*VD,T*(*RSEI*) = *a* × *VI,T*(*MAT*) + *b* × *VI,T*(*ATP*) + *c* × *VI,T*(*funds of the Grain to Green Project*) + *d* × *VI,T*(*funds of the Karst Rocky Desertification Restoration Project*) (13)

where *VD,T* denotes the dependent variables at a specific time, *VI,T* denotes the independent variables at a specific time, *a* and *b* are multivariate regression standardization coefficients of climate conditions, while *c* and *d* are multivariate regression standardization coefficients of human activities. Formula (13) was used to calculate the predicted RSEI value during project implementation (only affected by four factors) with a 95% confidence interval; the residual between the observed RSEI mean value and the predicted RSEI mean value was calculated. If the residual value was positive, PAR was assumed to have a positive impact on EEQ over time. If the residual was negative, PAR was assumed to harm the local ecology.

(4) Analysis of coupling coordination degree. To test the degree of spatial and temporal correlation between PAR and EEQ, this study used the coupling coordination degree to analyze and describe the spatial characteristics between PAR and ecological environment variables. The range of values was [0, 1], where a larger coupling coordination degree indicated a stronger correlation between the two variables [39].

$$C = \sqrt{\frac{\mathcal{U}\_1 \times \mathcal{U}\_2}{\left[ (\mathcal{U}\_1 + \mathcal{U}\_2)/2 \right]^2}}\tag{14}$$

$$D = \sqrt{\mathbb{C} \times T}; \ T = \mathfrak{a} \times \mathcal{U}\_1 + \mathfrak{f} \times \mathcal{U}\_2 \tag{15}$$

Using formula *C* in particular, the system coupling degree was calculated. Here, *U*1 and *U*2 are the RSEI variables, and the data were normalized for the size (number of people) of the relocated population. *T* is the comprehensive evaluation index of *U*1 and *U*2; *α* and *β* are the undetermined coefficients, where *α* = *β* = 0.5; *D* is the coupling coordination degree. The coupling coordination degree is based on SPSS software, which normalizes and standardizes the analysis data. The conclusion of calculations is between 0 and 1, which the software automatically divides into 10 levels. According to the trend in the level distribution and the existing method of dividing the research results, the coupling coordination level is divided into 4 levels: extreme detuning (0–0.3), general detuning (0.3–0.5), general coordination (0.5–0.7), and extreme coordination (0.7–1) [40].

#### **3. Results**


*4.1. Curvature Parameter Effects*

Figure 2 demonstrates a deteriorating impression of velocity *F* (*η*) when *A* becomes larger. Physically, the increasing curvature of the sheet significantly increases the resistance of the flow which slows down the velocity. The same figure shows a similar decreasing pattern for temperature *θ*(*η*); as the resistance towards fluid flow is larger for the improving curvature parameter, the concentration of the fluid also improves, which slows the temperature evolution. Figure 3 shows how the nanofluid concentration *φ*(*η*) and the concentration of motile micro-organisms *χ*(*η*) behave as the radius of curvature, *A*, increases. As this parameter is increased, the CBL and MMBL are seen to decrease, which is also reflected in the reduction of concentration *φ*(*η*) and concentration of motile micro-organisms *χ*(*η*).

**Figure 2.** Influence of the curvature parameter on *F* (*η*) and *θ*(*η*).


**2. Forest biomass**

seen as one of the greatest environmental problems facing the twenty-first century [1–3]. The impacts resulting from this period of great change begin to take place, are felt and will affect the whole world, every ecosystem, every nation, and every human endeavor [4]. Scientific consensus points to emissions of greenhouse gases, largely from the burning of fossil fuels, as the primary culprit behind this problem [5]. In this regard, one important effort to reduce greenhouse gases in the atmosphere is to immediately replace fossil fuels with renewable

