Keywords

1 Introduction

Controlling the volumetric error caused by thermal effects remains a major challenge in the field of machine tools. This is due to the increasing demand for more accurate products, which require tighter tolerances and higher quality, while maintaining high productivity [1].

Thermal distortions, caused by various heat sources, are among the primary contributors to the accuracy degradation of machine tools. These heat sources can arise from various elements of the machine, such as motors, spindles, and ball screws, as well as from the machining process itself, such as tool/workpiece heating, heat/cooling from lubrication, and external sources such as ambient temperature and radiation. Despite the efforts to control these heat sources, some may remain unknown or difficult to regulate.

Another important factor is the transient thermal state of the machine during the machining process, as the time constants of the machine elements involved are of the order of hours, which are longer than the typical operation times. This makes it challenging to reach a thermal steady-state for the machine.

Analytical approaches, such as Thermal Modal Analysis, have been employed to study the thermal behavior of machine tools [2, 3], model thermal errors [4, 5], and determine optimal sensor placement for machine error estimation [6, 7]. However, since machine operations occur within the machine volume, it is necessary to accurately characterize the volumetric error of the machine [8], which is impacted by the thermal distortions.

With these considerations in mind, this work proposes a methodology for analyzing the volumetric error due to thermal effects, induced by the thermal modes of the machine tool bodies. The methodology includes an in-depth explanation of thermo-elastic modeling and Thermal Modal Analysis, followed by the presentation of a Volumetric Thermo-Elastic Modal Analysis method for a multi-axis milling machine. The most relevant volumetric errors caused by thermal modes are then analyzed and discussed. Finally, the conclusions of this work and future steps are outlined.

2 Modelling

In this section, a comprehensive explanation of the thermo-elastic modeling and Thermal Modal Analysis (TMA) formulation is provided. The formulation outlined here applies to the finite element model of a single body, but can easily be extended to cover multiple interconnected bodies.

2.1 Thermo-Elastic Model

The equation for the calculation of the thermal field can be expressed in matrix form as follows.

$$ {\varvec{C}}^{{\varvec{t}}} \dot{\user2{\theta }}\left( t \right) + {\varvec{K}}^{{\varvec{t}}} {\varvec{\theta}}\left( t \right) = {\varvec{q}}\left( t \right) $$
(1)

where \({\varvec{C}}^{{\varvec{t}}}\) represents the specific heat or thermal inertia matrix,, \({\varvec{K}}^{{\varvec{t}}}\) represents the conductivity matrix, \({\varvec{q}}\left( t \right) \) denotes the thermal load vector, and \({\varvec{\theta}}\left( t \right)\) denotes the temperature vector. The equation that couples the thermal and elastic behavior, in the absence of mechanical loads, can be represented as follows.

$$ {\varvec{K}}^{{\varvec{u}}} {\varvec{u}}\left( t \right)\user2{ } + \user2{ K}^{{{\varvec{ut}}}} \left( {{\varvec{\theta}}\left( t \right)\user2{ } - \user2{ \theta }_{{{\varvec{ref}}}} } \right) = 0 $$
(2)

where \({\varvec{K}}^{{\varvec{u}}}\) represents the stiffness matrix, \({\varvec{K}}^{{{\varvec{ut}}}}\) represents the thermo-elastic stiffness matrix, \({\varvec{u}}\left( t \right)\) denotes the displacement vector, \({\varvec{\theta}}\left( t \right)\) denotes the temperature vector, and \({\varvec{\theta}}_{{{\varvec{ref}}}}\) represents the initial temperature vector.

The equation for a coupled thermo-elastic analysis, which results from the combination of Eqs. (1) and (2), can be expressed in matrix form as follows:

$$ \left[ {\begin{array}{*{20}c} {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {{\varvec{C}}^{{\varvec{t}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\dot{\user2{u}}\left( t \right)} \\ {\dot{\user2{\theta }}\left( t \right)} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {{\varvec{K}}^{{\varvec{u}}} } & {{\varvec{K}}^{{{\varvec{ut}}}} } \\ {\varvec{0}} & {{\varvec{K}}^{{\varvec{t}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varvec{u}}\left( t \right)} \\ {{\varvec{\theta}}\left( t \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\varvec{K}}^{{{\varvec{ut}}}} {\varvec{\theta}}_{{{\varvec{ref}}}} } \\ {{\varvec{q}}\left( t \right)} \\ \end{array} } \right] $$
(3)

