Dissertation for M.Sc master's degree
Mechanical engineering-applied design trend
Abstract
Bearings are widely used in various industries, including automobile and aircraft industry. Therefore, it is important to check the bearings, therefore, in this research, the hardness matrix of the angular contact ball bearings has been investigated. In this research, the equations governing the two-row stiffness matrix have been developed, and in addition to examining the axial force, the effect of preload on the vibration characteristics of the bearing has been investigated. At first, the analytical model was developed and subjected to axial load and preload. The governing equations lead to non-linear equations that are solved using numerical methods. The method of calculations and the flowchart of obtaining the stiffness matrices are presented at the beginning of the fourth chapter. The obtained results indicate that the size of the matrix elements increases with the increase of the preload. It was also observed that the natural frequency of the investigated system increases with the increase of preload.
Key words: double-row bearing, preload, stiffness matrix.
Chapter One
Introduction
1-1- Preface
In 2400 BC, the Egyptians used fluid to reduce the number of manpower needed to pull large stone blocks. After that, around 1100 BC, Australians used wooden rollers for similar blocks, but they realized the concept of rolling instead of sliding, and this led to the development of rolling bearings. Therefore, the invention of bearings, or what is known as bearings in the industry, goes back thousands of years. At that time, mankind realized that rotational movement is much easier than sliding movement, so if in tools such as wheels of chariots and carts or old millstones made of stone or wood in the form of pellets and placed between the parts that move relative to each other, much less energy will be consumed and it will overcome the frictional force between said parts. In this way, gradually over time, the bearings were completely removed and by taking the inner and outer rings and parts such as the rack, they took the shape of today's bearing.
Most of the advances that have taken place in the field of industry and industrial engineering in the world, which are an integral part of the current century, without the existence of bearings, which largely affect the friction force in all machinery and electrical and domestic machinery and industries and in the automotive industry. overcomes, is not possible, and with the increasing progress in the machine and automobile industries, the importance of bearings is increasing day by day.
From the historical point of view, the time of the invention of bearings is not precisely known, but despite the discoveries and historical evidence, it can be said that the manufacture of bearings goes back to the ancient Roman era, when the pioneers of the bearing industry lived at that time, because when in 1928, the lake It did not dry. On one of the two rowing ships that remained from the time of the great Roman emperor Caligula, a bearing was found, the supports of which were made of wood and had bronze balls, which are also similar in shape to modern bearings. Probably, this bearing was placed under a very large statue to carry or rotate it.
1500 years later, Leonardo da Vinci, an Italian philosopher and mathematician, during his lifetime, made calculations on rolling bearings, according to which the calculations of bearings were initially made, and it can be said that until the end of the 19th century, until the invention of bicycles and sewing machines, steam engines, textile machines, etc. There were not many actions in completing and manufacturing bearings.
At the end of the 19th century, with the expansion of the bicycle industry, the first demand was for frictionless bearings, so in this field, factories in England and Germany began to make steel balls, which at the same time were very few bearings for engineering works in America..
At the end of the 19th century, with the expansion of the bicycle industry, the first demand was for frictionless bearings, so in this field, factories in England and Germany began to make steel balls, and almost at the same time, very few bearings were made for engineering works in America. Although human knowledge in the field of bearings was at the elementary level, the need for objects that could have the relevant characteristics and overcome the friction force was felt more. Later, a scientist named Robert Stribek [2] did basic research on bearing engineering. It was at this time that the expansion of the automobile industry was the best demand for this product. Therefore, within ten years, that is, from 1902 to 1912, more than seven types of bearings were made that are still in use today.
With the fundamental transformation in technology during the 19th and 20th centuries in order to expand human needs and the rapid progress of technology in the fields of machine tools and equipment, automotive industries, household appliances, space structures and many other industries, this product also evolved day by day. Because this product has a close relationship with the efficiency of manufactured devices and cars and is closely related to the improvement of human technology.
Of course, the production of bearings in a mass, modern and standard way goes back to 1907. In this year, an engineer in a textile factory, due to the problems caused by the breaking of the bearings due to the bending of the shaft on the production line, thought of making a bearing that could adapt to the bending of the shaft. This person was Dr. Svan Winquist, the founder of SKF Bearings, Sweden, who invented self-adjusting bearings. Today, SKF companies produce various types of bearings in a wide range of sizes all over the world.
Guaranteeing the proper quality and efficiency of bearings under cycles and complex conditions of applied loads and exceptional conditions such as working in critical temperatures, vacuum and corrosive environments, etc. It is.
1-2- Statement of the problem
In this research, it is assumed that the shaft can rotate freely around the z axis. In the vibration analysis, it is assumed that the range of vibrations transmitted to the bearing is much smaller than the displacement of the bearing due to preload and external loads. With the above assumptions, the time-varying linear system model is introduced as follows:
where M, C, and K are the mass, damping, and stiffness of the bearing, respectively, and the vectors of the total force and total displacement of the bearing, respectively. After extracting the various components of the stiffness matrix of the system, the effect of these components on the vibration characteristics such as natural frequencies, the shape of the system modes and so on. will be investigated.
Therefore, the objectives of this research will be as follows:
Development of a new stiffness matrix for double-row angular contact ball bearings
Development of a simple analytical model for the axis-bearing system
Calculation of stiffness matrix components and Examining their role in the vibration characteristics of the axis-bearing system
Investigating the effect of preload on the vibration characteristics of the system
1-2-1- Introduction of the types of bearings
Each bearing has its own characteristics It has a design that makes it suitable for some applications. For example, deep groove wedge bearings transfer both axial and radial loads. They have low friction and can provide high precision without noise. Therefore, they are suitable for small or medium-sized electric motors. Spherical bearings and roller bearings can transfer very heavy loads and adjust themselves with axis rotation. This feature makes them suitable for heavy engineering work. They adjust the misalignments and the axis rise.