A method for calculating a suspension function of a multi-wheel suspension chain system

Abstract

The invention relates to a kind of multi-wheel chain suspension system suspension function calculation method, and belongs to the technical field of engineering machinery design.The calculation method of the invention is suitable for chain transmission trajectory movement in the same plane;the polygon effect of chain wheel and chain is not obvious, and can be calculated by using the index circle;the chain system presents rigid property and does not occur obvious deformation;the number of chain wheel is greater than or equal to 2;by establishing accurate geometric, mechanical and total length constraint equation set, a closed nonlinear system is formed.After reasonable initial prediction of the lowest point tension of catenary and the direction angle of chain wheel meshing point, the accurate value is obtained by iterative solution.The accurate equation of catenary of each section of chain in the system, the tension distribution along the chain and the sag at any position can be further calculated.The method has high precision and strong universality, can process chain wheel system with any number and arrangement, and provides an effective theoretical calculation tool for the design, tension optimization and state analysis of chain transmission.

Description

Technical Field

[0001] This invention belongs to the field of engineering machinery design technology, specifically relating to a method for calculating the catenary function of a multi-wheel chain catenary system. Background Technology

[0002] The computational importance of multi-wheel chain drive systems is becoming increasingly prominent in chain drive design. This is especially true for long-distance transport systems supported by multiple wheels, tension wheel systems for controlling tension, and idler wheel systems for controlling transmission guidance; how to design them rationally becomes a major issue. Traditional design and simulation methods are time-consuming and costly. Therefore, designing an algorithm applicable to the calculation of catenary morphology in various multi-wheel chain drives would allow for in-depth analysis of the entire ideal system and determine whether it meets design expectations.

[0003] The existing methods for analyzing catenary systems have the following problems:

[0004] 1. Traditional chain drive chain morphology calculation methods mainly fall into two categories: one is simplified calculation tools based on empirical formulas, and the other is general-purpose finite element analysis software. The former lacks accuracy, while the latter is complex to use and computationally expensive. High-precision dedicated analysis tools specifically for multi-sprocket chain systems are currently lacking.

[0005] 2. Traditional chain drive chain shape calculation methods make too many simplification assumptions: Existing methods are usually based on simplified geometric assumptions and cannot accurately handle the spatial geometric relationships in complex multi-wheel systems, especially when the sprocket positions are irregular;

[0006] 3. Traditional chain drive chain shape calculation methods suffer from inaccurate catenary calculations: Firstly, using a straight line instead of the tight side of the chain for calculation can accurately simulate the tight side under high load conditions, but under low load conditions, the discrepancy with reality is too large, easily leading to a series of design errors. Secondly, using a parabola instead of the loose side of the chain for calculation often only simulates the catenary shape under specific conditions, and the difference from reality increases rapidly after changing the boundary conditions.

[0007] 4. Lack of systematic solution methods in multi-sprocket chain drive systems: For systems containing multiple sprockets, there is a lack of effective numerical methods to simultaneously solve the interaction and tension distribution between all sprockets. Summary of the Invention

[0008] The purpose of this invention is to propose a method for calculating the suspension function of a multi-wheel chain suspension system, in order to solve the problems of insufficient accuracy, high calculation cost, and inaccurate calculation of multi-wheel chain transmission systems in the existing technology.

[0009] To achieve the above objectives, the present invention provides a method for calculating the catenary function of a multi-wheel chain catenary system, applicable to chain drive trajectories moving within the same plane; where the polygonal effect between the sprockets and chain is not significant, and calculation using the pitch circle is employed; where the chain system exhibits rigidity and does not undergo significant deformation; and where the number of sprockets is greater than or equal to 2; the method includes the following steps:

[0010] Step 1: Determine the boundary conditions and parameters of the chain drive system. The boundary conditions and parameters include at least: the number of sprockets n, the center coordinates and pitch circle radius of each sprocket in the calculation plane, the total length of the chain, the chain linear density, the gravitational acceleration under actual conditions, and the tension exerted by the sprockets on the chain in the observation plane; the sprockets include driving sprockets, driven sprockets, tension sprockets, and idler sprockets; the tension includes driving force, load force, and centrifugal force.

[0011] Step Two: Based on the boundary conditions and parameters determined in Step One, establish the geometric constraint equations between sprocket and chain based on the meshing geometry of the chain and sprocket. Establish the mechanical equilibrium equations between chain and sprocket based on the force balance relationship. Finally, establish the total length constraint equations, including all catenary segments and wrap angle arc segments, based on the total chain length. These equations together constitute a constraint equation containing 3... n A system of nonlinear equations with independent equations, where the unknowns are... n The lowest point tension of the catenary n The tangential angle of the chain at each sprocket engagement point and n The tangential angle of the chain at each sprocket engagement point;

[0012] Step 3: Predict initial values ​​for the unknowns in the nonlinear equation system described in Step 2;

[0013] Step 4: Substitute the unknowns predicted in Step 3 into the nonlinear equations from Step 2 and perform iterative numerical solutions until convergence, obtaining the desired result. n The lowest point tension of the catenary n The tangential angle of the chain at each sprocket engagement point and n The exact solution to the tangent angle of the chain at each sprocket engagement point;

[0014] Step 5: Using the exact solution obtained in Step 4, calculate and output the catenary function, tension distribution function, line function connecting the catenary to the two end contact points, and sag distribution function between each adjacent sprocket.