In general, biomass is the total weight or volume of organisms in a given area or volume. Biomass is defined as the total amount of living matter above the surface of a tree and is expressed by tons of dry weight per unit area [6]. For forest, biomass itself is defined as the overall volume of living things of all species at a given time and can be divided into three main groups, viz. trees, shrubs, and other vegetation [7]. Forest biomass is highly relevant to climate change issues. Forest biomass plays an important role in the biogeochemical cycle, especially in the carbon cycle. From the total forest carbon, about 50% is stored in forest vegetation. As a consequence, forest damage, fire, logging or illegal logging, and so on will increase the amount of carbon in the atmosphere. In general, the dynamics of carbon in nature can be explained simply by the carbon cycle. The carbon cycle is a biogeochemical cycle that includes the exchange/transfer of carbon between the biosphere, the pedosphere, the geosphere, the hydrosphere, and the earth's atmosphere. The carbon cycle is actually a complex process and every process interacts with other processes. Plants

store it in plant tissue. Until the time carbon is refluxed into the atmosphere, the carbon will occupy one of a number of carbon pools. All the components of the vegetation of trees, shrubs, lianas, and epiphytes are part of above biomass. Below the soil surface, plant roots are also carbon stores other than the soil itself. For example, in peat soils, the amount of carbon stores may be greater than the carbon deposits on the surface. Carbon is also stored in dead organic materials and biomass-based products such as wood products both when used or already in the landfill. Carbon can be stored in carbon pools for long periods or only briefly. Increasing the amount of carbon stored in this carbon pool represents the amount of carbon absorbed from the atmosphere [8]. The role of forest biomass is greater after having financial value in mechanism of carbon trading. Carbon markets need a unit of trade. For carbon, this is one ton of greenhouse gas emissions expressed as carbon dioxide

through multiplication by the global warming potential of the gas. This is the physics of the gas in the atmosphere that results in energy being absorbed rather than radiated out

That is why there is much research on the measurement of forest biomass from all forest components. In its development, forest biomass measurements include all living biomass aboveand below ground, such as trees, shrubs, palms, saplings, and other undersea plants, creeping plants, lianas, epiphytes, etc., and, in addition, biomass from dead plants such as dead wood and litter. Since carbon in the forest can be traded in the carbon market, an accurate mechanism for measuring forest biomass is required. Therefore, the purpose of this chapter is to inform the importance of measuring forest biomass as it forms the basis for carbon accounting

e). Besides that, each greenhouse gas can be converted to a ton of CO2

) through the process of photosynthesis and

e

energy sources.

6 Renewable Resources and Biorefineries

equivalents (tCO2

on carbon trading.

to space.

will reduce the carbon in the atmosphere (CO2

In general, forest biomass is the mass of the above-ground portion of live trees per unit area. It is a basic forest property linked to the productivity and processes of the forest ecosystem, and is an important indicator of the carbon stock that will help determine the contribution of forests to the global carbon cycle. Methods for estimating forest biomass have been largely undertaken from models or using allometric methods, forest inventory, applications of remote sensing data and geographical information system (GIS). The method has been widely practiced in various pilot areas in almost all countries in the world.

The allometric method for biomass assessment was first discovered by Kittredge [9] in the form of a logarithmic formulation as follows:

$$\mathbf{Y} = \mathbf{a}\mathbf{X}^{\flat} \tag{1}$$

where Y = dependent variable (in this case, biomass content); X = independent variable (in this case, may be the trunk diameter or height of tree, root, wide tree canopy, etc.); and a, b = constants. Allometric method is a method of measuring plant growth expressed in terms of exponential relationships or logarithms between plant organs that occur harmoniously and changes proportionally [10]. The methods used to measure carbon content in forest biomass can be done in three ways as follows:

	- **a.** carbon content of vegetation trees = 0.5 × biomass weight [11]
	- **b.** forest carbon content = 80% × charcoal weight [12]
	- **c.** stem biomass = stem volume × wood density
	- **d.** total above-ground biomass (tree biomass above ground) = biomass stem × BEF (Biomass Expansion Factor).

