Machine Structure – Kinematic link, pair and chain – Grueblers criteria – Constrained motion. – Degrees of freedom - Slider crank and crank rocker mechanisms. Kinematics (cont.) open chain - a kinematic chain where one link (the unitary link) is connected to a single joint. parallel chain - a mechanism with open or closed. 𝗣𝗗𝗙 | The assumption that open chain/lever kinematic models can represent the mechanics of biologic bodies has stood unexamined for several centuries; it is.
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Nov 18, PDF | Creation of novel kinematic structures of mechanisms has been a challenging issue which is usually conducted on experience and. A type of kinematic chain is one with constrained motion, which means that a definite motion of any link produces unique motion of all other links. Thus motion of. IJSTE - International Journal of Science Technology & Engineering | Volume 2 | Issue 08 | February ISSN (online): X Mechanisms of 10 Links.
In our example, the book would not be able to raise off the table or to rotate into the table. Figure A cylindrical pair C-pair A cylindrical pair keeps two axes of two rigid bodies aligned.
Two rigid bodies that are part of this kind of system will have an independent translational motion along the axis and a relative rotary motion around the axis. Therefore, a cylindrical pair removes four degrees of freedom from spatial mechanism. Figure A revolute pair R-pair A revolute pair keeps the axes of two rigid bodies together.
Two rigid bodies constrained by a revolute pair have an independent rotary motion around their common axis. Therefore, a revolute pair removes five degrees of freedom in spatial mechanism.
Figure A prismatic pair P-pair A prismatic pair keeps two axes of two rigid bodies align and allow no relative rotation. Two rigid bodies constrained by this kind of constraint will be able to have an independent translational motion along the axis.
Therefore, a prismatic pair removes five degrees of freedom in spatial mechanism.
Figure A screw pair H-pair The screw pair keeps two axes of two rigid bodies aligned and allows a relative screw motion. Two rigid bodies constrained by a screw pair a motion which is a composition of a translational motion along the axis and a corresponding rotary motion around the axis. A kinematic diagram is a schematic of the mechanical system that shows the kinematic chain.
The modern use of kinematic chains includes compliance that arises from flexure joints in precision mechanisms, link compliance in compliant mechanisms and micro-electro-mechanical systems , and cable compliance in cable robotic and tensegrity systems.
The degrees of freedom , or mobility, of a kinematic chain is the number of parameters that define the configuration of the chain. This frame is included in the count of bodies, so that mobility does not depend on link that forms the fixed frame. Joints that connect bodies impose constraints. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom.
The constraint equations of a kinematic chain couple the range of movement allowed at each joint to the dimensions of the links in the chain, and form algebraic equations that are solved to determine the configuration of the chain associated with specific values of input parameters, called degrees of freedom. The constraint equations for a kinematic chain are obtained using rigid transformations [Z] to characterize the relative movement allowed at each joint and separate rigid transformations [X] to define the dimensions of each link.
In the case of a serial open chain, the result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain to its end link, which is equated to the specified position for the end link.
A chain of n links connected in series has the kinematic equations,. These equations are called the forward kinematics equations of the serial chain.
Kinematic chains of a wide range of complexity are analyzed by equating the kinematics equations of serial chains that form loops within the kinematic chain. These equations are often called loop equations.
The method is explained with the help of examples of planar kinematic chain having simple joints. Undetected isomorphism results in duplicate solutions and unnecessary effort. Therefore, the need for a reliable and efficient algebraic method for this purpose is necessary. Identifying isomorphism among kinematic chains using characteristic polynomials of adjacency matrices of corresponding kinematic chains are simple methods [Uicher and Raicu , Mruthyunjaya and Raghavan , Yan and Hall, ].
But the reliability of these methods was in questions as several counter examples were found by Mruthyunjaya [Mruthyunjaya, ].
The elements of the degree matrix were sum of the degree of vertices degree or type of links or unity in a link-link adjacency matrix. Later on this test was also found unreliable. Krishnamurthy [Mruthyunjaya, ] proposed the representation polynomial for detecting isomorphism between two kinematic chains.
The representation polynomial is the determinant of the generalized adjacency matrix, called representation matrix of the kinematic chain. But the representation matrix requires the use of a large number of symbols, the calculation and comparison of the representation polynomials is not as easy as that of the characteristic coefficients of the adjacency matrix.
Balasubramanian and Parthasarthy [Balasubramanian and Parthasarthy ] proposed the procedure based on the concept of the permanent of a matrix for the purpose. Tang and Liu [Tang, Liu, Tyng, ] presented a method based on degree code as mechanism identifier. The monic polynomial of degree n is given by equation 2.
The two important properties of the characteristic polynomials are The sum of the absolute values of the characteristic polynomial coefficients SCPC is an invariant for a [JJ] matrix.
To make these [JJ] matrix characteristic polynomial coefficients as a powerful single number characteristic index, new composite invariants have been proposed.