[0015] Step two, which involves establishing the geometric constraint equations between the sprocket, chain, and sprocket based on the meshing geometry of the chain and sprocket, specifically includes the following cases:

[0016] When the sprocket-chain-sprocket relationship is such that the chain cuts from the top of the preceding sprocket to the bottom of the following sprocket, the geometric constraint equation is:

[0017] ;

[0018] When the sprocket-chain-sprocket relationship is such that the chain cuts into the upper side of the preceding sprocket from the upper side of the following sprocket, the geometric constraint equation is:

[0019] ;

[0020] When the sprocket-chain-sprocket relationship is such that the chain cuts into the underside of the preceding sprocket from the underside of the following sprocket, the geometric constraint equation is:

[0021] ;

[0022] When the sprocket-chain-sprocket relationship is such that the chain cuts from the underside of the preceding sprocket to the topside of the following sprocket, the geometric constraint equation is:

[0023] ;

[0024] in:( x i , y i () represents the coordinates of the center point of the previous sprocket in the coordinate system;

[0025] ( x i+1 , y i+1 ( ) represents the coordinates of the center point of the next sprocket in the coordinate system;

[0026] R i The pitch circle radius of the previous sprocket;

[0027] R i+1 The pitch circle radius of the next sprocket;

[0028] φ out,i To create a chamfer at the point of engagement with the previous sprocket;

[0029] φ in,i+1 The angle is cut at the point of engagement with the next sprocket.

[0030] The system of equations is established with the sprocket with the smallest x-coordinate as sprocket 1, and then sequentially to the next sprocket connected to the chain. If the x-coordinate of the contact point of the chain disengaging from the previous sprocket is less than the x-coordinate of the contact point of the chain engaging with the next sprocket, the equations are established in the forward direction; if the x-coordinate of the contact point of the chain disengaging from the previous sprocket is greater than the x-coordinate of the contact point of the chain engaging with the next sprocket, the equations are established in the reverse direction.

[0031] Step two, which involves establishing the mechanical equilibrium equations and corresponding envelope angle equations between the chain, sprocket, and chain based on the force balance relationship, specifically includes:

[0032] The formula for the chain enveloping the sprocket from the top is:

[0033] ;

[0034] The formula for the chain enveloping the sprocket from the right side is:

[0035] ;

[0036] The formula for the chain to envelop the left side of the sprocket is:

[0037] ;

[0038] The formula for the chain enveloping the sprocket from below is:

[0039] ;

[0040] in: Cut an angle at the sprocket engagement point;

[0041] Cut an angle at the point where the sprocket engages;

[0042] θ It is the envelope angle;

[0043] F The sprocket applies the total force to the chain;

[0044] Tension at the engagement point;

[0045] The tension at the disengagement point.

[0046] The chain equation set includes a pairwise combination based on the sprocket equations:

[0047] Combination 1 is a combination of two types of equations that engage from the bottom of the sprocket and two types of equations that disengage from the top of the sprocket; Combination 2 is a combination of two types of equations that engage from the top of the sprocket and two types of equations that disengage from the top of the sprocket; Combination 3 is a combination of two types of equations that engage from the bottom of the sprocket and two types of equations that disengage from the bottom of the sprocket; Combination 4 is a combination of two types of equations that engage from the top of the sprocket and two types of equations that disengage from the bottom of the sprocket.

[0048] In the cases of combination 1, combination 2, and combination 3, when the x-coordinate of the sprocket engagement point is less than the x-coordinate of the chain entry point into the next sprocket engagement point, the equation is established in the positive direction; when the x-coordinate of the sprocket engagement point is greater than the chain entry point into the next sprocket engagement point, the equation is established in the reverse direction.

[0049] In combination 4, if the x-coordinate of the sprocket exit contact point is less than the x-coordinate of the chain entering the next sprocket contact point, the equation is established in reverse. If the x-coordinate of the sprocket exit contact point is greater than the x-coordinate of the chain entering the next sprocket contact point, the equation is established in the forward direction.

[0050] Step two, which involves establishing a total length constraint equation based on the total chain length that includes all catenary segments and wrap angle segments, specifically involves:

[0051] The equation for calculating the length of each catenary segment is:

[0052] ;

[0053] in: L i For the first i Total length of the catenary segment;

[0054] The angle between the catenary segment and the point where it engages with the previous sprocket is cut.

[0055] The angle between the catenary segment and the point where it engages with the next sprocket is cut.