**4.** The types of allometric equations formulated include allometric equations based on types of forest, types of tree, and the parts of trees. For example: *Swietenia mahogany*, Y = 0.9029 (D2 .H)0.6840; *Dalbergia latifolia*, Y= 0.7458 (D2 .H)0.6394; and Tectona grandis, Y= 0.0149 (D2 .H)1.0835 where Y is the total biomass of trunk, branch, and leaf; D is the diameter at breast height; and H the total height. The total biomass of five combined tree species (*Swietenia mahogany, Dalbergia latifolia, Tectona grandis, Paraserianthes falcataria, and Acacia auriculiformis*) Y = 0.0219(D2 .H)1.0102 [13]; Y = 0.262 + 1.934 D where Y is the total liana biomass in tropical primary and secondary forests and D is the liana diameter; Y = 2.55 + 0.416 L, where Y is the stem biomass and L is the liana length [14]; for young rain tree or *Albizia saman,* Y = −10,31 0.50 + 1820.89X1 + 10.89X2 where X1 = Diameter and X2 = Height [15]. Besides that, there are also the relationships between above-ground dry plant biomass and stem diameter of liana [16] and bamboo [17–19]. Generally, several generalized allometric equations for tropical forests have been established and also widely used. Unfortunately, application of such generalized equations to individual sites may lead to large errors in biomass estimates especially when the species concerned is poorly represented by the generalized models. In this case, local allometric models are needed to give an accurate estimation [20].

and a digital representation conducive to image processing. This is supported by satellite sensors capable of recording the reflectance spectra of the stands which is a combination of soil reflection spectrum, trees and ground vegetation. Stand reflectance depends on the relative

From Forest Biomass to Carbon Trading http://dx.doi.org/10.5772/intechopen.80395 9

In relation to the search for renewable energy sources for the future, actual forest biomass such as felled and low-value trees can be an alternative in determining renewable energy sources or bioenergy. Thriving markets for these materials will add value to the working forests and provide an important tool for addressing a number of natural conservation goals, including hazardous fuel reduction, degraded forest restoration, habitat management, etc.

Available forest areas are limited by a number of non-market factors, such as environmental regulations, conservation efforts, the value of non-timber forest, and the behavior of landowners. In addition, economic factors will determine where biomass is available and its quantities. National policies of developed countries and global market mechanisms can improve the demand for woody biomass; then, the prices for these materials will tend to increase as well. The given high biomass prices will certainly benefit forest landowners and increase the bottom line for sustainable forest management. In terms of biomass, buyers will increase the cost for existing biomass users. In terms of the pricing process through supply and demand mechanisms in the market, increasing demand will lead energy producers into competition with forest products for timber and residues, or encourage timber harvesting to unsustainable levels. On the other hand, the forest products industry can afford higher prices for wood fibers than most energy producers can meet, due to the high value of wood, pulp/paper, and other wood products relative to energy values. Thus, the availability of biomass at low costs will limit where and to what extent bioenergy is seen as cost-effective. This is especially true as other renewable energy costs (such as wind, geothermal, solar, and water technology) continue to decline.Actually, the economic feasibility of bioenergy will depend on the supply of reliable and affordable raw materials. In this case, bioenergy has more in common with oil, natural gas, or coal than any other form of renewable energy, such as wind, geothermal, and solar. However, unlike fossil fuels, forest biomass is a living resource, subject to biological forces, climates, and natural disasters. Also, unlike fossil fuels, forests are much appreciated, more than just their energy content. People depend on forests for clean water, biodiversity, timber products, recreational opportunities, essential ecosystem services, and for their esthetic and spiritual appeal. The challenge is to build an infrastructure for cost-effective harvesting of reliable biomass supplies without negatively impacting these other values. The following recommendations address the need to develop infrastructure and an atmospheric biomass market that prioritizes conservation goals, ecosystem restoration, and other forest stewardship objectives.

