Vibration Extraction for Melting Plastic Hydraulic Injection System with Stick Slip Vibration Analysis

A hydraulic power injection machine is designed to use a driving screw to inject melted plastic into a specified mold. This machine can be found at an automotive spare parts factory. The cantilever-style heavy-duty screw injector is supported by one roller and secured at the end. An obvious need for vibration analysis on the roller support is essential. A mass spring damper model is proposed for deeply investigating the friction induced vibration mechanism for this injection system to well understand and analyze its vibration behavior. A mechanical mode of two degrees-of-freedom (DOF) is designed to improve research on the dynamic features of the Plastic Hydraulic Injection System (PHIS) mechanism. Experimental investigation and analysis of this mechanism are explored to obtain the instability speed and critical stick slip (SS) speed. The numerical imitation results of this work will help with the design and development of the PHIS mechanism. The stability of the system and SS behavior are next examined by determining the critical variability speediness and critical SS speed. A simulation study is carried out to evaluate the effect of various parameters of the system on its stability and on the behavior of the SS motion.


Introduction
Nowadays, the majority of mechanical devices suffer from self-excited noise and vibration caused by the resistance of the friction in the operating process. Friction self-excited oscillation [1] is a type of these vibrations that can be found in mechanical equipment such as mechanical decelerating tools. The phenomenon of vibration is well-known in science and it has a significant impact on engineering [2], in which the vibrations caused by friction surfaces can wear out mechanical structures and lead them to fail. In extreme cases, it may even cause mechanical equipment to fail. According to statistics, friction-induced vibration is directly or indirectly responsible for the failure of approximately 80% of mechanical components [3]. The nature of frictional vibration and strategies to reduce it, have long been a hot topic of discussion in mechanical engineering.
Many different vibrational models have been investigated in the past few years [4][5][6][7][8][9][10][11][12] due to their significance in various engineering applications. The dynamics of the motion of a damped spring and rigid body pendulums are examined in [4][5][6][7][8] and [9][10][11][12], respectively. Lagrange's equations were used to gain the EOM (Table 1), and the approach of multiple scales [13] is used to achieve the solutions of the regulating system. The requirements of solvability and the equations of modulations for different resonance cases are 1 3 derived. Furthermore, the nonlinear Routh-Hurwitz [14] criteria are used to investigate various stability and instability regions for the works [7, 8, 11, and 12].
Previously, researchers investigated friction-induced vibrations and their associated challenges using a variety of numerical calculations and experimental analytic approaches [15][16][17]. All of the component stiffness, material qualities at the frictional interface, damping process, surface topographies, and working conditions have an impact on frictioninduced oscillations. As a result, analyzing self-excited vibrations induced by friction is particularly challenging [18]. Currently, there is no complete paradigm that fully explains the phenomenon of friction self-excited vibration.
Many scientists have undertaken various studies on how to detect crucial variables that lead to the formation of vibrations induced by friction during some decades of scientific research, see [19,20]. The vibrating mechanism caused by friction is highly valued in both technical practice and scientific research.
Understanding friction types is a practical method for analyzing machine instabilities, and it's useful for building regulators that eliminate friction instabilities [21]. Various friction types are currently being used to demonstrate various friction phenomena, which include pre-sliding motion [22], Stribeck impact [23], SS motion [24], and hysteretic effects [25,26]. When the machine is inside the immobile phase, pre-sliding motion indicates that there may be tiny motion. The Stribeck effect is a phenomenon in which the friction pressure exhibits a drooping function even when the machine is powerful and the speed is slow. In general, there are two types of SS motion; micro-SS and macro-SS [27]. Micro SS describes a phenomenon in which a microvibration system's running velocity alternates between fast and slow. In a macroscopic sense, the running speed will vary when the system's innervation speed falls below a certain level. The previously described non-stationary motion is known as SS motion.
When the process is stable and under the existence of rough sliding circumstances, we can find that the friction force is a single magnitude property. However, owing to the instability of the system, friction forces have diverse meanings. The sort of sliding state hysteresis loop is called "friction memory hysteresis" and is separated into two types; clockwise [23,28] and counterclockwise [26].
Grinding patterns are divided into two categories, which are Phenomenological model [29,30] and material science ones [31,32]. Previous attempts have been used in heuristic analysis to derive lattice bending, also known as heuristics or experimental models [30]. This type of predictive capability is often dependent on the test environment and is not suitable for all applications. The materials science model seeks to determine grinding properties on a global scale using local physical data on contact surfaces. These models are mostly dependent on position, which may recognize sliding friction The speed at which the dynamic friction coefficient approaches its minimum The difference between the static friction and dynamic forces The stiffness of the system v b0 The critical instability of system's speed v b1 The critical stick slip speed behavior. On the other hand, the phenomenological model is distinguished principally by its sliding speed and many components. As a result, they can utilize the same criteria that describe the varied erosion behavior at various phases of sliding. The aim of this work is to investigate a dynamical model of mass spring damper representing the PHIS. This model has 2DOF and is designed to examine the behavior of stick slip vibration. Vibration analysis for the model is done to get the optimum design parameter. An experimental test is constructed to validate the numerical investigation. This paper is structured as follows: In Sect."Mechanical model of melting PHIS", we will organize a mechanical model of 2DOF to study the vibration mechanism of a hydraulic molten plastic injection system. In Sect. "Analysis of the speed's critical instability and the SS critical speed", the critical unstable velocity and critical SS velocity of the system are derived from fictitious calculations. Next, Sect. "Influences of system's parameters" focuses on the impact of various system parameters on system stability and SS behavior. Section "Experimental setup" explains the experimental setup. Section "Conclusion" the achieved results are presented.
To make the solution procedure easier, the majority of the study assumed a linear connection between the spring and the mass damper, as shown in the preceding survey. For this study, the motivation was to create more accurate assumptions in order to get closer to reality. Non-linear springs and equations in two dimensions are among the most fundamental of these assumptions, bringing a complete spring mass damper system closer to reality.