[0056] a i The catenary parameters calculated for this segment of the catenary;

[0057] The equation for calculating the total length of the chain is:

[0058] ;

[0059] in: L total This is the total length of the chain;

[0060] θ i The envelope angles on each pitch circle;

[0061] R i Let be the radius of each pitch circle.

[0062] Step three, which involves predicting the initial values ​​of the unknowns in the nonlinear equation system described in step two, specifically includes:

[0063] The n The tangential angle of the chain at each sprocket engagement point is approximated by the angle between the line connecting the center point of the engaging sprocket and the center point of the previous sprocket and the horizontal direction.

[0064] n The tangential angle of the chain at each sprocket engagement point is approximated by the angle between the line connecting the center point of the engagement sprocket and the center point of the next sprocket and the horizontal direction.

[0065] In the static state of chain drive, n The tension at the lowest point of the catenary segment is approximated by the gravity value associated with the chain segment.

[0066] In the dynamic state of chain drive, n The lowest point tension of the catenary segment is approximated by applying force to the tight side segment, and by applying gravity value related to the chain segment to the loose side segment.

[0067] Step five involves calculating and outputting the catenary function, tension distribution function, line function connecting the catenary to the contact points at both ends, and sag distribution function between adjacent sprockets.

[0068] According to the formula T / qg = a Calculate the catenary parameters a ;

[0069] in: T The tension at the lowest point of the catenary;

[0070] q These are the actual parameters of the chain linear density determined earlier;

[0071] g These are the actual parameters for gravitational acceleration;

[0072] The catenary function is:

[0073] y = a cosh(( x - x 1) / a )- a + y 1,

[0074] in: x 1 represents the horizontal movement size;

[0075] y 1 represents the vertical movement size;

[0076] The tension distribution function is:

[0077] y = aqg cosh(( x - x 1) / a )- aqg + y 1* qg ;

[0078] The connection function is:

[0079] y=( y in - y out ) / ( x in - x out ))*( x - x 1)+ y 1;

[0080] in:( x in , y in () represents the coordinates of the engagement point;

[0081] ( x out , y out ( ) represents the coordinates of the disengagement point;

[0082] The vertices distribution function is:

[0083] y =( y in - y out ) / ( x in - x out ))*( x - x 1)- a cosh(( x - x 1) / a )+ a .

[0084] The beneficial effects of this invention are as follows: The method for calculating the catenary function of a multi-wheel chain catenary system uses the accurate equation of the catenary curve, avoiding errors caused by approximations of parabolic or straight lines. It is applicable to sprocket systems of any number, arrangement (except horizontal, inclined, and vertical), and under any force (driving, load, tension), exhibiting strong versatility. By establishing and simultaneously solving the equations of the entire system, the mutual coupling effects between multiple sprockets can be handled simultaneously. It provides a precise quantitative analysis tool for chain drive design, tension adjustment, wear assessment, and life prediction. Traditional chain drive calculation methods lack the necessary precision. This invention utilizes a chain drive system that considers the catenary morphology, making it better suited for chain drive system design and enabling reasonable analysis under multiple conditions based on changes in driving force. Using the solved catenary equation, on the one hand, the sag equation at any point in the segment can be obtained by calculating the difference between the contact point line equation and the catenary equation; on the other hand, the catenary equation can be converted into a smoothly varying tension equation on the chain segment based on conditions such as gravitational acceleration and chain linear density. This invention provides a reference for theoretical design and offers a series of precise data for analysis of problems such as chain elongation and the formation of new catenaries under motion conditions and after wear. Attached Figure Description

[0085] Figure 1 This is a schematic diagram of the method for calculating the catenary function of a multi-wheel chain catenary system according to the present invention;

[0086] Figure 2 This is a schematic diagram of the principle of the chain suspension function calculation method and the sprocket-chain-sprocket judgment method for a multi-wheel chain suspension system according to the present invention;

[0087] Figure 3 This is a schematic diagram of the chain-sprocket-chain judgment method for a multi-wheel chain suspension system according to the present invention.

[0088] Figure 4 This invention provides a solution example diagram for a method of calculating the catenary function of a multi-wheel chain catenary system under one specific condition.

[0089] Among them: 1. First sprocket, 2. Second sprocket, 3. Third sprocket, 4. Fourth sprocket. Detailed Implementation

[0090] The embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0091] See Figure 1 This invention discloses a method for calculating the catenary function of a multi-wheel chain catenary system, applicable to chain drive trajectories moving within the same plane; where the polygonal effect between the sprockets and chain is not significant, and calculation using the pitch circle is employed; where the chain system exhibits rigidity and does not undergo significant deformation; and where the number of sprockets is greater than or equal to 2; the method includes the following steps:

[0092] Step 1: Determine the boundary conditions and parameters of the chain drive system. These boundary conditions and parameters include at least: the number of sprockets (n), the center coordinates and pitch circle radius of each sprocket in the calculation plane, the total chain length, the chain linear density, the gravitational acceleration under actual conditions, and the tension exerted by the sprockets on the chain within the observation plane. The tension exerted by the sprockets on the chain can be directly measured by a chain tension sensor or calculated by dividing the sprocket torque by the pitch circle radius. The remaining boundary conditions are actual parameters under real-world conditions and do not require calculation. The sprockets include a driving sprocket, a driven sprocket, a tensioning sprocket, and an idler sprocket. The tension includes driving force, load force, and centrifugal force.

