# Parameter Estimation of the Thermal Network Model of a Machine Tool Spindle by Self-made Bluetooth Temperature Sensor Module

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Thermal Characteristics of Spindle

_{l}), viscous shear (Q

_{v}), and spinning motion (Q

_{s}) [23].

_{l}is the applied-load-dependent term, M

_{v}is the viscous-shear-dependent term, and M

_{si}and M

_{so}are the spinning-motion-dependent terms, n denotes the rotational speed of bearing, n

_{si}and n

_{so}denotes the spin speeds of the inner and outer rolling element, respectively, and Z is the number of rolling elements.

_{l}) is obtained by Equation (2). In their research, the predicted thermally induced preload would reach steady over time and its magnitude grew with the rotational speed. The nonlinear relationship between thermally induced preload and rotational speed is like the power function.

_{t}is the thermally induced preload, F

_{n}is the contact force from external load, r

_{i}and r

_{b}is the radius of inner ring and rolling element, μ

_{l}is the friction coefficient due to load. The empirical Equation (3) explains the frictional torque that is caused by lubricant shear viscosity (M

_{v}) within the bearing.

_{0}is a factor depending on bearing type and lubrication, ν

_{L}is lubricant kinematic viscosity. For angular contact ball bearing, the gyroscopic moment of the rolling elements necessarily leads to boring motion that would cause the friction torque on the contact surface, and thus also introduce heat generation. The friction torque related to spinning motion (M

_{s}) is obtained by Equation (4) [25].

_{roll}is the contact load, a represent the major axes of the elliptical contact area, e is the elliptical integral of the second kind, and μ

_{s}is the friction coefficient.

^{5/3}) and the remaining term (ω), as described in Equation (5).

^{0.5}. In this case, we take the rotational speed as the leading term and express the thermal resistance of forced convection around moving surface as a function of ω

^{−0.5}.

_{g}, P) are obtained. Note that the geometry factors (Fg and P) depend on the inner and outer radius of annulus only; therefore, it can be viewed as a constant value.

_{i}and r

_{o}are the inner and outer radius. Namely, the coefficient of forced heat convection can be derived in Equation (11), which is proportional to ω

^{0.482}. Thus, it is appropriate to formulate the thermal resistance of forced convection near annulus as a function of ω

^{−0.482}in Equation (12).

_{s}) and ambient temperature (T

_{a}) particularly. It would be difficult to simulate the free convective phenomenon according to the temperature difference. Alternatively, the rotational speed has strong correlation to the temperature difference. We assume that the free convective coefficient can be expressed as the function in Equation (14). Hence, the free convective thermal resistance can be derived as two terms, a constant term and a speed-related term. Especially, all of the assumptions must be verified by experimental results, and it will be further discussed in Section 5.

_{s}) and ambient temperature (T

_{a}). Due to the extremely tiny value of Stefan-Boltzmann constant (σ) and the emissivity (ε), the influence of surface temperature variation would be limited. Consequently, the radiative thermal resistance can be considered as a constant value, which would be further verified by experiment.

#### 2.2. System Identification Technique

## 3. Parameterization Methodology for the Thermal Network Model of Spindle

#### 3.1. Parameterization Strategy

- (1)
- lumped element model assumption is valid, Bi << 0.1;
- (2)
- temperature distribution of spindle is axisymmetric;
- (3)
- heat generation and forced convective resistance are assumed according to the theoretical and empirical Equations (5), (9), and (12);
- (4)
- heat transfer through radiation and conduction are considered as a constant thermal resistance, including thermal contact resistance; and,
- (5)
- free convection coefficient is assumed as a function of ${h}_{free,c}+{h}_{free,s}{\omega}^{-0.5}$.