where \({\varvec{C}}^{{\varvec{t}}}\) is the specific heat or thermal inertia matrix, \({\varvec{K}}^{{\varvec{t}}}\) is the conductivity matrix, \({\varvec{K}}^{{\varvec{u}}}\) is the stiffness matrix, \({\varvec{K}}^{{{\varvec{ut}}}}\) is the thermo-elastic stiffness matrix, \({\varvec{q}}\left( t \right) \) is the thermal load vector, \({\varvec{u}}\left( t \right)\) is the displacement vector, and \({\varvec{\theta}}\left( t \right)\) is the temperature vector. It is important to note that in Eq. (3), only thermal loads are considered and structural loads are not taken into account.

2.2 Thermal Modal Analysis

The discrete heat equation derived from Eq. (1) can be written as follows:

$$ {\varvec{C}}^{{\varvec{t}}} \user2{ }\frac{{d\user2{ \theta }\left( t \right)}}{dt} + {\varvec{K}}^{{\varvec{t}}} {\varvec{\theta}}\left( t \right) = {\varvec{q}}\left( t \right). $$
(4)

Equation (4) represents a system of n first-order linear differential equations with constant coefficients. Hence, the general solution of this system is the sum of the general solution of the homogeneous system plus a particular solution of the entire system.

The homogeneous system can be expressed as:

$$ {\varvec{C}}^{{\varvec{t}}} \user2{ }\frac{{d\user2{ \theta }\left( t \right)}}{dt} + {\varvec{K}}^{{\varvec{t}}} {\varvec{\theta}}\left( t \right) = 0 $$
(5)

Its solution is given by:

$$ {\varvec{\theta}}\left( t \right) = {{\varvec{\Phi}}} e^{ - \lambda t} , $$
(6)

By substituting this solution into the homogeneous system, the following result is obtained:

$$ \left( {{\varvec{K}}^{{\varvec{t}}} \user2{ } - \lambda {\varvec{C}}^{{\varvec{t}}} } \right){{\varvec{\Phi}}}\user2{ } = \user2{ }0 $$
(7)

This system constitutes a general problem of eigenvalues and eigenvectors. The characteristic equation of this system is given by \(\left| {{\varvec{K}}^{{\varvec{t}}} \user2{ } - \lambda {\varvec{C}}^{{\varvec{t}}} } \right| = \user2{ }0\). Solving this equation results in n eigenvalues, \(\lambda_{1} , \lambda_{2} , ..,\lambda_{n}\). These eigenvalues represent the inverse of the time constants of each mode, \(\tau_{1} ,\tau_{2} ,...,\tau_{n}\). By replacing the eigenvalues in the system and solving it, a series of eigenvectors, \({{\varvec{\Phi}}}_{{\varvec{i}}}\), associated with them is obtained. As a result, the general equation of the homogeneous system is derived.

$$ {\varvec{\theta}}\left( t \right) = \mathop \sum \nolimits_{i = 1}^{n} {\varvec{D}}_{{\varvec{i}}} {{\varvec{\Phi}}}_{{\varvec{i}}} e^{{ - \lambda_{i} t}} $$
(8)

where \({\varvec{D}}_{{\varvec{i}}}\) represents a vector of coefficients.

The eigenvectors \({{\varvec{\Phi}}}_{{\varvec{i}}}\) form a set of n linearly independent vectors that make up a basis. The temperature vector \({\varvec{\theta}}\left( t \right)\) can be represented in relation to this basis with new coordinates \({\varvec{\xi}}\left( t \right)\), as follows:

$$ {\varvec{\theta}}\left( t \right) = \mathop \sum \nolimits_{i = 1}^{n} {{\varvec{\Phi}}}_{{\varvec{i}}} {\varvec{\xi}}_{{\varvec{i}}} \left( t \right) = \user2{ }{{\varvec{\Phi}}}{\varvec{\xi}}\left( t \right). $$
(9)