[0093] Step Two: Based on the boundary conditions and parameters determined in Step One, establish the geometric constraint equations between sprocket and chain based on the meshing geometry of the chain and sprocket. Establish the mechanical equilibrium equations between chain and sprocket based on the force balance relationship. Finally, establish the total length constraint equations, including all catenary segments and wrap angle arc segments, based on the total chain length. These equations together constitute a constraint equation containing 3... n A system of nonlinear equations with independent equations, where the unknowns are... n The lowest point tension of the catenary n The tangential angle of the chain at each sprocket engagement point and n The tangential angle of the chain at each sprocket engagement point;

[0094] Step 3: Predict initial values ​​for the unknowns in the nonlinear equation system described in Step 2;

[0095] Step 4: Substitute the unknowns predicted in Step 3 into the nonlinear equations from Step 2 and perform iterative numerical solutions until convergence, obtaining the desired result. n The lowest point tension of the catenary n The tangential angle of the chain at each sprocket engagement point and n The exact solution to the tangent angle of the chain at each sprocket engagement point;

[0096] Step 5: Using the exact solution obtained in Step 4, calculate and output the catenary function, tension distribution function, line function connecting the catenary to the two end contact points, and sag distribution function between each adjacent sprocket.

[0097] Step two, which involves establishing the geometric constraint equations between the sprocket and chain based on the meshing geometry of the chain and sprocket, includes four cases: the chain equations on the upper side of the front sprocket and the rear sprocket, the chain equations on the upper side of the front sprocket and the lower side of the rear sprocket, the chain equations on the lower side of the front sprocket and the upper side of the rear sprocket, and the chain equations on the lower side of the front sprocket and the rear sprocket. Specifically:

[0098] When the sprocket-chain-sprocket relationship is such that the chain cuts from the top of the preceding sprocket to the bottom of the following sprocket, the geometric constraint equation is:

[0099] ;

[0100] When the sprocket-chain-sprocket relationship is such that the chain cuts into the upper side of the preceding sprocket from the upper side of the following sprocket, the geometric constraint equation is:

[0101] ;

[0102] When the sprocket-chain-sprocket relationship is such that the chain cuts into the underside of the preceding sprocket from the underside of the following sprocket, the geometric constraint equation is:

[0103] ;

[0104] When the sprocket-chain-sprocket relationship is such that the chain cuts from the underside of the preceding sprocket to the topside of the following sprocket, the geometric constraint equation is:

[0105] ;

[0106] in:( x i , y i () represents the coordinates of the center point of the previous sprocket in the coordinate system;

[0107] ( x i+1 , y i+1 ( ) represents the coordinates of the center point of the next sprocket in the coordinate system;

[0108] R i The pitch circle radius of the previous sprocket;

[0109] R i+1 The pitch circle radius of the next sprocket;

[0110] φ out,i To create a chamfer at the point of engagement with the previous sprocket;

[0111] φ in,i+1 The angle is cut at the point of engagement with the next sprocket.

[0112] The sprocket equation set includes four types of equations based on the chain contact state between the driving sprocket, driven sprocket, tensioner sprocket, and idler sprocket. Each equation set is established by substituting the angle between the tangent direction of the contact point and the positive horizontal direction to ensure consistency. The equation set is established sequentially from the sprocket with the smallest horizontal coordinate (sprocket 1) to the next sprocket connected to the chain. If the horizontal coordinate of the chain disengaging from the previous sprocket is less than the horizontal coordinate of the chain engaging with the next sprocket, the equation is established in the forward direction. If the horizontal coordinate of the chain disengaging from the previous sprocket is greater than the horizontal coordinate of the chain engaging with the next sprocket, the equation is established in the reverse direction. This sprocket equation set ensures that the entire system is a closed loop and that the number of equations is twice the number of sprockets. n .

[0113] See Figure 2 These are four methods of contact between sprockets, chains, and sprockets as listed in this invention.

[0114] Formula 1:

[0115] ;

[0116] Formula 2:

[0117] ;

[0118] Formula 3:

[0119] ;

[0120] Formula 4:

[0121] ;

[0122] in:( x i , y i () represents the coordinates of the center point of the previous sprocket in the coordinate system;

[0123] ( x i+1 , y i+1 ( ) represents the coordinates of the center point of the next sprocket in the coordinate system;

[0124] R i The pitch circle radius of the previous sprocket;

[0125] R i+1 The pitch circle radius of the next sprocket;

[0126] φ out,i To create a chamfer at the point of engagement with the previous sprocket;

[0127] φ in,i+1 The angle is cut at the point of engagement with the next sprocket.

