Thermal Growth of Motor Spindle Units

Abstract

The paper deals with strategies for numerical compensation of thermo-mechanical deformation of machine tool spindles and the TCP, respectively. Methods for digital modelling and simulating the temperatures and thermo-elastic deformation are presented. This is done by considering the geometry, material data, drive signals and temperature values. The topic of compensating thermo-elastic effects in spindle units is an important topic in manufacturing. Analytical equation and function block methods for measuring and predicting thermal spindle growth are compared. The heat flow model converts variable spindle load, speed, coolant and ambient temperature into local temperatures followed by elastic deformations of the spindle unit. The simulation results were verified for different types of motor spindles by experiments on a spindle test rig at SPL GmbH. A thermal stiffness value [W/µm] is characterized by the energy losses of the spindle, which result in thermal growth. Different strategies for digital reduction of a thermal spindle growth were developed.

1 Causes for Thermal Growth

The linear theory of thermoelasticity of materials was established by Cauchy and Timoshenko before 1914 and relates a heat flow to the local temperature gradient, a relationship known as Fourier’s law. Under mechanical and thermal loading, an elastic spindle unit undergoes stresses, strains, and deformations that continue in spindle growth. Thermal growth of motor spindle units leads to significant manufacturing errors and risks due to lack of predictive applications and essential energy consumption until a thermostable state is reached.

The motor spindle units of SPL GmbH [1] are customized for machining processes (turning, milling, grinding, dressing, test bench) based on technical requirements including thermal spindle growth. Customized motor spindles require an approved design concept for mechanical and thermal stiffness, resource efficiency, service life reserve, precision and quality assessment at the test bench. A motor spindle unit, see Fig. 1, integrates the electric motor, shaft with bearing, tool holder, clamping, encoder and the housing with the stator, water cooling.

Fig. 1.

Motor spindle unit SPL2782.1

The heat generation bases on the energy consumption of the spindle unit, the losses of electrical energy and the mechanical-fluidic friction losses under mechanical load. The causes for thermal growth are heat generation in the motor spindle unit by the electric motor, cooling water, process and ambient temperature, and thermal stiffness of the tool adapter, shaft, bearings, and housing.

In [2], a method for analytical calculation of heat transfer on spindle bearings was developed with the objective of axial and angular misalignment compensation.

[3] gives an overview of research activities in the field of thermal errors in machine tools, e.g., advances in the measurement of thermal errors and temperatures, methods for calculating thermal errors in machine tools, and reduction of thermal errors.

In [4,5,6], the state of the art on methods for controlled thermal deformations in machine tools is analyzed and compared, and mathematical methods for thermoenergetic-thermoelastic models and thermal stiffness are advanced.

Previous thermostable measurements, require lengthy investigations and have been extended by non-stationary thermal analytical methods [7].

Paper [8] gives an overview of various thermal failure modeling methods, including least squares method, multivariable regression analysis, Grey system, neural network, support vector machine, and hybrid model. One can measure a number of local temperatures and compare them with the component displacements, but the compilation of rigid bodies does not lead to correct thermal growth values.

There are two basic thermal deformation shapes, thermal elongation and thermal bending, shown in [9: Fig. 3.1] … There are nine candidate temperature sensor locations for the thermal elongation, whereas there are eighteen, nine each in the upper and lower surfaces, for the thermal bending [9].

The International Organization for Standardization (ISO) has developed several standards for test methods for geometric accuracy under static or unloaded conditions: ISO 230-3 (Determination of thermal effects [10]); ISO 16907-2015 (Numerical compensation of geometric errors [11]).

The model-based prediction of thermoenergetic power transformations and heat flows, the influence of spindle thermoelasticity under process loads and the influence of the environment are not yet solved in the State of the Art. Compared to the research efforts on measurement, simulation and compensation strategies related to thermal growth of motor spindles, this paper proposes a digital prediction and numerical compensation under real-time process load.

2 Prediction Methods

A quality assessment of the geometric accuracy and kinematic errors in a real tool, part or machine tool is usually defined by a series of standards (see Fig. 2 right) [10, 11]. Quality assessment of a thermal spindle growth is necessary for real and digital systems, where digital prediction methods are required. Digital models base on 1-deterministic relations (temperature, position sensors), 2-input-output relationships and 3-data correlations. Therefore, digital quality assessment methods of a thermal spindle growth will apply analytical equations, or parameterized function block models, or black boxes (see Fig. 2 left). The digital models require new virtual validation and verification methods (structural parametric method), like IEEE Standard for System, Software, and Hardware Verification and Validation [12].

Fig. 2.

Prediction and quality assessment methods

Digital prediction and numerical compensation methods are necessary for the thermal spindle growth, because of missing real temperature and position sensors on the shaft at high spindle speed and process load. This paper targets methods for a digital prediction and numerical compensation of thermal spindle growth by both methods for analytical equations and functional blocks.

2.1 Analytical Equation Method

The initial scientific and methodological assumptions for creating a thermal and thermoelastic model of a motor spindle is the idea of the thermal system of a machine tool [13]. A thermal system is a set of interconnected thermoactive elements (parts and assemblies) with quasi-stable thermal and thermoelastic relationships between them. In this work, the same systematic approach is applied to motor spindle units. A spindle assembly (motor spindle) has inherent elements (see Fig. 1 and Fig. 3). Heat-active elements (parts and assemblies) are structurally interconnected with quasi-stable thermal and thermoelastic links. Heat carriers and heat sources are the main elements of the thermal system of the spindle unit. The heat released in the sources is partly transferred and redistributed between the heat-generating elements of the spindle unit. Partly it goes into heating the heat-generating sources themselves, and partly it is dissipated to the environment. Thus, there are thermal interactions or thermal connections between the main elements of the thermal system of the spindle unit and between the main elements of the thermal system and the environment.

