Degree Of Freedom Calculation Method Based On Constrained Normal Line

The degree of freedom(DOF)analysis of the mechanism is the basis of the mechanism design.It can help the mechanism designer to clarify the constraints of the mechanism,explore the motion characteristics of the mechanism,and predict the working range of the mechanism.The traditional mechanism DOF analysis method has problems in calculation.The fundamental reason is that the existence of virtual constraints affects the calculation of mechanism DOF.However,due to the unclear formation mechanism of virtual constraints,the specific geometric conditions and general rules for the formation of virtual constraints have not been found so far.The consensus has long been reached that ’ several specific geometric conditions for the existence of virtual constraints in the mechanism ’ is still the only way to identify virtual constraints in the mechanism through experience.Therefore,the theoretical method of accurately judging the virtual constraints in the mechanism is a very important research topic.In this paper,aiming at the problem that the virtual constraint affects the calculation of the DOF in the mechanism,on the basis of clarifying the specific geometric conditions and general laws of the formation of the virtual constraint,the balance equation between the DOF of the mechanism,the total number of degrees of freedom of each component,and the total number of degrees of freedom of the actual constraints between the components is established,and a method for calculating the DOF of the mechanism with sufficient theory and clear physical meaning is obtained.The main contents are as follows :(1)In order to judge the relationship between the DOF of the rigid body and the constraint between the reference object,the concept of the constraint normal and the constraint normal plane is proposed by focusing on the DOF of the rigid body and the over-constraint problem.The constraint at the contact point is simplified as the constraint normal to represent the constraint of the rigid body.By exploring the limitation of the geometric relationship between the number and distribution of normals on the DOF,a constraint normal geometry theorem is established to judge the DOF and its characteristics,overconstraint and its number of rigid bodies.It can accurately analyze and judge the DOF of rigid bodies,and provide a theoretical basis for the judgment of the DOF and overconstraint of rigid bodies.(2)In order to judge the constraint relationship between the motion pair and the DOF of the component,all kinds of motion pairs in the mechanism are simplified into normal lines and their distribution.According to the motion transmission of components,the definition of dynamic and static normals is proposed and the transmission law is explored,and the geometric theorem of dynamic and static constraint normals is established.Through the theorem,the constraint characteristics of the kinematic pair on the DOF of the component can be accurately analyzed and judged,and the mechanism of the constraint of the kinematic pair on the component can be clarified.It can also analyze and judge the DOF of the kinematic chain in the mechanism to the parallel component and its characteristics,the number of overconstraints and their properties.(3)In order to accurately identify the number of virtual constraints in the mechanism and eliminate the influence of virtual constraints on the analysis and calculation of the DOF of the mechanism,the necessary conditions for judging the formation of virtual constraints by components are studied and proposed.By analyzing and verifying the formation conditions of virtual constraints in the mechanism,the formation mechanism of virtual constraints in the mechanism is clarified.By deducing the relationship between the number of components in the mechanism,the number of constraint normals of all kinematic pairs and the number of virtual constraints,a general calculation method of DOF of the mechanism based on constraint normals is established.(4)In order to verify the accuracy and versatility of the new method,the DOF analysis of typical planar mechanisms,spatial mechanisms and coupling mechanisms is carried out.The new method can effectively overcome the influence of virtual constraints on the DOF analysis and calculation of mechanisms,accurately identify the number of virtual constraints in mechanisms,and obtain accurate DOF of mechanisms.Especially for the spatial parallel 3-RRC mechanism,through analysis and comparison,it is verified that the DOF analysis method based on the constraint normal line proposed in this paper is intuitive,rigorous and accurate.In the analysis of the DOF of the multi-loop coupling mechanism,a method of simplifying the rod group is proposed.Combined with the judgment theorem of the constraint normal line,the new method can effectively solve the problem of the DOF analysis and calculation of the coupling mechanism.