Bipartitle Graphical Integration of Grinding Process Models
Abstract
In this paper the key grinding process models and relationships that were discovered by previous research efforts have been unified in the form of a bipartite graph representation, which is part of body of knowledge in constraint theory. The common terminology resulting from the representation helps in consolidating grinding modeling research works available in literature. The grinding bipartite graph has been used to design the structures of real time process control models, as well as the process instrumentation and experimentation strategies.
1. Introduction
The grinding process has historically been considered a complex domain by the manufacturing researchers and practitioners because of the complicated characteristics of the underlying dynamics (Bukkapatnam and Palanna, 2004). For a long time, industries have been extensively using several configurations of these grinding processes for the precision grinding of outer diameter (OD) of shafts and other radial symmetrical features. Grinding of the outside diameter of shafts has numerous critical applications. The surface finish of the shafts determines to a large extent the quality and durability of many complex rotary products. Even though the shaft might cost only a small fraction of the entire product cost, it is a major determinant of the product performance. Nowadays the realization of the desired surface finish with minimum consumption of energy at optimum cost is one of the main technological challenges. For example, Air Bearings is one of the key technologies that give a competitive edge to aerospace companies like Honeywell. Air bearings allow the rotating machines to reach speeds of around 100,000 RPM (more than 3 times faster than what conventional bearings allow) because there is minimal friction between the rotating assembly and the stationary assembly. Higher speeds are a definite technological advantage in high technology rotating assemblies like aircraft ECS because they can lead to more powerful and lighter aircraft. An air bearing assembly consists of mainly two component groups: the foil bearings that are made of sheet metal and are part of the stationary portion of the system, and a shaft which forms the rotating portion of the system. If the surface finish is too fine (Ra<5in[*]), the “take-off” torque required for the air bearing is increased. If the surface finish is too rough (Ra>10in*), then the load carrying capacity of the air bearing is reduced. Hence, a fine balance has to be struck while specifying the surface tolerance. Usual surface tolerances have significantly tightened over the last 60 years (see Figure 1). In the forties and fifties, surface tolerances specified on prints were usually in 250 in Ra. Later, in the 1970s they shrank to 64 in Ra. Today, 5 in Ra are not uncommon. Yet, for all the tightening of design tolerances, machine tools have not changed drastically over the years [1]. Realizing consistent surface quality from grinding operations is among the major industrial challenges today. The present efforts towards this end has met with limited success; the major reason being the complexity of the process.
Broadly, the following four major phenomena, as summarized in Figure 2, influence surface quality in grinding operations:(1) wheel friability and degradation, (2) plastic deformation and other material removal sources that are distributed randomly in space and time, (3) the resulting response of the machine structure that in turn is an assemblage of many moving and rotary components, and (4) complicated thermomechanics (heat transfer) pattern resulting from these distributed material removal sources. In order to effectively control surface finish, one must first understand the vast set of relationships and interactions connecting the process state variables and parameters that underpin these four interacting phenomena. Over the past 80 years or so researchers have addressed these various aspects of the grinding process and have provided several quantitative models (both empirical and analytical) as well as qualitative relationships. In this paper, the authors report a review of the previous grinding process modeling efforts, specifically focusing on those research works relevant for surface finish control. The review has captured the relevant multifarious relationships between grinding process parameters, state variables and performance characteristics. Bipartite graph representation is introduced as a means to graphically capture these relationships. A bipartite graph is a concept borrowed from constraint theory body of knowledge. These graphs are generally used to consolidate and graphically represent relationships and evaluate their suitability for devising appropriate solutions (i.e. various aspects of solvability). A grinding bipartite graph will provide guidelines for developing real time control strategies. One of the applications of this consolidation using bipartite graph has been in the selection and design of appropriate model structures in order to facilitate the implementation of “Model Based Tampering” (MBT) of precision manufacturing operations [1,2].
Figure 2: The four major phenomena contributing to the complexity in overall grinding process mechanics.
Bipartite representation can also lay the groundwork for experimental studies, particularly for sensor selection and design of experiments. We therefore anticipate that the presented bipartite graph approach will be of immense value to many applications involving grinding process sensing, control and experimentation. Although the presented research focuses on MBT application to shaft OD grinding, the relevant modeling efforts in surface grinding are also included. For future research and applications involving grinding process sensing and control, the presented bipartite graph may, however, needs to be slightly adjusted by including or excluding relevant models.
This paper is organized as follows: Section 2 provides an overview of the bipartite graph technique, Section 3 describes the methodology for the bipartite graphical integration of the various grinding process models. Section 4 shows the potential of this bipartite graph representation for surface finish control.
2. A Concise Treatise on Bipartite Graphs
George J. Friedman has developed an analytic foundation called constraint theory for the systematic determination of mathematical model consistency and computational allowability [3]. Constraint theory’s primary application is in the construction and use of heterogeneous, multidimensional mathematical models, especially when interdisciplinary technical teams, systems analysts, and managers are involved.
