Modelling and motion characteristics of a new multistage lead screw telescopic mechanism

This paper presents a novel multistage electric telescopic mechanism. The mechanism is composed of multistage lead screw mechanism and multistage telescopic sleeve mechanism. It has the advantages of large stroke, large load and anti-sticking. Moreover, it has superior lateral bearing capacity and can be used in both transverse and longitudinal heavy load conditions. The force of the mechanism is analysed and the motion principle of the mechanism is expounded. Its mathematical model is established and its motion characteristics are studied. The speed, displacement and force data of the lead screw are analysed to verify its reliability and stability. The appropriate design parameters are obtained through the combination of experiment and simulation.


Introduction
Electric telescopic mechanisms are widely used in various industries and fields.However, traditional lead screw-based mechanisms may encounter challenges in maintaining concentricity and avoiding mechanical failures due to offset center of gravity and increased friction and axial locking force.This limits their application in important fields that require lateral load capacity.Therefore, this paper proposes a new multi-stage self-locking anti-stick electric telescopic mechanism that can withstand lateral loads and has a wide range of applications.
Lu Kaiyuan, Wu Weimin [1] proposed a novel electromagnetic lead screw for wave energy applications; KK Varanasi, SA Nayfeh [2] develop a model of lead-screw system dynamics; W Zhang, X Zhang, et al. [3] studied the machine tool with both ends fixed ball screw; Fengqing Tian, Jianmin Zhu, et al. [4] analyzed screw feed drive system; Shen Xin, Zheng Zhi et al. [5] optimized the screw parameters; Wu Honghui, Liu Jianke et al. [6] studied variable-lead rotor and established thermal process simulation model considering leakage; Cao Lin, Wu Jianjun et al. [7] established a precise friction torque model based on double-nut 2-point contact ball screws; Blanchet TA. [8] modeled an elastic screw body; Li Y, Zhang L et al. [9] simulated the forming process of lead screw high-speed cold roll-beating; Vahid-Araghi O, Golnaraghi F. [10] designed a sliding mode controller; Meruva K, Bi Z, Mueller D, et al. [11] modeled and validated the wear and fatigue life of a lead screw actuator; Wu DS, Lai HY, et al. [12] presented a systematic modelling procedure for producing high precision elliptic helicoid screw transmission systems; Ma S, Zhang T, Liu G, et al. [13] developed an analytical model of the planetary roller screw mechanism; Lai HY, Chen CK, et al. [14] presented a new circular helicoidal surface meshing model for producing high transmission performance lead screws; Liu Y, Feng X, et al. [15] studied the transmission performance of micro-nano feed system of hydraulic screw; Jintanawan, T., et al. [16,17] used MATLAB/Simulink to study the dynamic model of the lead screw; Hojjat, Y. and M.M. Agheli [18] studied the capabilities and limitations of roller-screw; Minh-Thuan, H., Y. Wu and T. Van-Quyet [19] proposed mathematical model that used for screw rotor honing with a variable lead.
But the electric telescopic mechanism using the lead screw as a telescopic carrier, when the working state of horizontal motion is needed, the center of gravity of the whole will be offset to the extended side as the lead screw gradually extends, and the concentricity of the lead screw will deviate under the action of gravity.This will increase the friction force between the lead screw threads and the axial locking force between the adjacent two stages of lead screw, resulting in lead screw wear, thread damage, lead screw sticking and other mechanical failures.In addition, this will not only reduce the effective working time of the telescopic mechanism, but also reduce the reliability and life of the telescopic mechanism.The currently used lead screw expansion mechanism can generally only withstand axial load, unable to withstand lateral load, which limits the application of electric expansion mechanism in many important fields.Therefore, this paper proposes a new type of multistage self-locking anti-stick electric telescopic mechanism with large stroke and large lateral load.It can work in a variety of conditions and has a wide range of applications.

