A blended empirical shot stream velocity model for improvement of shot peening production

Peening intensity and coverage are vital measurement outputs to quantify the quality of a peening process in the surface enhancement operation of metal parts. In practice, these parameters can only be measured offline upon process completion, which is not suitable for online tracking and operation. Instead, shot stream velocity can be used as a real-time monitoring parameter to bridge operational inputs to the outputs. As such, a robust and accurate shot stream velocity model is needed for real-time tracking. In this study, we propose a blended practical model for shot stream velocity to address the issues. The model is constructed using a regression algorithm based on the blended candidate functions, which are developed from the experimental data and nature of the particle-air flow inside the system. The obtained model is validated against the experimental data for ASR 70 media type for different operating conditions of the inlet airflow pressure and media flowrate. Calculated velocities are in good agreement with the measurements. In addition, the developed model is applied to predict the shot stream velocity for ASR 230 media type, as well as to evaluate the peening intensity and coverage for different media types under different operating conditions. The predicted results are comparable to the measurement data under the same operating conditions. The maximum relative error of the predicted shot stream velocity and measurement velocity is about 5%, while the maximum error in peening intensity is about ±0.0065 mmA. Furthermore, a single-input and single-output model-based control is developed based on the proposed shot stream velocity model. The developed control system is robust, accurate, and reliable. It implies that the developed model can be used to provide the necessary information, as well as to develop the optimal process control system to improve and accelerate the peening processes for cost and time reduction of actual production.

3 surface methodology to optimize the response variables (maximum compressive residual stress, the depth of residual stress field, and concentration factor) with dependent variable of shot velocity, shot diameter and coverage ratio based on the obtained FEM results using quadratic polynomial function.
Unal [26] performed optimizations for shot peening parameters using response surface methodology by selecting the surface roughness and hardness as the responses to classify the shot peening processes. In addition, Kirk [27] performed curve fitting for shot peening data analysis. In general, the numerical methods based the first principle model are often performed in 3-dimensional simulations, which require huge computational resources and time consuming. It is often not suitable for real-time monitoring and/or feedback control. And most of the existing developed data models are often relied on polynomial curve fittings (linear, quadratic, cubic polynomial approximation, etc.), and did not account for physical characteristics of the process. As such, the obtained data models often have no relevant physical meaning or even invalid operating conditions. Thus, an incorporation of nature and characteristics of the particle-air flow into the data model is vitally important to obtain a robust and accurate data model for output prediction, online tracking, and real-time feedback control system.
In reality, both the intensity and coverage cannot be tracked or measured during peening operation.
Therefore, an alternative real-time measurable parameter/variable is requested for both process monitoring and optimal feedback control applications. Among feasible variables, shot stream velocity upon impact considered as a great candidate for real-time tracking and feedback control because the shot stream velocity can play a role of a bridge to link operational inputs to peening intensity and/or coverage [28][29][30]. Currently, both ShotMeter [29,[31][32][33] and high-speed camera [29,[34][35][36] are widely used to measure shot stream velocity in shot peening application. The high-speed camera however often requires an open and large space for installation. Using high-speed camera also often requires an external software that takes time for image-processing to evaluate the shot stream velocity; therefore, it might not suitable for real-time monitoring or feedback control. In addition, the highspeed camera also requires special safety requirements to protect the camera. Whist, the ShotMeter uses a simple method of particle illumination to measure the shot stream velocity and does not require special safety features to protect this device. In which, two electro-optical sensors with a fixed spacing measures the shot stream as they travel toward downstream from the shot peening nozzle exit. The signals from these two sensor are compared to obtain a phase shift that is used to evaluate the shot stream velocity with an accuracy of about 1% [37]. The latest version of ShotMeter is a low-cost & rugged technology, which enables real-time measurement of shot stream velocity for process development as well as troubleshooting activities. With integrated software, it allows the ShotMeter to continuously record data as a function of time for easy retrieval using spreadsheet [38]. Thus, with its great advantages, the ShotMeter is utilized in this study to record the shot stream velocity for model development, as well as online tracking later.
Thus, in this study, a practical shot-stream velocity model is developed using regression algorithm, which is based on experimental data and candidate functions of separated operational inputs. For data collection, the averaged shot stream velocity is measured at a small region upon the impact location using ShotMeter for different operational inputs, while intensity data is obtained from other sets of experiments for the same input conditions. For applications, the shot stream velocity model is developed to (1) predict shot stream velocity of different peen types under different operating conditions; (2) predict peeing intensity and coverage for different operational inputs; (3) develop a closed loop optimal control system using process model. This paper is organized as the followings.
Section (1) introduces purpose and objectives of this study. Section (2) describes experimental setup and data collection. Section (3) expresses the process construction of candidate functions. Section (4) demonstrates the applications of the shot stream velocity model. Finally, Section (5) concludes the contents of this paper.

