Abstract The design theory of higher-order stopping mechanisms is an important branch of mechanism design theory. Mathematical analysis shows that the first derivatives of two related functions in a composite function are equal to zero at the same moment, resulting in the first to third derivatives of the composite function being zero at the corresponding moments. This principle has pioneered a new method for designing higher-order stopping mechanisms. Research shows that this mathematical principle applies to dozens of planar or spatial mechanisms where the driven member stops up to the third order at extreme positions. This paper only introduces the design of a planar seven-bar linkage based on a crank-rack slider that stops up to the third order at single or double extreme positions, and studies its higher-order transmission characteristics. This mechanism has a simple structure, is easy to design, and has a large transmission angle. Keywords : Mechanism, Third-order stopping, Composite function, Seven-bar linkage Introduction In textile machinery, packaging machinery, and logistics automation machinery, there are many operations that require the actuator to stop at high orders in single or double extreme positions. Since the basic linkage mechanism does not have such transmission characteristics, if the Watt-type planar six-bar mechanism is adopted, it can be regarded as a series connection of crank-rocker mechanism and double rocker mechanism, and it is impossible to realize the transmission characteristics of the driven part making high-order pauses at both ends of the displacement. If the Stephenson-type planar six-bar mechanism is adopted, although it utilizes the approximate arc feature of the connecting rod curve of the front end mechanism, it is not easy to realize the mechanism design of the driven part making high-order pauses at both ends of the displacement, whether using geometric drawing method or optimization design method, and there may be speed fluctuations during the pause. If the planetary gear train is adopted, although it can realize the driven part making third-order pauses at single or double limit positions[1], the load-bearing capacity of this type of mechanism is relatively small. This situation regarding the mechanism design of the driven part making high-order pauses at single or double limit positions has prompted people to carry out in-depth research on the design theory of high-order pause mechanisms. This paper proposes a design theory for third-order stopping mechanisms based on the fact that the first derivatives of two related functions in a composite function are equal to zero at the same moment, leading to the first to third derivatives of the composite function being zero at the corresponding moments. This transforms the design of higher-order stopping mechanisms into a problem of selecting, constructing, and independently designing two sub-mechanisms. Research shows that this mathematical principle applies to dozens of planar or spatial mechanisms where the driven member stops up to the third order at extreme positions. This paper only introduces the design of a planar seven-bar linkage based on a crank-rack slider that stops up to the third order at single or double extreme positions, and studies its transmission characteristics. The design of this type of stopping mechanism only involves the series connection of basic mechanisms and the analytical design of dimensions. The design is simple, the transmission angle is large, and the transmission and stopping characteristics are superior to the existing loom shedding mechanism. [sup][2][/sup] 1. Design principle of higher-order zeros of composite functions and third-order stopping mechanisms. Let the composite function represent the zero-order transmission function of a class of combined mechanisms, where represents the zero-order transmission function of the input terminal mechanism, and represents the zero-order transmission function of the output terminal mechanism. Let the second-order and higher derivatives with respect to time be zero, that is, the driving member of the combined mechanism rotates at a constant speed. If the value at the same moment is equal to zero, and corresponds to the respective limit positions of the two sub-mechanisms, then the value of is equal to zero at the same time. This indicates that the output member of the combined mechanism has transmission characteristics up to the third-order stopping at one or two limit positions. Based on this mathematical principle, the design of a combined mechanism in which the driven member stops at one or two limit positions can be realized by using the combination of basic mechanisms. 