Principle of multistage lead screw telescopic mechanism
As shown in Figure 1, the mechanism comprises a motor, a lead screw drive mechanism, a telescopic sleeve mechanism, a support flange, a shell, a force sensor and a connecting flange.The motor is connected to the drive shaft and fixed on the support flange to provide power for the whole system.The connecting flange connects the lead screw drive mechanism and the telescopic sleeve mechanism as a whole and is fixed on the supporting flange to move together.The force sensor is mounted on the connecting flange to monitor the axial pressure and tension during the movement.The mechanism constitutes the movement carrier of the whole system which can greatly improve the travel of the telescopic mechanism.As shown in Figure 2, this schematic diagram illustrates the screw drive mechanism and the telescopic sleeve mechanism.Lead screw drive mechanism includes: trapezoidal groove, ball, ball frame, lead screw nut.Telescopic sleeve mechanism includes: wedge guide block, rectangular slot, plane key.The telescopic sleeve mechanism is connected with the lead screw drive mechanism through the connecting flange and the common movement.It has rectangular slots meshing with the plane key of the retainer ring.The expansion sleeve is designed with a wedge guide block and a conical locking device.The wedge-shaped guide block and the locking device can lock each other and ensure the concentricity of the mechanism in the process of movement.At the same time, there is sufficient overlap between the adjacent lead screw and the adjacent telescopic sleeve.The lead screw and the telescopic sleeve cooperate with each other to enhance the stiffness and lateral load bearing capacity of the mechanism.A force sensor is installed at the connecting flange to monitor the axial force during movement.A ball bearing device is installed between the adjacent screw nuts to prevent them from sticking.Working process of large stroke multi-stage self-locking anti-stick electric telescopic mechanism: during extension, the motor rotates the drive shaft, which transfers torque to the lead screw transmission mechanism.All levels of the lead screw are integrated through nut matching.When the primary lead screw is fully extended, the trapezoidal groove end face of the lead screw reaches the end face of the upper primary lead screw nut and is tightened to extend as a whole.All lead screws are fully extended through this circular extension.The lead screw drive mechanism through the connecting flange and telescopic sleeve mechanism is connected as one of the common rotations, the telescopic sleeve is provided with a rectangular slot, and the plane key on the baffle-ring to cooperate with each other to prevent the mechanism from rotating and rotating motion into linear motion, until the telescopic sleeve mechanism is completely extended; When retracted, the motor provides reverse rotation torque.The drive shaft drives the screw drive mechanism and the telescopic sleeve mechanism to rotate, and the rectangular groove on the telescopic sleeve and the plane key on the baffle-ring force the rotating motion into a straight line.When the first level of the lead screw is completely recovered, the end face of the nut is pushed to the end face of the upper level of the lead screw nut to form a whole for rotation.When the expansion sleeve of the first level is completely recovered, the retaining ring will hold the upper level of the retaining ring for recovery movement.Lead screw nut and trapezoidal groove against the recovery.All lead screws and telescopic sleeves are fully recovered in this circulation mode.

Force analysis of kinematic pair of mechanism
The motion pair of the mechanism is mainly screw pair, which can be equivalent to the horizontal force F acting on the middle diameter to push the slider to move uniformly along the thread.Therefore, as shown in Figure 3, the rectangular thread is expanded along diameter d 2 to get an inclined plane.Figure 3(a) shows the screw tightening state.F acting on the slider is the driving force, the axial load FQ is a resistance, and the total reaction force is Fr.When the nut is rotated at a constant speed relative to the lead screw, it can be seen as a slide block (nut) along the slope of the thread diameter d2, and the slope of the thread angle L. When the nut is tightened at a constant speed, it is equivalent to pushing the slide block upward along the inclined plane with a horizontal force F. The reaction force is FN, the friction force is f, and the friction angle is ρ.Since the friction force is downward when the slider rises along the inclined plane, the angle between Fr and FQ is + λ ϕ .The balance equation of tightening can be obtained as shown in Equation (1).
As shown in Figure 3(b), when the lead screw is released, it is equivalent to making the sliding block slide along the inclined plane at a constant speed.The axial load FQ becomes the driving force and F becomes the balancing force required to maintain the constant speed motion of the sliding block.The force balance equation of decline is obtained as shown in Equation ( 2).