Experimental trials
In actual peening operation, the both inlet air pressure and peen (or media) flowrate are dynamically manipulated to achieve target setting intensity and/or coverage. In previous study [20], we have studied the relation of shot stream velocity and peening coverage. Thus, this study only focuses on the development of shot stream velocity model to link up the inlet air pressure and media flowrate to peening intensity.

Experimental setup
To build the shot stream velocity model, the shot peening machine from Abrasive Engineering Singapore Lt. Co, which is shown in Fig. 1 (left), and schematic diagram of experimental setup is shown in Fig. 1 (right). In particular, a straight bore nozzle with internal diameter of 14.0 mm, which is widely used in the industry, is used in the experimental trials. The shot types ASR 70 with averaged size of 0.1778 mm (or 0.007 inch) is utilized to measure shot stream velocity and intensity. The shot stream velocity upon the impact is experimentally measured using ShotMeter, while Almen system with saturation curve is used to evaluate measurement intensity. Two piezoresistive pressure sensors from KISTLER are mounted at inlet (before shot injected point) and on the nozzle to measure the air pressure at inlet ( ) and outlet ( ), respectively. The pressure difference of the two sensors is determined as a pressure loss along the system and energy transferring to the peens. To hold the strips during the tests, an Almen strip holder is designed to fix up to 8 Almen strips on both sides (4 strips for each side) for each test (see Fig. 2).
In  For intensity measurement, the shot stream is directly shot on the Almen strips from stand-off distance of 120 mm and peening angle of 70º for measuring intensity. The system is calibrated to ensure that the settings are similar to the previous experimental set. To ensure the model is relevant and consistent, the operating conditions are assumed to be valid as both pressure and mass flow rate did not vary drastically (e.g., ~ ± 1 psi of pressure and ± 0.2 kg/min of mass flow rate) in both experimental sets. The measurement values of arc height for each operating condition are used to construct a saturation curve to determine peening intensity and saturation time. Normally, a minimum of four points with four corresponding exposure times is required to plot a saturation curve. In addition, each corresponding intensity value is experimentally repeated for at least 4 times with tolerance of ±0.02mmA to ensure the measurements are accurate and repeatable. The experimental setup is shown in Fig. 2.

Shot stream velocity measurement result and discussion
For shot stream velocity measurement, each experiment is performed for about 120 seconds, while spherical shots are released for a period of about 80 seconds. ShotMeter measures and saves measurement data at every 0.1 seconds. Measurement data of three typical cases are shown in Fig. 3.
Obtained results shows that the shot stream velocity is higher for a higher inlet air pressure for the same media flowrate. Inversely, the shot stream velocity is lower for higher media flowrate for the same inlet air pressure. The measurement results are more accurate and stable when the media flowrate is higher, and inversely. After reaching each stable state, the shot stream velocity attains a constant value, which is shown by the dashed lines in Fig. 3. As such, the averaged value of the shot stream velocity for each operating condition is used in the later analysis for model development. The averaged measurement data of shot stream velocity for all cases are shown in Fig. 4. Each averaged value is evaluated from four times of repeating experiments for each operating condition.
The obtained results clearly show that the averaged shot stream velocity increases with the increase in air pressure, and decreases with the increase in media flowrate. Physically, for the same peening system and media flowrate, a higher inlet air pressure can bring a higher airflow velocity inside system and directly provide a higher energy transferred from the air stream to media, which leads to a higher shot stream velocity. For a similar inlet air pressure, the momentum energy transferred from airflow to individual shot particle is lesser as number of shot particles increases, as such the averaged shot stream velocity is lower. The obtained results also show that the shot stream velocity is more sensitive to the inlet air pressure than media flowrate. In addition, the measurement data also indicates that the shot stream velocity slightly fluctuates at the region of high inlet air pressure. The fluctuation is caused by some factors such as the media flow control valve fluctuation, calibration drift of pressure valve and compressor stability, shockwave formation and propagation, etc.