Thermal interaction between two elements or between an element and the environment is understood to be such an interaction where a change in the thermal state of one element or the environment leads to a change in the thermal state of the other element. The thermal interaction between the main elements of the thermal system of the spindle assembly forms an internal thermal link. The thermal interaction between the main elements of the thermal system and the environment forms an external thermal connection. Thermal relations between the main elements in spindle units are realized by conduction (1), convection (2), radiation (3) and heat exchange (4). The heat flows between the thermal masses lead to physical phenomena of heat conduction, heat convection, heat radiation, heat exchange and lead to local spindle temperatures compared to environmental and coolant temperatures.

$$\dot{Q}=\frac{\lambda At\varDelta T}{L};$$

(1)

$$\dot{Q}=\alpha At\varDelta T;$$

(2)

$$\varPhi =\alpha {\varPhi }_{s};$$

(3)

$${c}_{1}{m}_{1}\cdot \left({T}_{1}-{T}_{m}\right)={c}_{2}{m}_{2}\cdot \left({T}_{m}-{T}_{2}\right).$$

(4)

The set of internal thermal relations between the main elements of the thermal system forms the thermal structure of the spindle unit (or assembly or part) [4,5,6]. A thermal model (structure) of a spindle assembly is created by idealizing the structure of the heat-active elements of the spindle assembly, the connections between them and the heat transfer processes taking place within it. A mathematical model of the thermal structure of the spindle unit is created based on the concept of the thermal system and the cause-effect relationship diagram. To create a mathematical model, it is necessary to schematize the determining factors and proceed to their quantitative description.

The heat flow brought to the heat-active element from the sources is withdrawn into the environment and other heat-active elements of the spindle unit, as well as goes to the heating of the element itself. On the basis of the law of conservation of energy, the equation of heat balance of the thermoactive element i of the system will be written in the form of

$${Q}_{i}+{Q}_{ienv}+{\sum }_{i=1,i\ne j}^{n}{Q}_{ij}={\sum }_{i=1}^{n}{Q}_{iall};$$

(5)

where ${Q}_{i}$ – power losses, going for heating the thermoactive element i (accumulation of heat energy); ${Q}_{ienv}$ – power losses transferred to the environment; ${Q}_{ij}$ – power losses transferred to the thermoactive element i; ${\sum }_{i=1}^{n}{Q}_{iall}$ – total power losses brought to this element from heat sources; $n$ – number of thermoactive elements of the system.

According to the first law of thermodynamics, heat energy accumulation in the element i is described by the equation

$${\dot{Q}}_{i}={c}_{i}\cdot {m}_{i}\cdot \frac{d{\tilde{T }}_{i}}{dt};$$

(6)

where ${c}_{i}$ – specific heat capacity of the thermoactive element; ${m}_{i}$ – mass of the thermoactive element; ${\tilde{T }}_{i}$ – average temperature.

Similar equations can be written for other thermoactive elements of the system, then we get a system of thermal balance equations:

$${c}_{i}\cdot {m}_{i}\cdot \frac{d{\tilde{T }}_{i}}{dt}+{\alpha }_{i}\cdot {A}_{i}\cdot \left({\tilde{T }}_{i}-{T}_{ienv}\right)+{\sum }_{i=2,i\ne j}^{n}{\alpha }_{ijcont}\cdot {A}_{ij}\cdot \left({\tilde{T }}_{i}-{\tilde{T }}_{j}\right)={\sum }_{i=1}^{n}{Q}_{iall};$$

(7)

where ${\alpha }_{i}$ – heat convection or radiation coefficient, ${A}_{i}$ - heat transfer surface area, ${\alpha }_{ijcont}$ - heat conductivity coefficient of the junction between the contacting i and j elements (if the intermediate medium between the elements is air, oil, strips, inserts, wedges ${\alpha }_{ijcont}=({\alpha }_{icont}^{-1}+h/\lambda +{\alpha }_{jcont}^{-1}{)}^{-1}$; ${A}_{ij}$ - contact area of mating element surfaces, ${T}_{ienv}$ – ambient temperature (air).

Then a system of equations can be written for the spindle unit (part, unit) in matrix form:

$$\left[\mathrm{C}\right]\cdot \frac{d\left[\mathrm{T}\right]}{dt}+\left[\mathrm{K}\right]\cdot \left[\mathrm{T}\right]=\left[\dot{\mathrm{Q}}\right];$$

(8)

where $\left[T\right]$ is the matrix column of average element temperatures,

$$\left[\mathrm{C}\right]=\left(\begin{array}{ccc}{c}_{11}\cdot {m}_{11}& \dots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & {c}_{\mathrm{nn}}\cdot {m}_{\mathrm{nn}}\end{array}\right);$$