( ) ( )
The axial displacement l and axial velocity v of the lead screw are represented by Equation (3).
where S represents the lead of lead screw, w represents the angular velocity of the lead screw, λ (2.1°) represents the helix angle, and n represents the rotational speed of the lead screw.The transmission efficiency of the lead screw is shown in Equation ( 4).tan 0.95 tan( ) where coefficient 0.95 represents the efficiency of the bearing and ρ = 8°.So, the efficiency of the lead screw is 19.5%.

Force analysis of lead screw
As shown in Figure 4, this is the connection section between the drive shaft and the primary lead screw.The nut and the lead screw on each rod are fixed as one through the pin, in which case the force on the fixed surface can be ignored.When the lead screw runs at a constant speed and the shoulder of the primary lead screw is not in contact with the nut of the drive shaft, the force of the lead screw at this time is respectively the contact force between the screw nut and the threaded surface of the primary lead screw, the static friction force between the primary lead screw nut and the secondary lead screw, and the sliding friction force between the primary lead screw and the inner wall of the drive shaft.When the primary lead screw extends outward, it is equivalent to the drive shaft nut sliding downward.In this case, the model of equivalent slider sliding down the inclined plane can be obtained.According to the force model of rectangular thread, it can be equivalent to sliding block down the inclined plane.The axial load FQ becomes the driving force, and F becomes the balance force required to maintain the constant speed motion of sliding block.When there is relative slide between the primary lead screw and the drive shaft, the primary lead screw and the second, third and fourth lead screw are a whole.The sliding friction of the drive shaft against the lead screw is the driving force f01.The axial load force of the drive shaft against the lead screw thread is Q01 F .The sliding friction force f1 between the shoulder of the lead screw and the inner wall of the drive shaft and the load F1 on the lead screw.When the lead screw rotates at uniform speed, the force balance equation of the primary lead screw (Equation ( 5)) can be listed.
The axial thrust is transmitted through the contact between the nut and the shoulder and the energy loss due to collision and friction between them is ignored.The torque is transmitted through the friction torque of the thread.The material of each lead screw is the same, so the coefficient of friction between them is the same.Therefore, the friction torque generated by the drive shaft is only related to the middle diameter of the lead screw thread.Then tightening friction torque T1n and unscrewing friction torque T2n of all levels of lead screw can be obtained, as shown in Equation ( 6).
( ) where ρ is the friction angle; β is the lead angle; n d Is the middle diameter of the thread.

Model and verify
In the process of the lead screw extension, the axial pressure is mainly generated by the axial pressure between the lower end face of the nut between the adjacent two stages of the lead screw.When replacing the extended lead screw, the axial pressure and speed of the lead screw will fluctuate.This paper builds a model from servo motor to reducer, lead screw and load.Its schematic diagram is shown in Figure 5, and its experimental device is shown in Figure 6.The servo motor is used in this experiment, its voltage is 220V, the rated speed is 1800 r/min, and the output torque is 10 N M ⋅ .The reduction ratio of the reducer is 20, so the output torque applied to the lead screw (150 mm) is 200 N M ⋅ .The friction coefficient of the lead screw is 0.15, and the pressure of the load is 2500 N, so the maximum friction force of the lead screw is 375 N. The displacement sensor collects the displacement information of the lead screw.The force sensor is installed at the front end of the lead screw and transmits the collected information to the industrial computer.The rotation speed of the motor is collected through the motor driver.After establishing the model of the telescopic mechanism, the actual rotation speed and displacement data of lead screw expansion mechanism are obtained through experiments.Then its simulation data of the rotation speed and displacement are obtained by the simulation model.The comparison of experimental data and simulation data is shown in Figure 7.In Figure 7, the experimental displacement curve of the lead screw telescopic mechanism is located below the simulation displacement curve.And the maximum difference between experimental data and simulation data of lead screw displacement is 0.23m.Considering the gap error between the lead screw and the error of the displacement sensor, the overall error is within 0.25% and is in the acceptable range.Similarly, because of the gap between lead screws, the experimental curve of lead screw rotation speed is slightly behind the simulation curve.And the fluctuation of the experimental curve and simulation curve of the screw rotation speed is within the allowable range of system error.Therefore, the model of lead screw telescopic mechanism established is correct.

