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Abstract

The main transmission system, the key component for transferring energy in machine tools, is essential for studying the energy efficiency of machine tools, highlighting the need for manufacturers to measure the input power of the main transmission system to develop energy-saving technology. In response to such a need, this article proposes a new approach that can be applied to calculate the input power of the main transmission system fed by a converter. To develop this approach, the influences of the converter and frequency variations on motor loss are analyzed to build the idle and loaded loss motor models for manufacturers. Based on these two models, the relationships between the input power of the main transmission system, the motor, and its frequency are investigated to establish a calculation approach that is convenient because only the motor input power measurement is required. Finally, 20 group experiments are conducted to validate the proposed approach, and its applications are discussed. The applications demonstrate that this method provides a high level of accuracy and validity for calculating the input power of the main transmission system driven by a converter and could be widely used to study high-efficiency machine tools and monitor the energy efficiency of machine tools.

Keywords

Main transmission system energy efficiency machine tools sustainable manufacturing

Introduction

Improving energy efficiency in manufacturing systems is vital to achieving critical global targets in energy efficiency via low-carbon and low-emissions operations.^1,2 Many strategies have been specified through rules of conduct and legal directives to promote energy efficiency.³ Two examples are the regulatory frameworks European 2020 and European 2030, which aim to improve energy efficiency by 20% by 2020 and 27% by 2030, respectively.⁴ Machine tools, as the key manufacturing equipment of machining systems, are abundant⁵ and consume substantial amounts of energy with low efficiency during the manufacturing process.^6,7 Thus, research on the energy efficiency of machine tools is critical. For instance, the International Organization for Standardization (ISO) published the ISO 14955-1:2014⁸ international standard, which focuses on design methodologies for energy-efficient machine tools. Similarly, the European Commission has issued the Ecodesign⁹ directive, which seeks to develop green machine tools. Therefore, improving the energy efficiency of machine tools has been an important strategy for sustainable manufacturing.

The energy efficiency of machine tools has been extensively studied and reported to date. The efforts can be largely divided into the following three categories. (1) Energy consumption modeling and assessment: studies at this level are designed to know where to apply energy-saving technologies and how much energy reduction is needed. For example, Kara and Li¹⁰ established the specific energy consumption (SEC) model and found that the energy requirement of machining processes was inversely proportional to the material removal rate (MRR); Schudeleit et al.¹¹ described and compared four general energy efficiency evaluation methods and explicitly recommended the component benchmark method for machine tool design purposes. Therefore, both energy consumption models of machine tools and their components consumption models are important for assessing the energy efficiency of machine tools.¹² (2) Energy efficiency monitoring and management: in this category, energy efficiency on-line acquisition, energy efficiency prediction, and multi-objective optimization problems are discussed to provide essential support to the high-efficiency working of machine tools.¹³ Hu et al.¹⁴ developed an energy monitoring method and systems for machine tools fed by a variable speed drive (VSD). However, due to structural limitations, this method cannot be easily applied to certain machine tools on which a cutting–measuring instrument cannot be installed. Liu et al.¹⁵ developed an on-line monitoring method for the energy efficiency of machine tools using the component energy transmission model in the spindle system. Although this method cannot be applied on certain machine tools fed by a VSD, it provides an idea for the acquisition of energy efficiency. (3) Design high-efficiency machine tools: in this category, lightweight component and high-efficiency component selection are investigated.¹⁶ For instance, Kroll et al.¹⁷ investigated the effects of lightweight components on machine tools and summarized lightweight strategies and effects. Chen et al.¹⁸ investigated the structural design of machine tools reformed with artificial granite material. Meanwhile, the energy consumption modeling of components can provide technical support for engineers to select high-efficiency components in the machine tool design stage.¹⁶ Therefore, the energy consumption modeling and assessment, energy monitoring and management of components, and selection of high-efficiency components are essential to achieving energy-efficient machine tools.

As the key component for transferring energy in machine tools, the main transmission system (MTS) is essential for studying the tools’ energy efficiency. However, because the MTS and motor are integrated into machine tools during the assembly phase, it is difficult for machine tool users to measure the input power of the MTS, which is essential for calculating the energy efficiency of the motor, MTS, and machine tools. Many years ago, Liu and Xu¹⁹ proposed the Loss K_{load rate} Power (LKP) method to calculate the input power of the MTS fed by normal power. Based on LKP, Liu et al.²⁰ established a method to separate the power information of the spindle system during the machining process that can be widely applied to monitor and assess the energy consumption and efficiency of the spindle system and its components. However, with the application of electrical VSDs in the machine tool field, the LKP method^19,20 cannot be applied to calculate the input power of the MTS fed by a converter because the power loss of a motor driven by a converter is a function of the power frequency and not a constant, as considered in the LKP method. To solve this problem, Shi²¹ investigated the base frequency loss of the spindle motor and the harmonic loss caused by the converter in machine tools; however, they did not establish the relationship between the power loss of the MTS and the spindle motor. Based on Shi’s studies, Hu et al.²² studied the load loss characteristics of the MTS and motor to determine the relationship between the input power of the MTS and the motor by conducting numerous cutting experiments. It is not convenient for machine tool users because cutting-force-measuring instruments are required, and it is time-consuming to install sensors and perform experiments.

In conclusion, previous studies have addressed the problem of calculating the input power of a MTS driven by normal power but have not considered MTSs fed by a converter. Specifically, the following issues have not been elucidated: (1) the influence of the converter and frequency variations on spindle motor loss and (2) the relationships between the input power of the MTS, the input power of the motor, and frequency. This article aims to address those deficiencies to develop a new approach for calculating the input power of the MTS that will have wider applicability for MTSs fed by a converter.

Model for calculating the input power of the MTS

The spindle system, which is the power source for machining, has complex energy consumption and energy transmission characteristics, which are shown in Figure 1. In this figure, $U, i$ are the input voltage and current of the spindle system, respectively; $U_{m}, i_{m}$ are the input voltage and current of the motor, respectively; $n_{m}, T_{m}$ are the output speed and torque of the motor, respectively; $n_{o}, T_{o}$ are the output speed and torque of the spindle system, respectively; $U_{r}$ is the rated power of the motor; and $f_{r}$ is the rated frequency of the motor. CPS is the abbreviation for constant power stage, and CTS is the abbreviation for constant torque stage.

Figure 1.

(a) Schematic of the inputs and outputs of the spindle system, (b) Schematic of the inputs voltage of the spindle motor and (c) Schematic of power transmission in the spindle system.

As shown in Figure 1, because of the use of a VSD in motors,²³ the input voltage and current of a motor driven by a frequency converter is a function of the power frequency^24,25 (illustrated in equation (1a)), which is different if the motor is fed by normal power directly (detailed in equation (1b) and Figure 1(b))

i = I_{1} \sin (2 π f_{o} t + θ_{1})

(1a)

i_{c} = I_{1} \sin (2 π ft + θ_{1}) + \sum_{k = 2}^{\infty} I_{k} \sin (2 k π ft + θ_{k})

(1b)

where $I_{1}$ is the effective value of the fundamental wave current, $f_{o} = 50 or 60 Hz$ ; $θ_{1}$ is the initial phase of the fundamental wave current; $f$ is variable and $f_{\min} \leq f \leq f_{\max}$ , $θ_{k}$ is the initial phase of kth-order harmonic current.

Thus, because of frequency variations, the motor loss (such as idle loss) is variable and not a constant, as considered by Liu et al.¹⁵ Therefore, to calculate the input power of the MTS, a new method (as shown in Figure 2) with respect to the frequency variation caused by a converter is required.

Figure 2.

Flowchart of the proposed method for calculating the input power of an MTS.

Due to the structural limitations of machine tools, it is difficult for users in the workshop to measure the output speed and torque of the motor (which can be used to calculate the input power of the MTS by formula $P_{t} = n_{m} T_{m} / 9550$ ). Due to the measurement convenience of the input current and voltage of the motor, the input power of the MTS is typically calculated by equation (2)

P_{t} = P_{m} - P_{ml}

(2)

The power loss model of a motor driven by a converter is critical for calculating MTS input power.

Therefore, there are three stages of developing the new approach for calculating the input power of MTSs. First, the influence of the converter and frequency variations on the motor loss must be considered, and the quantitative relationship model between the input power of the MTS and motor loss must be established. Second, the quantitative relationship model between the input power of the MTS, the motor input power, and the frequency of the motor must be established (PPF model). Finally, the input power calculation for the MTS can be proposed based on the PPF model.

Toward machine tool users: modeling motor loss

The motor has complex power loss characteristics that are composed of copper loss, core loss, friction loss, windage loss, and stray loss.²³ Many previous studies have focused on motor loss. For example, Hildebrand and Roehrdanz²⁴ investigated a loss calculation model based on the structural parameters of a motor, such as the height and width of the stator tooth. Machine tool users only have motor nameplate information and do not have motor structural data. Therefore, the existing method is rarely applied to calculate the input power of the MTS in machine tools. From another perspective, the input power of the MTS is considerably larger than the motor loss; thus, there has been no requirement to develop a precise model for calculating motor loss in machine tools.

Therefore, an approximate estimation model for calculating the power loss of the motor fed by a converter is developed for machine tool users, as described below.

Based on the existing studies, motor loss can be divided into idle loss $P_{il}$ and variable loss $P_{vl}$ , as shown in equation (3), where $P_{vl}$ is caused by the motor load

P_{lm} = P_{il} + P_{vl}

(3)

A previous study²⁵ reviewed the difference in motor loss between a motor fed by a converter and normal power. According to that study, the loss caused by a converter is highly complex, but it is irrelevant to the motor load. Thus, it is regarded as a portion of the idle loss, and the idle loss and variable loss can be expressed as

P_{il} = {mI}_{0 s}^{2} R_{s} + m \sum_{k = 2}^{\infty} I_{k}^{2} R'_{rk} + {mI}_{0 s 1}^{2} R_{m} + P_{w} + P_{FW}

(4a)

P_{vl} = {mI}_{r}^{2} (R_{s} + {R'}_{r})

(4b)

where $I_{0 s}$ is the unloaded current of the stator, $I_{0 s 1}$ is the unloaded fundamental current of the stator, $I_{k}$ is the kth-order unloaded harmonic current, $I_{r}$ is the current of the rotor, $R_{s}$ is the stator resistance, $R_{m}$ is the equivalent resistance of iron, $R'_{rk}$ is the kth-order harmonic current related to the equivalent resistance of the rotor, $R'_{r}$ is the equivalent resistance of the rotor, $P_{w}$ is eddy current loss, and $P_{FW}$ is mechanical loss.

Then, the estimation model for idle loss and loaded loss are developed as detailed below.

Idle loss

The rotor current is relatively small and is omitted when the motor is unloaded; then, the fixed loss can be expressed as follows

P_{il} = {mI}_{0 s 1}^{2} (R_{s} + R_{m}) + m \sum_{k = 2}^{\infty} I_{k}^{2} (R_{s} + {R'}_{rk}) + P_{w} + P_{FW}

(5)

Because $R_{s} + R_{m} << X_{s} + X_{m}$ , where $X_{s}$ is the stator leakage resistance and $X_{m}$ is the excitation resistance, $I_{0 s 1}$ is typically calculated as

I_{0 s 1} = \frac{U_{m}}{\sqrt{{(R_{s} + R_{m})}^{2} + {(X_{s} + X_{m})}^{2}}} \approx \frac{U_{m}}{X_{s} + X_{m}}

(6)

$\sum_{k}^{\infty} I_{0 k}^{2}$ is typically calculated as

I_{h} = \sqrt{\sum_{k = 2}^{\infty} I_{k}^{2}} \approx \frac{M U_{m}}{X_{s} + X_{m}}

(7)

where M is a constant.²⁶

Then, the expression for fixed loss is

P_{il} = \frac{{mU}_{m}^{2}}{{(X_{s} + X_{m})}^{2}} (R_{s} + R_{m} + M^{2} (R_{s} + {R'}_{rk})) + P_{w} + P_{FW}

(8)

In this article, the friction and windage loss $P_{FW}$ is simplified to increase in proportion to the square of the frequency. The eddy current loss $P_{s}$ is divided into two parts: the first part, which is zero in the idle state, is proportional to the motor output power, and the second part, which is added to $P_{FW}$ (yielding $P'_{FW}$ ), is unrelated to the motor output power. Meanwhile, in the CPS, the stator voltage is constant and equal to the rated voltage of the motor. During the CTS, the stator voltage of the spindle motor is proportional to the frequency of the input current. Then, the following relationship can be obtained:

In the CPS

P_{il} (f) = P_{il} (f_{r}) γ^{- 1} + P'_{FW} (f_{r}) (γ - γ^{- 1})

(9)

where $γ = f^{2} / f_{r}^{2}$ and $P_{il} (f_{r})$ and $P'_{FW} (f_{r})$ are the power losses under the rated frequency $f_{r}$ .

In the CTS

P_{il} (f) = P_{il} (f_{r}) + P_{FW}^{'} (f_{r}) (γ - 1)

(10)

When the motor works on the minimum allowed frequency, the friction and windage loss and the eddy current loss are sufficiently small to be neglected. Therefore, the following relation is obtained: $P'_{FW} (f_{r}) \approx P_{il} (f_{r}) - P_{il} (f_{\min})$ , and the idle loss of the motor driven by a converter can be expressed as follows

P_{il} (f) = {\begin{matrix} P_{il} (f_{r}) \frac{f^{2}}{f_{r}^{2}} - P_{il} (f_{\min}) (\frac{f^{2}}{f_{r}^{2}} - \frac{f_{r}^{2}}{f^{2}}) \cdot f_{r} \leq f \leq f_{\max} \\ P_{il} (f_{r}) \frac{f^{2}}{f_{r}^{2}} + P_{il} (f_{\min}) (1 - \frac{f_{r}^{2}}{f^{2}}) \cdot f_{\min} \leq f \leq f_{r} \end{matrix}

(11)

According to equation (11), the idle loss of a motor in other frequencies can be determined by its idle loss under the minimum and rated frequency.

Loaded loss

When a motor is loaded, the rotor current $I_{r}$ increases and can be expressed as follows¹⁵

I_{r} = s_{r} \frac{U_{r}}{{R'}_{r}} \frac{U_{r}}{U_{m}} \frac{P_{t}}{P_{r}}

(12)

where $s_{r}$ is the rated slip of the motor and $P_{r}$ is the rated output power.

Combining equation (12) with equation (4) yields the following

P_{vl} (f) = {mI}_{r}^{2} (R_{s} + {R'}_{r}) = \frac{{ms}_{r}^{2} U_{r}^{2} (R_{s} + {R'}_{r})}{{({R'}_{r})}^{2}} \frac{U_{r}^{2}}{U_{m}^{2}} \frac{P_{t}^{2}}{P_{r}^{2}}

(13)

When the motor is operating under the rated condition, $P_{t} = P_{r}$ , $U_{m} = U_{r}$ , and the energy efficiency of the motor is $η_{r}$ . Then, the motor’s power loss can be obtained as

P_{ml} (f_{r}) = P_{il} (f_{r}) + \frac{{ms}_{r}^{2} U_{r}^{2} (R_{s} + {R'}_{r})}{{({R'}_{r})}^{2}} = P_{r} (\frac{1}{η_{r} - 1})

(14)

Then, the variable loss can be expressed as

P_{vl} (f) = [P_{r} (\frac{1}{η_{r}} - 1) - P_{il} (f_{r})] \frac{U_{r}^{2}}{U_{m}^{2}} \frac{P_{t}^{2}}{P_{r}^{2}}

(15)

Thus, in the CPS

P_{ml} (f) = P_{il} (f) + [P_{r} (\frac{1}{η_{r}} - 1) - P_{il} (f_{r})] \frac{P_{t}^{2}}{P_{r}^{2}}

(16)

and in the CTS

P_{ml} (f) = P_{il} (f) + [P_{r} (\frac{1}{η_{r}} - 1) - P_{il} (f_{r})] \frac{f^{2}}{f_{r}^{2}} \frac{P_{t}^{2}}{P_{r}^{2}}

(17)

Let $A = [P_{r} (1 / η_{r} - 1) - P_{il} (f_{r})] / P_{r}^{2}, B = P_{il} (f_{r})$ , and $C = P_{il} (f_{\min})$ ; then, the motor loss driven by a converter can be expressed as

P_{ml} (f) = {\begin{matrix} {AP}_{t}^{2} + B \frac{f^{2}}{f_{r}^{2}} - C (\frac{f^{2}}{f_{r}^{2}} - \frac{f_{r}^{2}}{f^{2}}), f_{r} \leq f \leq f_{\max} \\ A \frac{f^{2}}{f_{r}^{2}} P_{t}^{2} + B \frac{f^{2}}{f_{r}^{2}} + C (1 - \frac{f_{r}^{2}}{f^{2}}), f_{\min} \leq f \leq f_{r} \end{matrix}

(18)

According to equation (18), the motor loss driven by a converter can be determined by the input power of the MTS, the frequency, and the basic data $(A, B, C)$ of the motor.

Equation (18) can provide a new way to investigate the comparison between electrical and mechanical losses under various conditions that have also been studied in the literature.^27,28

Input power relationship between the MTS and motor

According to Figure 1, when the motor frequency is $f_{x}$ , the following relationship can be obtained

P_{ml} (f_{x}) = P_{m} - P_{t}

(19)

The quantitative relationships among the input power of the MTS $P_{t}$ , the input power of the motor $P_{m}$ , and the motor frequency $f$ can be obtained by combining equations (18) and (19)

When f_{r} \leq f \leq f_{m a x}, P_{t} = \frac{1}{2 A} [- 1 + \sqrt{1 + 4 A [P_{m} - B \frac{f^{2}}{f_{r}^{2}} + C (\frac{f^{2}}{f_{r}^{2}} - \frac{f_{r}^{2}}{f^{2}})]}]

(20)

When f_{m i n} \leq f \leq f_{r}, P_{t} = \frac{1}{2 A} \frac{f_{r}^{2}}{f^{2}} [- 1 + \sqrt{1 + 4 A \frac{f_{r}^{2}}{f^{2}} [P_{m} - B \frac{f^{2}}{f_{r}^{2}} - C (1 - \frac{f_{r}^{2}}{f^{2}})]}]

(21)

According to the one-to-one relationships between the motor frequency, motor speed, and spindle speed, the relationship $f_{r}^{2} / f^{2} \approx n_{r}^{2} / n^{2}$ can be obtained, where $n$ is the motor speed or spindle speed and $n_{r}$ is the rated speed. Therefore, the quantitative relationships among the input power of the MTS $P_{t}$ , the input power of the motor $P_{m}$ , and the motor speed or spindle speed $n$ can be obtained as

When n_{r} \leq n \leq n_{m a x}, P_{t} = \frac{1}{2 A} [- 1 + \sqrt{1 + 4 A [P_{m} - B \frac{n^{2}}{n_{r}^{2}} + C (\frac{n^{2}}{n_{r}^{2}} - \frac{n_{r}^{2}}{n^{2}})]}]

(22)

When n_{m i n} \leq n \leq n_{r}, P_{t} = \frac{1}{2 A} \frac{n_{r}^{2}}{n^{2}} [- 1 + \sqrt{1 + 4 A \frac{n_{r}^{2}}{n^{2}} [P_{m} - B \frac{n^{2}}{n_{r}^{2}} - C (1 - \frac{n_{r}^{2}}{n^{2}})]}]

(23)

Calculation approach

According to equations (20) and (21), the PPF model is composed of the input power of the motor $P_{m}$ , the input power of the MTS $P_{t}$ , the motor frequency $f$ , and the basic data of the motor ( $P_{r}$ , $η_{r}$ , $f_{r}$ , $P_{il} (f_{r})$ , $P_{il} (f_{\min})$ ). Therefore, only the input power of the motor $P_{m} (t)$ must be measured to calculate the input power of the MTS $P_{t} (t)$ . Thus, the approach for calculating the input power of a machine tool’s MTS is shown in following procedures:

Step 1. Preparation for basic data. Obtain the basic data, including $P_{r}$ , $η_{r}$ , $f_{r}$ , $f_{\min}$ , $f_{\max}$ , $P_{il} (f_{r})$ , $P_{il} (f_{\min})$ , and then the coefficients $A, B, and C$ need to be calculated.

Step 2. Measure the input power of motor by power meter.

Step 3. Combine with the motor frequency, calculate the input power of MTS using equations (20) and (21).

Those basic data (nameplate values of the motor and its idle loss under minimum and rated frequencies) in procedures are available for machine tool users via specification books or predictions. After obtaining basic data during the preparation stage, the only necessary steps are to measure the input power of the motor and combine it with the frequency or speed of the motor, which is convenient for machine tool users. According to equations (22) and (23), the motor speed and spindle speed can be used as substitutes for the motor frequency.

This approach is based on the acquisition of the motor input power $P_{m}$ , MTS input power $P_{t}$ , and motor frequency $f$ ; thus, it is abbreviated as the PPF approach in this article.

Verification

Experimental methods and apparatus

Because the MTS and motor are integrated in machine tools during the assembly phase, it is difficult for machine tool users to install sensors on the input terminal of the MTS. To validate the proposed PPF approach, the converter and motor of the machine tools are employed to form a new testing platform (see Figure 3) with the help of Chongqing Machine Tools Works Co., Ltd, China. The input power of the MTS is also the output power of the motor; thus, only the output power of the motor needs to be tested.

Figure 3.

Experimental apparatus.

As shown in Figure 3, a 7.5-kW ABB motor (QABP132M4A) is driven by a Siemens MM440 converter and loaded by a CZ20-type magnetic powder brake and ZH07-B-type torque speed sensor. The HIOKI3390C power analyzer is used to measure the input power of the motor. The nameplate values of the QABP132M4A motor are as follows: $P_{r} = 7.5 kW, η_{r} = 89 %, f_{r} = 50 Hz, f_{m a x} = 100 Hz, f_{m i n} = 5 Hz,$ $P_{il} (f_{r}) = 489 W,$ and $P_{il} (f_{\min}) = 300 W$ .

Accuracy analysis

Motor load experiments with 20 different frequencies were conducted to investigate the accuracy and validity of the PPF approach. The experimental results are illustrated in Appendix 1, where $P_{mm}$ is the input power of the motor measured by HIOKI3390C, $P_{tm}$ is the input power of the MTS measured by ZH07-B, $P_{tc}$ is the input power of the MTS calculated by the PPF approach, and $Err = | P_{tm} - P_{tc} | / P_{tm}$ is the relative error. The power loss of the motor and the input power of the MTS at 50 Hz are shown in Figure 4 to demonstrate the accuracy of the PPF method.

Figure 4.

Accuracy of the PPF approach (a) power loss of the motor at 50 Hz and (b) transient input power of the MTS at 50 Hz.

Although there are small fluctuations in the transient input power of the motor, the PPF method is excellent for calculating the input power of the MTS by measuring the input power of the motor.

Furthermore, the power loss of the motor calculated by PPF is smaller than the measured value, and the input power of the MTS calculated by PPF is considerably greater than the measured value in Figure 4(a) and Appendix 1 when the motor is under a low frequency or in the overload state. This trend occurs because the PPF method is developed by investigating and formulating the power loss characteristics of the motor under a regular or commonly used state, and those characteristics change dramatically when the motor is working under a low frequency and in the overload state.²⁵ For example, the mechanical loss of the motor will increase dramatically if the motor output torque is over its rated value. As a result, the PPF method underestimates the power loss of the motor under low frequencies and in the overload state, and the input power of the MTS is overestimated in those cases. Thus, the error will increase when the motor is working under low frequencies or in the overload state.

The statistical results of the relative errors presented in Appendix 1 are shown in Figure 5, indicating that 84.13% of the errors are under 3%, and over 95% of the errors are under 5%.

Figure 5.

Chart of the relative errors.

Therefore, although certain estimated models in the electrical field calculate the power loss of the motor, because the spindle motors of machine tools rarely working under low frequencies or in the overload state, the accuracy of the PPF method is acceptable in the mechanical field.

Application and discussion

The proposed PPF approach for calculating the input power of the MTS is available and convenient for machine tool users because only the motor input power is required, allowing a greater and wider practicability than the methods proposed by Liu et al.²⁰ and Hu et al.²² Furthermore, the method can be implemented without any use of a torque (or moment) sensor, as is required in the traditional method. Furthermore, the PPF method can be widely used to study the energy efficiency and energy-saving technologies of machine tools, as discussed below:

1. The PPF method can be used to study the energy efficiency of machine tools and their components. Figures 4 and 5 illustrate that the PPF method provides a high level of accuracy and validity for calculating the input power of the MTS. In addition, F Liu¹⁵ established a method for calculating the power loss of the MTS. According to Figure 1

P_{o} (t) = P_{t} (t) - P_{tl} (t)

(24)

Then, the expressions for the energy efficiency of machine tools, the spindle motor, and the MTS are as follows

η_{MACH} = \frac{\int_{0}^{T} [P_{t} (t) - P_{tl} (t)] dt}{\int_{0}^{T} P_{MT} (t) dt}

(25)

η_{MOT} = \frac{\int_{0}^{T} P_{t} (t) dt}{\int_{0}^{T} P_{m} (t) dt}

(26)

η_{MST} = \frac{\int_{0}^{T} P_{o} (t) dt}{\int_{0}^{T} P_{t} (t) dt}

(27)

where $P_{MT}$ is the total input power of the computer numerical controlled (CNC) machine tools.

Therefore, the input power of the machine tool MTS is essential for calculating the energy efficiency of machine tools, the spindle motor, and the MTS. The PPF method can be used to study the energy efficiency of machine tools and their components.

2. The PPF method can be used to study machine tool energy-saving technologies, including energy saving in the use of machine tools and the design of high-efficiency machine tools.¹² As the main energy consumption component of the machine tool spindle unit, the spindle unit is an important consideration when investigating the energy-saving technologies of machine tools.^29,30 According to equation (26)

η_{m} = \frac{P_{t}}{P_{t} + P_{ml}} = \frac{n_{m} T_{m}}{n_{m} T_{m} + 9550 P_{lm}}

(28)

$n_{m} = (60 f / p) (1 - s)$ , $p$ is the pole number of the motor, and $s$ is motor slip. Then, only the motor basic data $A, B, C$ need to be obtained to obtain the relationship between the energy efficiency of the motor, the output torque of the motor, and frequency. According to equation (28), $A = 7.86 \times 10^{- 6}, B = 485, C = 300$ . The contour map of the motor energy efficiency can be easily obtained, as shown in Figure 6, indicating that an increased energy efficiency of the motor, with $900 r / \min \leq speed \leq 1050 r / \min$ and $40 N m \leq torque \leq N m$ .

Figure 6.

Contour map of the QABP132M4A motor energy efficiency.

For manufacturers of machine tools, high-efficiency motors are required to design high-efficiency machine tools. Assume that there are N different total motors from which to choose. If the motor working conditions $(n_{m}, T_{m}, P_{t})$ are obtained, the manufacturer would calculate each motor’s efficiency based on equations (18) and (28). Then, it is easy to select the highest efficiency motor for machine tools.

Therefore, the PPF method can be used to obtain the optimal working frequency and working torque of a motor, which can be used to determine the optimal machine tool working frequency and working torque as well as to select the spindle motors for designing high-efficiency machine tools.

The proposed PPF approach can be applied for machines to calculate the input power of the MTS (the same as the output power of the motor) fed by a converter if those machines, including milling and grinding, have the structure shown in Figure 1. However, it may be difficult to apply the PPF approach if the input power of the motor cannot be measured due to the physical structure of the machine tools. An improved approach may be required to address this deficiency by establishing the relationship model between the input power of a converter and the MTS.

Conclusion

This article presented a new approach (PPF) for calculating the input power of the MTS of machine tools. The influences of the converter and frequency variations on the motor loss were considered in developing this approach. Three input power models of the MTS and motor loss were established based on the nameplate values of the motor and its idle loss under the minimum and rated frequencies. These models include the model that describes the relationship between the idle loss of the motor and frequency, the motor loss model that can be used to determine the motor power loss using the input power of the MTS and frequency, and the PPF model that describes the relationship between the input power of the MTS, the frequency, and the input power of the motor.

A total of 20 group experiments were conducted to validate the proposed approach, and the applications of the PPF method were discussed. To calculate the input power of the machine tool MTS, the PPF method need to measure only the input power of the motor after obtaining basic motor data in the preparation stage. The verification and applications demonstrate that this method provides a high level of accuracy and validity for calculating the input power of an MTS driven by a converter and can be widely used to study the energy efficiency and energy-saving technologies of machine tools. Based on this study, future research will investigate high-efficiency machine tools and energy efficiency monitoring of machine tools.

Footnotes

Appendix 1

Table 1.

Experimental verification data.