This theory establishes a rigorous basis for the notion of a constraint. Constraint theory separates a model from computations. It charts the flow of constraints throughout the model and detects instances of over-constraint and under-constraint. Over-constraint is a situation where more variables exists than that required for solving a group of equations and under-constraint is a situation where fewer variables exists than that required for solving a group of equations.
A bipartite graph is one of the representation tools given in the constraint theory. A bipartite graph has two sets of vertexes: (1) nodes, representing the model’s relations and (2) knots, denoting model variables. A knot will be connected to a node by an edge if and only if the corresponding variable is present in the corresponding functional relationship of a model. In this research nodes are represented using squares and knots are represented using circles. A model graph can be thought of merely as the circuit diagram of an analogue computer hood-up of the mathematical model where the nodes are the function (or relation) generators and the knots are merely wired connections that permit the values (voltages of variables) to pass from one function generator to another. The constraint matrix method [3] can be used to develop elegant solution strategies using bipartite graphs.
Based on the foregoing, one of the benefits of representing model equations group in bipartite graph format is that it helps to develop a common language for unifying the various previous models, likely developed independent of one another, and it also shows the relationships between the various governing equations in the application domain. It provides an insightful metamodel for both the mathematical model and all of its potential computations.
The bipartite graph representation exhibits many computability characteristics far more clearly than the original set of equations. Of course, we must return to the original equations to solve the model and to perform computations, but a much better visualization and more efficient control of the computations involving a large set of equations can be accomplished by resorting to bipartite graphs.
3. Bipartite Representation of Grinding Process Models
The literature contains a vast body of efforts related to grinding process modeling. The types of models ranged from analytical, physics-based models to empirical and phenomenological relationships. The original terminology used in the reviewed papers has been modified in order to establish a common language that integrates and consolidates the various research works. The nomenclature used in this paper is provided in Appendix 1.
The variables used in the grinding bipartite graph, shown in Figure 3, can be categorized as output variables, state variables and process parameters. The output variables are shown shaded in Figure 3. These include Temperature (), Normal and Tangential Force vector (), Acceleration, Acoustic Emission (Ae), and Grinding Power (P), which may be measured in real-time using appropriate sensors. These variables comprise state-output models as well as on-line observers, which are necessary for real-time sensing and control of processes. State variables such as those related to surface finish, wheel condition, etc., could be tied together using common equation numbers and their dependent variables. For example, a relationship to surface roughness (Ra) can be derived from knowing or measuring four different output variables as well as from estimating different state variables as seen in Figure 3. This segment of the graph is of significant relevance whenever the purpose is to determine the factors affecting degradation of process that impact surface finish.
Since it is practically impossible for any research work to address the entire gamut of issues in grinding process, many researchers tend to focus on a few specific aspects of the process so that they can develop rigorous models. For convenience and easier comprehension, this section categorizes the research works into four zones. Models in each zone concentrate on variables within the zone or the equations that connect the different zones. While the equations that tie the parameters together within the zones help to provide a solid foundation for the integrated model, it is the equations that span multiple zones that help to transgress between the zones and develop a holistic grinding process model.
1
1
The bipartite graph and the corresponding model equations are partitioned into zones as follows (also see Figure 4): The relationships ultimately connecting surface finish with wheel degradation, material removal processes, forces and structural response and temperature, respectively, are given in equation sets (1)-(10), (11)-(18), (19)-(33), (34)-(37).
3.1. Zone 1 Models (Wheel degradation): The equations in this zone relate wheel degradation to surface finish. Several researchers have developed empirical relationships connecting surface finish with process parameters including material removal rate, surface speed of the wheel, grinding depth, as well as parameters related to properties of the wheel, including the dressing lead, dressing angle, and dressing depth. All models available in literature are predominantly empirical in nature. For example, Malkin[4] describes the following four equations for surface finish:
………………..(1)
……………….(2)
……………….(3)
……………….(4)
Eranki et al., related the material removal rate QW to surface finish Raand wheel grade g0as [5];
,
………………………(5)
g0, with Raand grinding ratio Gas
……………………….(6)
QW was related to Raand G, as
……………………….(7)
and dressing leads Sdwas related to Ra as
………………………(8)
Fawcett and Dow provided the following relationship between Raand feed rate fr [6];
……………………….(9)
where k is an empirical constant. They also gave a relationship between Raand nose radius Nras;
……………………….(10)
3.2. Zone 2 Model (Plastic deformation and material removal): The equations in this zone largely pertain to material removal processes, and their relationship to the various process parameters process state variables including those describing forces and energy. We note that we have included a few elementary equations hereunder for the sake of completeness of the bipartite graph.