Peening intensity measurement result
To quantify the peening intensity, an Almen gage is employed to measure the arc height of the Almen strip before and after peening process to determine the change in arc height caused by the peening process. In these experimental trials, each operating condition is experimentally repeated for four times to ensure that all variation falls within the industrial threshold of ±0.02 mmA. Each averaged value is evaluated from four measurement values for each operating condition. Fig. 6 shows averaged measurement values of intensity for different operational conditions. Generally, the obtained results show that the peening intensity is greater as the inlet air pressure is higher, while the peening intensity is smaller as the media flowrate is higher. As the peening intensity represents the kinetic energy of the peens transferred to the substrate, which is proportional to quadratic relation of peen velocity ( 2 /2) and linear form of media flowrate. As mentioned in previous subsection that the shot stream velocity is higher as the inlet air pressure is higher, while it is lower as media flowrate is higher. And, in fact, the change of shot stream velocity is much greater than the change of media flowrate that explain why the peening intensity is more sensitive to the inlet air pressure than the media flowrate.

Development of candidate functions for library construction
To build an accurate and suitable candidate functional library, we first analyze the obtained experimental data to gain understanding physical insights of the peening process as well as dependent parameters. In fact, the dependent parameters include the inlet air pressure, media flowrate, pressure loss inside the system, piping system, nozzle geometry configuration, shot type, stand-off distance, and impinging angle, etc. However, in this study, we only focus on main parameters of inlet air pressure, media flowrate and pressure lost, while other parameters are worth to consider in other study in a future plan. Followings are the details of analysis and candidate functions for functional library construction.

Candidate function of shot stream velocity for inlet air pressure
To study relation of inlet air pressure and shot stream velocity, a pressure sensor mounted at a location on the air pipe just before the peen-released point is used to record inlet air pressure, while the ShotMeter is employed to measure shot stream velocity upon impact region. The measurement results show that the increase of inlet air pressure leads to an increase of the shot stream velocity exponentially. In fact, the shot stream velocity only reaches to a maximum value although the inlet air pressure keeps increasing. It can be explained that the increase of the inlet air pressure results an increase in airflow velocity from subsonic to supersonic. However, the airflow velocity also reaches to a certain maximum value due to compressibility effects, as the result the shot stream velocity can only reach to a certain maximum value. Mathematically, a chosen candidate function must be valid that the shot stream velocity will be zero as inlet air pressure ( ) is zero, and the velocity reaches to a maximum value as inlet air pressure reaches to a certain value or higher as nature of the flow in the system. Thus, a candidate function with the form of 1 ( ) = 0 (1 − exp(− 1 * )) satisfies the requirements. In this expression, 0 and 1 are the model parameters, while is inlet air pressure.  fitting of ( 1 = 0 (1 − exp(− 1 * ))).

Candidate function of shot stream velocity for pressure drop
To gain understanding on the relation of the shot stream velocity and the pressure loss, a pressure sensor is placed at the inlet and another pressure sensor are mounted near nozzle exit (see Fig. 1) to measure the pressure loss along peening system, ∆ = − . As mentioned before that the momentum transferred from airflow to the peens is one of the factor among others (e.g., the turbulence, friction, and piping system, etc.) that causes pressure loss inside the peening system. The obtained results show that an increase in pressure loss can lead to an increase in shot velocity (refer to Fig. 8). We have tried different fitting functions, however, we found that the exponential function is the best fit for the measurement data. Fig. 8 shows the plot of averaged measurement shot stream velocity against the pressure loss (∆ ) using the proposed function of 2 (∆ ) = ( 1 exp( 2 * ∆ ) ). In this function, 2 is a model parameter. The plot shows that the selected candidate function can be fit well for the pressure loss and average shot stream velocity. Thus, in this study, this 2 (∆ ) candidate function is also selected and added to the function library for blended model development later.

Candidate function of shot stream velocity for media flow rate
Physically, to ensure that the model is valid and consistent, the model has to satisfy some constraints of particle-air flow nature; e.g, for a certain inlet air pressure, the stream shot velocity must be zero as the media flowrate increases to a certain maximum value, while the shot stream velocity achieves a maximum value as the media flowrate reduces to a certain minimum value. A mathematical model in the form of 3 ( ) = 3 ( 1 exp( 4 + 5 * + 6 * 2 ) ) is derived to satisfy the above contrainsts and fit well with the measurment data. In this function, 3 , 4 , 5 and 6 are model parameters. Fig. 9 shows the plot of shot stream velocity against the media flowrate for certain inlet air pressure values using the derived fitting function. It is clear that the chosen candidate function can fit well with the experimental data. In addition, the plot also shows that the change of the media flowrate does not significantly influence the change of the shot stream velocity as compared to the change inlet air pressure. In another word, the shot stream velocity is more sensitive to the change of the inlet air pressure compare to the media flowrate. It is consistent with the analysis in previous section. Thus, similarly, this candidate function is also chosen and stored in the candidate functional library for model development. Fig. 9: Averaged shot stream velocity plots against peen mass flowrate for ASR 70 peen type using fitting function.

