoidal (epitrochoidal) movement. 2

3 Two identical curves are presented in figure 1, one generated with epitrochoidal and other with peritrochoidal movement. Fig. 1. The generation of the identical cyclic curves using epitrochoidal and peritrochoidal movements (intermediate positions). The second possibility is to obtain a profile consisting of circular arcs and curved portions which have a sinusoidal variation in radial direction. Considering the cam has six lobes, corresponding a 60 central angle, internal cam profiles can have the shapes presented in figure 2, pure sinusoidal curve or a so called modified sinusoidal curve, which in this case consists of arcs of circle combined with sinusoidal curves. Fig.2. Different sinusoidal profiles of a cam In both cases the cartesian and polar coordinates of these curves were determined as well as the cartesian and polar coordinates of the equidistant curves described by the center of the roller and the study of relative 3

4 displacements, speeds and accelerations of the pistons with respect to the rotor, was performed. All numerical results were obtained using programs written in C. Chapter 3. Contributions to determine the coordinates of the equidistant curve to the cam profile and to the calculation of the relative speeds and accelerations of the radial pistons. Regardless of the type of curve used (shortened cycloid or a combination of circular arcs and curves with sinusoidal variation in radial direction) following questions should be addressed: - to determine the inner equidistant curve to the profile traveled by the center of the roller which is at the end of the hydraulic piston, - to determine the law governing the movement of the radial piston relative to the rotary group comprising the eight ( or ten or six) cylinders, - to determine the speed and acceleration of the piston in his radial movement considering different angular speeds of the cylinders and pistons. To determine the radial speed and acceleration the numerical derivation method is used, the obtained values being approximate. This procedure is being repeated because analytical relations to determine the coordinates of the curve points can t be used. The numerical results presented in the form of diagrams were obtained with a program written in C. Using this program one can determine how different values taken for angles corresponding to circular arcs of the profile as well as the radius of the roller at the end of the piston which is in contact with the cam, influence the values of the relative piston velocity and acceleration in regards to the rotor. Chapter 4. Contributions to determine the forces of interaction between pistons and cam and the torque produced by the radial piston motors depending on the cam profile. The purpose of this chapter is to establish an accurate mathematical expression for the torque acting on the cylinder block during the active stroke and its average value, for cams having the inner profile shortened epicycloids and for cams having the inner profile arcs of circle combined with sinusoidal curves in radial direction. The torque depends on the cam dimensions, the roller radius, the piston surface and the pressure of the hydraulic fluid. 4

5 Determining the exact relations between the parameters of the motor and the obtained torque, enables us to find an optimum variant which leads to a maximum output torque. Based on the procedure presented in this chapter, the optimal parameters can be determined for the design of a hydraulic motor, to obtain the average values for the resulting output torque. If a comparison is being made between the values of the average output torque of two hydraulic motors having identical dimensions (the diameter of the roller, of the piston, the dimensions of the cam, the piston surface and pressure of the hydraulic fluid) it can be observed that the values for the resulting torque when using a cam having the inner profile a shortened epicycloid are higher compared to those having the inner profile arcs of circle combined with sinusoidal curves. Chapter 5. Contributions to determine the radius of curvature of the cam profiles. In order to determine the contact pressure between the roller and the inner profile of the cam, it is necessary the values of the radii of curvature of the inner cam profile to be known, values which occur in formulas of Hertz. The determination of the values for the radius of curvature for each point of the profile can be done using the known formula from analytic geometry (which involves calculating the derivates for the functions defining the coordinates of the profile) or using another method, that of determining the coordinates of the circles circumscribed on three successive points of the profile, covering all points of the profile. The two cases presented are: - when the three points occupies a certain position in plan (no two points are on the same horizontal or vertical). In this case the coordinates are determined for each middle of segment, the angular coefficient and the angular coefficient of the perpendicular to the respective segment. - if two of the points are on the same vertical or same horizontal, here 12 possible scenarios intervene which are all presented in detail. A program in C was written, containing the function previously presented, with which were calculated the radii of curvature corresponding to the first lobe of the cam (the parameter φ between 0 and 60 ), for both profiles (shortened epicycloids and arcs of circle combined with sinusoidal curves) corresponding to hydraulic motors manufactured by Bosch Rexroth, forming data files to be used in other programs written in C to determine the contact stresses. Graphs showing the radius of curvature are presented for epicycloids profile and sinusoidal profile of a cam corresponding to a MCR3 hydraulic motor (name of a range of motors manufactured by Bosch Rexroth). 5

6 Chapter 6. Calculations and verifications of contact pressure. The life of a cam is very important and plays a primary role when choosing the inner profile of the cam. Not only because of the large numbers of hydraulic motors where these cams are used but also because they play an important role in the multitude of applications, including operational safety. The failures occur mainly due to high contact stresses between the roller and the inner surface of the cam. In this chapter are presented calculations and verifications of different contact stresses at the surface of the cam. With the help of a powerful software named BEARIX (developed by INA/FAG, Germany) the tensions which arise in the area of contact and under the surface were calculated and the longitudinal profile of the roller was determined which leads to a decrease of the contact stresses (or contact pressure) between the roller and cam. Chapter 7. The design of the radial pistons hydraulic motor and its components. Experimental research. This chapter presents the design and the manufacturing of the radial pistons hydraulic motors. Designing is done using the software Pro/Engineer Wildfire 2.0 and the verifications (for sizing) of certain components of the motor are done using finite elements method. For this purpose a package, part of Pro/E called ProMechanical is being used. Components which affect the safety of operation (output shaft) or defining the lifetime (cam) require complex verifications, for torsion and bending for the output shaft or contact stresses which occur under the surface in case of the cam, for this purpose a more efficient and complex software was used, called Cosmos Design Star (produced by Structural Research and Analyses Corp.) which allows also the definition of knots under the surface. The author participated in carrying out experiments and verifications on a test bench owned by Bosch Rexroth, where the hydraulic motors manufactured are subjected to tests of output speeds, torque and sound and where the mechanical and volumetric efficiency is measured. Hydraulic motors with cams having characteristics similar to those presented in this thesis were tested and the results in form of graphs are presented as well in this chapter. 6

7 Chapter 8. Conclusions and prospects for the future. Original contributions. Different inner cam profiles were presented in this thesis and the influence of their shape to the radial speed and acceleration of the pistons, to the obtained output torque and contact stresses. I consider that in the future these studies can be continued also with reference to other inner profiles with similar shapes, like spline functions of different degrees (at least of third degree). For now and for the coming years these studies will form the cornerstones in designing and manufacturing of hydraulic motors with radial pistons. Improvements could be brought in two directions, to increase the starting efficiency and to extend the lifetime of the motor. In most applications it is desirable to have a high starting efficiency. This can be obtained by reducing the friction between and inner leakage. It was found that friction between the roller and piston seat greatly influences the overall efficiency and that the lifetime of the motor can be extended by improvements in heat treatment of the inner cam profile. The possibility to predict the starting efficiency will be a great advantage and the finding of such a theoretical method is what companies are looking for. As for the use of the radial pistons motors in new applications, a shift from off-highway to on-highway can be observed as well as other new areas like in the wind industry or as an alternative to planetary gear units. 7