Keywords

12.1 Introduction

The main tasks of fixtures are the precise locating and reliable clamping of workpieces during the machining on the machine tools. Fixtures are an integral part of the closed-loop technological system “machine tool–fixture–cutting tool–workpiece”. Fixtures considerably effect on the quality and the cost of final products, as well as productivity and flexibility of production [1, 2]. The statistical data [3, 4] confirmed the importance of fixtures for products manufacturing. Fixture design is complicated and time-consuming process [5, 6]. It is important to find the optimal solution between flexibility and productivity, as well as to ensure the required accuracy [7]. Therefore, fixture requirements were identified and should be considered during design process [8,9,10,11,12,13,14,15,16,17]. The problem of fixture design is very urgent and relates to the multidisciplinary tasks on ensuring the stable locating of workpiece in the fixture during the machining process.

12.2 Literature Review

For many years fixture design has been the focus of academic research with significant progress in both theoretical and practical studies.

Research paper [18] presents the methodology of modeling of the system “fixture–workpiece”. In addition, the influence of the previous load from the impact of clamping and cutting forces on the error of the work surface was determined [18]. Friction between elements of the mechanical system “fixture–workpiece” has been researched and have been determined the deformations, which appear in the contact points [19]. 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 [20]. The influence of modes of cutting and fixture compliance on the workpiece stability was analytically calculated and researched [21]. The simplified analytical model of the contact interaction between workpiece and clamping elements was developed [22]. The finite element model for determination of the stability of workpiece locating in fixture and the methodology of optimization of the previous loading were developed [23]. The mathematical model of fixture interaction with workpiece and analysis of deterministic positioning of fixture was developed [24]. Many researches in fixture accuracy and analysis of the fixture stability were realized, based on the consideration of 2D problem and introduction of “operative factor,” which considers the friction forces [25]. The methodology on determination of points and clamping forces for ensuring the stable locating of workpiece in fixture has been developed [26]. 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 [27]. 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 [28]. The influence of material removal on the dynamic state of the system “fixture–workpiece” was considered as opposed to quasistatic approach of predecessors [29]. Moreover, scientific and methodological approach for identification of mathematical models of the mechanical system “fixture–workpiece” using artificial intelligent systems is proposed in paper [30]. Paper [31] focuses on the investigation of the affect that an automated flexible fixture has on the vibrational characteristics of parts in a reconfigurable manufacturing environment, through modeling the modal characteristics of test parts, of differing geometries, for various fixturing setups.

The goal of the proposed work is the mathematical modeling and numerical simulation 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.

12.3 Research Methodology

12.3.1 Mathematical Model

The fixture for multiaxis machining of parts is developed based on the analysis of requirements for fixture designing and fork-type parts manufacturing (Fig. 12.1). The main advantage of the proposed fixture configuration is the possibility of readjusting its functional elements when changing the workpiece dimensions.

Fig. 12.1
figure 1

The mechanical system “fixture–workpiece”

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The conservative mechanical system including a workpiece and clamped by supports is considered in a local orthogonal coordinate system xyz with the origin in the mass center C. The relative coordinates of the current point Bj, forces Fxj, Fyj, Fzj and moments Mxj, Myj, Mzj for modeling space movement of a workpiece are presented on the related design scheme (Fig. 12.2).

Fig. 12.2
figure 2

The local coordinate system for modeling spatial movement of the workpiece

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The displacements xi, yi, zi of the workpiece supporting surfaces in a fixture are determined as static displacement. They can be expressed by geometric dependencies with respect to six independent parameters (degrees of freedom of the workpiece): x, y, z, φ, ψ, θ [32].

All the components of supports displacements are calculated by the following formulas:

$$ {\displaystyle \begin{array}{l}{x}_i={x}_C-{b}_i\theta +{h}_i\phi; \\ {}{y}_i={y}_C+{l}_i\theta +{h}_i\psi; \\ {}{z}_i={z}_C+{l}_i\phi +{b}_i\psi, \end{array}} $$
(12.1)

where xC, yC, zC are coordinates of mass center and li, bi, hi are local longitudinal, transverse, and vertical coordinates.

The spatial movement of the workpiece in a fixture can be determined by the center-of-mass theorem and the momentum theorem. The related mathematical model of dynamic analysis is realized by the system of 12th order differential equations with respect to the abovementioned independent parameters [33].

