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Design of linkage mechanism: motion design of trajectory generation mechanism

1 Atlas method

this method uses the compilation and collection of linkage curve atlas to design planar linkage mechanism. Here is an example to illustrate as follows: for example, in production, it is necessary to design a mechanism with stop motion (this mechanism is often used in some machines such as packers). First, consult the connecting rod curve atlas, and find a hinged four-bar mechanism close to the arc on the connecting rod curve, as shown in the figure. The size of each short line of the connecting rod curve in the figure is equivalent to the distance depicted by point m on the connecting rod when the crank AB rotates 50. The whole connecting rod curve consists of 72 short lines. Take the length of the crank as the benchmark and take it as 1. The length of other components is proportional to the length of the crank. Therefore, when the mechanism is enlarged or reduced in proportion to the length of the rod shown in the atlas, the characteristics of the connecting rod curve will not be changed. From the figure, it can be found that the part from point P to point Q on the connecting rod curve is close to the arc, and its radius of curvature f=1.26. This arc is composed of eighteen short lines, so when point m moves through this arc, the crank turns 900, and its curvature center g remains stationary. Then one end of the other component MF is hinged with point m on the connecting rod, and the other end f is hinged with the slider at point g. the length of the component is equal to the radius of curvature (the output part at G can be a slider or a rocker for the convenience of our customers' material research needs, depending on the actual needs). Thus, in the mechanism shown in the figure, when point m moves from point P to point Q, the slider f is stationary; When point m to point Q moves to point R, slider f moves downward to load the sample; The load cell is installed on the upper beam; When point m to point R moves to point P, slider f moves back. The stroke h of slider f is 1.48. By adjusting the inclination B of slider guide, the size of slider stroke h and the time ratio of round-trip stroke can be changed. However, it should be noted that the minimum transmission angle of the mechanism should not be less than the allowable value

it can be seen from the above that the connecting rod curve that is very close to the required trajectory can be found from the connecting rod curve atlas by using the atlas method, so as to determine the parameters of the mechanism and greatly simplify the design process

2 analytical method

for the illustrated hinged four-bar mechanism, a rectangular coordinate system ax'y'is established with point a as the origin and frame ad as the X' axis. If the position coordinates of a point m on the connecting rod in the coordinate system are X', Y', then there is

or:

f is deleted from equations (7.26) and (7.27), and then:

y is deleted from equations (7.28) and (7.29), and then:

B is deleted from equations (7.30) and (7.31), then the curve equation of point m represented in the coordinate system ax'y'is obtained:

where:

equation (7.32) is a sixth degree algebraic equation about X', Y'

when the given trajectory is reproduced by the connecting rod point m of the hinged four-bar mechanism, the given trajectory is usually represented in another coordinate system oxy. As shown in the figure, if the position coordinates of a in oxy are Xa and ya, the included angle from the positive direction of X axis to the positive direction of X 'axis in the counterclockwise direction is F0, and the coordinates of m point in oxy are x and y, then there is

which determines the size of fixture structure and the labor intensity of fixture operation. Replace the above formula into formula (7.32), and get the sixth degree algebraic equation about X and Y

which has nine undetermined size parameters, That is, the connecting rod points of the hinged four-bar mechanism can pass through the selected nine points on the given trajectory accurately at most. If the coordinate values of nine points on a given trajectory in the coordinate system oxy are known to be XMI and ymi (i=1,2,..., 9), replace them into equation (7.34) to obtain nine nonlinear equations. By solving these equations numerically, nine undetermined dimensional parameters of the mechanism can be obtained. When the number of trace points to pass is less than nine, some mechanism parameters can be selected in advance to obtain a unique solution; When the number of track points is greater than nine, due to the limitation of the number of undetermined size parameters, the connecting rod points of the hinged four-bar mechanism can only approximate the given requirements. At this time, the optimization method can be used for track approximation

3 Lopez theorem

when using the above method to design and realize the plane four-bar mechanism with known trajectory, if the obtained mechanism size cannot meet the transmission angle and other geometric conditions (such as the installation position of the mechanism is not suitable, etc.), it is necessary to calculate another plane four-bar mechanism so that it can realize the same connecting rod curve. Lopez theorem can help solve this problem

Lopez's theorem can be expressed as: the trajectory of any point of the connecting rod of a hinged four-bar mechanism can be realized by three different hinged four-bar mechanisms. When the first hinged four-bar mechanism realizing the known trajectory is obtained, the practices of the other two mechanisms are as follows

as shown in the figure, if a certain point m on the connecting rod of the hinged four-bar mechanism ABCD can achieve the known trajectory, the other two hinged four-bar mechanisms that can achieve the same trajectory can be based on the hinged four-bar mechanism ABCD. First make two parallelograms ABME and cdfm, and then Δ GEM∽ Δ MBC∽ Δ HMF, and finally parallelogram gmhk

the degree of freedom of the ten bar mechanism formed in this way remains unchanged. Moreover, Yicheng Xinneng has previously laid out the cathode material industry. When the mechanism moves, the hinge K will always remain stationary relative to the frame. Therefore, K can be fixed on the frame without affecting the movement of the mechanism. In this way, the original planar ten bar mechanism becomes three hinged four-bar mechanisms with point m as the common point: ABCD, aegk and DFHK. They have the same trajectory at point M. Therefore, when the originally designed hinged four-bar mechanism ABCD cannot meet the requirements, a better scheme can be selected from the other two hinged four-bar mechanisms aegk and DFHK. (end)

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