Efforts to address climate change have been made by the international community through greenhouse gas (GHG) emission reduction programs with avoided deforestation and forest degradation through the afforestation/reforestation clean development mechanism (A/R CDM), reducing emission from deforestation and degradation (REDD+), Joint Implementation

amounts of these components within a ground resolution cell.

**3. Handling climate change**

(JI), and voluntary carbon market (VCM) schemes.

In a forest carbon inventory, a calculated carbon pool contains at least four pools of carbon. The pools of carbon are above- and below-ground biomass, dead organic matter, and soil organic carbon.

Remote sensing satellites have been used in many studies on forest biomass successfully. The use of remote sensing is increasingly widespread after supported by the use of spatial analysis in the geographical information system (GIS). That is why the making of forest biomass maps and other thematic maps has been done for many purposes. Remote sensing applications have been able to estimate forest structure and biophysical parameters such as land cover, crown closure, stand height, leaf area index, biomass, volume, etc. The advantages of remote sensing applications include systematic repetition scope with spectral and spatial consistency, the ability to monitor areas of interest over time, suitability for large area coverage, and a digital representation conducive to image processing. This is supported by satellite sensors capable of recording the reflectance spectra of the stands which is a combination of soil reflection spectrum, trees and ground vegetation. Stand reflectance depends on the relative amounts of these components within a ground resolution cell.

In relation to the search for renewable energy sources for the future, actual forest biomass such as felled and low-value trees can be an alternative in determining renewable energy sources or bioenergy. Thriving markets for these materials will add value to the working forests and provide an important tool for addressing a number of natural conservation goals, including hazardous fuel reduction, degraded forest restoration, habitat management, etc.

Available forest areas are limited by a number of non-market factors, such as environmental regulations, conservation efforts, the value of non-timber forest, and the behavior of landowners. In addition, economic factors will determine where biomass is available and its quantities. National policies of developed countries and global market mechanisms can improve the demand for woody biomass; then, the prices for these materials will tend to increase as well. The given high biomass prices will certainly benefit forest landowners and increase the bottom line for sustainable forest management. In terms of biomass, buyers will increase the cost for existing biomass users. In terms of the pricing process through supply and demand mechanisms in the market, increasing demand will lead energy producers into competition with forest products for timber and residues, or encourage timber harvesting to unsustainable levels. On the other hand, the forest products industry can afford higher prices for wood fibers than most energy producers can meet, due to the high value of wood, pulp/paper, and other wood products relative to energy values. Thus, the availability of biomass at low costs will limit where and to what extent bioenergy is seen as cost-effective. This is especially true as other renewable energy costs (such as wind, geothermal, solar, and water technology) continue to decline.Actually, the economic feasibility of bioenergy will depend on the supply of reliable and affordable raw materials. In this case, bioenergy has more in common with oil, natural gas, or coal than any other form of renewable energy, such as wind, geothermal, and solar. However, unlike fossil fuels, forest biomass is a living resource, subject to biological forces, climates, and natural disasters. Also, unlike fossil fuels, forests are much appreciated, more than just their energy content. People depend on forests for clean water, biodiversity, timber products, recreational opportunities, essential ecosystem services, and for their esthetic and spiritual appeal. The challenge is to build an infrastructure for cost-effective harvesting of reliable biomass supplies without negatively impacting these other values. The following recommendations address the need to develop infrastructure and an atmospheric biomass market that prioritizes conservation goals, ecosystem restoration, and other forest stewardship objectives.


**4. Interventions**

The management of older people should be multimodal and multidisciplinary, especially for those with or at risk of malnutrition [147], in order to improve di fferent conditions (i.e., oral problems and sarcopenia). From a practical point of view, comprehensive geriatric assessment (CGA) is the multidimensional, interdisciplinary diagnostic and therapeutic process aimed at determining the medical, psychological, and functional problems of older people. The CGA's objective is the development of a coordinated and integrated plan for treatment and follow-up in order to maximize overall health with aging [148]. To date, increasing evidence suggests that prosthodontic treatment in combination with personalized dietary counselling may improve the nutritional status of patients [51]. Here, we provide an overview on the management of oral problems, malnutrition, and sarcopenia.