Local components of the related dynamic forces Fx, Fy, Fz and moments Mx, My, Mz have the following form:

$$ {\displaystyle \begin{array}{l}{F}_x=-\varSigma {k}_x^i\left({x}_C-{b}_i\theta +{h}_i\phi \right);\\ {}{F}_y=-\varSigma {k}_y^i\left({y}_C+{l}_i\theta +{h}_i\psi \right);\\ {}{F}_z=-\varSigma {k}_z^i\left({z}_C+{l}_i\phi +{b}_i\psi \right);\\ {}{M}_x=-\varSigma \left[{k}_y^i{h}_i\left({y}_C+{l}_i\theta +{h}_i\psi \right)+{k}_z^i{b}_i\left({z}_C+{l}_i\phi +{b}_i\psi \right)\right];\\ {}{M}_y=-\varSigma \left[{k}_x^i{h}_i\left({x}_C-{b}_i\theta +{h}_i\phi \right)+{k}_z^i{l}_i\left({z}_C+{l}_i\phi +{b}_i\psi \right)\right];\\ {}{M}_z=-\varSigma \left[-{k}_x^i{b}_i\left({x}_C-{b}_i\theta +{h}_i\phi \right)+{k}_y^i{l}_i\left({y}_C+{l}_i\theta +{h}_i\psi \right)\right],\end{array}} $$
(12.2)

where m is workpiece mass and JCx, JCy, JCz are central moments of inertia.

12.3.2 Modal Analysis

To prevent a resonance phenomenon during the process of workpiece machining, the oscillation frequency for fixture elements should not coincide with the main frequency of the cutting process. One of the ways to avoid this problem is to set another cutting mode. For this purpose, eigenfrequencies of the mechanical system “fixture–workpiece” are determined using the ANSYS software. As a result, the minimal value of the obtained frequencies is compared with the related frequency during the cutting process.

In the case of zero value of forces and cutting moments, the mathematical model takes the following matrix form:

$$ \left[M\right]\left\{{\ddot{X}}_C\right\}+\left[K\right]\left\{X\right\}=\left\{0\right\}, $$
(12.3)

where {0} is zero column vector of external impact, {X} is column vector of displacement, [M] is matrix of inertia, [K] is stiffness matrix, the components of which depend on the contact stiffness coefficients kxi, kyi, kzi of supporting surfaces:

$$ {\displaystyle \begin{array}{l}{k}_{x x}=\varSigma {k}_x^i;\kern0.5em {k}_{y y}=\varSigma {k}_y^i;\kern0.5em {k}_{z z}=\varSigma {k}_z^i;\kern0.5em {k}_{\psi \psi}=\varSigma \left({k}_y^i{h}_i^2+{k}_z^i{b}_i^2\right);\\ {}{k}_{\phi \phi}=\varSigma \left({k}_x^i{h}_i^2+{k}_z^i{l}_i^2\right);\kern0.5em {k}_{\theta \theta}=\varSigma \left({k}_x^i{b}_i^2+{k}_y^i{l}_i^2\right);\kern0.5em {k}_{x\phi}=\varSigma {k}_x^i{h}_i;\\ {}{k}_{x\theta}=\varSigma {k}_x^i{b}_i;\kern0.5em {k}_{y\psi}=\varSigma {k}_y^i{h}_i;\kern0.5em {k}_{y\theta}=\varSigma {k}_y^i{l}_i;\kern0.5em {k}_{z\psi}=\varSigma {k}_z^i{b}_i;\\ {}{k}_{z\phi}=\varSigma {k}_z^i{l}_i;\kern0.5em {k}_{\psi \phi}=\varSigma {k}_z^i{l}_i{b}_i;\kern0.5em {k}_{\psi \theta}=\varSigma {k}_y^i{l}_i{h}_i;\kern0.5em {k}_{\phi \theta}=\varSigma {k}_x^i{b}_i{h}_i.\end{array}} $$
(12.4)

Six eigenfrequencies of the mechanical system “fixture–workpiece” are determined as natural values of the matrix of dynamic stiffness, which is equal to the roots of the following polynomial equation:

$$ \mid \kern0.125em \left[K\right]-{\omega}^2\left[M\right]\kern0.125em \mid =0. $$
(12.5)

12.3.3 Harmonic Analysis

Spatial movement of the workpiece in a fixture as a 6-degrees-of-freedom system is determined analytically by the following matrix form:

$$ \left[M\right]\left\{{\ddot{X}}_C\right\}+\left[K\right]\left\{X\right\}=\left\{F\right\}, $$
(12.6)

where {X} is column vector of displacements and {F} is column vector of external dynamic impact.