(9)

$$\left[\dot{\mathrm{Q}}\right]=\left(\begin{array}{ccc}{\sum }_{i=1}^{n}{Q}_{1all}+{\alpha }_{1}\cdot {A}_{1}\cdot {T}_{1env}& \dots & 0\\ \vdots & \ddots & \vdots \\ {\sum }_{i=1}^{n}{Q}_{\mathrm{n}all}+{\alpha }_{\mathrm{n}}\cdot {A}_{\mathrm{n}}\cdot {T}_{\mathrm{n}env}& \cdots & 0\end{array}\right);$$

(10)

$$\left[\mathrm{K}\right]=\left(\begin{array}{ccc}{\alpha }_{1}\cdot {A}_{1}+{\sum }_{i=2,i\ne j}^{n}{\alpha }_{1jcont}\cdot {A}_{1j}& \dots & {\alpha }_{1\mathrm{n}cont}\cdot {A}_{1\mathrm{n}}\\ \vdots & \ddots & \vdots \\ {\alpha }_{\mathrm{n}1}\cdot {A}_{\mathrm{n}1}& \cdots & {\alpha }_{\mathrm{n}}\cdot {A}_{\mathrm{n}}+{\sum }_{i=2,i\ne j}^{n}{\alpha }_{\mathrm{n}jcont}\cdot {A}_{\mathrm{n}j}\end{array}\right);$$

(11)

Solving the obtained system (8) or (7), we obtain the average temperatures of heat-active elements of the spindle unit (unit, part). Using the calculated average temperatures of thermoactive elements, we also obtain the distribution of heat flows in the thermal structure. A thermo-energetic model can substitute the installed temperature sensors by virtual ones.

The thermoelastic model subsystem is implemented based on the representation of the thermoelastic structure of the motor-spindle (by analogy with a spindle unit [13]) as an interconnected set of homogeneous thermoactive elements that form and determine their spatial and temporal relative position. Over time and due to thermal influence, the thermoactive elements change their thermoelastic properties, and the relations between the two thermoactive elements of the system are also subject to the conditions of reciprocity and jointness of linear and angular motions, that is, each thermoactive element performs only those displacements that are allowed by their mutual link. In order to construct the thermoelastic structure of a motor-spindle, it is necessary to define in more detail the types of typical quasi-thermostable links, interacting thermoactive elements and thermal behavior functions. As the simplest thermoactive element, it is reasonable to choose [5] a rod of arbitrary cross-section, the movement of the ends of which is limited by the links imposed on them. The connection of the thermoactive elements by quasi-thermally stable links can be serial, parallel, or a combination thereof. If a thermoactive element transmits a thermoelastic action through an intermediate link, then such a thermoelastic structure consists of three links and functions of thermal behavior. Then a thermoelastic model of one thermoactive element with two quasi-thermostable links will describe its thermal behavior, the general solution of which will be defined by an expression of the following form:

$${\delta }_{fr}={\int }_{0}^{x}\beta \cdot {T}_{1}(x)\cdot dx+{C}_{1}\cdot x+{C}_{2};0<x<{L}_{fr};$$

(12)

$${\delta }_{L}={\int }_{\mathrm{L}-{L}_{fr}}^{x}\beta \cdot {T}_{2}(x)\cdot dx+{C}_{3}\cdot x+{C}_{4};L-{L}_{fr}<x<L;$$

(13)

where ${C}_{1}$, ${C}_{2}$, ${C}_{3}$, ${C}_{4}$ are determined from the conditions of reciprocity ${\delta }_{T}-{\delta }_{P}={\delta }_{0}^{*}+{\delta }_{fr}^{*}$ and jointness of linear and angular motions in the places of quasi-thermostable links of the motor-spindle elements, ${\delta }_{0}^{*}$, ${\delta }_{fr}^{*}$ - displacements in the links, $L$ – length of the motor-spindle shaft, ${L}_{fr}$ – length between the front and rear supports, $(L-{L}_{fr})$ – length of the free part of the spindle shaft, ${T}_{1}(x)$, ${T}_{2}(x)$ – temperature distribution between the supports and the free part of the spindle.

A TCP is located at the tool tip. If a tool is not clamped, the spindle front is the target. Then the axial displacement of the front end of the spindle will be defined as the vector sum of the elements of the thermoelastic structure of the motor-spindle.

2.2 Function Block Method

Analytical equation method for a shaft temperature over length in (14) uses closed-form solutions and offers a clear view into how variables and interactions between variables affect the result.

$$T(x)=\frac{{Q}_{1}}{A\cdot \lambda \cdot m}\cdot \frac{\mathit{cos}h[m\cdot (l-x)]}{\mathit{sin}h(m\cdot l)}+\frac{{Q}_{2}}{A\cdot \lambda \cdot m}\cdot \frac{\mathit{cos}h(m\cdot x)]}{\mathit{sin}h(m\cdot l)};$$

(14)

A function block (see Fig. 3) graphically defines a physical function with input-output variables and parameter sets. The function block method uses lumped thermal masses linked to virtual temperature sensors by conductive, convective blocks. The function block model in Fig. 3 [14] applies the shaft model, parameters and equations of [6]. The friction heat generation was set to ${Q}_{1}=15\,\text{W}$ for the rear bearing A and ${Q}_{2}=45\,\text{W}$ for the front bearing B, spaced by the shaft length 0,5 m. The temperature sensor T represents one of 11 virtual sensors, spaced by 1/10 over the shaft length.

Fig. 3.

Function block method for a spindle shaft (1-Rear bearing A, 2-Environment temperature, 3-Front bearing B, T-Temperature sensor)

Figure 4 shows a temperature comparison between the analytical equation T(x) (14) and function block simulations Tsim of Fig. 3 aiming at a thermostable state.