Modeling of the Examined System
To gain a clear understanding of the examined system's operation, Fig. 1a depicts the machine's schematic, whereas Fig. 1b represents the mass spring damper model for this machine. With this equipment, the primary goal is to inject the melted plastic into a stamp and to create a spare part for vehicles. As shown in Fig. 1a, this process is entirely reliant on a hydraulically controlled system, including a hydraulic pump and a screw for injection. The machine's rolling bearings and other installation components were subjected to vibration as a result of the hydraulic and injection systems.
The description of the mechanical components, as seen in Fig. 1a, to the corresponding model, as in Fig. 1b, has the following form: The block is represented by the mass of the front moving part that performs the injection process. The spring is represented by the bearings. The rest of the machine block restores and maintains the moving part in its place. As for the damper, it is represented by the hydraulic oil and grease in the bearing and the bearing itself, which work to dampen a large part of the vibration.
It is clear that the system is primarily composed of the injection rod head, hydraulic system, and roll support. The injection rod is a large component with dimensions of 1.5 m, 0.3 m, and 0.3 m, that is supported by the foundation's support rollers.
The injection is moved on the support within the injection site with the melting plastic in the hopper during the plastic injection process. Table 2 presents a complete description of the machine.
Developing an adequate friction model is an efficient way to investigate several basic friction phenomena. Because there is no suitable friction type for the injection process, A spring mass damper (SMD) friction model is established, as illustrated in Fig. 1b. Let v 0 denotes the speed of a conveyor belt and m represent a block's mass on the conveyor belt, in which it will produce a force N which is normal on the conveyor belt. Two linear springs and a nonlinear one with stiffness k 1 , k 2 , k 3 , and are kept the block in place. Moreover, we consider a damper with viscous damping c , see Table 1. Considering F is the friction force between the block and the belt as defined by the Stribeck friction type [33], in which the friction acts as a catalyst for the block.