[0128] The four contact methods, when properly designed, can cover the vast majority of multi-wheel chain drive design models, including: multi-wheel output systems with multiple driven wheels sharing the driving force, complex path transmission and long-distance transmission systems with multiple tension wheels, and guide systems with multiple idler wheels.

[0129] To accommodate most system designs, four contact methods between sprockets, chains, and sprockets are introduced. Considering that the sprocket radius may be much larger than the center distance or the contact point between the chain and sprocket may change under various conditions, the starting point of the catenary is determined by the contact point instead of the center point. Four tangent lines are drawn between adjacent sprockets, and the calculation equation is determined based on the horizontal coordinates of the tangent points. Theoretically, this method can perform catenary calculations for all chain systems except those containing vertical straight lines.

[0130] See Figure 3 Step two, establishing the mechanical equilibrium equations between the chain, sprocket, and chain based on the force balance relationship, includes calculating the relevant equations for the wrap angle on the upper side, the right side, the left side, and the lower side; and the corresponding envelope angle equations, specifically including:

[0131] The formula for the chain enveloping the sprocket from the top is:

[0132] ;

[0133] The formula for the chain enveloping the sprocket from the right side is:

[0134] ;

[0135] The formula for the chain to envelop the left side of the sprocket is:

[0136] ;

[0137] The formula for the chain enveloping the sprocket from below is:

[0138] ;

[0139] in: Cut an angle at the sprocket engagement point;

[0140] Cut an angle at the point where the sprocket engages;

[0141] θ It is the envelope angle;

[0142] F The sprocket applies the total force to the chain;

[0143] Tension at the engagement point;

[0144] The tension at the disengagement point.

[0145] The chain equation set includes a pairwise combination based on the sprocket equations:

[0146] Combination 1 is a combination of two types of equations for meshing into the sprocket from the bottom and two types of equations for meshing out from the top of the sprocket; Combination 2 is a combination of two types of equations for meshing into the sprocket from the top and two types of equations for meshing out from the top of the sprocket; Combination 3 is a combination of two types of equations for meshing into the sprocket from the bottom and two types of equations for meshing out from the bottom of the sprocket.

[0147] In the cases of combination 1, combination 2, and combination 3, when the x-coordinate of the sprocket engagement point is less than the x-coordinate of the chain entry point into the next sprocket engagement point, the equation is established in the positive direction; when the x-coordinate of the sprocket engagement point is greater than the chain entry point into the next sprocket engagement point, the equation is established in the reverse direction.

[0148] Combination 4 is a combination of two types of equations that engage from the top of the sprocket and two types of equations that disengage from the bottom of the sprocket.

[0149] In combination 4, when the x-coordinate of the sprocket disengagement contact point is less than the x-coordinate of the chain entering the next sprocket engagement contact point, the equation is established in reverse. When the x-coordinate of the sprocket disengagement contact point is greater than the x-coordinate of the chain entering the next sprocket engagement contact point, the equation is established in the forward direction.

[0150] The chain equation set consists of the mechanical equations related to all sprockets except for sprocket 1, and the number of mechanical equations established in the closed system. n -1.

[0151] The four contact methods between chain and sprocket are combined in pairs according to the four contact methods between sprocket and chain, and the coordinates of the chain's engagement point with the sprocket and the engagement point with the next sprocket are also considered. Since using only the previous equation combination for judgment cannot determine whether the calculation direction in the calculation process is followed, which may lead to errors in the determination of the driving force direction, the horizontal coordinates of the chain's engagement point with the sprocket and the engagement point with the next sprocket are used for judgment. This can cover the calculation of catenary systems for all chain systems except those containing vertical straight lines.

[0152] Step two, which involves establishing a total length constraint equation based on the total chain length that includes all catenary segments and wrap angle segments, specifically involves:

[0153] The equation for calculating the length of each catenary segment is:

[0154] ;

[0155] in: L i For the firsti Total length of the catenary segment;

[0156] The angle between the catenary segment and the point where it engages with the previous sprocket is cut.

[0157] The angle between the catenary segment and the point where it engages with the next sprocket is cut.

[0158] a i The catenary parameters calculated for this segment of the catenary;

[0159] The equation for calculating the total length of the chain is:

[0160] ;

[0161] in: L total This is the total length of the chain;

[0162] θ i The envelope angles on each pitch circle;

[0163] R i Let be the radius of each pitch circle.

[0164] The establishment of the chain's total length equation involves equations between all sprockets and equations between chains:

[0165] In the sprocket equation, the length of the catenary segment is calculated from the x-coordinate of the point where the chain engages the previous sprocket to the x-coordinate of the point where the chain engages the next sprocket, with the x-coordinates calculated from smallest to largest. n ;

[0166] In the chain equation, the forward calculation from the chain engaging the gear to disengaging the gear is performed, and the product of the wrap angle arcs of all sprockets including wheel 1 and the corresponding pitch circle radii is used. n ;

[0167] The equation for the total length of the chain is an equation established by summing all the chain segments and the total length of the chain.