Replacing Eq. (9) in the original system of Eq. (4) and multiplying by \({{\varvec{\Phi}}}^{{\text{T}}}\)

$$ {{\varvec{\Phi}}}^{{\varvec{T}}} {\varvec{C}}^{{\varvec{t}}} {{\varvec{\Phi}}}\dot{\user2{\xi }}\left( t \right)\user2{ } + {{\varvec{\Phi}}}^{{\varvec{T}}} {\varvec{K}}^{{\varvec{t}}} {{\varvec{\Phi}}}{\varvec{\xi}}\left( t \right) = {{\varvec{\Phi}}}^{{\varvec{T}}} {\varvec{q}}\left( t \right). $$
(10)

then, if \({{\varvec{\Phi}}}^{{\varvec{T}}} { }{\varvec{q}}\left( t \right){ } = { }{\varvec{\psi}}\left( t \right)\), Eq. (10) may be re-written as

$$ \dot{\user2{\xi }}\left( t \right)\user2{ } + {{\varvec{\uplambda}}}{\varvec{\xi}}\left( t \right) = {\varvec{\psi}}\left( t \right), $$
(11)

since \({{\varvec{\Phi}}}^{{\varvec{T}}} {\varvec{C}}^{{\varvec{t}}} {{\varvec{\Phi}}} = 0\) and \({{\varvec{\Phi}}}^{{\varvec{T}}} {\varvec{K}}^{{\varvec{t}}} {{\varvec{\Phi}}} = {{\varvec{\uplambda}}}\), which are due to the orthogonality of the eigenvectors with respect to the matrices \({\varvec{K}}^{{\varvec{t}}}\) and \({\varvec{C}}^{{\varvec{t}}}\), and \({{\varvec{\uplambda}}}\) is a diagonal matrix, it is immediately evident that Eq. (11) represents a system of n uncoupled equations in the form:

$$ \dot{\user2{\xi }}\left( t \right)\user2{ } + {\uplambda }_{i} { }{\varvec{\xi}}\left( t \right) = {\varvec{\psi}}_{{\varvec{i}}} \left( t \right). $$
(12)

Therefore, by introducing the natural coordinates \({\varvec{\xi}}\left( t \right)\), the system of n linear differential equations with constant coefficients is reduced to n uncoupled equations of a single variable. The eigenvectors \({{\varvec{\Phi}}}_{{\varvec{i}}}\) are known as natural thermal modes, and the process of decoupling the thermal balance equations is referred to as Thermal Modal Analysis (TMA). Each natural thermal mode has a time constant defined as \({\uptau }_{i} { }\) = 1/\({\uplambda }_{i}\).

3 Volumetric Thermo-Elastic Modal Analysis

In this section, a methodology is presented to assess the impact of thermal modes on the volumetric error of a multi-axis machine tool. Volumetric error refers to the deviation in the volume of the machine tool resulting from structural effects, thermal effects, and errors caused by positioning and acting elements. The contribution of thermal effects to volumetric error is significant and difficult to identify.

As volumetric error significantly affects the accuracy of machine tools, this study focuses on evaluating its possible thermal sources, through the examination of thermal modes as heating patterns of the machine bodies.

3.1 Machine Finite Element Model

A case study of a cantilever-style traveling column milling machine with three linear axes is presented to illustrate the proposed methodology (as depicted in Fig. 1). The machine consists of moving parts such as the column (X), console (Z), and ram (Y), which are supported by a bedtable that is fixed to the ground. The kinematic chain of the milling machine is Tool-Y-Z-X-Bedtable and the working volume of the machine is defined as X: 0–4000 mm, Y: 0–1200 mm, and Z: 0–1500 mm.

The model was developed using an in-house software implemented in MATLAB. This software allows for the input of body matrices (exported from ANSYS), followed by the definition of links (such as linear guideways, ballscrews, linear scales, etc.), application of structural and/or thermal boundaries and loads, and ultimately, the solution of the model.

The structural part of the machine model consists of four bodies (bedtable, column, console, and ram) that are connected by appropriate links such as linear guideways, ballscrews, and linear scales. The linear guideway links were simplified and modelled as ensembles of springs with equivalent stiffness, while the ballscrews were simplified and modelled as linear springs with position-dependent stiffness. The linear scales were included in the model as constraint equations.

Fig. 1.
figure 1

Machine model.

Full size image

The developed finite element model allows for easy evaluation of the volumetric error by changing the machine position. This is achieved by moving the bodies to the desired position, linking the body matrices considering simplifications of the mechanical connections, and generating the structural model matrices (\({\varvec{K}}^{{\varvec{u}}}\) and \({\varvec{K}}^{{{\varvec{ut}}}}\))) for each position for use in Eq. (3).

3.2 Volumetric Thermo-Elastic Modal Analysis

The purpose of this paper is to examine the impact of thermal variations on machine tools on their accuracy. To achieve this goal, the methodology presented in this paper uses normalized thermal modes and assesses their effect on the volumetric error. The normalized thermal modal loads have been selected as they are unique to the bodies and independent of the loads.