Simulation and analysis
After the model is established, the lead screw rotation speed, displacement and axial pressure curves are obtained by simulation.Figure 8 shows the curve of the speed of the lead screw, and Figure 9 shows the curve of the lead screw displacement.As can be seen from figures, the first stage lead screw rises to 30 r/min in a few seconds and stabilizes.When the first lead screw extends to its maximum stroke of 1.5m, it reaches the limit and stops rotating.At this time, the speed of the second stage lead screw increases to 35 r/min instantaneously at about 190 s, and then it becomes stable after two oscillations and stabilizes at 30 r/min.Similarly, when the second stage lead screw extends to its maximum limit of 2.8m, the third stage lead screw extends back.When the third stage lead screw extends to the maximum limit, the fourth stage lead screw will extend.As can be seen from Figure 9, the first three lead screws will stop moving when they reach the limit.In this case, the adjacent lead screws move as a whole.The fourth stage lead screw is the last stage lead screw, and its displacement curve is approximately a straight line.In the process of the lead screw extension, the axial pressure is mainly generated by the axial pressure between the shoulder and the lower end face of the upper nut between the adjacent two stages of lead screw.During the movement, the axial pressure curve of the lead screw is shown in Figure 10.When moving, when replacing the extended lead screw, the axial pressure will fluctuate and the pressure will decrease with the increase of the lead screw series.This is because in the front of screw need to take the weight of the screw.In the normal single stage lead screw extension process, the overall axial pressure is basically stable.The screw speed curves under 50kg, 75kg and 100kg loads are shown in Figure 11.It can be obtained that with the increase of load, the lead screw start time is later and later.But the overall trend and amplitude of speed change is basically the same.The axial pressure curve of the lead screw under different loads is shown in Figure 12.The greater the load, the greater the axial pressure the lead screw will suffer.Through the above simulation analysis, the appropriate lead screw parameters are finally obtained, including the middle diameter, lead and length of each lead screw, as shown in Table 1.

Conclusions
The proposed multistage screw telescopic mechanism is a promising solution for applications that require high extension stroke and precision positioning.The paper provides a comprehensive analysis of the mechanism's force, motion, and friction characteristics, which allows for the optimization of the design parameters.
The force balance equation and mathematical model developed in this paper are essential for understanding the behaviours of the mechanism and predicting its performance.The simulation results obtained using the model are in good agreement with experimental data, which confirms the accuracy and reliability of the model.
The appropriate lead screw parameters obtained from the simulation can guide the selection of screw pairs and ensure the optimal performance of the mechanism.The proposed model and analysis can be used as a reference for the design and optimization of multistage screw telescopic mechanisms in various engineering applications.

Figure 2 .
Figure 2. Structure diagram of lead screw and telescopic sleeve.

Figure 3 .
Figure 3. Mechanical principal diagram of screw pair.

Figure 4 .
Figure 4. Force diagram of lead screws.

Figure 7 .
Figure 7. Rotation speed and displacement curves of lead screw expansion mechanism.

Figure 8 .
Figure 8. Rotation speed curves of lead screws.

Figure 10 .
Figure 10.Axial pressure curves of lead screws.

Figure 11 .
Figure 11.Lead screws rotation speed under the load of 50kg, 75kg and 100kg.

Figure 12 .
Figure 12.Lead screws axial pressure under the load of 50kg, 75kg and 100kg.