Mechanical Model of Melting PHIS
In this section, we'll figure out how to get the model's equations of motion. It has usually been an essential research approach to create precise types for some significant friction atmospheres. Seeing that there is a shortage of mechanical type in PHIS, we built a 2DOF spring mass damper system on a treadmill to provide a deep guide to the SS vibration mechanism of a molten plastic hydraulic injection process. Use the SMD mechanism to investigate system stability and SS behavior. In addition, we will investigate the impacts of different parameters of the system on the stability and SS motion of the molten PHIS and it is revealed in Fig. 1b. This demonstrates that the conveyor belt goes in one direction at v 0 and is pushed away from the block by a constant regular force N , which is considered a point mass m . The springs have stiffness k 1 and k 2 are utilized to keep the block in place. Moreover, the block is connected to a horizontal position at its fixed end with the help of the viscous damping c and the non-linear spring The kinematic regulating equations of motion (EOM) can be written as follows where Here, v r =̇x − v 0 represents the motion's relative speed of block and conveyor belt, s denotes the static friction's coefficient, m points to the kinetic friction's coefficient, N indicates the block's and conveyor belt's normal force, and v m denotes the speed at which the dynamic friction coefficient approaches its minimum value [34]. As a result, the function of friction is satisfied.
It should be noted that if the block remains stationary in relation to the conveyor belt, the relative velocity is equal to zero. After the block starts to slide, the friction tendency decreases first and then increases with the increase of speed.

Analysis of the Speed's Critical Instability and the SS Critical Speed
If 0 is given by the formula k 3 = m 2 0 , then we can write the dimensionless parameters and variables as follows Therefore, the dimensionless form of the EOM can be written as where; Based on the ordinary differential equations (ODE) theory, the usual form of a set of higher-order differential High injection speed is required for thin-walled packing containers, although medium injection speed is frequently more acceptable for other types of plastic items Screw speed An excessive screw speed (1.3m∕s) is allowed. As lengthy because the cooling time has elapsed, the plasticizing technique is finished. The screw torque requirement is low Operating speed The operating speed is defined as a percentage of total hydraulic pressure with respect to the hydraulic system with a range of 10% to 50%. Therefore, the range of the working speed is from 0.025 m/s to 0.15 m/s equations can be replaced by another equivalent one of firstorder ODE. As a result, the following formulas can be used.
The substitution of (8) into Eqs. (5) and (6), produces the following system of differential equations Providing that the excitation speed is high enough, the block is in static equilibrium as a result of the interaction of friction and spring forces [35]. Therefore, the equilibrium conditions sgn(v r ) = −1,Ẋ = 0, and Ẏ = 0 of the system's parameters must be satisfied. The parameters of the system can be selected in the forms s = 0.85, m = 0.65, and v m = 0.48m∕s. Utilizing the stability Lyapunov's theory, we can examine the equilibrium point's stability of the considered system. At Ẋ = 0,Ẏ = 0, we can write the system's equilibrium points in the forms The following equations can be obtained by relocating the point of equilibrium to the origin zero point.
Then, the aforementioned Eq. (9) become (8) At Ẑ 1 =Ẑ 2 =Ẑ 3 =Ẑ 4 = 0, and from the Eq. (12), it is possible to acquire the Jacobian matrix of the approximate equation of the first-order as in the form where After inserting the following values of the variables into the Jacobi matrix as follows, then, the characteristic equation of the matrix A has the form where Based on the theory of Hurwitz [36,37], the basic critical instability speed (CIS) of the considered system may be gained from the solution of = −0.5999999999999999 + 2.7126736111111107 v 2 0 = 0, , and the critical instability of the system's speed will be v b0 = 0.4703m∕s. Theoretical calculations demonstrate that the system has unstable equilibrium points, and SS motion (12)  can readily arise whenever the speed of excitation is less than the CIS. Moreover, when the speed of excitation is greater than the CIS, the system reaches a stable position. The system's stability will improve when the speed of excitation increases, and the likelihood of SS motion will be reduced. The critical SS speed v b1 can be obtained according to [38,39].
where (F S − F m ) denotes the difference between the static friction and dynamic forces, C signifies the damping of the system, represents the stiffness of the system, and m is the block's mass.
For the examined system in the present work, the critical SS speed of the system will be 0.1688 at F S − F m = 0.2, C = 0.025, = 80 and m = 1 . If the feed speed is less than 0.1688 the block may adhere to the belt and the SS motion will be produced. In practice, we observe that the feeding speed in the PHIS pushing system is generally lower than the speed of critical instability and the crucial SS speed, resulting in the PHIS experiencing SS motion. The numerical simulation will be analyzed in the next section for further investigation of the impact of the system parameters on the stability and the SS motion.