Monoharmonic external impact can be described by the expression {F} = {Fa}·sinωt, where {Fa} is column vector of dynamic amplitudes; ω = ω0·z—frequency as a product of the spindle’s rotation frequency and a number z of teeth of the operating cutting tool.

The general solution of the Eq. (12.6) can be presented as a column vector {Xa} of displacement amplitudes of center of weight and rotation angles around the coordinate axes:

$$ \left\{{X}_a\right\}={\left(\left[K\right]-{\omega}^2\left[M\right]\right)}^{-1}\left\{{F}_a\right\}. $$
(12.7)

The achieved dependence of {Xa} components on the cutting frequency ω determines the amplitude-frequency response characteristics of the considered mechanical system “fixture–workpiece.”

12.4 Results

12.4.1 Modal Analysis

The objects of the numerical simulation are free and forced oscillations of the mechanical system “fixture–workpiece”. The fixture has the adjustment mechanisms for ensuring its readjustment when changing the object of machining. In this case, the workpiece needs to be machined from different sides.

The results of modal analysis for the first eigenfrequency are presented in Fig. 12.3 and allow detuning from a resonance. The related data is summarized in Table 12.1.

Fig. 12.3
figure 3

The results of modal analysis

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Table 12.1 Comparative analysis of eigenfrequencies for the mechanical system “fixture–workpiece” in the case of fork-type parts machining, Hz
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Additionally, the abovementioned data allows concluding that a fixture for the proposed manufacturing process is less compliant in comparison with the same fixture for the typical manufacturing process, since the first eigenfrequency increases in 1.64 times.

12.4.2 Harmonic Analysis

Determining the oscillation amplitudes for the machined surfaces under the deviation of cutting forces and moments allows ensuring the sufficient rigidity of the mechanical system “fixture–workpiece”. For this purpose, the following objectives are stated for the case of fork-type parts machining:

  1. 1.

    obtaining the amplitude-frequency response under the most loaded operating step,

  2. 2.

    evaluating the resonance frequency and comparing with the related cutting frequency.

Harmonic analysis is realized by using the ANSYS software. Amplitudes of the dynamic components of cutting forces are equal 20% from their nominal values. The proposed model takes into account the Coulomb friction between the contact surfaces [34], which have approximately the same roughness Ra = 1.6 μm with the friction coefficient 0.1. A range of oscillation frequencies is chosen for the reason of providing coverage of the first three eigenfrequencies obtained above. So, the maximum oscillating frequency is equal to 1550 Hz.

The amplitude-frequency response is obtained (Fig. 12.4), as well as pre- and under resonance modes are determined. For ensuring equal conditions, the comparison analysis is carried out for displacements under the resonance frequency. The maximum displacement for the proposed fixture design is equal to 25 μm. The result of harmonic analysis is presented in Fig. 12.4, and the related numerical simulation data is summarized in Table 12.2.

Fig. 12.4
figure 4

The amplitude-frequency response for the fixture design while drilling holes in fork plugs

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Table 12.2 The results of numerical simulation for the fixture design while drilling holes in fork plugs
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12.5 Conclusions

In the research of recent challenges in fixture design for manufacturing complex parts, the existing approaches for designing CAx systems are systematized and generalized. As a result, a brand-new approach is developed based on a comprehensive analysis of free and forced oscillations of the mechanical system “fixture–workpiece” under the impact of clamping and cutting forces.

The mathematical model of free and forced oscillations for research of the dynamic states of the comprehensive mechanical system is proposed. It is proved on the example of fork-type parts considering contact stiffness of functional elements, required clamping forces, and dynamic components of cutting forces and moments. This model allows determining natural frequencies of free oscillations with the aim of detuning from a resonance. It should be noted that the first eigenfrequency 1025 Hz for the proposed manufacturing process is 1.64 times more than the same frequency for the typical manufacturing process. Consequently, the dynamic stiffness of the new design is significantly increased, and required deturning 20% from a resonance frequency 1271 Hz is ensured.

The research of forced oscillations of the proposed system under the impact of dynamic forces and cutting moments is presented. As a result, the amplitude-frequency response is obtained using the ANSYS software. It should be noted that the maximum displacement 25 μm does not exceed all the required tolerances and related manufacturing errors.

Further research will be aimed at developing the research methodology and related engineering approach for estimating parameters of the proposed mathematical model by the results of numerical simulation and experimental data using both of a linear regression procedure and a nonlinear regression analysis by means of artificial intelligent systems.