Fig. 4.

Shaft temperature over length (T(x)-analytical Eq. 5; T_sim-Function block)

Analytical equation method has a continuous high accuracy, when the function block method is using Fourier’s law for linear approximations by 5 sections in Fig. 4. The application of one of both methods depends on the spindle design and thermo-physical boundary conditions.

A thermal model of the spindle unit growth bases on linked physical phenomena of thermoenergetics and thermoelasticity, see Fig. 5. Parametrized function blocks reflect the motor spindle design and working conditions. Because of the model complexity, this paper describes a general structure following Fig. 3 and Fig. 5. Parametrized function blocks of the motor spindle unit SPL2782.1 were approved by MATLAB/Simulink, SimulationX and OpenModelica projects. The number of Simulink elements is 526, and 128 parameters define the element properties.

Fig. 5.

Thermoenergetic and thermoelastic models

Power balance model (see Fig. 5): Process loads, motor power consumption $P$, spindle speed $n$, environmental and coolant temperatures are the variable input values. Motor spindle drive units provide digital values of the power consumption, speed.

Analytical equations calculate the motor heat power and bearing heat power from these variables and electric motor parameters. This deterministic model is parametrized for a special type out of a wide range of spindle modifications. Elements for asynchronous and permanent magnet synchronous motors are included. Motor heat power is available by analytical equations or from the SIEMENS calculation tool SIMOTICS M-1FE1 “Total power loss” [15]. Friction heat of different spindle bearing types and configurations is calculated under pre- and process loads. Friction and power loss calculation is available by analytical equations from the SKF product catalogue rolling bearings or an SKF application tool [16]. The mechanical spindle power minus heat power at the output in relation to the electrical power consumption at the input defines the actual energy efficiency of the motor spindle.

The thermoenergetic model (see Fig. 5) applies a lumped mass method, consisting of each five thermal masses along the shaft and housing axis. Heat flows between thermal masses are linked by conductive, convective elements. Variable environmental and water-cooling temperatures are reference temperatures for heat flows. Therefore, thermophysical and thermomechanical parameters support the parametrized model elements. Following Fig. 1, the motor spindle CAD geometry provides geometrical parameters for the rotor, shaft, bearing shields, and stator housing. Referring to [6] the obligatory number of temperature elements was found at 11. Respectively 11 virtual temperature sensors elements monitor the shaft and housing temperature over the spindle length, extended by 2 bearing temperatures. A thermo-energetic model can substitute the installed temperature sensors. Two alternative methods are included: whether the thermo-energetic model calculates virtual housing, bearing, shaft temperatures, or real sensors measure them.

The thermoelastic model (see Fig. 5) also bases on lumped masses linked by elastic Hook (spring) and Newton (damper) elements. The thermoelastic parameters define mechanical stiffness, damping and thermoelastic elongation values. The model output value generates the axial spindle growth at the shaft front end relative to the housing fixation below the front bearing. Simulated thermo-energetic temperatures and thermo-elastic displacements were verified by installed sensor values, in order to monitor failure states of the spindle unit and/or assess the simulation. The simulation hardware was an Intel Core i5-7200U processor with 16 MB RAM and SSD under OS Windows 10. The input variables and output axial spindle growth of the simulation model build digital time series for monitoring and reporting issues. A typical simulation in variable step mode continues 12 s for a real 16 h spindle test cycle.

3 Verification of the Simulation by Test Bench Results

The simulated axial spindle growth bases on Fig. 5 and was verified by test results of motor spindle units. Main geometrical dimensions of the motor spindle unit base on CAD data in Fig. 6: shaft length 0.5295 m with HSK80C; tool dummy length 0.2 m; together with a mass of 36.7 kg, additional rotor 1FE1084-4WN31 with a mass of 6.2 kg. The dynamic load torque ${M}_{L}$ is known from the motor torque $M$ following (15)

$$M-{M}_{L}=J \cdot \frac{d\omega }{dt},$$

(15)

where $J$ is the moment of inertia, $\omega$ – the angular velocity (see Fig. 6). A motor spindle test at different speeds is required during 16 h under no load conditions for a bearing lubrication run and a quality assessment and actual spindle power [kW], see Fig. 7.

Fig. 6.

Shaft of the motor spindle SPL2782.1

Fig. 7.

Speed and actual spindle power during test cycle

Table 1. Internal sensor types

Several temperature sensors (Table 1) were installed inside the spindle unit for data sampling. Installed sensors can monitor temperatures causing a thermal spindle growth. PT1000 with a resistance of 1000 Ω measures the stator temperature, caused by heat power loss (16)

$${P}_{cu}={I}^{2}R\left(T\right);$$

(16)

where the resistance $R\left(T\right)$ depends on the winding temperature in (17) and thermal elongation coefficient of copper is ${\beta }_{20^\circ {\rm C}}=0.00393 \frac{1}{K}$ [17]

$$R\left(T\right)={R}_{20^\circ {\rm C}}\left(1+{\beta }_{20^\circ {\rm C}}\left(T-20\,^\circ {\rm C}\right)\right).$$

(17)