Influences of System's Parameters
In this section, we will largely examine the experimental results obtained through the numerical solutions of the EOM of the PHIS system in the framework of Eqs. (5) and (6). In addition, we examine the influence of alternative system parameters on stability and SS behavior in general. Therefore, we will divide this section into some subsections to understand the implications of each parameter on the system's response.

Effect of Excitation Speed
The following data for the numerical simulations is taken into account  Figure 2 shows the phase-plane graphs that reveal the impact of the excitation speed v 0 on the response of the system, as depicted in the above section, the speed of critical instability v b0 is 0.4703m∕s and the critical SS speed v b1 is 0.1688 m∕s . It is clear from Fig. 2 that the phase-plane plots view are varied with the variation of excitation speed. If the excitation speed meets with the rela- and v b1 < v b0 < v 0 , the phase-plane plots in the corresponding parts (a,b), (c,d), and (e,f) of this figure determines that the system is stable and the SS phase lodges an important part of the entire cycle. Moreover, the stage of sticking is still present, in which the sticking phase gradually dissolves with increasing speed, as in Fig. 2b and f. An inspection of parts of Fig. 3 shows that the matching Poincaré maps of the corresponding parts if Fig. 2. All of the Poincaré cuts in Fig. 3, indicate that the system exhibits quasi-periodic motion at various excitation rates and that there is no chaotic motion.
When the excitation speed satisfies the relation 1.1 < v 0 < 1.3, the phase-plane graphs and the Poincaré sections in Fig. 4b and e demonstrate that the system is unstable and the SS phase lodges.

Impact of System Damping
In this subsection, we demonstrate the system's behavior according to the phase-plane plots when has distinct values as explored in Fig. 5. The previous values of the parameters in subsection (5.1) and the same initial conditions are used besides the certain value of the speed of excitation v 0 = 0. 35 . The values of the damping system are set according to 0.001, 0.01, 0.1, 0.5, 1, and 2 . It is noted in Fig. 5 that the sticking phase becomes shorter while the value of grows. Moreover, at = 2 the sticking phase is unobserved.
We may see from parts of Fig. 5 that as the value of grows, the sticking phase becomes shorter, and the sticking phase is hidden at = 2.
The behavior described above demonstrates that system's damping has a significant impact on whether the sticking phase increases or decreases. As a result, boosting system damping can reduce the sticking phase to some extent. A closer look to the portions (a-c) of Fig. 5 Indicates that the outer loops go closer to the limit cycles as the value of increases, and the limit cycles get smaller. In general, the limit cycles are nearing the equilibrium point, as seen in Fig. 5d-f. Each of these shows that the system stabilizes after a period of instability and finally reaches an asymptotically stable state.
In parts (a-f) of Fig. 6, one can find that all of the examined Poincaré sections are closed curves, which reveals quasi-periodic motion of this system at various damping settings and no chaotic motion.  The static and dynamic friction coefficient variations are specified to be 0.17, 0.12, 0.04, 0.022, 0.012, and 0.002.

