Research Article  Open Access
Xiaodong Wang, Bin Liu, Xuesong Mei, Jun Yang, Xialun Yun, Xiao Li, Jian Li, "A Novel Method to Reduce the Laser Drilling Time for Hole Cluster considering the Influence of Residual Vibration of the Sample Stage", Shock and Vibration, vol. 2020, Article ID 8342041, 15 pages, 2020. https://doi.org/10.1155/2020/8342041
A Novel Method to Reduce the Laser Drilling Time for Hole Cluster considering the Influence of Residual Vibration of the Sample Stage
Abstract
In large areas of laser drilling, the residual vibration occurs when the sample stage moves in PTP (Point to Point) movement. It affects the surface quality and processing efficiency of the holes. The common solution for this problem is to set the laser irradiation delay time by the controller to wait for the residual vibration attenuation, but the whole drilling circle will increase. In this paper, a new method is introduced to reduce the laser drilling circle. By setting the allowable threshold of the residual vibration for the subsequent process, the sum of the time in deceleration segment of the trapezoidal moving profile, and the time when the residual vibration attenuates below the amplitude threshold (ST) can be minimized as the optimization goal. The results show that for a given operating speed, there is always an optimum acceleration value for the deceleration segment of the trapezoidal moving profile, which minimizes the ST value. Further, the delay time for laser irradiation can also be estimated according to the optimal acceleration during laser drilling.
1. Introduction
Laser processing holds the characteristics of noncontact, low pollution, and high efficiency. In recent years, with the development of laser technology, laser processing has become ever more indepth in the fields of welding [1–3], hard or brittle material cutting [4–7], drilling [8–12], and surface structure texturing [13–15]. The performance requirements of laser processing products are higher and higher, such as ultracomplex structure, super surface of hydrophobic and oleophobic and high aspect ratio microholes, etc., which induces higher performance requirements for laser processing equipment. Laser processing equipment is usually integrated by laser systems, motion control systems, optical path systems, and galvanometer systems. These subsystems are usually produced by different manufacturers and are relatively independent, which is necessary to be effectively integrated. Laser processing is achieved through cooperative control between these subsystems. The delay problem in the cooperative control process determines the performance of the equipment and affects the processing efficiency. It is of profound significance to reduce the delay problem in the cooperative control process. For a better understanding, Table 1 provides an explanation of the abbreviations in this paper.

Figure 1(a) shows a common laser processing equipment that is equipped with a laser source, a galvanometer, and a motion control system. Figure 1(b) shows the common PTP path for laser drilling. Figure 1(c) is the schematic diagram of the vibration of the laser beam relative to the workpiece caused by residual vibration. Usually, the sample stage achieves motion positioning and then the galvanometer controls the laser drilling. When the galvanometer is positioned, the controller triggers the laser shutter to emit several laser pulses for drilling. However, the completion of positioning action is always accompanied by the generation of residual vibration, as shown in Figure 1(d). The effect of vibration on holes’ quality is not negligible if the laser is turned on immediately, especially when the design size of the hole is small. The common solution to this problem is to set the laser irradiation delay so that the laser pulses are irradiated after the attenuation of the residual vibration [16]. When the motion actuator is a robot arm, the residual vibration tends to be larger and the decay time is longer, which seriously affects the entire drilling efficiency [17, 18].
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Nowadays, in the field of laser drilling, many motion controllers still use the simple trapezoidal moving profile to control the motion of the sample stage due to the complexity of motion control algorithms and product cost. Although this type of move profile is simple and easy to be operated, residual vibration will inevitably occur when the sample stage stops, which affects the processing quality and processing efficiency [19–21]. The residual vibration is related to the deceleration section of the trapezoidal moving profile. So how to effectively control the acceleration value of the deceleration section to make the motion positioning time and the delay time for laser irradiation shortest is of great significance for shortening the whole drilling cycle. At present, most scholars' researches focus on how to eliminate the residual vibration and they have done a lot of meaningful researches. Some scholars studied the highorder trajectory smoothing algorithm, such as the fifthorder polynomial trajectory to make the acceleration and jerk (the derivative of acceleration with time) continuous, so as to reduce the acceleration fluctuation during the motion [22]. Some scholars study the optimization algorithm to minimize the energy of the jerk for the whole running trajectory under certain speed and acceleration constraints. Then the obtained jerk trajectory is used to generate smooth acceleration and velocity trajectory, which make the sample stage run smoother [23–25]. Some scholars use filtering methods such as lowpass filtering, bandpass filtering, input shaping, etc. to make the spectrum of the acceleration trajectory have lower energy near the natural frequency of the sample stage, thereby reducing the vibration during the motion of the table [26–29]. There are also some scholars using feedforward plus feedback control to improve servo performance. The residual vibration is regarded as the disturbing term to the command position. The filter model needed to eliminate the residual vibration is obtained by solving the inverse model of the system dynamics model [30–32].