[0168] Step three, which involves predicting the initial values ​​of the unknowns in the nonlinear equation system described in step two, specifically includes:

[0169] The n The tangential angle of the chain at each sprocket engagement point is approximated by the angle between the line connecting the center point of the engaging sprocket and the center point of the previous sprocket and the horizontal direction.

[0170] nThe tangential angle of the chain at each sprocket engagement point is approximated by the angle between the line connecting the center point of the engagement sprocket and the center point of the next sprocket and the horizontal direction. Multiple sets of values ​​between the minimum minus π / 3 and the maximum plus π / 3 are taken and substituted into the calculation. The tilt angle values ​​within this range are close to the actual values, and there is a large success rate of iteration.

[0171] In the static state of chain drive, n The lowest point tension of a catenary segment is approximated by the gravity value associated with that segment, typically taken as 1 / 3 of the total chain weight. n Approximate substitution;

[0172] In the dynamic state of chain drive, n The tension at the lowest point of the catenary is affected by the driving or load force applied by the sprocket. When the force applied by the sprocket to the chain segment is much greater than its own weight, the tight side chain segment is approximated by the applied force, while the loose side chain segment is approximated by the gravity value related to the chain segment.

[0173] The engagement and disengagement angles are calculated using the angle between the line connecting the center points of the front and rear sprockets and the horizontal direction as the initial value. If the x-coordinate of the current sprocket disengagement contact point is less than the x-coordinate of the rear sprocket engagement contact point, the angle value is used for calculation. If the x-coordinate of the current sprocket disengagement contact point is greater than the x-coordinate of the rear sprocket engagement contact point, the negative angle value is used for calculation.

[0174] The initial tension estimate for the catenary segment takes into account that, under conditions where no lateral tension is applied, the tension at the lowest point of the catenary segment is much less than the weight of that segment. The initial tension estimate includes a minimum of 1 / 80 of the total weight of the chain segment divided by the number of sprockets, which can be used for calculations of most multi-wheel catenary systems. The maximum initial tension estimate is the theoretical driving force exerted by the sprockets on the corresponding chain in the chain drive system.

[0175] Step five involves calculating and outputting the catenary function, tension distribution function, line function connecting the catenary to the contact points at both ends, and sag distribution function between adjacent sprockets.

[0176] According to the formula T / qg = a Calculate the catenary parameters a ;

[0177] in: T The tension at the lowest point of the catenary;

[0178] q These are the actual parameters of the chain linear density determined earlier;

[0179] g These are the actual parameters for gravitational acceleration;

[0180] aThe shape of a catenary segment can be determined, but its fixed coordinates in the coordinate system cannot be determined. The tangent angles at different points on the same catenary segment are different, and the angle size gradually increases along the horizontal coordinate. Therefore, by using the tangent angle on the fixed coordinate pitch circle that coincides with the chain engagement point, we can determine which point on the catenary should coincide with the engagement point, thus determining the fixed coordinates of the catenary. Step four requires substituting the tangent direction angle at the engagement point to form a complete mechanical equation with the next catenary segment for calculation. The tension distribution function needs to be multiplied by the catenary function. qg The conversion is performed; the line connecting the calculated engagement point coordinates and disengagement point coordinates yields the connecting line function; the difference between the connecting line function and the catenary function yields the sag distribution function.

[0181] The catenary function is:

[0182] y = a cosh(( x - x 1) / a )- a + y 1,

[0183] in: x 1 represents the horizontal movement size;

[0184] y 1 represents the vertical movement size;

[0185] The tension distribution function is:

[0186] y = aqg cosh(( x - x 1) / a )- aqg + y 1* qg ;

[0187] The connection function is:

[0188] y =( y in - y out ) / ( x in - x out ))*( x - x 1)+ y 1;

[0189] in:( x in , yin () represents the coordinates of the engagement point;

[0190] ( x out , y out ( ) represents the coordinates of the disengagement point;

[0191] The vertices distribution function is:

[0192] y =( y in - y out ) / ( x in - x out ))*( x - x 1)- a cosh(( x - x 1) / a )+ a .

[0193] See Figure 4 This is a multi-wheel chain system calculation case provided by an embodiment of the present invention, covering the calculation of a four-wheel internal chain, and the results show the output equations and value ranges of the four chain segments.

[0194] The four wheels are designated as first sprocket 1, second sprocket 2, third sprocket 3, and fourth sprocket 4.

[0195] Wherein: the position of the first sprocket 1 is (0.00, 0.00)m; the radius is 0.50m;

[0196] The second sprocket 2 is positioned at (0.00, 0.00)m; its radius is 0.50m.

