Fundamental Approach for Analysis of Dynamic Characteristics of Fixtures

Present work is aimed at research of locating and clamping process of prismatic parts in fixtures during the machining on drilling-milling-boring machines. The fundamental approach for developing mathematical modeling of the system “fixture – workpiece” has been proposed. It allows to determine critical frequencies of functional elements oscillations and amplitude-frequency response characteristics, as well as dynamic component of fixture deformations at operating frequency of the machining process. The developed mathematical model is evaluated by numerical calculations of spectrum of critical frequencies considering the stiffness of cutting tool. The further researches are directed on the development of the methodology of calculation of the system “fixture – workpiece” by computer means of numerous realization and development of practical recommendations regarding its use during fixture design.


Introduction
Fixtures are used for ensuring the precise locating and reliable clamping of work-pieces during the machining on the metal-cutting machines and are an integral part of the closed-loop technological system "machine toolfixturecutting toolwork-piece".Fixtures considerably effect on the output of the competitive products, ensuring the steady state accuracy and quality of the work surfaces, sufficient stiffness of the technological system, flexibility of production and initial cost of the final product.It is confirmed by the percentage number of fixtures that makes 70-80% from the total quantity of tooling, 10-20% of the total cost of manufacturing system, 80-90% of the total time for production planning, 40% of the rejected parts after the machining due to imperfection fixture design, 70% of new fixtures are the modification of the existing [1][2][3][4][5].
Besides, most of existing methods of calculation do not include the dynamic character of the cutting process and interaction of the workpiece with the elements of fixture.Therefore, the problem of designing of fixtures is very urgent and relates to the basic multidisciplinary scientific and practical tasks on ensuring the stable locating of workpiece in the fixture during the machining process of the surfaces of parts.

Literature Review
At present, different approaches to the design and research of fixture for carrying out the manufacturing operations of machining of parts of different types are being developed.With the purpose of determining of the present state of researches in fixture designing and determining of the urgency of the research, the basic approaches to the designing of fixture from the point of view of interaction between fixture and workpiece have been considered.
Friction between elements of fixture and workpiece has been researched and have been determined the deformations, which appear in the points of contact [6].The methodology of modeling of the system "fixture -workpiece" was developed and the influence of the previous load from the impact of clamping and cutting forces on the error of the work surface was determined [7].The methodology of analysis of stability of the system "fixture -workpiece" has been developed and calculation of the minimum clamping force, required for machining process, has been presented; also, the influence of sequence of workpiece clamping has been researched [8].The influence of modes of cutting and fixture compliance on the workpiece stability was analytically calculated and researched [9].The simplified analytical model of the contact interaction between clamping elements of fixture and workpiece and the developed finite element model allows evaluating the contact deformation in contact points between clamping elements and workpiece were presented [10].The finite element model of determination the stability of workpiece locating in fixture and the methodology of optimization of the previous loading were developed [11].The mathematical model of fixture interaction with workpiece and analysis of deterministic positioning of fixture was developed [12].Many researches in fixture accuracy and analysis of the fixture stability were realized, based on the consideration of two-dimensional problem and introduction of "operative factor", which considers the friction forces [13].The methodology on determination of points and forces of clamping for ensuring the stable locating of workpiece in fixture has been developed [14].The methodology of control of clamping force considering the contact interaction between workpiece and clamping elements by means of the methods of nonlinear programming was presented [15].The problem of the temporariness of the stability of fixture locating considering the restriction of the force and direction of its action in the system "fixture -workpiece" has been researched [16].The influence of material removal on the dynamic state of the system "fixture -workpiece" was considered as opposed to quasi-static approach of predecessors [17].An essential experience of research of mechanical system "fixture -workpiece" has been obtained in terms of ensuring the condition of workpiece stability in fixture [18], [19].
Considering that the existing information on determine the contact points of functional elements of fixture, which are locating and clamping elements, is of advisory character and do not base on the analytical researches; and basing on that fact that the existing methodologies of calculation of clamping forces do not include the condition of stability of workpiece and provide the unreasonably overestimated value, performing of the researches is the relevant objective.
The present work is focused on locating and clamping of prismatic parts in fixtures during the machining on drillingmilling-boring machines.The characteristic feature of this scheme is that three flat surfaces are used as a datum (3-2-1 locating chart), ensuring the stability, convenience of locating and preventing of oscillations during the machining [20].
The goal of the proposed work is the development of the fundamental approach for mathematical modeling of the system "fixture -workpiece" based on manufacturing features of the process of locating and clamping of the workpiece in the fixture considering dynamic analysis of the workpiece in fixture under the effect of spatial system of cutting and clamping forces.