## *4.1. Oral Management*

The stomatognathic system is very vulnerable over time, but with special care, it can be preserved throughout the lifetime [30]. Nevertheless, one of the major challenges in providing both restorative and preventive care for older adults is to check dental status on a regular basis [34]. Prevention is pivotal to detecting oral disease as soon as possible and requires regular patient contact. However, since it has been reported that older people frequently fail to achieve a good oral hygiene, both patients and caregivers should be made more aware about the importance to check dental status as well as oral hygiene.

The oral health-care professionals should develop a personalized program, in order to prevent all the problems related to the aging process. In some cases, it is di fficult to provide dental care in the hospital setting in a short time, since in many countries there are long waiting lists (especially in publicly funded hospitals) [149]. Therefore, private dentists also need better awareness concerning the complexity of older people. There is, first and foremost, a need to understand the level of dependency, the medical condition, and the physical or cognitive impairment of the patient. Secondly, it is important to establish an oral healthcare plan that includes both professional and self-care elements [150].

The oral management of older people usually involves di fferent aspects:

#### *4.2. Nutritional Interventions*

As discussed above, nutrition is an important determinant of health in older people. Thereby, it is pivotal to provide adequate amounts of energy, proteins, fluid, and micronutrients in order to prevent or treat excess or deficiencies, and therefore improve several health-related outcomes in terms of morbidity and mortality. A personalized approach is pivotal in order to respect individual

preferences, needs, and to increase compliance to the diet. Nutritional status should be assessed before each intervention, and the amount of energy and proteins should be individually adjusted with regard to nutritional status, physical activity level, disease status, and tolerance [152]. The European Society for Clinical Nutrition and Metabolism (ESPEN) [152], in its guidelines on clinical nutrition and hydration in geriatrics, recommends a guiding value for energy intake of 30 kcal/kg of body weight/day. However, as stated above, it should be adapted individually. Both ESPEN [153] and the PROT-AGE study group [147] recommend providing a protein intake of at least 1.0 g/kg body weight/day in older people to maintain muscle mass, increasing the intake up to 1.2–1.5 g/kg body weight/day in presence of acute or chronic illness. Additionally, it seems that the per-meal anabolic threshold of protein intake is higher in older individuals (i.e., 25 to 30 g protein/meal, containing about 2.5 to 2.8 g leucine) than young adults [147]. However, since older people may experience di fficulty of ingesting large amounts of proteins in a single meal, supplementation should be considered. Since serum vitamin D levels decline gradually with aging [154,155] and have been associated with reduced muscle mass and strength, supplementation should thus be considered in those who are deficient.

Food texture should be adapted depending on the chewing and swallowing condition in order to avoid choking risk [10]. Harder foods may be modified to soft consistencies (i.e., bite-sized, minced, pureed) requiring little chewing, as well as liquids, which may be thickened to render the swallowing process slower and safer [10,156,157]. Controlling the intake of simple sugars is pivotal to prevent both dental caries [101] and metabolic complications [158]. World Health Organization recommends to limit the intake of free sugars to less than 10% of total energy intake to minimize the risk of dental caries [159].

Fruit and vegetables are major sources of minerals and vitamins with antioxidant properties; therefore, their consumption should be promoted both for oral and general health. It has been documented that excessive antioxidant supplementation could compromise both the mechanism of adaption to exercise and have even pro-oxidant e ffects. Thus, supplementation in people who are not deficient should be regarded carefully [160]. Dietary consumption of fatty fish (i.e., salmon, mackerel, herring, lake trout, sardines, albacore tuna, and their oils), which are a major source of omega-3 fatty acids, has been associated with a greater fat-free mass [161]. Given their antioxidant role, omega-3 fatty acid supplementation has been suggested to improve inflammatory status both in periodontal disease [162] and sarcopenia [163]. However, more studies are needed to further elucidate the exact time and dosage of supplementation as well as long term e ffects [164]. Nevertheless, consumption of foods rich in omega-3, such as as fatty fish, should be promoted.