A PTC-thermistor compares the winding temperature to a temperature limit relating to an electrical overload, before a switch-off by the power unit occurs. PT100 measures the bearing ring temperature from friction heat, caused by lubrication or bearing overloads. Magnetizing loss in the stator ${Q}_{h}$ is determined by hysteresis coefficient ${k}_{h}$, frequency $f$, and magnetic induction $\Psi$ (18):

$${Q}_{h}={k}_{h}\cdot f\cdot {\Psi }^{2};$$

(18)

and eddy current loss ${Q}_{e}$ [14] by eddy current coefficient ${k}_{e}$, frequency $f$, and magnetic induction $\Psi$ (19):

$${Q}_{e}={k}_{e}\cdot {f}^{2}\cdot {\Psi }^{2}.$$

(19)

Methods for the measurement of thermal spindle growth $\varDelta L$ base on thermal elongation coefficient $\beta$, temperature difference $\varDelta T$, and length $L$ (20):

$$ \varDelta L = \beta \cdot \varDelta T \cdot L; $$

(20)

Table 2. Additional sensors

Additional measurements are necessary for temperatures of the water coolant, air-oil lubricant flow, rotor temperature and for the shaft displacement. While the stator sensors are measured by the drive control, the water coolant, lubrication flow and shaft position ask for a programmable logic controller function. A rotor temperature sensor at a rotating shaft with spindle speed is commonly not available. A digital eddy current measuring system for tool spindles type eddyNCDT SGS 4701 tests a spindle front position against the bearing shield.

Figure 8 presents the test scheme of the motor spindle SPL2782.1, including temperature T and position P sensors. Both front and rear bearings have a grease lubrication by the manufacturer. The motor spindle housing at the front end has a labyrinth ring with compressed air for sealing and bearing protection.

Fig. 8.

Test scheme of the motor spindle SPL2782.1

Figure 9 shows a photo of the motor spindle 2782.1 with tool dummy at test bench. The SPL test bench is equipped with Sinamics® spindle power unit, PLC control of the water cooler, compressed air supply and PC measurement systems. Multichannel test data were measured in time periods of milliseconds and stored by the PC as raw data files. Without any thermostable climate in the test bench room, the environmental temperature changes. Following the test cycle in Fig. 7, a spindle unit is heating up to thermostable states and cooling down at the end.

Fig. 9.

Motor spindle 2782.1 with tool dummy at test bench

Fig. 10.

Test of the water coolant T_w and environment temperatures T_env

Measured water coolant T_w and environment temperatures T_env are shown in Fig. 10 and build variables for the function block simulation.

Figure 11 presents T1 simulation and test temperatures of the front bearing pairs (see Fig. 8). The peak values are 32.1 ℃ for T1 temperature simulation drawn up as dash line, and 32.4 ℃ for the T1 Test temperature as continuous lines at sensor T1.

Fig. 11.

T1 Simulation and T1 Test temperature of the front bearings

Figure 12 presents P1 simulation (dashed line) and P1 test results of the thermal spindle growth (see Fig. 8; continuous line). The peak values are 8 µm for simulation, and 6.5 µm for the measurement.

Fig. 12.

P1 simulation and P1 test of the thermal spindle growth

Fig. 13.

P1 simulation ±1.5 µm tolerance and P1 test for thermal spindle growth assessment

Figure 13 bases on Fig. 11 and comprises a ±1.5 µm tolerance range to P1 simulation. Following Fig. 13, this digital assessment of a thermal spindle growth bases on the method of a function block simulation and a quality assessment of P1 test compared to P1 simulated values. Deviations between virtual and real sensor values between the tolerance limits indicate a good quality assessment, see Fig. 2. Otherwise, deviations outside the range are quality issues and may be assigned following the thermoenergetic and thermoelastic model in Fig. 5.

Another simulation run of the motor spindle 2782.1 was completed for spindle tests under nominal load power 38 kW/4300 rpm with 5146 W motor heat power losses and shaft temperature 41.6 K, housing temperature 46.1 K above the environmental temperature. Therefore, the axial spindle growth increases to 79.7 µm.

In addition, SPL checked the front-end position of the shaft of another 2176.0 spindle unit relative to the bearing shield with an internal eddyNCDT SGS 4701 digital eddy current measurement system. Figure 14 shows this internal position measurement “P1 intern” in the range (0.265:0.275) mm compared to an external position sensor “SPL extern” in the inverse range (0.275:0.265) mm. The internal readings do monitor a thermal spindle growth and dynamic shaft position relative to the bearings, visible as axial vibration amplitudes ±1 µm in the time domain (26:37) min, caused by switching on a speed of 1000 rpm. However, the thermal growth sensor produces stable submicron values at a speed of 8000 rpm for up to 3 h. Therefore, the internal sensor measures the thermal spindle growth including dynamic errors for digital compensation.

Fig. 14.

Test of the thermal spindle growth with internal and external position sensor

Methods for predicting the thermal growth of motor spindle units based on virtual or real sensors, function block models and parameters were verified by test bench experiments. The function block models with parameter sets were verified for different SPL spindle unit types with asynchronous and synchronous motors and different bearing types. Small temperature deviations of 1 K and position deviations of ±1.5 µm are acceptable for a wide range of precision spindle units at no load and under process load, even when the axial spindle growth increases to 79.7 µm under nominal load. A simulation-based reduction of the thermal spindle growth under nominal load from 79.7 µm to 1.5 µm is equivalent to 1.9%.

Digital prediction and numerical compensation of thermal growth in motor spindle units under process loads and environmental influences becomes a reality in a digital system such as the digital twin through a predictive model or through internal temperature and position sensors.