All of the above researches are considered to reduce residual vibration completely by modulating the acceleration during motion. There is no guarantee that these methods will make the whole processing time smaller, but it may make the time larger. Moreover, the methods above researches need to be written to the motion controller before used, which is timeconsuming and laborious and very difficult for the laser application. Most motion controllers in the laser drilling field are closed to the user at the control algorithm level and do not support userdefined motion control algorithms. Otherwise, it may cause security risks [33].
In actual laser drilling, the trapezoidal moving profile is the most common velocity profile and almost all controllers support this velocity profile. In fact, when the residual vibration is attenuated to a certain amplitude which does not affect the quality of the sample, the laser can be irradiated. The decay time of the residual vibration is related to the magnitude of the excitation force when the motion stage is stopped, that is to say, it is related to the magnitude of the acceleration value of the deceleration section. Therefore, there may be an acceleration value to let the ST get the minimum, so as to shorten the whole drilling circle. In this paper, the problem is deeply studied from the perspective of shortening the whole drilling cycle rather than from the perspective of eliminating the residual vibration. The ST value is minimized as the optimization goal, and the optimal acceleration is obtained. The results demonstrate that the optimum acceleration of the deceleration section is only related to the dynamic parameters of the sample stage at a given operating speed. When the dynamic parameters are obtained, the optimal acceleration of the deceleration section is just set according to the speed. The optimal delay time before laser irradiation can also be estimated according to the optimal acceleration.
The content of this paper is organized as follows. The second part gives the model of the impulse response for the secondorder underdamped system and the system parameters identification method based on the ARMA (autoregressive moving average) method. The third part gives the method for reducing the ST value proposed in this paper. The fourth part introduces the experimental design and experimental equipment of this paper and explains the system parameters identification results based on ARMA. Then the experimental results of the method for reducing the whole laser drilling circle in this paper are given. The last part gives the conclusions of this paper.
2. Residual Vibration Description and Identification of System Dynamics Parameters
2.1. SecondOrder Underdamped System Dynamics Model
Figure 2 is a schematic diagram of the mechanical dynamics such as the sample stage, which can be described by a typical secondorder system.
The transfer function of a typical secondorder system can be described as follows:where ω_{n} is the undamped natural frequency and ξ is the damping ratio. For a secondorder underdamped system, 0 < ξ < 1, and ξ determines the degree of vibration attenuation of the system, and ω_{n} determines the period of vibration attenuation [33].
When the sample stage moves in a trapezoidal moving profile, the residual vibration occurs at the moment the sample stage stops. The form of the residual vibration usually appears as the attenuation vibration of a secondorder underdamped system for the impulse input. By identifying the parameters of the system model, the residual vibration can be mathematically described, which provide conditions for the planning algorithms of optimal move profile in further time.
The response of the secondorder system to the input can be expressed as follows:
When U(s) = K, that is, the input is a pulse signal, the output Y(s) = KG(s), then the time response signal of the secondorder system for the pulse input is obtained by inverse Laplace transform:
Let
Then
Using system identification algorithms such as ARMA, STD (sparse time domain), etc., the model parameters can be obtained through the measurement and analysis of the residual vibration signal.
2.2. ARMA Time Series Analysis Method for Solving System Parameters
In the process of obtaining the dynamic parameters of the sample stage, the selfexcited force generated by the highspeed impact of the sample stage is used to generate the residual vibration, and the residual vibration signal is collected. The ARMA time series analysis method is used to obtain the inherent spectrum and damping coefficient of the sample stage. Compared with the traditional hammer method, the parameter identification method based on residual vibration signal analysis can better describe the system state in the actual working environment [34, 35].
The ARMA time series analysis method is a method for describing the ordered random vibration data by using model parameters, thereby obtaining system parameters; the details of this method being available in the literature [36, 37]. The parametric model includes an autoregressive (AR) model and a moving average (MA) model. The AR model describes the dynamic characteristics of the system and the MA moving is related to external disturbances and inputs. Usually, a system model composed by a porder AR model and a qorder MA model can be expressed as ARMA(p, q), and its expression is as follows:
In the above expression, x_{t} is the time domain signal, ϕ_{i} is the autoregressive coefficient, θ_{l} is the moving average coefficient, and ɛ_{t} is the white noise.
Since the ARMA process {x_{t}} has a unique stationary solution expressed by the following:
Then
In equation (7), G_{j} is an impulse response function.
