1 Introduction The parallel mechanism has the advantages of large rigidity, strong bearing capacity, high position accuracy, fast response, etc., which are not available in many series mechanisms. Its application prospects are very broad. In recent years, extensive attention has been paid to the field of machine tool researchers and industrial circles. However, the kinematics and dynamics solving problems of such institutions are more complicated. It is one of the difficulties in institutional research. In addition, in the dynamic analysis of the mechanical structure system of the parallel mechanism, the system structure is generally treated as a rigid body, so that only the force analysis can be performed and the displacement under the force can not be analyzed. This paper introduces a new stiffness analysis method for a parallel mechanism that uses the system structure as a flexible body to directly solve the stiffness of the system and has unique advantages. The following takes the 3-RPS parallel mechanism as an example. We use this mechanism as an independent parallel mechanism for hybrid CNC machine tools. In the design process of 3-RPS mechanism, the stiffness of one mechanism depends on the stiffness of each component element (rod and joint) of the constituent mechanism. How to reasonably design the structural parameters of the various rods to make the mechanism in various postures and external forces The balance of stiffness under the effect is the key to the design.
Figure 1 3 Degrees of Freedom Parallel Mechanism Model
2 Mechanisms The 3-RPS parallel mechanism consists of a moving platform, a fixed platform, and two branches that connect the two platforms. As shown in Figure 1, three pairs of motions connected to the fixed platform are the rotation pair (R). The moving pair connected by the moving platform is a spherical pair (S), and the pair of R and S is a moving pair (P). It is known from the kinematic analysis of the organization. The mechanism has a movement along the Z axis and a rotation around the X axis and the Y axis (equivalent instantaneous rotation axis). Three degrees of freedom When the length of the movement of the mechanism changes, the posture of the motion platform changes. 3 Joint boundary element method synthesis The boundary element method converts the controlled differential equation of an object into an integral equation on the boundary, and then discretes the integral equation and solves it. Because the boundary element method only deals with the boundary of the object, the dimension of the problem can be reduced by one dimension. For the bar parts, because the field is a one-dimensional field, its boundary is a point. Compared with the finite element method, the number of boundary elements and the number of nodes are small, the data input preparation is simple, the required computer capacity is small, and the calculation speed is fast.
Fig. 2 Schematic diagram of child nodes
The parallel robot mechanism is a rod mechanism in which rods are connected through various joints and can be used as a mechanical structure system for performance analysis. A single rod is a substructure of the system. The boundary equation of each substructure can be obtained by the boundary element method, and then the boundary relationship between the connected substructures can be found, that is, the combination of conditional expressions. Finally, the boundary equations of each substructure are solved by combining conditional expressions to obtain the boundary forces and displacements of the entire system, and then the static stiffness of the system is obtained. The process of synthesis of the two substructures shown in Figure 2 is as follows. The substructure equation {Fa} =[K1] {Xa} +{P1} Fb Xb (1) {Fc} =[K2] {Xc} +{P2} Fd Xd (2) Where: K1, K2 are The stiffness coefficient matrix of substructures 1 and 2 (including tension, compression, bending, and torsion) depends on the length of each substructure, structural parameters such as cross-section and material properties, and P1 and P2 are the external forces acting on the substructure. The force vectors; Fa, Fb, Fc, Fd, and Xa, Xb, Xc, Xd are the force vector displacement vectors at the boundary points of the two substructures, respectively. They include force, moment or displacement, and rotation angle in six coordinate directions, respectively. . For example, Fa={fa1 Fa2 Fa3 Fa4 Fa5 Fa6}T combined condition When the two substructures are rigidly coupled, the forces at the joint point are equal in magnitude, opposite in direction, and the displacement is equal, so the conditional expression is: Displacement binding condition: [0 -IC I 0]{Xa Xb Xc Xd}T=0 (3) Force combining condition: [0 -IC I 0]{Fa Fb Fc Fd}T=0 (4) where: I represents a unit matrix; C represents Coordinate transformation matrix. When the substructures are articulated with each other, the degree of freedom around their joint axes is unconstrained, that is, the moment around the joint axis is zero. The displacement of the rigid body causes the boundary equation of the substructure to have no solution in the simultaneous solution, so it cannot be The combination of conditional expressions includes the corresponding moments and displacements. Since the moment around the joint axis is a known quantity, this element in the substructure equation can be used as a boundary condition. As shown in Fig. 2, the binding condition of the two substructures rotating around the X1 axis at the joint is as follows: [0 - IC' I 0] {Xa Xb Xc Xd} T = 0 (5) [0 - IC' I 0] {Fa Fb Fc Fd}T=0 (6) where: C' is the coordinate transformation matrix associated with the non-joint motion direction. In the division of children in the joint direction, the corresponding moment of zero displacement is unknown. After the synthesis of the system boundary equations, the boundary equations obtained by equations (5), (6) and (1) and (2) are combined into {Fa} = [K] {Xa} + {P} Fd Xd ( 7) In the formula: Xa, Fa are the boundary equations of the rod 1 containing the joint direction components, and [K] is the stiffness coefficient matrix of the entire mechanical structural system after synthesis. By solving all known boundary conditions in equation (7), the unknown boundary conditions in the system can be solved to obtain the stiffness of the system.