2. Design and Analysis of a Planar Seven-Bar Linkage with Crank-Rack and Slider Extreme Position and Third-Order Stop: In the combined mechanism shown in Figure 1, let the length of crank 1 be ψ, the length between C on guide rod 2 be S2, the angular displacement be δ, the length between DC be R, the length between AD be d, the diameter of gear 5 be d, E be a point on gear 5, DE = R, and the angular displacement of DE be β. The geometric conditions and kinematic relationships of the input-end crank-rack mechanism and the output-end crank-slider mechanism simultaneously reaching their extreme positions are studied. The angular displacement β of gear 5 is generated from two aspects: the linear displacement ΔS2 of rack 2 relative to gear 5 causes the angular displacement ΔS2/(0.5d5) of gear 5; the change in angular displacement Δδ of rack 2 is directly transmitted to the angular displacement Δδ of gear 5. Therefore, the function expression of the angular displacement β of gear 5 is the angular velocity WL5 of gear 5, and the first and second derivatives of the angular velocity are respectively shown in Figure 1. When gear 5 reaches the two extreme positions, the instantaneous center of velocity of the crank-rocker mechanism is shown in Figures 2a and 2b. Figure 2a shows the position of gear 5 swinging to the right limit. Let the length of P12 and P24 be HR. Combining equations (18) and (19), we get the equation for HR. In Figure 1, when R3 = 0.5d5, if βB = or 2π, then slider 7 reaches the upper and lower limit positions respectively. From the functional relationship of equations (1) to (3) (let S = 0), we can see that slider 7 stops at the upper and lower limit positions until the third derivative is zero. The stroke of slider 7 is 2R5, which is independent of the magnitude of other parameters. Then slider 7 only stops at the position where β = 0 until the third derivative is zero. It can be seen that under this geometric condition, the size design of the mechanism is very simple. 3. Transmission characteristics of a three-order planar seven-bar linkage with crank rack and pinion at extreme position of slider. In Figure 1, let R3 = R5 = 0.5d5, 2r1/(0.5d5) = π, 1 = 0.0628m, d4 = 0.2m, 6 = 0.120m, then d5 = 0.080m, R3 = R = 0.040m, H = 0.080m. Thus, the transmission characteristics of slider 7 with respect to crank 1 are shown in Figure 3a. In Figure 1, let R3 = R5 = 0.5d5, 2r1/(0.5d5) = 2π, r1 = 0.1256m, d4 = 0.2m, r6 = 0.120m, then d5 = 0.080m, R3 = R5 = 0.040m, H7 = 0.080m. Thus, the transmission characteristics of slider 7 with respect to crank 1 are shown in Figure 3b. 4. Conclusion Based on the mathematical principle that the first derivatives of two related functions in a composite function are equal to zero at the same time, resulting in the first to third derivatives of the composite function being zero at the corresponding time, a design principle for the third-order pause of the follower of a class of combined mechanisms at single or double limit positions is proposed. By adjusting the slider based on the crank rack at the limit position until the three equations are used, D — — diameter of the series disk (m) γ — — specific gravity of the material (N/m ) µ — — friction coefficient of the wall surface to the material k — k = tanψ, ψ is the internal friction angle of the material h — material height on a single series disk (m) H — — conveying height of the entire series disk conveyor (m) L — — spacing between series disks (m) During the operation of the series disks, the loading time of each series disk is 3 The determination of the rotation speed n[sub]2[/sub] of the receiving drive motor 2 is to ensure the constant power operation of the drive motor 1. The calculation result P of equation (7) should be approximately equal to the rated power P[sub]1[/sub] of the motor 1. Taking the logarithm of both sides and rearranging, it can be seen that the speed of the drive motor n and the speed of the receiving motor n2 are nonlinear. 4 Conclusion 1) The calculation method of the amount of material transported by the series disk conveyor is determined. 2) To meet the requirements of the transportation task, the calculation method of the output shaft torque T[sub]M[/sub] of the drive motor was derived; it was found that T[sub]M[/sub] is not only related to the mechanical friction of the system, but also to the speed n of the receiving system motor, and the relationship between them is nonlinear. To ensure that the drive motor runs under constant power, the control of the speed n[sub]2[/sub] of the receiving motor needs to adopt an intelligent control method. According to the characteristics of this system, the PID neural network control method should be considered. References 1 Wen Bangchun. 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