3
The sticking phase increases longer when the disparity between dynamic and static friction coefficients becomes less, as shown in Fig. 7. This result focuses on the decreasing of the difference between the static and dynamic coefficients, and will not assist in decreasing the sticking phase.
Whenever the difference gets smaller, the outer loops go closer to the limit cycles, and the limit cycles get smaller as well. As a result, reducing the disparity between the dynamic and static coefficients of friction is an effective strategy to increase system stability.

Effect of Equivalent Stiffness of Nonlinear Spring Along the x Axis
For the numerical simulations, we set the used parameters as well as the initial values, such as  Along the x axis, the comparable stiffness values of the nonlinear spring are set to: 20, 50, 70, 100, 150, and 200. Examination of the curves of Fig. 8 shows that the effect of on the phase plane graph. It is observed from parts of Fig. 8 that the equivalent stiffness of the nonlinear spring along the x axis increase, the adhesion phase becomes shorter until eliminates. Both the horizontal magnitude of the limiting cycle and the magnitude of the innermost cycle decrease with increasing . As shown in portions (d-f) of Fig. 8, the inner loops are getting closer to the limit cycles. Thus, the system transitions from an unstable to a stable state as increases.

Experimental Setup
The spontaneous jerking motion that can happen as two objects slide over one another is referred to as the stick-slip phenomenon, sometimes known as the slip-stick phenomenon or simply stick-slip. In another way, the surfaces alternately slide over and stick to one another, changing the force of friction in the process [40]. According to this definition, a machine that achieves this principle will be a good example for the proposed theoretical modeling to make a better machine design. Self-excited vibration and noise due to friction in operation is a common problem with most mechanical devices today. It can also occur owing to friction. Because of this, we focus on actual examples of the phenomenon and find one when we study the mechanism of a PHIS: the observable occurrence of vibration. To get a good understanding of this phenomenon, one can look into the vibrational behavior of this material.
The PHIS system may become unsteady due to a variety of causes. Furthermore, determining what causes vibrations in the PHIS system is an urgent problem. However, there are still glaring research problems in the implementation of PHIS, such as a lack of a dynamic model, a superficial understanding of vibration theory, and a lack of system systematicness in theoretical research. As a result, there is substantial theoretical and practical value in investigating PHIS system vibrations. This research will lay the theoretical basis for making PHIS machines that are less prone to vibration and have better designs. To deeply study the SS vibration mechanism of the PHIS system, through this belt spring-block model, we analyze the stability and the SS behavior of the system. The influences of different system parameters on the stability and the SS motion of the PHIS system are also analyzed. Figure 9a and b reveal the location of the definite machine and sensor setup. Two vibration sensors are located on either side of the roller support. The response curves between time and frequency in the y and z axes are extracted to assess the comprehensiveness of the data. The sensors are located horizontally, without any vibration amplitude in the vertical x direction. It must be noted that the predictable vibration will occur in both the moving y direction and the perpendicular z direction caused by bending impact.

Results and Discussion
It is known that the speed is affected by three comparable characteristics, namely stiffness, damping of the system, and force difference at constant geometry, according to the above-mentioned equation of critical SS speed. Figure 10a shows the variation of critical speed at different equivalent stiffness. It is obvious that the change in the critical speed is nearly linear. The value of critical speed increases with the increase of the equivalent stiffness, which has a closing value at a low force difference. Figure 10b explores the variation of critical speed at different values of the system's damping, such as stiffness effect, in which the damping has a linear increase with force difference increase. According to the critical SS equation, Fig. 10a and b give the specific value of critical SS at a specific testing parameter of equivalent stiffness, damping system, and force difference with a variable range of values for each one at constant specific mass. Figure 10 proves that the critical speed of the mass spring damper system is influenced and changed by the variation of the main two components of this system, which are the spring and the damper. Moreover, it shows that the natural of this change at different values of each other, in which the force is considered to be a constant.