[0197] The third sprocket is positioned at (1.00, 2.00) m; its radius is 0.50 m.

[0198] The fourth sprocket is positioned at (6.00, 1.00) m and has a radius of 1.50 m.

[0199] The formula for the catenary line between the first sprocket 1 and the second sprocket 2 is:

[0200] y=8.28349149*cosh((x-2.10300250) / 8.28349149)+0.24689980-2.10300250;

[0201] x∈[0.11759774,3.08237914];

[0202] The formula for the catenary line between the second sprocket 2 and the third sprocket 3 is:

[0203] y=8.15610528*cosh((x-1.11771576) / 8.15610528)+1.49909502-1.11771576;

[0204] x∈[2.85325506, 0.99231294];

[0205] The formula for the catenary line between the third sprocket 3 and the fourth sprocket 4 is:

[0206] y=7.82022097*cosh((x-3.42935356) / 7.82022097)+2.14250398-3.42935356;

[0207] x∈[1.14220165,5.59492298];

[0208] The formula for the catenary between the fourth sprocket 4 and the first sprocket 1 is:

[0209] y=7.22189599*cosh((x-3.11712650) / 7.22189599)-1.23385778-3.11712650;

[0210] x∈[6.68634045,-0.21564549。

Claims

1. A method for calculating the catenary function of a multi-wheel chain catenary system, characterized in that, Applicable to chain drive trajectories moving in the same plane; where the polygonal effect of the sprocket and chain is not obvious, the pitch circle is used for calculation. The chain system exhibits rigid properties and does not undergo significant deformation; The number of sprockets is greater than or equal to 2; the following steps are included: Step 1: Determine the boundary conditions and parameters of the chain drive system. The boundary conditions and parameters include at least the number of sprockets. n The calculation plane includes the center coordinates and pitch circle radius of each sprocket, the total length of the chain, the chain linear density, the gravitational acceleration under actual conditions, and the tension exerted on the chain by the sprockets in the observation plane; the sprockets include a driving sprocket, a driven sprocket, a tension sprocket, and an idler sprocket; the tension includes driving force, load force, and centrifugal force. Step Two: Based on the boundary conditions and parameters determined in Step One, establish the geometric constraint equations between sprocket and chain based on the meshing geometry of the chain and sprocket. Establish the mechanical equilibrium equations between chain and sprocket based on the force balance relationship. Finally, establish the total length constraint equations, including all catenary segments and wrap angle arc segments, based on the total chain length. These equations together constitute a constraint equation containing 3... n A system of nonlinear equations with independent equations, where the unknowns are... n The lowest point tension of the catenary n The tangential angle of the chain at each sprocket engagement point and n The tangential angle of the chain at each sprocket engagement point; Step 3: Predict initial values ​​for the unknowns in the nonlinear equation system described in Step 2; Step 4: Substitute the unknowns predicted in Step 3 into the nonlinear equations from Step 2 and perform iterative numerical solutions until convergence, obtaining the desired result. n The lowest point tension of the catenary n The tangential angle of the chain at each sprocket engagement point and n The exact solution to the tangent angle of the chain at each sprocket engagement point; Step 5: Using the exact solution obtained in Step 4, calculate and output the catenary function, tension distribution function, line function connecting the catenary to the two end contact points, and sag distribution function between each adjacent sprocket.

2. The method for calculating the catenary function of a multi-wheel chain catenary system according to claim 1, characterized in that, Step two, which involves establishing the geometric constraint equations between the sprocket, chain, and sprocket based on the meshing geometry of the chain and sprocket, specifically involves: When the sprocket-chain-sprocket relationship is such that the chain cuts from the top of the preceding sprocket to the bottom of the following sprocket, the geometric constraint equation is: ； When the sprocket-chain-sprocket relationship is such that the chain cuts into the upper side of the preceding sprocket from the upper side of the following sprocket, the geometric constraint equation is: ； When the sprocket-chain-sprocket relationship is such that the chain cuts into the underside of the preceding sprocket from the underside of the following sprocket, the geometric constraint equation is: ； When the sprocket-chain-sprocket relationship is such that the chain cuts from the underside of the preceding sprocket to the topside of the following sprocket, the geometric constraint equation is: ； in:( x i , y i () represents the coordinates of the center point of the previous sprocket in the coordinate system; ( x i+1 , y i+1 ( ) represents the coordinates of the center point of the next sprocket in the coordinate system; R i The pitch circle radius of the previous sprocket; R i+1 The pitch circle radius of the next sprocket; φ out,i To create a chamfer at the point of engagement with the previous sprocket; φ in,i+1 The angle is cut at the point of engagement with the next sprocket.