Shot stream velocity model development with regression method
In this subsection, the shot stream velocity model is developed using regression method based on the blended candidate functions that developed in previous section. The regression method is widely used in mathematical and statistical representation for modeling, analyzing, and optimizing a process where the outputs response to the change of input variables [39][40][41][42].
In this expression, 1 , 2 , 3 , … , are model parameters of the process system. In this study, the The term ‖̂‖ 1 term promotes sparsity in the coefficient vector . And is measurement shot stream velocity or intensity. This optimization can be solved using the sequentially threshold least squares procedures as in the following algorithm: In this formulation, ( , ∆ , ) is shot stream velocity upon impact region, while , ∆ , and are inlet air pressure, pressure loss along the system, and media flowrate, respectively. Moreover, in this study, we introduce that is a constant parameter representing for peen type (e.g., ASR 70 or = 70).
Coefficient is system parameter that represents for typical shot peening machine. A might be different for other machines. Parameter is a coefficient for inlet air pressure. Parameter describes a coefficient for pressure loss along the peening system. In addition, and are parameters related to convection and diffusion energy transfer to the shot peens, respectively.
These coefficients and parameters are evaluated using proposed regression method with measurement experimental data as mentioned in previous subsection. In this study, the coefficients ( , , , and ) are 1250, 1.0, 1.1, 0.015 and 0.00015, respectively. The root means square fit of this regression method is 0.9687, and the maximum root mean squared error is about 7 m/s (about < 5%). The obtained blended model and the residual fit are presented in Fig. 10. It also can see that the developed model can accurately predict the shot stream velocity for any given set of operational input of inlet pressure and media flowrate with measured pressure loss. In addition, recall that the shot stream velocity is more sensitive to the inlet air pressure than the media flowrate. It implies that it is possible to control the peen velocity by adjusting the inlet air pressure, while it is difficult to success by manipulating the peen mass flowrate.

Applications of developed model of shot stream velocity
In this section, the developed model of shot stream velocity is applied to (1) predict shot stream velocity of different peen type for different operating conditions, (2) predict peening intensity for a given operational inputs, and (3) develop optimal feedback control system for automated shot peening machine. Followings are the details.  Fig. 11. It can be seen that the prediction results are in good agreement with experimental data for all operating conditions. It implies that the developed model can be used to predict shot stream velocity of different peening types for different operating conditions of the same shot peening machine by only changing peen size and its ratios.

Prediction of shot stream velocity for other shot type
Furthermore, it can be applied to predict shot stream velocity of other shot peening machine by choosing a suitable system parameter ( ) and peen parameters, accordingly.

Prediction of peening intensity for different peen types
In this subsection, the shot stream velocity developed in Eqn. (3) with relevant parameters is used to predict peening coverage and intensity for a given peen type and operating conditions. In particular, for peening coverage, the impact velocity of the shot is used to evaluate the peen work potential ( = peening coverage area for known-operational inputs. However, this study only focuses on intensity, thus, details of coverage prediction is not shown here.
For peening intensity, which is often represented by kinetic energy transferred from shot stream to material substrate, which is proportional to the shot stream velocity and media flowrate. The intensity can be evaluated using the analytical formulation as in the following form [8,15]: In which, the shot stream velocity ( ) is expressed in Eqn. And it also can be used to determine an optimal set of operational input from a target setting intensity using optimization method(s) to solve for inverse problem. In another word, the optimization method can be applied for the developed model to find a suitable input operating condition from a desired intensity. As such, it can further help to reduce cost, time, labor, and wasted material for building Almen system to guide for actual practical operation.

Model predictive controller design application
As mentioned, intensity is a static parameter, which only can be measured upon the peen process completion. So that shot stream velocity is considered as an intermediate variable, which can be used to track for peening intensity in real time. Thus, in this section, the developed model of shot stream velocity is applied to develop a real-time optimal model-based control system for achieving the target setting intensity. For simplification, this study mainly focus on control system development with one control variable (inlet air pressure) and control soft-launch scenarios, while multiple inputs and onsite control is carrying on in the future development.