The following two equations relate the specific grinding energy UE and the specific Q’w[4]:
……………….(11)
…………………(12)
The specific grinding energy UE and the specific Q’w are related with the cutting force as [7];
……………………….(13)
G can be determined from knowing Sd and Q’w as given by Eranki et al., [5];
………………………(14)
Where for equation (14) we have,
Q’w(vw dc)
……………………….(15)
and
……………………..(16)
Mann et al. demonstrated the relationship between force F and Q’w [7];
……………………….(17)
The grinding power is a function of grinding width w, depth of cut d, Q’w and work speed vw as given by Eranki et al., 1992 [5];
……………………….(18)
3.3. Zone 3 Models (Structural response and forces): The models contained in this zone largely pertain to capturing the dynamics of the process, especially the manifestation of wheel and work vibrations and the various forces under various process conditions.
Kannappan and Malkin [8] have found that the main grinding forces are related to the number of passes and g0 as;
……………………….(19)
They have also developed a relationship between F and wear flat area [8];
……………………….(20)
Thompson [9] in his seminal investigation of grinding dynamics provided a model of the form;
……………………….(21)
where Fd is the normal grinding force.
The relationship betweenFand deflection of the loading tool is given by Nakayama et al. [10];
……………………….(22)
where, H and E are constant for a given grindingprocess setup.
Mann et al. [7] have characterized the relationship connecting d andF;
……………………….(23)
The relationship between acoustic emission amplitude IAeI vs. truing depth dt and d is given by Hundt et al. [11] as;
……………………….(24)
The relationship between maximum temperature and F is given by Mann et al., [7];
……………………….(25)
The set of equations that are related to grinding force is given below. The force is determined using a series of variables according to the equation given by Hundt et al., [11].
……………………….(26)
where
and
For Range A:
Chip thickness is 0.
For Range B:
For Range C:
For Range D:
Chip thickness is 0.
Now the cutting forceF can be determined by Kinzle’s equation;
where
and
where and
The relationship between the grinding time versus infeed velocity ui and radial displacement r is given by Malkin and Koren, [12];
……………………….(27)
The relationship between chip thickness T and number of active cutting edges Nkin, vw, vs, d and work diameter ds is given by Nakayama et al., [10];
……………………….(28)
The chip area Ac is a function of height dc and the w and is given by:
……………………….(29)
P = (vs)
………………………(30)
As given by Kannappan and Malkin [8],
……………………….(31)
and the number of active grinding edges na is a function of number of passes and g0 as given by;
……………………….(32)
The wear flat area Aw is a function of length L of contact and the w and is given by;
……………………….(33)
The relationship between forcing on work piece Fd, Force F , forcing frequency and time t is given by Thompson, [9] as;
……………………….(34)
According to Thompson [9], the relationship between Force F, force damping and time t is given by;
……………………….(35)
the dynamic wheel movement is governed by;
……………………….(36)
wheel wear x3 is a function of Force F given by;
……………………….(37)
wheel wear x3 is a linear function of Area of wear Aw;
……………………….(38)
the work-piece wear (reduction in radius) x4 is given by Thompson, 1986 [9];
……………………….(39)
the regenerative form for work displacement considering its abrasion is given by;
……………………….(40)
and the relation between x1, x2, x5 and x3 was given as
……………………….(41)
3.4. Zone 4 Models (Thermomechanics): The model in this zone mainly focus on temperature relationships.
The temperature is related to d [7] as;
……………………….(42)
The quasi steady state temperature distribution is given as a function of dimensionless length as given by Kohli et al., [13];
……………………….(43)
The max is related to depth of energy partition Z, as given by Kohli et al., [13];
……………………….(44)
The average temperature is related to the material removal rate as given by Mann et al., [7];
……………………….(45)
4. Application of Bipartite Representation to Modeling and Control
One may develop experimental as well as modeling strategies for grinding process sensing and control using bipartite graph representation. One may note that every acyclic path within a bipartite graph is equivalent to traversing a set of equations, and hence a specified set of variables. One notes that while a few of these variables and parameters are either set or measurable (eg: F, Ra, vw) the others are not (eg: Aw). Therefore, an effective sensing strategy for estimating an unknown variable must involve traversing a path connecting the variables of interest to one or more measurables. Evidently, researchers need to explore multiple paths in order to integrate multi-sensor information so as to circumvent practical experimental obstacles arising from measurement errors and uncertainties and to increase the robustness of the estimates by determining the values in multiple ways.
For example, as shown in Figure 5, Path 1 relates force signals () to surface finish. Equation (25) relates force () to temperature (max). Using equation (37), we are able to relate temperature to material removal rate (QW). With QWand knowing the wheel grade (g0) we can use equation (5) to determine surface roughness (Ra).
Path 2 shown in Figure 6, is an alternate path between force signals () and surface roughness (Ra). Equation (17) relates force () to material removal rate (QW). Equation (7) uses QWand grinding ratio (G) to give surface roughness (Ra).
1
Path 3 shown in Figure 7 is the third path connecting forces () and surface roughness (Ra). Force signals are related to material removal rate (QW) through equation (17). With QW, grinding wheel grade (g0), and grinding ratio (G) along with the use of equation (14) we can get dressing leads (Sd). With QW , Sd, and velocity of the work-piece (Vw) along with equation two we can get surface roughness (Ra).