Dynamic equation
Consider the conservative mechanical system, which consists from workpiece, clamped in 6 supports 1, 2, …, 6 in the global orthogonal coordinate system XYZ with the center in the point O (Fig. 1).Consider the space movement of the workpiece, calculation scheme of which is presented on the Fig. 2. Displacement of xi, yi, zi of the supporting surfaces of work-piece in the fixture regarding the stationary location, that is characterized by static displacement, determined because of static calculation, could be expressed by geometric relation through 6 independent parametersdegrees of freedom of workpiece (Fig. 3): • components of displacement of the center of weights xC, yC, zC; • rotation angles φ, ψ, θ of three mutually perpendicular planes y-z, x-z, х-у around axes х, у, z of the local coordinate system, which pass through the center of the weight of workpiece C: , ; ; where Δxi /, Δxi //, Δyi /, Δyi ///, Δzi //, Δzi ///additional displacements of sup-ports through the rotation angles φ, ψ, θ, determined by trigonometric ratios:  .; ; For relatively small values of deformation of supports the statements (2) could be expanded into a Maclaurin series, holding members of the first order of smallness: ., ; , ; , Herewith, the components of displacements of supports (1) take the form: .; ; The space movement of the workpiece in the fixture could be described by means of the theorem on movement of the center of weight and the theorem on replacement of the angular momentum of mechanical system in the projections on the axis of the local coordinate system xyz: where mweight of workpiece; JCx, JCy, JCzinertia moments of workpiece relative to the axis, which pass through the center of weight; z projections of acceleration of the center of weight on the coordinate system;   ,   ,   angular acceleration of the workpiece relatively to the coordinate axes.
Components of the spring forces and their moments (Fig. 2 b) relative to the coordinate axes considering the relations (5) are defined by the statements (7).Statements (7) allow presenting the equations set in other form (8).
where mweight of the workpiece; JCx, JCy, JCzinertia moments of workpiece relative to the axis, which pass through the center of weight;   ,   ,   angular acceleration of the workpiece relatively to the coordinate axes; Fx, Fy, Fzdynamical components of the cutting forces (Fig. 2 а).
Thus, the mathematical model of dynamic analysis of the workpiece in fixture is described by the system of differential equations of the 12-th order relative to 6 changeable in time independent parameters xС, yС, zС, φ, ψ, θ.

Research of the free oscillations
For the case of zero value of forces and cutting moments, the equations set (8) in matrix form takes the following form: and 9 consolidated coefficients of the сross stiffness: , ; ; ; ; ; ; ; ; which depend on the stiffness and locating of the supporting surfaces.
During the machining of the prismatic workpiece with the center of weight С (L/2, B/2, H/2) for locating chart 3-2-1 in three planes the stiffness coefficients make (14) and another 8 coefficients of the cross stiffness make zero.
In order to consider the stiffness of the cutting tool and clamping element, the coefficient of stiffness of the clamping element k3 should be taken into account, and also the coefficients of longitudinal (along the axis z), transverse (along the axes x, y) and angular (around the axis z) stiffness of the cutting tool (an example is end mills), which could be roughly estimated by the following approximate dependencies that determine the appropriate state of stress (compression, bending and torsion): where d, Lp -diameter and length of the mill; Е, νelasticity modulus of the first grade and Poisson's ratio of mill's material.The consolidated coefficients of the direct and cross rigidities with the consideration of rigidities of the cutting instruments and clamping device receive the following form:  For free oscillations of the mechanical system "fixture -workpiece", the solution of the equation ( 9) is a harmonic movement described by the following statement: Fundamental Approach for Analysis of Dynamic Characteristics of Fixtures Vitalii Ivanov, Ivan Pavlenko which includes an unknown own frequency ω, and also vector-column of the peak values of displacements and rotation angles around the local system of coordinates {Xa} = {xСa, yСa, zСa, φa, ψa, θa} T .The last determines the form of free system oscillations with the accuracy up to the constant multiplier.
Equation ( 9) with the consideration of the statements (10) receives the following form: The condition of existing of the nontrivial solutions of equation ( 18) allows recording the frequency equation: from which 6 eigenfrequencies of the mechanical system are determined: Under condition (19) the equation ( 18) has the nontrivial solutions, which determine 6 forms of free oscillations with the accuracy up to the constant multiplier: where   n j i d , -the main diagonal minor of the matrix of dynamic stiffness [D], which elements ).6 ..., , are determined for the definite eigenfrequency ωn (n = 1, 2, …, 6).