$f = 85 - 100 Hz$ , P (W)
85 Hz				90 Hz				95 Hz				100 Hz
P_mm	P_tm	P_tc	E_rr (%)	P_mm	P_tm	P_tc	E_rr (%)	P_mm	P_tm	P_tc	E_rr (%)	P_mm	P_tm	P_tc	E_rr (%)
739	360	359	0.28	760	353	352	0.28	942	488	486	0.38	892	440	448	1.91
1106	728	722	0.70	1197	773	785	1.57	1219	743	760	2.35	1196	730	750	2.68
1897	1493	1500	0.49	1892	1455	1468	0.90	1978	1485	1506	1.43	1938	1485	1479	0.42
2728	2280	2307	1.19	2655	2190	2210	0.90	2781	2265	2286	0.92	2697	2205	2216	0.52
3439	2963	2989	0.90	3608	3083	3124	1.34	3554	3000	3028	0.93	3532	2985	3018	1.12
4244	3690	3754	1.74	4333	3758	3811	1.43	4299	3690	3735	1.23	4417	3788	3858	1.85
5200	4545	4651	2.33	5191	4538	4616	1.73	5189	4493	4570	1.73	5206	4493	4597	2.34
6020	5295	5411	2.19	5968	5228	5337	2.09	6029	5235	5350	2.20	6086	5250	5414	3.12
6850	6030	6172	2.35	6814	5970	6113	2.40	6900	6008	6148	2.35	7081	6105	6325	3.60
7640	6713	6888	2.61	7724	6750	6939	2.80	7545	6563	6734	2.62	7821	6728	6994	3.96
8620	7523	7767	3.24	8566	7440	7694	3.41	8782	7583	7844	3.45	8757	7478	7832	4.75
9422	8198	8478	3.42	9354	8085	8393	3.81	9594	8243	8563	3.89	9912	8393	8854	5.49
10516	9068	9437	4.07	10432	8940	9340	4.47	10611	9053	9454	4.44	10932	9165	9744	6.32
$f = 65 - 80 Hz$ , P (W)
65 Hz				70 Hz				75 Hz				80 Hz
P_mm	P_tm	P_tc	E_r _r	P_mm	P_tm	P_tc	E_rr	P_mm	P_tm	P_tc	E_rr	P_mm	P_tm	P_tc	E_rr
704	383	381	0.30	682	368	366	0.29	682	345	344	0.27	694	345	344	0.27
1139	818	812	0.63	1075	750	757	0.87	1122	773	780	0.93	1165	810	811	0.10
2142	1800	1796	0.24	1836	1493	1504	0.75	1760	1403	1407	0.32	1868	1500	1501	0.09
2767	2400	2401	0.03	2608	2228	2254	1.17	2672	2273	2293	0.91	2612	2213	2224	0.50
3438	3038	3044	0.20	4130	3698	3707	0.27	3465	3008	3054	1.55	3513	3038	3089	1.68
4194	3750	3762	0.31	5103	4635	4621	0.30	4289	3780	3836	1.49	4290	3758	3826	1.81
5405	4898	4895	0.05	5795	5265	5263	0.03	5096	4515	4593	1.73	5142	4530	4625	2.10
5999	5423	5445	0.41	6699	6075	6093	0.30	5934	5273	5370	1.85	5973	5288	5395	2.03
6643	5993	6035	0.71	7384	6690	6716	0.38	6823	6053	6185	2.19	6814	6030	6166	2.26
7471	6728	6787	0.89	7551	6750	6867	1.73	7613	6750	6902	2.25	7643	6750	6918	2.49
8403	7538	7625	1.16	8481	7545	7701	2.07	8574	7575	7763	2.49	8533	7500	7716	2.88
9335	8348	8452	1.25	9296	8243	8424	2.20	9409	8280	8504	2.70	9495	8310	8569	3.12
10211	9060	9222	1.79	10309	9083	9313	2.54	10361	9060	9339	3.08	10473	9098	9425	3.60
$f = 45 - 60 Hz$ , P (W)
45 Hz				50 Hz				55 Hz				60 Hz
P_mm	P_tm	P_tc	E_rr	P_mm	P_tm	P_tc	E_rr	P_mm	P_tm	P_tc	E_rr	P_mm	P_tm	P_tc	E_rr
837	375	374	0.24	834	345	344	0.27	750	360	359	0.28	663	353	352	0.28
1235	773	769	0.49	1242	750	749	0.19	1150	750	756	0.74	1101	780	786	0.79
1988	1515	1511	0.23	2060	1553	1552	0.06	1924	1508	1515	0.53	1917	1575	1587	0.77
2708	2213	2214	0.08	2759	2228	2230	0.13	2698	2265	2268	0.11	2637	2273	2285	0.57
3548	3008	3027	0.65	3588	3030	3027	0.10	3514	3045	3051	0.19	3491	3090	3105	0.49
4345	3758	3791	0.89	4281	3698	3685	0.34	4307	3795	3803	0.22	4213	3765	3790	0.67
5211	4560	4614	1.17	5143	4500	4495	0.11	5089	4523	4537	0.32	5010	4515	4538	0.51
5946	5243	5304	1.18	5992	5288	5283	0.08	5896	5273	5286	0.25	5900	5340	5364	0.45
6821	6030	6121	1.50	6845	6068	6066	0.02	6764	6068	6083	0.25	6732	6090	6127	0.61
7600	6713	6840	1.90	7629	6788	6778	0.13	7518	6750	6768	0.27	7516	6788	6838	0.75
8551	7560	7711	1.99	8407	7478	7478	0.01	8513	7628	7661	0.44	8486	7643	7709	0.86
9367	8235	8450	2.62	9300	8258	8273	0.18	9251	8273	8317	0.54	9265	8318	8400	0.99
10251	9000	9245	2.72	10161	9000	9031	0.35	10144	9038	9102	0.72	10201	9105	9223	1.29
$f = 5 - 40 Hz$ , P (W)
25 Hz				30 Hz				35 Hz				40 Hz
P_mm	P_tm	P_tc	E_rr	P_mm	P_tm	P_tc	E_rr	P_mm	P_tm	P_tc	E_rr	P_mm	P_tm	P_tc	E_rr
738	360	364	1.19	741	330	341	3.23	807	383	391	2.25	821	375	374	0.14
1129	750	755	0.70	1169	758	767	1.26	1189	765	772	0.89	1427	975	977	0.18
1518	1126	1142	1.44	1506	1088	1102	1.36	1940	1500	1516	1.08	2204	1725	1743	1.07
1931	1515	1553	2.52	1959	1523	1552	1.94	2688	2213	2254	1.87	2833	2325	2359	1.48
2356	1905	1976	3.71	2382	1920	1971	2.66	3131	2625	2688	2.40	3230	2700	2747	1.73
2827	2325	2443	5.05	2921	2415	2503	3.65	3646	3098	3191	3.03	3550	3000	3058	1.93
3163	2618	2775	6.01	3498	2933	3072	4.74	4019	3435	3555	3.50	3954	3375	3448	2.17
				3831	3225	3398	5.38	4447	3818	3971	4.03	4442	3825	3919	2.46
				4265	3600	3823	6.20	4838	4852	4200	4313	2.70	6105	6325	3.60
								5295	4560	4792	5.08	5266	4575	4709	2.94
												6104	5325	5506	3.40
20 Hz				15 Hz				10 Hz				5 Hz
696	360	371	3.08	708	368	367	−0.03	650	338	337	−0.14	654	375	375	0.08
1011	662	695	4.87	1105	743	764	2.91	1089	750	772	2.96
1409	1035	1104	6.64	1909	1493	1567	4.98
1651	1253	1351	7.90
1959	523	1667	9.49

Academic Editor: ZW Zhong

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is funded by the National Natural Science Foundation of China (grant no. 51375513), the National Hi-Tech R&D Program (grant no. 2014AA041506), and the Special Research Foundation for the Doctoral Program of Higher Education of China (grant no. 20120191110001).

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