#### *4.3. Exercise and Rehabilitative Strategies*

Physical inactivity is considered one of the main causes of sarcopenia [165] because it determines a resistance to muscle anabolic stimuli [166]. Moreover, it has been proposed that physically inactive individuals may have a greater risk of periodontal disease [167]. In particular, resistance training seems to be the most e ffective type of exercise to counteract sarcopenia [168]. Furthermore, since sarcopenia is a systemic process [15,21], it has been recommended to perform a holistic training involving all muscle groups [15]. In fact, it has been documented that both masticatory and swallowing functions can be improved through muscle-strengthening exercises [169,170]. Several studies reported enhancements in subjective chewing ability, swallowing function, salivation, relief of oral dryness, and oral-health quality of life. Indeed, the synergistic e ffect of nutritional interventions coupled with physical exercise may improve both muscle [164] and oral health [167]. Recently, Kim et al. [171] reported an improvement in oral function following an exercise program which included stretching of the lip, tongue, cheek, masticatory muscle exercise, and swallowing movements. Several studies have been focused on swallowing rehabilitation. To date, a positive e ffect of expiratory muscle resistance training has been documented in improving suprahyoid muscle activity [172,173]. Furthermore, head lift exercises showed a beneficial impact on swallowing movements [174,175], and tongue strengthening exercises have been reported to enhance tongue strength [176,177]. Yeates et al. [178] demonstrated that isometric tongue strength exercises and tongue pressure accuracy tasks improved isometric tongue

strength, tongue pressure generation accuracy, bolus control, and dietary intake by mouth. It has also been reported that tongue exercises prevented general sarcopenia [178,179]. Indeed, swallowing muscles training, despite its focus on swallowing function, may exert its beneficial effects systemically.


*2.1.2. Stiffness*

The total static stiffness of machine tools is, in almost all cases, limited by the stiffness of the weakest parts. Amongst all the elements, the Spindle-Bearings System of the machine tool plays the most important role.

From results of structural analyses, the headstock can be considered as the heart of the whole machine tool. The design and quality of the machine tool must respect the quality of the drives and their features.

The headstock (as tool, or workpiece carrier), has a direct influence on the static and dynamic properties of the cutting process. The Spindle-Bearing System's stiffness also influences the final surface quality, profile, and dimensional accuracy of the workpiece.

The problem here is how quickly the headstock stiffness can be calculated with sufficient precision. The headstock stiffness must be calculated according to the deflection at the front end of the spindle, because the deflection at this point directly affects the precision of the finished product. The deflection at the spindle front end is the accumulation of various other, more or less important, partial distortions. The radial headstock stiffness can be calculated as follows:

$$K\_{rc} = \frac{F\_r}{y\_{rc}} \tag{1}$$

The individual headstock parts, (spindle and bearing arrangement), create a serial spring arrangement and it is evident that the resulting stiffness *Krc* is limited by the stiffness of the weakest part. An expert can see which part should be improved, and which partial distortions need to be minimized.

### **2.2. Simplified method of calculation**

52 Performance Evaluation of Bearings

issues.

bearing nodes

> running accuracy durability stiffness

high - speedability temperature

1. Externally - by shortening working time - within a working cycle

**Figure 3.** Modular structure of theoretical research, [5]

clamping elements

spindle noses

and cooling conditions.

plays the most important role.

*2.1.2. Stiffness*

on tool life and on the dynamic stability of the cutting process.

