Dynamic Simulation of Crank-Group Driving Mechanism with Clearance

For the crank-group driving mechanism, the assembly clearances can offer the possibility to move overcoming the processing errors. But the clearances inevitably affect the dynamic characteristics of the mechanism. In this paper, the dynamic simulation model of the crank-group driving mechanism is established by NX/Motion Simulation which is based on the mathematical model of planar link mechanism. The comparative analysis is made about the dynamic characteristics of the mechanism which is influenced by different clearances, different angular velocity, different damping. The simulation results also show that appropriately increasing the angular velocity, increasing stiffness and damping of the components can effectively inhibit the adverse influences of clearance on the dynamic characteristics of the mechanism.


INTRODUCTION
In some mechanisms, transmitting small torque, or having a larger center distance, such as packaging machinery, tobacco machinery, using the crank-group driving mechanism instead of common transmission mode such as gears, chains can simplify the structure and cut the cost [1,2].The general form of crank-group driving mechanism is shown in Fig. (1).Driving crank and driven cranks have the same phase, structure and size, therefore, driving crank, any driven crank and the rod between them can be seen as a parallelogram mechanism.Under ideal conditions (no size error, no stress deformation, and no installation error), each crank doe's rotary movement around its own axis, and the rod does translational motion around its own center [3].The redundant constraints are often transformed into statically indeterminate structure because of the errors and material deformation in processing and installation.The appropriate clearance fit between sizes of the crank pin and rod pin hole are selected to avoid statically indeterminate structure and ensure the feasibility of the mechanism motion [4].But the mechanism's output parameters are adversely affected by the clearances.The dynamic simulation of the crank-group driving mechanism with clearance can help to grasp the mechanism's dynamic characteristics, know the real motion model of each crank, and design the structure.

CLEARANCE MODEL
For simplicity, a schematic diagram of the crank-group driving mechanism based on parallelogram mechanism is shown in Fig. (2).A 1 B 1 is the drive crank.A 2 B 2 and A 3 B 3 are the driven cranks.r is the radius of the pin, and R is the radius of the pin hole on B 1 .The collision is caused by the clearance between the pin and the pin hole.The non-linear spring-damper model is used to describe the clearance between the pin and the pin hole, that is, the collision force can be shown by formula 1 when two objects come into contact [1,2,5].In the formula 1, F n represents the normal force, q = 1.5 represents the collision index, δ represents the penetrating amount on the normal direction of the two objects' contact surfaces, k represents equivalent stiffness coefficient [6][7][8], respectively the Poisson's ratio of two objects, E 1 , E 2 represent respectively the elastic modulus of the two objects, c max represents the maximum damping, δ max represents the maximum penetrating amount on the normal direction of the two objects' contact surfaces, v τ represents the relative velocity on the tangential direction of the two objects' contact surfaces, and Step(*) represents the damping function shown by the formula 2.

Fig. (2).
A diagram of the crank-group driving mechanism.

SIMULATION MODEL OF CRANK-GROUP DRIVING MECHANISM WITH CLEARANCE
The simulation model shown in represent the rod length, and their value are respectively 600 mm, 527 mm and 592 mm. the density of all parts is 7830 kg/m 3 .The nominal diameter of the pin and pin hole is ϕ20 .
In order to simplify the model, only the clearance at point B 1 associated with the drive crank A 1 B 1 is considered, and the clearance fit is ϕ20H8/h7 , that is, the clearance maximum is 0.054 mm.

The Influence of the Clearance on the Angular Acceleration of the Driven Crank
The angular acceleration curves of the driven crank are shown in Fig. (4) when the clearance between the pin and pin hole are taken 0.054 mm and 0.027 mm.From Fig. ( 4), when the drive crank angular velocity is 600 r/min and other conditions unchanged [9][10][11], the maximum amplitude of the driven crank angular acceleration when the clearance is 0.054 mm is much larger than that when the clearance is 0.027 mm.

The Influence of the Angular Velocity of Drive Crank on the Angular Acceleration of the Driven Crank
The angular acceleration curves of the driven crank are taken 300 r/min and 600 r/min.When the clearance is 0.054 mm and other conditons unchanged, the maximum amplitude of the driven crank angular acceleration when the angular velocity of drive crank is 600 r/min is much larger than that when the angular velocity is 300 r/min.

The Influence of the Material Damping on the Angular Acceleration of the Driven Crank
When material dampings are taken 10 N-s/mm and 15 Ns/mm.When the clearance is 0.027 mm and other conditions unchanged, increasing the material damping can significantly reduce the magnitude of the angular acceleration of the driven crank.

1)
The impact caused by the clearance is mainly in the initial phase of the movement, and the influence of the clearance on the angular acceleration of the driven crank is great.But the movement is stable when crank turns to 50°.

2)
Increasing angular velocity increases the magnitude of the angular acceleration of the driven crank in the initial phase of the movement, but angular acceleration is relatively stable when the movement stability.
3) Increasing the material damping not only can effectively reduce the angular acceleration magnitude of driven crank in the initial movement, but can further reduce the impact cause by the clearance in the steady phase of the movement.

4)
The simulation of the crank-group driving mechanism with clearance based on NX/Motion Simulation can avoid a lot of programming.In the simulation platform, modifying the simulation parameters can achieve the purpose of understanding the movement of the crank-group driving mechanism with clearances.
Fig. (3).Simulation model of crank-group driving mechanism with clearance.driving mechanism with clearance is built on its diagram shown in Fig. (2).A 1 B 1 , A 2 B 2 , A 3 B 3 all represent the crank length, and their value are all 300 mm.B 1 B 2 , B 2 B 3 , B 1 B 3 all