The Time-dependence of the Wheel's Vibration
In this subsection, we examine the time history of the wheel's vibration when the injector moves forward to inject the molten plastic. The practical results of the vibrations in the directions y and z were plotted at a different value of the total pressure of the machine, as drawn in parts of Fig. 11. The tested values of operating injector  Fig. 11, where the vibration values for velocities less than critical differ from those for velocities greater than critical, particularly in the y direction. The magnitude of vibration's amplitude at low speeds below the critical speed is found to be greater than at higher speeds. It may be deduced and confirmed what was found in previous theoretical findings regarding the extent to which Based on the theoretical and experimental studies, several key points can be distilled regarding the correlation between observed and predicted outcomes as follows: 1-To investigate the vibrational properties of the PHIS, a spring-mass-damper mechanical model is constructed as the PHIS. 2-According to theoretical calculations, the system's critical instability speed is 0.4703 m.s −1 , and its critical SS speed is 0.1688 m.s −1 . The system will operate in an unstable mode as long as the excitation speeds are less than 0.4703 m.s −1 . The SS motion occurs as the excitation speeds drop below about 0.1688 m.s −1 . 3-The average running speed of the PHIS device is lower than 0.1688 m.s −1 , which demonstrates that the PHIS is easy to vibrate when it is working as its running speed ranges from 0.025 to 0.15. 4-The injector rod's head and body could experience significant vibration deformation during PHIS device operation, jeopardizing the device's stability and reliability. 5-The vibration form of the PHIS device is low-frequency vibration. 6-Theoretical analysis results show that the block will be at rest on the conveyor belt as the excitation speed of the system is less than its critical value. That is to say, the SS motion of the system occurs. In practice, we observe that the average running speed of the PHIS device is less than the values of the critical instability and critical SS speeds. Therefore, there will be unstable SS motion in the pushing process. 7-The scientific aspects of this study revealed the following suggested ways to eliminate and control the SS phenomenon: (a) The derived optimal solution from the investigated theoretical analysis involves optimizing damping and minimizing friction, as shown in Fig. (5). This can be done through several methods, such as placing a lubricant between the injector and the roller support, improving the quality of the surface of them, or choosing a type of less-friction material. (b) Another roller support can be placed before this support to show the effect of the spring, as its effect is not clearly visible.
According to the preceding points, the importance of the practical study and its application with the theoretical study of the system is emphasized.

Conclusion
A 2DOF mechanical simulation model is used to examine the stability and the SS behaviors of the PHIS. The influence of various system characteristics, such as the speed of excitation, equal stiffness along the x axis, damping of the system, and the coefficient of friction, are investigated. The obtained results will be useful in expanding the model of the Plastic Hydraulic Injection mechanism, controlling the working circumstances, and reducing the SS vibration in the Plastic Hydraulic Injection mechanism process. The following are the main points of this paper: -The SMD mechanical model is used to investigate the dynamic properties of hydraulic plastic injection mechanisms. According to theoretical results, the critical instability of the system and the velocity of the SS are 0.4703m∕s and 0.1688 m∕s , respectively. The system is unstable as long as the excitation velocity falls below 0.4703m∕s. -The system's stability is investigated using Lyapunov stability theory; the speed of critical instability is determined to be 0.4703m∕s , and the critical SS speed of the system is calculated to be 0.1688 m∕s . If the excitation speed is greater than 0.4703m∕s , the system becomes stable. -The innervation and decay rate of the system have an important impact on the stability and behavior of SS plastic hydraulic injection. As the rate of excitation and decay of the system increases, the system reaches the asymptotic steady-state after the unstable state and shortens the adhesion phase until it vanishes completely. --There is good agreement between the practical results and the numerical solutions of the governing system. --The impact of various values of the stiffness of the nonlinear spring and the damping on the system's behavior has a good influence on the model's motion, as graphed in Fig. 10. Data Availability Data sharing is not appropriate for this research because no datasets were created or analysed.

Conflict of Interest
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