2. Internally - by reducing machining times (increasing the cutting width) - technological

Matematical models

**THEORETICAL RESEARCH**

statical models

statical stiffness dyn. stiffness

dynamical models

> natural frequencies spectrum

**MOUNTING ELEMENTS RESEARCH SH SYSTEM RESEARCH**

supporting elements

The philosophy of intelligent manufacturing systems applied to production processes minimise lost time. Further reducing lost time is expensive and has limited effectiveness at current levels of technological development. It has been shown that increased productivity can be achieved for example by changing the cutting speed. However this has a direct effect

The cutting speeds in machining processes depend on the technology applied, the cutting tool, and the workpiece material. The cutting speed also relates directly to the high-speed capability, and average diameter, of the bearings, the so-called factor *N = nmax.dmid..* Thus, from the point of view of the required cutting speed, the most important factor is the

The calculation of the headstock's maximum revolving speed is relatively simple. The highest revolving speed of a bearing node is calculated on the basis of the highest revolving speed of one bearing, multiplied by various coefficients reflecting the influence on the bearings, the bearing arrangement, bearing precision, their preloaded value, and lubrication

The total static stiffness of machine tools is, in almost all cases, limited by the stiffness of the weakest parts. Amongst all the elements, the Spindle-Bearings System of the machine tool

revolving frequency capacity of a spindle which is supported on a bearing system.

The calculation of spindle front end deflection, which takes into consideration all the important parameters, can only be achieved by using powerful computers. The analysis can be carried out by standard or custom software programs.

Calculating the many combinations of SBS arrangements is very demanding on time and money. Undertaking stiffness analysis using standard programs depends on the engineers' experience. The results can be open to questionable even when a suitable mathematical method is used (finite element method, boundary element method, Castilian's theorem, graphic Mohr's method, etc). This is because the headstock box, bearings or bearings nodes are statically indefinite systems which produce a nonlinear deformation of the node when under load.

Special software programmes are very expensive. They are developed using the most up-todate theoretical and practical knowledge. These programs have been developed by research institutions and bearing producers and the possibility of using such programs significantly influences their position on the SBS market. Taking the above into account, engineers would benefit from the existence of a simplified method of static analysis. Such a methodology would enable the engineer at the preliminary design stage to limit the number of possible spindle-bearing variants and determine the direction which would lead to the optimal SBS design, [6].

The main methodological advantage of computer analysis is the possibility of repeating single calculating algorithms in a matrix shape. To this end a special software package, "*Spindle Headstock"* was developed at the Department of Production Engineering in the Faculty of Mechanical Engineering at STU, Bratislava, [7].

The resulting radial deformation, *yrc*, of the front spindle end is shown in Figure 4.

Resulting static distortion of the front-end spindle equals

$$\mathbf{y}\_{rc} = \mathbf{y}\_0 + \mathbf{y}\_1 + \mathbf{y}\_t + \mathbf{y}\_a + \mathbf{y}\_v + \mathbf{y}\_{sb} + \mathbf{y}\_h \tag{2}$$

Radial Ball Bearings with Angular Contact in Machine Tools 55

(7)

4 4 4 4 and

(6)

64 64 *a aa L LL J Dd J Dd*

The definition of the quantities is shown in Figure 5. The individual headstock parts (spindle, bearing arrangement,) create a serial spring arrangement, and it is evident that the

At the same time, parameters *"Fr"*, *"a"*, *"L"* influence the value of both deflections. The spindle deflection caused by bending moments can be decreased by the following methods:

The resulting static distortion of the spindle front-end can be explicitly described by a multi-

F r a a a L L L A B, y f [E, F , a, L, J ,, D , d J D , d , K , K ]

There remains one significant problem with the calculation of the bearing nodes, and that is

resulting stiffness *Krc* is limited by the stiffness of the weakest part, [1] [2] and [9].

**Figure 5.** Cross section scheme of the spindle-bearing system

*"da", "dL"*.

parametrical equation in the form of:

that they are statically indeterminate systems.

Our experience has shown that whatever mathematical method and software is used, the spindle distortion caused by *bending moments y0* and by *bearing compliances yl* have the greatest influence on the resulting front end spindle distortion, [6].