4 Thermal Stiffness Strategy

Thermal stiffness [W/µm] is a key accuracy indicator, characterized by the power losses of the spindle, which result in thermal growth [5]. An innovative quality target is set by a minimal thermal growth or minimal power losses per thermal stiffness. Therefore, this chapter investigates methods for a maximal thermal stiffness and the impact potential from a reduced thermal growth by those methods.

Thermal spindle performance indicators for SPL 2782.1 spindle unit were calculated based on the methodology in [5] and simulation results under nominal load in Sect. 3:

• heat stiffness ${K}_{\alpha j}=0.00019\,\mathrm{K}/\mathrm{W}$,

• temperature stiffness ${K}_{Tj}=0.52\,\mathrm{ K}/\upmu\mathrm{m}$,

• thermal stiffness ${K}_{Qj}=64.53\,\mathrm{W}/\upmu\mathrm{m}$.

The definition of heat stiffness ${K}_{\alpha j}$ is given by Eq. (21) [5]:

$${K}_{\alpha j}=\frac{\left[{\theta }_{Q}+\left(1+\frac{{\psi }_{eq\alpha }}{{\psi }_{jeq}}\right)\right]}{\left[{\psi }_{\alpha j}+{\psi }_{eq\alpha }\cdot \left(1+\frac{{\psi }_{\alpha j}}{{\psi }_{jeq}}\right)\right]};$$

(21)

where ${\theta }_{Q}=\frac{{Q}_{eq}}{{Q}_{j}}$ - the ratio of the machine’s internal heat sources to the source of the first heat-active element; $\psi _{{eq\alpha }} = \sum\nolimits_{\begin{subarray}{l} k = 1 \\ k \ne j \end{subarray} }^{n} {\psi _{{ak}} }$ - thermal convection at the border between the body of the j - component and liquid (gas), [W/K]; $\psi _{{jeq}} = \sum\nolimits_{\begin{subarray}{l} k = 1 \\ k \ne j \end{subarray} }^{n} {\psi _{{jk}} }$ - thermal conductance at the borders of the j and k - component, [W/K]; ${\psi }_{aj}$ - thermal convection at the border between the body of the k - component and liquid (gas), [W/K].

The ability of a spindle unit to resist its heating under the action of internal and external sources is characterized and can be determined by the heat stiffness ${K}_{\alpha j}$ [K/W] as a special case of the heat stiffness of a machine tool, when the machine tool consists of one element – the spindle unit. The maximum and average spindle temperature can be determined as the products of this coefficient on the input power (heat loss in the bearings, stator, etc.). For this reason, there are two fundamentally different ways of ensuring the required or minimum heating of the spindle in both design and operation by controlling the heating. In both cases, a tool is needed to effectively evaluate these solutions. In particular, as tests have shown, the proposed model and its hardware-software solution (see Fig. 3) not only make it possible to determine, but according to the proposed method, it also makes it possible to evaluate solution options under various changing design and thermophysical parameters. A similar method also applies to the evaluation of axial thermoelastic deformations of the spindle, but in this case the inverse thermal stiffness ${1/K}_{Qj}$ [μm/W] of the spindle will be applied. Therefore, when designing a spindle, various design options and their thermal-physical characteristics can be considered, including heat transfer conditions both between the parts and the external environment.

For spindles that have a high coefficient of heat stiffness ${K}_{\alpha j}$, it is necessary to carry out measures aimed at reducing their average and maximum heating temperature. Assessing the effectiveness of these measures allows to choose the most rational way to reduce the spindle temperature. Evaluating the effectiveness of these measures allows to choose the most rational way to reduce the spindle temperature. It was shown that the coefficients of heat and thermal stiffness are universal thermal characteristics of both the machine and the spindle. Therefore, the effectiveness of measures can be assessed by changing the value of heat stiffness coefficients before and after the measure to increase it. To evaluate this, we introduce a heat stiffness ratio ${\lambda }_{tr}$ for the spindle (22):

$${\lambda }_{tr}=\frac{{K}_{\alpha 1}}{{K}_{\alpha 2}};$$

(22)

where ${K}_{\alpha 1}$, ${K}_{\alpha 2}$, - heat stiffness before and after the measure.

This heat stiffness ratio ${\lambda }_{tr}$ characterizes the relative increase of the spindle’s heat stiffness after the measure. Let’s estimate the limits of change in the coefficient of increase of heat stiffness of the spindle ${\lambda }_{tr}$. To do this, we need to establish a functional relationship between the entered coefficient and the value of the spindle temperature decrease as a result of the measure with the heat stiffness ratio ${\lambda }_{tr}$.

Let’s set the required value of the spindle temperature drop $\Delta T$ (23):

$$\Delta T={T}_{1}-{T}_{2};$$

(23)

where ${T}_{1}$, ${T}_{2}$, is the average spindle temperature respectively before and after the event.

Considering expressions (22), (23), and taking the temperature ${T}_{1}$ out of brackets, we obtain $\Delta T$ (24):

$$\Delta T={T}_{1}\left(1-\frac{{T}_{2}}{{T}_{1}}\right)={T}_{1}\left(1-\frac{1}{{\lambda }_{tr}}\right)={T}_{1}\left(\frac{{\lambda }_{tr}-1}{{\lambda }_{tr}}\right)={T}_{1}\left(\frac{{K}_{\alpha 1}-{K}_{\alpha 2}}{{K}_{\alpha 1}}\right);$$

(24)

The graph of dependence $\Delta T=f\left({\lambda }_{tr}\right)$ is shown in Fig. 15. The curve of the spindle temperature decrease has an asymptotic character, and the decreasing rate of the spindle temperature reduces with the increase of the heat stiffness ratio ${\lambda }_{tr}$.