Development of process model with control
In this study, a model The dynamical process model is developed based on sparse identification of nonlinear dynamical system (SINDy) algorithm with control [41,42] and obtained data of both inlet air pressure and shot stream velocity. Following is the general form of dynamical process model obtained by SINDy algorithm: ̇= ( ) + ( ( )) + ( ).
In this expression, is state variable, while is manipulated variable. Reader can refer to reference [41,42] for more details. In this study, the comparisons of the developed dynamical process model and the experimental data are shown in Fig. 13. In particular, Fig. 13(a) shows the comparison of ASR 70 peen type with media flowrate of 1 kg/min, while Fig. 13(b) shows comparison of ASR 230 peen type with media flowrate of 3 kg/min. It can be seen that the obtained process models can accurately represent peening processes for both ASR 70 and ASR 230 peen types. These obtained models are then used to develop the feedback MPC control system.  Fig. 14 (a) shows plot of RSM of shot stream velocity and its experimental residual versus peening intensity and media flowrate for ASR 70 peen type, while Fig. 14(b) shows plot of RSM of shot stream velocity for ASR 230 peen type. It can be seen that shot stream velocity has a stronger correlation and is more sensitive to intensity, while it has a weaker correlation to media flowrate.

Model predictive control development
As mentioned, a single input and single output (SISO) MPC controller is designed to automatically obtain any target setting value of peening intensity through tracking shot stream velocity reference setpoints. In which, the shot stream velocity reference set-point is evaluated using proxy model for the target intensity and media flowrate. The only control input variable is inlet air pressure, while the only output variable is shot stream velocity. Thus, the objective function for the model predictive controller design is often expressed in the following form: and are the minimum and maximum inlet pressure that the air compressor can supply to the peening system, respectively. ∆ and ∆ are the minimum and maximum step of air pressure that the regulator can adjust at each step, respectively. In process control, the controller determines a difference between the set-point and the actual measurement variable to optimally solve for a controller action (∆ ), which satisfy the constraints (7.5 and 7.6), using process model (7.2). The obtained control action ∆ is then sent to regulator (or inlet valve) to adjust the system for attaining the set-point. Fig. 15 shows control system design, while control action evaluation is performed using Algorithm 2 as bellow: Fig. 15: Schematic diagram of the optimal feedback model predictive control system for shot peening machine using ShotMeter for shot stream velocity measurement.

Algorithm 2: Model predictive control algorithm
Step 1: Obtain the measurement data from ShotMeter (shot stream velocity) Step 2: Solve for the state variables (7.2) Step 3: Solve optimization solver for and apply the constraints (7.5) Step 4: Retrieve control signal from optimal solution ∆ and apply constraints (9.6) Step 5: The final control action is sent to the physical machine for operation The control windows are determined by the constraints of the physical system. In which,

Control soft-launch demonstration
In this study, control soft-launches are performed to demonstrate for control application of the proposed control development. In particular, two in-silico control scenarios are used in the demonstrations for both ASR 70 and ASR 230 peen types to attain different values of target intensity, 20 respectively. Control scenarios of ASR 70 peen type for different intensities and other settings are listed in Table 2, while ASR 230 is shown in Table 3, as bellows:

Concluding remarks and recommendations
A blended practical model of shot stream velocity upon impact is developed based on both the experimental data and candidate functions to accurately describe the response of shot stream velocity to any change of inlet air pressure and media flowrate. The nature of particle-air flow inside the peening system is used to select a suitable candidate functions for functional library, while the regression technique is employed to determine the sparse coefficients of the candidate function for the final shot stream velocity model expression.
The developed model is employed to predict the shot stream velocity of different peen types for different operating conditions. The developed model is also used to predict peening intensity for different peen types and operating conditions. The obtained results are in good agreement with the experimental data for the same operating conditions. These achievements may provide great information for operational setup in the actual production. In another word, it can greatly reduce the cost, time, labor, and material wastage to perform experimental trials to build saturation curve for practical guidance.
In addition, the developed shot stream velocity model is also applied to develop the dynamical process model for model-based process control system. An optimal feedback model predictive control system has been developed for fully automated shot peening machine based on simulated shot stream velocity to attain targeted intensity. In-silico control scenarios are performed for two different peen types to automatically achieve desired intensities. The obtained results show that the process control performance is fast, stable, accurate, and reliable. It implies that the developed model-based control is suitable and can be utilized for actual peening operations.

 Conflicts of interest/Competing interests (include appropriate disclosures)
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

 Availability of data and material (data transparency)
The authors will make availability of any related data to this manuscript if requested (or data will available on request).