3. The method for calculating the catenary function of a multi-wheel chain catenary system according to claim 2, characterized in that, The system of equations is established with the sprocket with the smallest x-coordinate as sprocket 1, and then sequentially to the next sprocket connected to the chain. If the x-coordinate of the contact point of the chain disengaging from the previous sprocket is less than the x-coordinate of the contact point of the chain engaging with the next sprocket, the equations are established in the forward direction; if the x-coordinate of the contact point of the chain disengaging from the previous sprocket is greater than the x-coordinate of the contact point of the chain engaging with the next sprocket, the equations are established in the reverse direction.

4. The method for calculating the catenary function of a multi-wheel chain catenary system according to claim 1, characterized in that, Step two, which involves establishing the mechanical equilibrium equations and corresponding envelope angle equations between the chain, sprocket, and chain based on the force balance relationship, specifically includes: The formula for the chain enveloping the sprocket from the top is: ； The formula for the chain enveloping the sprocket from the right side is: ； The formula for the chain to envelop the left side of the sprocket is: ； The formula for the chain enveloping the sprocket from below is: ； in: Cut an angle at the sprocket engagement point; Cut an angle at the point where the sprocket engages; θ It is the envelope angle; F The sprocket applies the total force to the chain; Tension at the engagement point; The tension at the disengagement point.

5. The method for calculating the catenary function of a multi-wheel chain catenary system according to claim 4, characterized in that, The chain equations are divided into pairs based on the chain-to-chain equations: Combination 1 is a combination of two types of equations that engage from the bottom of the sprocket and two types of equations that disengage from the top of the sprocket; Combination 2 is a combination of two types of equations that engage from the top of the sprocket and two types of equations that disengage from the top of the sprocket; Combination 3 is a combination of two types of equations that engage from the bottom of the sprocket and two types of equations that disengage from the bottom of the sprocket; Combination 4 is a combination of two types of equations that engage from the top of the sprocket and two types of equations that disengage from the bottom of the sprocket. In the cases of combination 1, combination 2, and combination 3, when the x-coordinate of the sprocket engagement point is less than the x-coordinate of the chain engagement point of the next sprocket, a positive equation is established. When the x-coordinate of the sprocket engagement point is greater than the engagement point of the chain entering the next sprocket, the equation is established in reverse; In combination 4, when the x-coordinate of the sprocket's engagement point is less than the x-coordinate of the chain's engagement point on the next sprocket, the equation is established in reverse; when the x-coordinate of the sprocket's engagement point is greater than the chain's engagement point on the next sprocket, the equation is established in the forward direction.

6. The method for calculating the catenary function of a multi-wheel chain catenary system according to claim 1, characterized in that, Step two, which involves establishing a total length constraint equation based on the total chain length that includes all catenary segments and wrap angle segments, specifically includes the following cases: The equation for calculating the length of each catenary segment is: ； in: L i For the first i Total length of the catenary segment; The angle between the catenary segment and the point where it engages with the previous sprocket is cut. The angle between the catenary segment and the point where it engages with the next sprocket is cut. a i The catenary parameters calculated for this segment of the catenary; The equation for calculating the total length of the chain is: ； in: L total This is the total length of the chain; θ i The envelope angles on each pitch circle; R i Let be the radius of each pitch circle.

7. The method for calculating the catenary function of a multi-wheel chain catenary system according to claim 1, characterized in that, Step three, which involves predicting the initial values ​​of the unknowns in the nonlinear equation system described in step two, specifically includes: The n The tangential angle of the chain at each sprocket engagement point is approximated by the angle between the line connecting the center point of the engaging sprocket and the center point of the previous sprocket and the horizontal direction. n The tangential angle of the chain at each sprocket engagement point is approximated by the angle between the line connecting the center point of the engagement sprocket and the center point of the next sprocket and the horizontal direction. In the static state of chain drive, n The tension at the lowest point of the catenary segment is approximated by the gravity value associated with the chain segment. In the dynamic state of chain drive, n The lowest point tension of the catenary segment is approximated by applying force to the tight side segment, and by applying gravity value related to the chain segment to the loose side segment.

8. The method for calculating the catenary function of a multi-wheel chain catenary system according to claim 1, characterized in that, Step five involves calculating and outputting the catenary function, tension distribution function, line function connecting the catenary to the contact points at both ends, and sag distribution function between adjacent sprockets. According to the formula T / qg = a Calculate the catenary parameters a ; in: T The tension at the lowest point of the catenary; q These are the actual parameters of the chain linear density determined earlier; g These are the actual parameters for gravitational acceleration; The catenary function is: y = a cosh(( x - x 1) / a )- a + y 1, in: x 1 represents the horizontal movement size; y 1 represents the vertical movement size; The tension distribution function is: y = aqg cosh(( x - x 1) / a )- aqg + y 1* qg ; The connection function is: y =（（ y in - y out ） / （ x in - x out ）)*( x - x 1)+ y 1； in:( x in , y in () represents the coordinates of the engagement point; ( x out , y out ( ) represents the coordinates of the disengagement point; The vertices distribution function is: y =(( y in - y out ) / ( x in - x out ))*( x - x 1)- a cosh(( x - x 1) / a )+ a 。

Images