**Figure 4.** Factors influencing the resulting deflection

Then

$$\mathbf{y\_{rc}} = \mathbf{y\_0} + \mathbf{y\_1} \tag{3}$$

where the distortion caused by bending moments is as follows:

$$y\_o = \frac{F\_r a^2}{3E} \left[ \frac{a}{J\_a} + \frac{L}{J\_L} \right] \tag{4}$$

and the deflection caused by bearing compliance is as follows:

$$y\_I = \frac{F\_r}{L^2} \left[ \frac{a^2}{K\_B} + \frac{\left(L + a\right)^2}{K\_A} \right] \tag{5}$$

Increasing moments of inertia *"Ja"*, *"JL"* were calculated as follows:

$$J\_a = \frac{\pi}{64} \left[ D\_a^4 - d\_a^4 \right] \quad \text{and} \quad J\_L = \frac{\pi}{64} \left[ D\_L^4 - d\_L^4 \right] \tag{6}$$

The definition of the quantities is shown in Figure 5. The individual headstock parts (spindle, bearing arrangement,) create a serial spring arrangement, and it is evident that the resulting stiffness *Krc* is limited by the stiffness of the weakest part, [1] [2] and [9].

At the same time, parameters *"Fr"*, *"a"*, *"L"* influence the value of both deflections. The spindle deflection caused by bending moments can be decreased by the following methods:

The resulting static distortion of the spindle front-end can be explicitly described by a multiparametrical equation in the form of:

$$\mathbf{y}\_{\text{F}} = \mathbf{f} \begin{bmatrix} \mathbf{E} \ \mathbf{F}\_{\mathbf{r}} \ \mathbf{a} \ \mathbf{L} \ \mathbf{J}\_{\text{a}} \ \mathbf{J}\_{\text{a}} \ \mathbf{J}\_{\text{a}} \end{bmatrix} \begin{Bmatrix} \mathbf{D}\_{\text{a}} \ \mathbf{d}\_{\text{a}} \end{Bmatrix} \mathbf{J}\_{\text{L}} \begin{Bmatrix} \mathbf{D}\_{\text{L}} \ \mathbf{d}\_{\text{L}} \end{Bmatrix} , \begin{Bmatrix} \mathbf{K}\_{\text{A}} \ \mathbf{K}\_{\text{B}} \ \mathbf{J}\_{\text{B}} \end{Bmatrix} \tag{7}$$

54 Performance Evaluation of Bearings

Faculty of Mechanical Engineering at STU, Bratislava, [7].

Resulting static distortion of the front-end spindle equals

**Figure 4.** Factors influencing the resulting deflection

**"ya"**

**Deflection of the spindle from bending moments "yo"**

where the distortion caused by bending moments is as follows:

and the deflection caused by bearing compliance is as follows:

Increasing moments of inertia *"Ja"*, *"JL"* were calculated as follows:

2

*F a a L y EJ J*

Axial force **"yh"**

**Resulting deformation of spindle front end is influenced by "yrc"**

*a L*

2 2

*B A*

*F a L a y L K K*

3 *r*

2 *r l*

*o*

Deflection of headstock box **"yv"**

Then

greatest influence on the resulting front end spindle distortion, [6].

The main methodological advantage of computer analysis is the possibility of repeating single calculating algorithms in a matrix shape. To this end a special software package, "*Spindle Headstock"* was developed at the Department of Production Engineering in the

Our experience has shown that whatever mathematical method and software is used, the spindle distortion caused by *bending moments y0* and by *bearing compliances yl* have the

> **Bearing compliance "yl"**

0 l t a v sb h y y y y y y y *rc y* (2)

rc 0 l y y y (3)

Stiffening effect of bearings **"ysb"**

(4)

Drive force

Spindle deflection by transversal forces **"yt"**

(5)

The resulting radial deformation, *yrc*, of the front spindle end is shown in Figure 4.

There remains one significant problem with the calculation of the bearing nodes, and that is that they are statically indeterminate systems.

**Figure 5.** Cross section scheme of the spindle-bearing system