This research did not consider the thermal connections between the various structural components of the machine tool, as recalculating the thermal modes at each evaluated position within the workspace would be required in this case. It is assumed that the thermal connections between these components are weak, and as such, this assumption was made to simplify the simulation process.

To implement the proposed methodology, the first 50 modes of each body of the machine tool are calculated according to the procedure described in Sect. 2.2. It should be noted that the thermal modes of the machine bodies are only affected by the convection thermal boundary, as the thermal modes are only dependent on the bodies’ thermal matrices (\({\varvec{C}}^{{\varvec{t}}}\) and \({\varvec{K}}^{{\varvec{t}}}\)).). The convection boundary affects the conductivity matrix \({\varvec{K}}^{{\varvec{t}}}\), so it is important to mention that for the analysis conducted in this work, a convection boundary load (with \(h = 10\,{\rm W}/\left( { {\rm m}^{2} {\rm K}} \right)\))) was added to the external faces of the bodies.

To evaluate the volumetric error caused by the thermal modes of the bodies, the thermal modes are introduced as a thermal field in each body separately. The thermal modes are normalized with zero mean and in the range [−1, 1], and then introduced as the body temperature (\({\varvec{\theta}}_{{{\varvec{ref}}}}\) in ºC) in Eq. (3), as shown in Fig. 2, with only the structural equation considered.

Fig. 2.
figure 2

Machine model and error analysis plane (Plane ZY).

Full size image

The aim is to observe how the body thermal modes affect the volumetric error (in this case study only in the ZY plane). Equation (3) is then solved in a mesh of positions in the ZY plane with 100 mm steps in each direction and the volumetric error for each body mode is calculated.

4 Results and Analysis

In this section, an analysis of the most significant results obtained from the methodology described in the previous section is presented. As previously explained, the volumetric error on the ZY plane (discretized with 100mm mesh intervals) was evaluated for the first 50 modes of each body, resulting in 200 evaluations of the error on the ZY plane.

To evaluate the effect of the body modes on the volumetric error, only the modes with the largest magnitude or most significant shape are analyzed. In this work, two such modes are discussed.

The most significant volumetric error found in the analysis was due to mode 9 of the ram, which has a time constant of 0.9 h and is depicted in Fig. 3. In the first image of Fig. 3, the shape and magnitude of the thermal mode is represented, while in the second image, the error components (X, Y, Z translations, A, B, C rotations) for the ZY plane are shown. It is observed that the volumetric error changes significantly with Y axis movement, while Z axis movement also only has a relevant effect on EY.

Fig. 3.
figure 3

Mode 8 – Ram.

Full size image

Another mode that has a significant volumetric error is mode 9 of the column, which has a time constant of 2 h and is depicted in Fig. 4. In the first image, the shape of mode 9 of the column body is shown, while in the second image, the volumetric error for the ZY plane under study is shown, with magnitudes that are smaller than those caused by mode 8 of the ram, but still significant. In this case, it can be observed that the Z axis movement has a significant impact on the volumetric error due to the thermal mode shape, and the Y axis movement also has an impact on error EZ.

Fig. 4.
figure 4

Mode 9 - Column.

Full size image

From these results, it is evident that certain thermal loads on the bodies may have a significant impact on the volumetric error of a machine tool. This information is relevant in identifying the errors that can be caused by specific thermal shapes. In this work, thermal modes were used due to their physical significance, but the presented methodology can be used to identify thermal errors due to any thermal basis.

5 Conclusions and Future Steps

The study introduced a novel thermal modal volumetric error analysis for multi-axis machine tools. The purpose of this methodology is to gain a deeper understanding of the thermal behavior of the structural components of machine tools and to develop tools that can help improve machine tool accuracy.

The results obtained from this type of analysis can have several practical applications. Firstly, the results can be used to identify the most relevant positions to monitor the thermal field of the machine, which can help optimize sensor placement for thermal error compensation models, for example.

Lastly, this analysis can be used to identify avoidable temperature fields in machine tools and take mitigating measures at design stage. Reducing the thermal modal volumetric error, the machine tool can be designed to perform more accurately.

Overall, this research highlights the importance of understanding the thermal behavior of machine tools and the potential benefits that can be derived from a thermal modal volumetric error analysis. This information can be used to improve machine tool accuracy, which is critical for many industrial applications.