From equations (6) and (8), the following expressions can be derived:
In the above expression, the item E(x_{t}x_{t−k}) is the korder autocorrelation function of the time domain signal x_{t} and can be expressed by r_{k}. The item E(ɛ_{t−l}ɛ_{t−j−k}) is the korder autocorrelation function of the white noise and can be expressed by the following:
The σ^{2} is the variance of the white noise.
It can be further derived from equations (10) and (11)
In the above expression, when j < 0, G_{j} = 0. When l = k + j ∉ {1, 2, …, q}, θ_{l} = 0. The r_{k} is the korder autocorrelation function of the time domain signal. Usually, the difference order k of the ARMA model satisfies k ≫ max(p, q + 1), which means k + j ≫ q, then the right term in equation (12) is equal to 0, which gives the following:
Let the length of the autocorrelation function be L. Usually, L is much larger than M. Then, the following expressions can be derived:which is as follows:
Let R = R_{(L−M)×P}, R′ = , ϕ = ϕ_{(L−M)×1.} So the autoregressive coefficient ϕ can be expressed as follows:
When the autoregressive coefficient ϕ_{k} is obtained, the dynamic parameters of the system can be calculated by the transfer function expression of the ARMA model. The transfer function form of the ARMA model is expressed as follows:
The characteristic equation of the above formula can be expressed as follows:
The relationship between the transfer function poles and the system dynamics parameters can be expressed as in the following equation, in which z_{k} is the conjugate of :
The expression of the modal frequency and damping ratio ξ_{k} can be obtained from equation (19), as shown in the following equation, where s_{k} is the korder modal frequency and Δt is the sampling interval:
3. Time Optimal Trapezoidal Moving Profile under Residual Vibration Threshold
For the PTP motion using the trapezoidal moving profile, it is not true that the larger the acceleration value of the deceleration section is, the shorter the time for the PTP motion is, if the attenuation time of the residual vibration should be considered. For example, in the field of precision laser drilling, the residual vibration often has an influence on the shape of the hole [38]. Therefore, it is necessary to set a settling time to reduce the influence of residual vibration. Generally, the greater the acceleration of the deceleration section is, the shorter the time taken by the deceleration section will be, but the larger the amplitude of the residual vibration will be and the longer the settling time is required. It is of great significance to set the optimal acceleration value of the deceleration section to obtain the minimum ST value.
Figure 3 is a schematic diagram of the residual vibration of the trapezoidal move profile. The motion profile indicated by the red and blue curves is different only in the acceleration value of the deceleration section. The distance S_{des} of the PTP motion and the maximum speed V_{max} remain constant. Assuming that the positioning time ST is T_{−}, the following expression can be achieved when there is a constant velocity segment:
In equation (22), represents the deceleration period time, and represents the time required for the residual vibration to decay below the set threshold. Obviously, when the acceleration section adopts the same acceleration and the set operation speed is constant, the positioning cycle T takes the minimum value T_{min} when the T_{−} takes its minimum value. Thereby, for a complete trapezoidal move profile, it is only to require the minimum value T_{−} to minimize the total positioning cycle T.
As can be seen from Figure 3, at the moment of the sample stage stops, the acceleration will be abrupt, resulting in an instantaneous inertial force that acts on the sample stage and causing the residual vibration of the table.
The following expression can be obtained by Newton’s second law:
In the case of a certain V_{max}, the larger the a_{−} is, the smaller the will be. As can be seen from equation (23), the larger the F will be, which will cause the larger impact, then the longer the time when the residual vibration reaches the set amplitude threshold will be. Therefore, there should be an optimal acceleration to make T_{−} get the minimum value T_{_min}:
For the impulse response of the secondorder underdamped system, the envelope exhibits the characteristic of the exponential decay curve, as shown in Figure 4. The envelope of the oscillation can be expressed by the following equation:with .
Assuming that the amplitude threshold allowed by the process is S_{Vlimit} and the relationship between the proportional coefficient K for the impulse response and the acceleration value a_{−} of the deceleration section is as the following equation, λ is the proportional coefficient between K and acceleration a_{−}:
If
Then, the following expression can be obtained from equations (26) and (27):
Let = 0, the following equations can be obtained:
It can be seen from equation (30) that when takes the minimum value. The required acceleration a_{−} is represented by the following equation:
It can be seen that for the different speeds V_{max}, there is always an optimal acceleration a_{−} to guarantee the shortest execution time of PTP movement in the case of meeting the residual vibration threshold requirement. It also shows that the optimal value is dependent on the system dynamics parameters ω_{n} and ξ.