#### 3.2. Estimated Thermal Network Model in State-Space

_{f}

_{1}, Q

_{f}

_{2}. As a result, the heat balance equation with internal heat generation is expressed as

_{1}), midpoint of housing (T

_{2}), front bearing D (T

_{3}), midpoint of shaft (T

_{4}), respectively. The specific locations will be illustrated latterly in Section 4.2. TNM contains four types of components, thermal resistance, speed-related thermal resistance, thermal capacitance, and heat source. The authors classify the thermal behavior of spindle into two modes, operating mode and natural cooling mode. At operating mode, the spindle is operated at a constant rotation speed and the initial temperature is equal to the ambient temperature, illustrated in Figure 2a. At natural cooling mode, the spindle is stopped and releases the heat that is stored during operating mode, as illustrated in Figure 2b. The state-space representation of the TNM at operating mode is expressed as

_{av}

_{j}, r

_{24}) and the heat generation coefficient (q

_{f}

_{1}–q

_{f}

_{4}) are unknown parameters needed to be solved.

_{0}is 5000 rpm, thus the unit of the coefficients can be meaningful. At steady state, the thermal capacitances can be ignored due to the temperature derivative terms decay at steady state. Therefore, rewriting the heat balance equation as:

## 4. Experiment Setup

#### 4.1. Self-Made Bluetooth Temperature Sensor Module

#### 4.2. Experiment Setup

## 5. Results

_{1}) at steady state is used to simulate h

_{free}and h

_{rad}. Due to the temperature of rear bearing is the highest one among all of the other positions, the simulation results can indicate the significant influence. In Figure 7a, the approximate computation of R

_{rad}(the purple dashed line) from Equation (15) is almost a constant value. As for free convective heat transfer coefficient, the authors consider the spindle to be a homogeneous cylinder with the diameter (D) of 0.135 m and length (L) of 0.278 m, and its other material properties are based on 25 °C. Consequently, the approximately calculated R

_{free}(green dotted line) from Equation (13) can be curve-fitted with the formula $1/A\left({h}_{free,c}+{h}_{free,s}{\omega}^{-0.5}\right)$. For this reason, it is appropriate to adopt these assumptions. Figure 7b shows the predicted heat generations of rear and front bearings. One can estimate the dissipated heat through radiation (Q

_{rad.approx.}) and free convection (Q

_{free.approx.}) by means of the preceding approximated h

_{free}and h

_{rad}. Furthermore, Figure 7a shows the speed-dependent parameters with respect to the rotational speed. It indicates that the nonlinear behavior is more significant under lower rotational speed. For this reason, it is necessary to take the speed-dependent thermal behavior into consideration. In other words, due to the varying of the thermal parameters, the poles of the estimated TNM would slightly change with different operating speeds, which will be discussed latterly in Section 5.4.

#### 5.1. Steady State Self-Validation

#### 5.2. Transient State Self-Validation

_{4}) cannot be obtained, and the T

_{4}-est represents the estimated shaft temperature, as shown in Figure 9d.

#### 5.3. External Validation

#### 5.4. Model Order Reduction

_{f}

_{1}, Q

_{f}

_{2}) and the output temperature T

_{1}to T

_{4}at rotational speed of 6000 rpm. It indicates that the truncated model has equivalent performance to the original estimated TNM under low frequency (<10

^{−3}) condition, as shown in Figure 13.

#### 5.5. Short Circuit Time Constant

## 6. Conclusions

_{1}), midpoint of housing (T

_{2}), and front bearing (T

_{3}) attempt to grow along with the curve predicted by the estimated TNM. The TID-sys ensure that the machine tool spindle runs under normal condition. Once the significant temperature variation or abnormal temperature rise is detected, the operator can slow down the spindle or even stop for early protection. The full demonstration video is shown in supplementary materials Video S1.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. System Parameter Matrices

## References

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**Figure 2.**(

**a**) 4-node Thermal Network Model (TNM) at operating mode; (

**b**) 4-node TNM at natural cooling mode.

**Figure 3.**(

**a**) Circuit diagram of Bluetooth Temperature Sensor Module (BTSM); (

**b**) Printed Circuit Board (PCB) layout of BTSM.

**Figure 6.**(

**a**) Experimental result of 12 nodes on 6021 rpm; (

**b**) The transition region of reaching thermal equilibrium.

**Figure 7.**(

**a**) Simulated heat generation varying with rotational speeds; (

**b**) Estimated convective resistance varying with rotational speeds.

**Figure 8.**(

**a**) Self-validation between predicted and measured steady state temperature, T

_{1}to T

_{4}are the positions of the rear bearing A, midpoint of inner housing, front bearing D, and midpoint of shaft, respectively; (

**b**) Steady-state temperature validation.

**Figure 9.**Model self-validation at operating mode with various rotation speeds (

**a**) Rear bearing (T

_{1}) temperature validation (

**b**) Midpoint of housing (T

_{2}) temperature validation (

**c**) Front bearing (T

_{3}) temperature validation (

**d**) Midpoint of shaft (T

_{4}) temperature validation.

**Figure 10.**Model self-validation at natural cooling mode with various rotation speeds (

**a**) Rear bearing (T

_{1}) temperature validation (

**b**) Midpoint of housing (T

_{2}) temperature validation (

**c**) Front bearing (T

_{3}) temperature validation (

**d**) Midpoint of shaft (T

_{4}) temperature validation.

**Figure 11.**External validation between measured temperature and predicted temperature of estimated TNM under stepwise rotational speed (3001, 5018, 7028 rpm).

**Figure 12.**(

**a**) Hankel singular values; (

**b**) Short circuit time constant method with C2–C4 are short circuited; (

**c**) Pole location of estimated TNM when comparing with the Model Order Reduction (MOR) model, SCTC model and varying with rotational speed.

Sensor Element | Accuracy [°C] | Resolution [°C] | Measurement Range [°C] | Power [mW] | Module Size [mm^{3}] |
---|---|---|---|---|---|

RTD | ±(0.1 + 0.0029|ϑ|) | 0.00489 | −40~150 | 7 | Ø40× |

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

R_{12} [KW^{−1}] | 0.1263 | R_{a}_{1} [KW^{−1}] | 0.884 | R_{av}_{3} [KW^{−1}] | 1.0032 |

R_{14} [KW^{−1}] | 0.1537 | R_{a}_{2} [KW^{−1}] | 1.393 | q_{f}_{1} [W] | 37.162 |

R_{23} [KW^{−1}] | 0.5954 | R_{a}_{3} [KW^{−1}] | 3.071 | q_{f}_{2} [W] | 28.406 |

R_{24} [KW^{−1}] | 3.609 | R_{av}_{1} [KW^{−1}] | 1.498 | q_{f}_{3} [W] | 0.0043 |

R_{34} [KW^{−1}] | 0.48 | R_{av}_{2} [KW^{−1}] | 0.303 | q_{f}_{4} [W] | 0.0073 |

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

R_{12} [KW^{−1}] | 0.1263 | R′_{a}_{1} [KW^{−1}] | 18.765 | C_{1} [JK^{−1}] | 5375.8 |

R_{14} [KW^{−1}] | 0.1537 | R_{a}_{2} [KW^{−1}] | 1.393 | C_{2} [JK^{−1}] | 3545.8 |

R_{23} [KW^{−1}] | 0.5954 | R′_{a}_{3} [KW^{−1}] | 6.496 | C_{3} [JK^{−1}] | 10,931.7 |

R′_{24} [KW^{−1}] | 6.737 | R′_{a}_{4} [KW^{−1}] | 2.497 | C_{4} [JK^{−1}] | 625.4 |

R_{34} [KW^{−1}] | 0.48 |

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**MDPI and ACS Style**

Lo, Y.-C.; Hu, Y.-C.; Chang, P.-Z. Parameter Estimation of the Thermal Network Model of a Machine Tool Spindle by Self-made Bluetooth Temperature Sensor Module. *Sensors* **2018**, *18*, 656.
https://doi.org/10.3390/s18020656

**AMA Style**

Lo Y-C, Hu Y-C, Chang P-Z. Parameter Estimation of the Thermal Network Model of a Machine Tool Spindle by Self-made Bluetooth Temperature Sensor Module. *Sensors*. 2018; 18(2):656.
https://doi.org/10.3390/s18020656

**Chicago/Turabian Style**

Lo, Yuan-Chieh, Yuh-Chung Hu, and Pei-Zen Chang. 2018. "Parameter Estimation of the Thermal Network Model of a Machine Tool Spindle by Self-made Bluetooth Temperature Sensor Module" *Sensors* 18, no. 2: 656.
https://doi.org/10.3390/s18020656