Fig. 3 Mechanical model of 3-DOF parallel mechanism
4 Boundary Element Method Analysis of 3-RPS Parallel Mechanism Stiffness Figure 3 shows the boundary element model of 3-RPS parallel mechanism stiffness analysis. O-XYZ is the global coordinate system established on a fixed platform, and oi-xiyizi is based on each The local coordinate system on the substructure. F is the external force acting on the moving platform (it can be force or moment in any direction). The stiffness coefficient matrix [K] in the substructure equation (1) is obtained from the structural parameters such as the length and cross-sectional characteristics of each member in each branch; the coordinate transformation matrix is ​​obtained from the attitude of each member's local coordinate system with respect to the global coordinate system. C; Solve the unknown boundary conditions by using the known boundary condition values ​​for the equivalence equation (7). At this point, the deformation of the entire mechanical structural system under the action of the external force F, that is, the static stiffness of the system was obtained. To optimize the system structure and improve the stiffness of the system, it is necessary to rationally design the structural parameters of each sub-structure in the system so as to maximize the overall stiffness of the system. The boundary conditions of the entire mechanical structural system are converted into the boundary conditions of a single rod by coordinate transformation. Using the above analytical process again, the deformation states of all the members that make up the parallel mechanism and the force are obtained, and the structural design is provided. The basis. As shown in Fig. 3, the pose matrix of the 3-RPS parallel mechanism moving platform is T, and the force conditions and system stiffness of each rod are calculated when the force at the Om point is Fx=1000N and Fy=1000N, respectively, as shown in the following table. .
Forces and stiffness calculation results for each branch Table Force (N) (relative to O-XYZ) Loads to each branch (with respect to each branch's local coordinate system oi-xiyizi) (N) System stiffness (N/μm ) Aa branch Bb branch Cc branch Fx = 1000 Fz = -1515.5 Fz = 757.76 Fz = 757.76 112.8 Fy = 1000 Fz = 370.5 Fz = 370.5 Fz = 370.5 942.4 Note: Since the friction and the member's weight are not taken into account, the solution is obtained. The force of each branch in the other direction is zero, so only Fz is given. Among them, the pose matrix T=[ RP ]O 1 R=[ 1 0 0 ] is the direction cosine matrix of the motion platform coordinate system in the fixed platform coordinate system. Ie posture; 0 1 0 0 0 1 P={Px Py Pz}T is the position of the origin of the moving platform coordinate in the fixed platform coordinate system. 5 Conclusion In this paper, the 3-RPS parallel mechanism in the hybrid machine tool is regarded as a rod type mechanical structure system. Each rod of the constituent mechanism is used as a flexible rest. The static model of the system is established using the boundary method. Under the given pose and force, the system's deformation and stiffness are solved. The results show that the mechanism is reasonable. Combining the forward and inverse kinematics of the method described in this paper and the mechanism can solve the system stiffness of the moving platform in any position. And can simulate the analysis of the system during the movement of force changes.
Continuous Spray Bottle
China leading manufacturers and suppliers of Continuous Mist Spray Bottle,Continuous Water Mister Spray Bottle, and we are specialize in Continuous Spray Bottle For Hair,Continuous Water Spray Bottle, etc.WIDE USES: The powerful and consistent spray of these continuous mist spray bottle can be used for various household and beauty purposes including cleaning, ironing, hairstyling, watering plants, dispensing air fresheners, misting essential oils and much more.
CONSISTENT & EVEN SPRAY: Engineered with a robust pre-compression technology, spray bottle dispenses a stream of sustained mist that lasts for 1.2 seconds. The spray has a fixed output of 1.25cc per second and is dispensed consistently from first to last drop.
360-DEGREE SPRAYING: The continuous spray bottle allows 360-degrees spraying for those areas which are difficult to reach. You can even use it sideways and upside down. Mist spray bottle provides 98% water evacuation so you can utilize even the last drop of liquid.
Continuous Mist Spray Bottle,Continuous Water Mister Spray Bottle,Continuous Spray Bottle For Hair,Continuous Water Spray Bottle
NINGBO SUNWINJER DAILY PRODUCTS CO.,LTD , https://www.nblux-packing.com