Fig. 15.

Spindle temperature reduction $\Delta T$ as a function of the heat stiffness ratio ${\lambda }_{tr}$

This must be taken into account, because change is always associated with additional costs, and the effect of temperature reduction is not always proportional or effective to the measures taken. Therefore, for practical purposes, the following limits of change can be recommended $\mathrm{1,1}<{\lambda }_{tr}\le 5\dots 6$.

The spindle heating temperature is mostly influenced by the size and shape of the spindle structure parts and heat transfer conditions and thermophysical parameters.

The following strategies for a reduction of the thermal spindle growth can be distinguished by a heat stiffness ${K}_{\alpha 1}$ improvement (24):

• 1. changing the size and shape of the spindle assembly design parts, for example, increasing the area of the heat dissipating surface;

• 2. Change of spindle material thermal-physical properties (for instance, application of carbon fiber composite as in the paper [18] with thermal conductivity coefficient equal to 1,1 W/m²K – transversal, 186 W/m²K – longitudinal elements, a heat capacity coefficient of 850 J/kg K), and conditions of heat emission and heat transfer;

• 3. change in the structural size and shape of the spindle assembly design parts and thermophysical parameters to improve the conditions of heat transfer and heat dissipation.

In all of the following conclusions, the heat stiffness of the machine tool will be expressed in a simplified form with respect to the spindle as one of the heat-active elements of the thermal system.

The first direction, as an illustration of the method and principle, includes measures that improve the thermal characteristics of the spindle by introducing additional heat dissipating surfaces. In this case, let us give a notation of the heat stiffness ratio ${\lambda }_{tr}$ aiming at a heat stiffness improvement of the spindle in a rather simplified form will be equal to (25):

$${\lambda }_{rt}^{I}=\frac{{\sum }_{i=1}^{n}{\alpha }_{i}\cdot {A}_{i}}{{\sum }_{i=1}^{n}{\alpha }_{i}\cdot {A}_{i}+{\alpha }_{M1}\cdot {A}_{M1}};$$

(25)

where ${\alpha }_{i}$ is the coefficient of heat transfer on the i surface of a spindle; ${A}_{i}$ - area of the i surface; ${\alpha }_{M}$ - heat transfer coefficient on the additional surface; ${A}_{M}$ - area of the additional surface.

The denominator of expression (25) characterizes the thermal stiffness of the spindle after the measure ${K}_{\alpha 2}$, and the numerator is equal to the initial value of the spindle heat stiffness ${K}_{\alpha 1}$. In the particular case where the heat transfer coefficients are the same and equal to α, expression (25) will take the form (26):

$${\lambda }_{rt}^{I}=1+\frac{{A}_{M1}}{{\sum }_{i=1}^{n}{A}_{i}};$$

(26)

It follows from expression (26) that the additional surface area must be larger or at least commensurate with the heat dissipating surface area of the spindle. Otherwise, the value ${\lambda }_{tr}$ will be small and the efficiency of the measure is negligible. So, for example, with an area ratio of 1:1 the heat stiffness ratio ${\lambda }_{tr}$ will be 2 and hence the heating temperature reduction will be twofold.

The second direction includes measures that increase the thermal characteristics of the spindle by improving the heat dissipation conditions on the spindle surface while its structural shape and dimensions remain unchanged. In this case, the spindle’s ${\lambda }_{rt}^{II}$ will be equal (27):

$${\lambda }_{rt}^{II}=\frac{{\sum }_{i=1}^{n-k}{\alpha }_{i}\cdot {A}_{i}+{\sum }_{j=1}^{k}{\alpha }_{M2}\cdot {A}_{j}}{{\sum }_{i=1}^{n}{\alpha }_{i}\cdot {A}_{i}}=1+\frac{{\sum }_{j=1}^{k}\left({\alpha }_{M2}-{\alpha }_{j}\right)\cdot {A}_{j}}{{\sum }_{i=1}^{n}{\alpha }_{i}\cdot {A}_{i}};$$

(27)

where $k$ is the number of surfaces on which the heat transfer conditions are improved; ${\alpha }_{M2}$ - is the heat transfer coefficient after the heat transfer conditions are improved.

If the heat transfer coefficients are equal to α on the surfaces where the heat exchange conditions do not change, then

$${\lambda }_{rt}^{II}=1+\left(\frac{{\alpha }_{M2}}{\alpha }-1\right)\frac{{\sum }_{j=1}^{k}{F}_{j}}{{\sum }_{i=1}^{n}{F}_{i}};$$

(28)

where α is the heat transfer coefficient on those surfaces where the heat exchange conditions have not changed. It follows from expressions (27) and (28) that the effectiveness of measures of the second group depends on the surface area, on which the heat transfer conditions are improved, and on the ratio of values of the heat transfer coefficient on this surface before and after the measures.

The third direction includes measures that improve the thermal characteristics of the spindle by introducing additional heat transfer surfaces and increasing the heat transfer coefficient. In this case, ${\lambda }_{rt}^{III}$ of the spindle is determined by the following dependence:

$${\lambda }_{rt}^{III}=1+\frac{{\alpha }_{M1}\cdot {A}_{M1}}{{\sum }_{i=1}^{n}{\alpha }_{i}\cdot {A}_{i}}+\frac{{\sum }_{j=1}^{k}\left({\alpha }_{M2}-{\alpha }_{j}\right)\cdot {A}_{j}}{{\sum }_{i=1}^{n}{\alpha }_{i}\cdot {A}_{i}}={\lambda }_{rt}^{I}+{\lambda }_{rt}^{II}-1;$$

(29)

If the heat transfer coefficients on all spindle surfaces are equal (except those where the heat transfer coefficient is increased), then expression (28) will be written in the form:

$${\lambda }_{rt}^{III}=1+\frac{{A}_{M1}}{{\sum }_{i=1}^{n}{A}_{i}}+\left(\frac{{\alpha }_{M2}}{\alpha }-1\right)\cdot \frac{{\sum }_{j=1}^{k}{A}_{j}}{{\sum }_{i=1}^{n}{A}_{i}};$$

(30)

Comparison of expressions (29) and (30) allows us to draw an obvious conclusion that the measures of the third group will be much more effective, if additional surfaces improve the conditions of heat transfer.

An evaluation of influence of geometrical parameters is given for a spindle unit (d - diameter, $L$ - length, ${q}_{1}$ - front heat flow density, ${q}_{2}$ - rear heat flow density, $A$ - square area), and shown in Fig. 16, assuming invariable thermophysical parameters.

Fig. 16.

Influence of geometrical parameters on the heat stiffness ${K}_{\alpha j}$ of the spindle (where the abscise range A = [0 ÷ 2·10^–2] m²; L = [0 ÷ 1] m; a = [0 ÷ 60] W/m²·K; q = [0 ÷ 120] W)

Similarly, based on the complete description of the components for the values of ${K}_{\alpha 1}$, indicators are formed both in their full form by (21), which take into account all structural and thermophysical components, and in a simplified form (neglecting some parameters due to their low influence on the temperature change), as given for all three strategies for a reduction of the spindle heating temperature. To estimate the influence of the spindle thermal mode on its thermoelastic displacements, in particular axial displacements, the expression for ${\lambda }_{tr}$ will have the form (31):

$${\lambda }_{tr}=\frac{{K}_{\mathrm{Q}2}}{{K}_{\mathrm{Q}1}};$$

(31)

and the number of measures to manage the thermal deformation of the spindle will be supplemented (with respect to temperature reduction only) by redistributing the effects on the thermoelastic system of the spindle.

Considering the above, let us also introduce the reduction coefficient of excess spindle temperature $\upxi$ in (32)

$$\upxi =\left(\frac{{\lambda }_{tr}-1}{{\lambda }_{tr}}\right)100\mathrm{\%};$$

(32)

which shows by how many percent the spindle temperature will be reduced as a result of the measure. In addition, an inverse problem can also be solved, in which the value of $\xi$ is given and the value of ${\lambda }_{tr}$, is determined, and the effective and most rational option to improve the thermal resistance of the spindle is found.

5 Summary and Outlook

This paper presents and verifies digital methods for modeling and simulating a thermal growth of motor spindle units for digital prediction and numerical compensation of a thermal growth of motor spindle units. The digital model requires new virtual test and measurement methods, therefore models, parameters and variables are defined by data structures and simulated for test cycles. The State of the Art in thermal caused geometrical and kinematic errors is limited to quasi-static states and under no-load conditions. The realized model simulation and experiments conclude in a verification of thermal growth and innovative prediction under load. The best location of temperature and position sensors is not sufficient for a correct prediction of thermal spindle growth. So far, the rotating shaft temperature was not measured, but this paper did simulate multiple virtual sensors at the shaft and housing.

A system analysis for transfer functions requires thermal sources, sinks and knowledge of thermoelasticity in components. Correlations between multiple input and output values depend on system and environmental conditions. The Analytical Equation Method investigates deterministic cause-effect-relationships. Parametrized function block models of the motor spindle unit types SPL2782.1 were approved by MATLAB/Simulink, SimulationX and OpenModelica projects and verified by similar SPL test bench measurements. The parametrized Function Block Models provides a temperature and thermal spindle growth prediction for different motor spindle types. The model flexibility is given by modular blocks, as for asynchronous or synchronous motors and water cooling.

Compared to longtime FEM simulations, the Function Block simulation allows real-time applications and closed-loop control for a CNC compensation. The short simulation time enables real-time digital compensation applications in order to reduce thermal growth of spindle units and machine tools under load. Simulated bearing temperature values are comparable to integrated sensor values and predict thermal properties (temperature, position growth) of the spindle unit design SPL2782.1 for different load conditions. Temperature deviations of 1 K and position deviations of ±1.5 µm are acceptable for a wide range of precision spindle units under no load and process loads, reducing the thermal growth of spindle units. Simulation extends the test stand measurements for spindle tests under load power without a second load motor and power unit. Simulated thermal spindle growth is comparable to internal and external position sensor values, even when external sensors are not applicable for machining processes.

The spindle thermal stiffness strategies present three methods for a heat stiffness improvement for further spindle design variants. Different methods for a digital reduction of a thermal spindle growth were developed and ask further work on a real-time controller in a digital twin for a TCP coordinate shift. Further investigations are needed on model parameter optimization and on digital targets for a quality assessment regarding the thermal growth of motor spindle units.