Research of the forced oscillations
Spatial motion of the workpiece in fixture as a body with 6 degrees of freedom in projections on the axes of the local system of coordinate xyz has the following form of equation ( 23), which could be presented in matrix form: where {X}vector-column of an unknown displacements; {F}vector-column of external dynamical action: Herewith, the moments of dynamical components of cutting forces regarding axes of the local system of coordinates (Fig.
. ; ; Consider the case of monoharmonic external action: where {Fa}vector-column of amplitudes of external forces and moments; ω = ω0zfrequency, which is equal to the product of rotation frequency of the spindle and number of teeth z of the cutting tool.
Solution of equation ( 23) is sought in the form For vector-column {Xa}, elements of which are displacement amplitudes of center of weight and rotation angles around the axes of coordinates.Substitution of the statement (27) into the equation (23) allows writing down the equation  where ω1, ω2, …, ω6value of eigenfrequencies, determined because of modal analysis.
According to the formula of passing from displacement of the center of weight of workpiece to displacement of supports So, for locating chart 3-2-1 according to three planes the location of mill Y = B/2 the frequency-response characteristics of the system "fixture -workpiece" in the bottom supporting points have the following form: The analysis of the last dependencies confirms that frequencies correspond the appearance of the antiresonance effect, for which zero value of amplitudes z1a, z2a, і z3a are is observed.

Input data
The verification of adequacy of the proposed mathematical model is carried out on the example of calculation of the mechanical system "fixtureworkpiece" for the locating chart on three planes.The prismatic workpiece with the overall dimensions of 100×50×50 mm is chosen for further research.Milling of the shoulder is investigated with the width 18 mm and cutting depth 5 mm.The material of the workpiece is steel C45.The cutting tool is the solid end mill RF 100 VA DIN 6527 K with the following parameters: • diameter: 20 mm; • length of the cutting part: 42 mm; • total length: 92 mm; • number of teeth: 4; • material: hard alloy.

Conclusions
The study of the previous experience, generalization and systematization of the existing approaches to the research of mechanical systems allowed making a proposal on development a new approach, which is the comprehensive analysis of equilibrium state and oscillations relatively fixed position of the workpiece as an element of the mechanical system with considering the stiffness of the cutting tool.The mathematical model of static calculation system "fixture -workpiece" is being proposed, which ensures the defining of contact points of functional elements of fixture with the workpiece, the minimum required clamping force, components of cutting forces and places of application of these forces.Regression dependences for estimation of stiffness of functional elements of machine devices are being proposed.
The mathematical model of dynamic analysis, which is described by the system of differential equations of the 12 orders in matrix form relatively 6 variables in time independent parameters has been developed.The presented model allows research the eigenfrequencies of the system "fixture -workpiece" and determines the appropriate forms of free oscillations.The model also allows research the forced oscillations of the workpiece relatively the balanced position because of the action of dynamic component forces and cutting moments, and also determines frequency-response characteristics and the frequencies of resonance and antiresonance appearance.
The adequacy of the proposed mathematical model is approved by the example of numerical calculations.It is determined that the first critical frequency of oscillations of the workpiece is 6.4 times higher than the operating frequency of the cutting process, and the amplitude of forced oscillations is in a range of tolerances for the manufacture of the corresponding part.
The following researches are directed on the development of the methodology of calculation of the system "fixture -workpiece" by computer means of numerous realization and development of practical recommendations regarding its use during fixture design.

Figure 1 .
Figure 1.Calculation scheme of the mechanical system "fixture -workpiece"

EAIFigure 2 .
Endorsed Transactions on Industrial Networks and Intelligent Systems 01 2018 -03 2018 | Volume 4 | Issue 13 | e1 a b Calculation scheme (а) and the system of dynamic components of cutting forces (b)

EAI
Endorsed Transactions on Industrial Networks and Intelligent Systems 01 2018 -03 2018 | Volume 4 | Issue 13 | e1 of the component Xai of vector-column {Xa} on frequency of external action ω determines the frequency-response characteristics of the mechanical system «fixtureworkpiece».EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 01 2018 -03 2018 | Volume 4 | Issue 13 | e1For locating chart 3-2-1 according to 3 planes frequency values of displacements of the center of weight of workpiece and rotation angles around the axes of coordinates have the following form: