# Separation Method for Installation Eccentricity Error of Workpiece
## 1 Introduction  
As key core equipment in the field of precision machining, rotating machinery plays an irreplaceable role in various key industries such as aerospace, energy, and transportation and is an essential driving force for developing related technologies [1, 2]. Eccentricity failure is a critical factor affecting rotor stability and reliability during its operation [3, 4]. Currently, existing methods rely on detecting signals such as current, vibration, and noise, as well as analyzing the frequency spectrum of the signals to monitor eccentricity faults [5, 6]. Researchers also detect eccentricity faults by measuring magnetic field strength and unbalanced magnetic pulling force [7, 8, 9]. Most of these methods are based on spectral analysis, are susceptible to environmental influences, and are not conducive to generalization to other rotating machinery. Therefore, proposing a reliable and easily generalizable method to separate the eccentricity error is the current direction of work. As precision rotary machining equipment, the rotational accuracy of the spindle of a Computer Numerical Control (CNC) machine tool is determined by the assembly accuracy. High-precision measuring equipment is often used to control the spindle assembly accuracyduring installation [10, 11]. However, these detection devices have high costs and are not conducive to promotion. The rotational accuracy of the spindle is inevitably affected by installation eccentricity errors [12, 13]. The common accuracy detection method is to install a standard ball or cylinder at the machining end of the spindle and measure its displacement signal during rotation to evaluate the rotational accuracy of the spindle [14, 15]. In its operation, the installation eccentricity error of the workpiece exhibits the same frequency characteristics as the first harmonic component of the spindle motion error, which affects the extraction of the true vibration of the spindle [16]. During the installation process of standard equipment, it is inevitable to introduce installation eccentricity errors, which will be detrimental to the measurement of rotational accuracy. In addition, the installation eccentricity error of the workpiece can also cause cutting force fluctuations, accelerate tool wear, and reduce the surface integrity of the workpiece [17, 18, 19]. Researchers have also attempted to embed some sensors inside the spindle, but the internal space is limited, and the working conditions are complex, making it unsuitable for installing precision instruments [20, 21, 22]. Accurately separating the installing eccentricity errors is a prerequisite for compensation [23]. Based on this, there is an urgent need to propose a low-cost and computationally simple method for separating installation eccentricity errors. In addition, the angle measurement system, as a key subsystem in the advanced manufacturing ecosystem, is also affected by installation eccentricity errors [24, 25]. As the core of the angle measurement system, the installation eccentricity error of the grating encoder can lead to a deviation between the actual and ideal angular positions, reducing the measurement accuracy. Although encoders can achieve sub-arc second resolution, measurement errors caused by eccentricity during installation are still difficult to eliminate through traditional calibration methods [26, 27]. Although researchers have proposed many methods to compensate for installation eccentricity errors, such as adaptive methods and neural network methods [28, 29, 30], further research and improvement are needed to detect such errors effectively. Therefore, this article proposes a calculation method that combines PSO and the LSM to separate installation eccentricity errors. In this section, the combined motion law of the first harmonic of the spindle and the eccentricity of the workpiece installation is derived.