Heavy Bearings Exploitation Energy and Reduction Methods

The global trend of resource conservation so as “not to compromise the ability of future generation's development” is the fundamental basis of the concept of sustainable development. Concordant with this, the energy efficiency of products is increasingly discussed and frequently taken into account in the design stage. In more cases a product is more appreciated and more attractive as the energy consumption and its associated materials are lower. In the production stage, said consumption advantages primarily the manufacturer, particularly through low consumption thereof. In the operational phase, low energy and materials consumption represents an user advantage and it's a major argument in the decision to purchase and use a particular product. Heavy bearings are frequent products used in wind turbines that are producing non-conventional “clean” energy, as windmills. An enhanced energy efficiency bearing contributes to the enhancement of the overall efficiency of the wind turbines. Based on a suitable mathematical model, this paper identifies and recommends courses of action to reduce the operating energy of heavy bearing through the “cage” - which is the subject of a much larger research - with the highest priority. The identified actions may constitute from a set of requirements for the design stage of the heavy bearing predominantly oriented towards innovation-invention.


Introduction
In more cases a product is more appreciated and more attractive as the energy consumption and its associated materials are lower. In the production phase, reduced consumption advantages primarily the producer and it contributes to the reductions of costs. In the operational phase, low energy and materials consumption represents an user advantage and it's a major argument in the decision to purchase and use a particular product.
The energy efficiency of products is increasingly discussed and frequently taken into account at the design stage. Moreover, improving the energy efficiency of products can represent an objective to redesign a product, for its eco-design. The global trend of resource conservation so as not to compromise the ability of future generation's development [1] is the fundamental basis of the concept of sustainable development. Energy efficiency is part of the sustainable development concept [2], and eco-design is one of the specific means. In this context it's necessary to specify that the mission of Transilvania University of Brasov (institution within which the authors of this paper are activating), stated in the first article of the Charter of the University [3], is "to produce and transfer knowledge to society through: (1) Advanced scientific research, development, innovation and technology transfer in the field of Sustainable Development ...". Not incidentally the research institute of the university is called "Pro-DD" (in English "Pro-SD") and gives priority to research for sustainable development.
The quality of the bearings presented in many technical systems largely depends on their energy efficiency. Bearings are effective solutions used from a long time, already of old tradition. Scientific literature regarding bearings is particularly rich: hundreds or even thousands of books, thousands of patents, (tens of) thousands of articles published in specialized magazines and bulletins of professional conferences. Relatively few of them [4] [5] [6] [7] addresses to mathematical models to study bearings either in their entirety or on certain issues. Special mention must be given to doctoral theses in the field; major scientific papers have a higher consistency and originality. Only in Romania have been developed dozens of such work. It seems that everything has been studied until now and that something new can hardly be added.
Heavy bearings are frequent products used in wind turbines which produce unconventional "clean" energy [8]. A bearing with higher energy efficiency contributes to the overall efficiency of the wind turbines, contributing to the increase of the wind turbines efficiency and performance. This paper aims to identify courses of action to reduce the operating energy for heavy bearing, influence of the "cage" -which is the subject of a much larger research -which has higher priority. The identified actions may constitute a set of requirements for the design phase of the heavy bearings, predominantly oriented towards innovation-invention.

Movements of the bearing components
Representative components of a bearing are the two rings, inner and outer, intermediate rolling bodies (balls, rollers or needles) and the cage. In heavy bearings are used as rolling elements balls or rollers and the cage is often massive. At many heavy bearings, and not only, the cage is made of brass, a deficient and expensive material.
The model which describes the movements of the bearing components can be simple or complex depending on the multitude of details taken into account. In this case is sufficient a simple model, the purpose of this study is to identify courses of action -especially to redesign and/or develop new constructive solutions -that will contribute to reduce the necessary operating energy of heavy bearings. For this purpose it is sufficient that the model allows the expression of the bearing components revolutions depending on the rotary motion of the moving ring under ideal conditions of contact between the rings and the intermediate rolling bodies.
It is assumed that the components of the bearing do not have dimensional and shape deviations and all intermediate rolling bodies have identical dimensions and that none of the bearing components don't present elastic deformations.
According to the assembly into which a bearing is integrated the outer ring could be fixed and the inner ring would be into a rotary motion, rolling elements and the cage (figure 1a), or the inner ring could be fixed and the outer ring would be into a rotary motion, rolling elements and the cage (figure 1b).  n ii , n iethe revolutions of the inner ring and outer ring; n crthe revolution of rolling bodies (balls or rollers) around their own axes of rotation; n cthe revolution of the cage which is identical whit the revolution of the rolling element (ball or roller) around the rotation axis of the bearing; R i-iithe inner radius of the bearing inner ring; R e-iithe outer radius of the bearing inner ring, expressed for the theoretical point of contact with the rolling elements (balls or rollers); R i-iethe inner radius of the bearing outer ring, expressed for the theoretical point of contact with the rolling elements (balls or rollers); R e-iethe outer radius of the outer ring; R crradius of the rolling elements (balls or rollers); R mcthe average radius of the cage; R i-c , R e-cthe inner radius of the cage, respectively the outer radius thereof; B ii , B iewidth of the bearing inner ring and outer ring; B cwidth of bearing cage.
Obvious, ( Using the previous relationships and imposing the condition v A1 = v A2 the next expressions for revolutions n cr and n c are obtained: Using the relations (4) (5) (which remain valid for the specified case), (8) and (9), and imposing the condition v B4 = v B2 the next expressions for revolutions n cr and n c corresponding to case B are obtained:

Heavy bearing operating/exploitation energy
At any time t j , the kinetic energy E cj = E c (t j ) of the bearing is the sum of the kinetic energies of its components in motion of rotation: E c-ii or E c-ie -the kinetic energy of the inner ring, respectively of the outer ring, as the case may be; E c-cr -cumulated kinetic energy of the q rolling elements (balls or rollers); E c-c -the kinetic energy of the cage. Most often the inner ring of the bearing is the ring which is in motion of rotation, hereafter this case is approached. Taking into account the relations (6), (7), (10) and (11) the referred kinetic energies have the next expressions [9]: where k ii and k c are coefficients that take into account the particularities of the inner ring geometry and that of the cage in relation to the idealized rectangular section thereof. J ii , J c and J cr are the moments of inertia in relation to their own rotation axes of the inner ring, cage and respectively of an rolling element, m cr is the mass of an rolling element, ρ ii , ρ c and ρ cr are the materials densities of the inner ring, cage and rolling elements.
The k c coefficient takes into account the presence of the q pockets in the cage which are for the rolling elements.
If the rolling element is a full ball, then: and therefore the relation (14) becomes (17) and relation (14) becomes At any time t j revolution of the driven ring (the inner ring in the studied case) is n ii = n j = n(t j ), and the kinetic energy of the bearing is During its operation, the driven ring of the bearing has a variable speed; very evident in the case of wind turbine where the speed of the port-blade shaft is variable depending on the wind speed, figure 2. The main consequence is that all components of the bearing which are rotating have variable speed. The operating energy of the bearing is given by the amount of energy required to accelerate the moving parts thereof.
Either (n j ) min and (n j ) max two consecutive extreme local values from the port-blade shaft speed chart, corresponding to different moments of time: (n j ) min = n(t j1 ); (n j ) max = n(t j2 ), t j1 < t j2 .
Obviously the n j port-blade shaft speed becomes the speed of the engaged bearing ring. The typical case addressed in the paper is the one with the outer ring fixed and the inner ring rotating, so n j = (n ii ) j .
Reducing the operating energy of the bearing assumes the necessity to decrease the value of the K factor, the instantaneous n j speed of the ring engaged depends strictly by the external conditions.
The possibility to reduce the value of the K factor associated to the engaged ring, inner or outer ring, as the case may be, it is quite low.
It is possible to reduce the value of the K factor associated to the rolling elements either by reducing the radius R cr either by using hallow rolling elements [10] [11]. Reducing the radius R cr entail the necessity to increase the number of rolling elements and, to a large extent, this action causes a decrease of the bearing loading capacity. The use of hollow rollers or balls is possible only if their stiffness is not affected, radius R cr must be large enough. Hollow rolling elements have an important influence on the decrease of the K factor due to the subtraction of the mass thereof, and by decreasing the moment of inertia to their own axis of rotation.
The K factor can be reduced by reducing the value which corresponds to the cage. There are two possible courses of action: using a material with low density [12] (the effect would determine a simultaneous reduction, in the same proportions, of the cage mass as well as its moment of inertia) and the development of new constructive solutions with lower masses and/or inertia exclusively due to the cage geometry and not due to their realization of materials with low densities.
Obviously, it is possible to gain cumulative benefits/effects due to the use of less dense material and due to the adoption of favorable constructive solutions.

Conclusions
Heavy bearings constitute a frequent product often used in unconventional "clean" energy plants as wind turbines. A higher energy efficacy of the bearing increases the overall wind turbine efficiency.
At any time the kinetic energy of the bearing is the sum of the kinetic energies of its components in motion of rotation: inner or outer ring, as the case may be, the q rolling elements (balls or rollers) and the cage. At wind turbines, but not only, the engaged bearing ring has a variable speed; in consequence all components of the bearing which are engaged have variable speed. Operating energy of the bearing is given at least by the amount of energy required to accelerate the moving parts thereof.
It is unlikely to reduce the kinetic energy by changing the constructive solution of the bearing ring which is rotating.
One way to reduce the kinetic energy, which is evidenced, is to use hollow rolling elements, balls and rollers.
Reducing the kinetic energy of the bearing by reducing the kinetic energy of the cage is rarely addressed. The current paper demonstrates this possibility and identifies, with arguments, two courses of actions: using a material with low density and the development of new constructive solutions whit significantly lower masses and/or inertia compared with those of a reference cage.