The Geometry Theorem And Application Of Determining The Degree Of Freedom Of Workpiece

Machine tool fixture is an important process equipment in parts machining.The quality of fixture design affects the quality and efficiency of parts machining.The basic content of fixture design is the correct determination of degree of freedom(DOF)of the workpiece by using the traditional method,six-point positioning principle.The principle can only determine the DOF of the workpiece in the fixture perceptually and can’t easily solve difficult positioning problems,such as complex curved surface positioning and combined location in non-coplanar planes of the workpiece.Using the traditional method often misjudges the DOF of the workpiece due to lack of the specific rules to lead to the failure of fixture design.There were two problems that can’t be solved by using traditional method.The first was that the analysis and determination of DOF of the workpiece can’t be abstracted into mathematical rules,which was inconvenient to determine the DOF in computer aided fixture design.The second was that a variety of irregular positioning of the workpiece can often be encountered in engineering practice,in which the DOF of the workpiece constrained by locators wasn’t the DOF along the coordinate axis but the arbitrary direction in space.The problems above can’t be settled by traditional method.The research results showed that the DOF of the workpiece in fixture completely depended on the number of the positioning normal lines and the geometry relationship between them.Therefore,the concept of positioning normal line and positioning normal plane were proposed in this paper.A set of geometry theorems that could determine the DOF of the workpiece were derived according to the number of positioning normal lines and the geometry relationship between them.Determining the DOF of the workpiece rationally could be realized by these geometry theorems.The geometry theorems provided a practical and simple method for the production practice,teaching and computer aided fixture design.The main contents of this paper are as follows:(1)The fundamental reason of the DOF of the workpiece constrained by locators was discussed.The mechanics model and kinematics model were established based on the view of mechanics and kinematics respectively.(2)The positioning of the workpiece in production practice was transferred to the mechanics model and the kinematics model and the constraint of the degree of freedom of the workpiece was analyzed later.(3)The concept of positioning normal line,positioning normal plane and rotational positioning normal plane were proposed.Geometry theorems for determining the DOF of the workpiece were summarized and were proved in theory.Some specific methods to determine the over-position state of the workpiece were summarized by using geometry theorems,which greatly improved the efficiency of determining the position state of the workpiece.(4)The number of DOF of the workpiece that was constrained by common locators was obtained by using geometry theorems.Several specific examples were enumerated.The DOF of the workpiece in examples was determined and the corresponding improvement scheme was given for unreasonable positioning by using the geometry theorems.(5)The process to determine the DOF of the workpiece became programmatic based on the knowledge of matrix and function,which provided the corresponding basic theory of computer aid determining the positioning state of the workpiece in the fixtures.Some examples of workpiece location were enumerated to verify the correctness of the proposed theory.The main conclusions of this paper are as follows:(1)The DOF of the workpiece in the fixtures completely depended on the number of the positioning normal lines of the workpiece and geometric relationship between them.(2)It was easy to deal with all kinds of complex DOF analysis with geometric theorems,which could effectively avoid the under-positioning and over-positioning,and provide an important theoretical basis for improving the unreasonable positioning scheme.(3)By using the equivalent of the normal orientation,the complex spatial relation of the normal can be simplified to easily determine the DOF of the workpiece by using the geometry theorems.(4)It was more rational and easier to determine the DOF of the workpiece by using the geometric theorem,which was proper for computer aided determining the DOF of the workpiece.