4. Experiment
In order to obtain the dynamic parameters of the sample stage, two different move profiles are used to excite the sample stage, and the residual vibration signal after the sample stage stops is obtained by the signal acquisition device. The designed moving profiles are shown in Figure 5, which adopts a triangular motion profile and trapezoidal motion profile, respectively. The experimental setup is illustrated in Figure 6. The data acquisition equipment belongs to the NI company, and the data acquisition software is developed based on the LabVIEW environment. During the experiment, the sampling frequency is set as 1000 Hz. The acceleration sensor is ICP acceleration sensor. The measurement frequency range is 0.5–10000 Hz. The acceleration range is ±50 g. The machine tool is a threeaxis ultraviolet nanosecond laser processing equipment, including a sample stage, a galvanometer, and so on.
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Figure 7 shows the results of the residual vibration in the triangular moving profile, and Figure 8 shows the results in the trapezoidal move profile. It can be seen that the sample stage stops accompanied by the residual vibration, which is as shown by the curve FVSF. The larger the acceleration of the deceleration section is, the larger the amplitude of the residual vibration will be.
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It can be seen from the PSD diagram that the dominant mode frequency of the sample stage is about 7 Hz. In order to facilitate the identification of the system parameters, the bandpass filtering method is used to process the frequency band near the dominant mode. Then the frequency domain integration method is used to quadraticize the acceleration signal to obtain the displacement vibration signal of the residual vibration, which is used for the identification of system parameters. The ARMA method is used to identify the system model parameters. The results are presented in Figures 9 and 10. Figure 9 shows the ARMA residual vibration identification results under the triangular move profile and Figure 10 shows the ARMA residual vibration identification results under the trapezoidal move profile.
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The natural frequency and damping coefficient ξ of the sample stage identified in the two move profiles are shown in Tables 2 and 3. Finally, = 44.55, ξ = 0.04356, and λ = 0.004783 are taken as the dynamic parameters of the sample stage.


It can be obtained from equation (31), when = 44.55 and ξ = 0.04356, then a_{−} = 1.940598 V_{max}. Assuming V_{max} = 0.75 m s^{−1}, then the optimal acceleration of the deceleration section is a_{−} = 1.46 m s^{−2} and the deceleration section time is = 0.515 s. The evaluated optimal delay time for laser irradiation is = 0.5850 s, according to equation (28), which means the amplitude of the residual vibration will decay below the threshold when the decay time is greater than . Figure 11 shows the experimental results when a_{−} takes different values. The threshold of the residual vibration is set to 0.1 um. The results (Figures 11(c) and 11(d)) show that when = 0.515 s, i.e., a_{−} = 1.46 m s^{−2}, the experimental result of the is 0.5869 s, which is very close to the evaluated optimal delay time. The time T_{−} = + = 1.1019 s takes the minimum value, which proves the correctness of the conclusions in this paper. Obviously, this is useful for reducing the settling time and improving processing efficiency.
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Although the difference in the values for T_{−} between the three experiments is small, this is because the laser drilling machine of this paper uses a marble bed, which results in a smaller initial amplitude and a faster decay for the residual vibration. If the robotic arm or nonmarble bed structure of the laser processing equipment is used for positioning, the experimental results will be more obvious.
5. Conclusion
PTP motion is common in the industry. When the trapezoidal velocity trajectory is used in laser drilling, the sample stage will generate residual vibration when positioning completes. In order to reduce the influence of residual vibration on laser processing quality, it is necessary to set delay time to make the residual vibration attenuate, which causes a drop in overall machining efficiency. In this paper, based on the theoretical analysis, we have found that the acceleration value of the deceleration section for the trapezoidal moving profile can minimize the ST value, and the decay time after the sample stage positioning for laser irradiation can also be evaluated so that the whole laser drilling circle can be shorter.
The conclusions provided in this paper are simple and effective. Only the dynamic parameters of the sample stage are needed to obtain the optimal acceleration value of the deceleration section and the delay time for laser irradiation at a given operating speed, which effectively shortens the execution time of the whole laser drilling circle and improves the processing efficiency. In further research, the accurate acquisition methods of the dynamic parameters of the sample stage and the accurate time of the laser irradiation delay will be studied in more detail. At the same time, the conclusions of this paper will be applied to design the process instruction of largearea laser drilling and help to improve the overall drilling efficiency.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
Xiaodong Wang, Bin Liu, Xuesong Mei, Jun Yang, Xialun Yun, Xiao Li, and Jian Li declare that they have no conflicts of interest or financial conflicts to disclose.
Acknowledgments
This work was supported by the National Key Development Program of China (Grant nos. 2016YFB1102500 and 2016YFB1102602) and the National Natural Science Foundation of China (Grant no. 51735010).
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Copyright
Copyright © 2020 Xiaodong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.