One of the simplest and most useful mechanisms is the four-bar linkage. Mast of the following description will concentrate on this linkage, but the procedures are also applicable to more complex linkages.
We already know that a four-bar linkage has one degree of freedom. Are there any more that are useful to know about four-bar linkage? Indeed there are! These include the Grashof criteria, the concept of inversion dead-center position (branch points), branching, transmission angle and their motion feature, including positions, including positions, velocities and accelerations.
The four-bra linkage may take form of a so-called crank-rocker or a double-rocker or a double-crank(drag-link)linkage, depending on the range of motion of the two links connected to the ground link. The input crank-rocker type can rotate continuously through 360º,while the output link just “rocks”(or oscillates). As a particular case, in a parallelogram linkage, where the length of the input link equals that of the output link and the lengths of the coupler and the ground link are also the same, both the input and output link many rotate entirely around or switch into a crossed configuration called an antiparallelogram linkage. Grashof’s criteria states that the Sum of the shortest and longest links of a planar four-bat linkage cannot be greater then the sum of the remaining two links if there is to be continuous relative rotation between any two links.
Notice that the same four-bar linkage can be a different type, depending on which link is specified as the frame (or ground). Kinematic inversion is the process of a chain to create different mechanisms. Note that the relative motion between links of a mechanism does not change in different inversions.
Besides having knowledge of the extent of the rotations of the links, it would be useful to have a measure of how well a mechanism might “run” before actually building it. Hartenberg mentions that “run” is a term that means effectiveness with which motion is imparted to output link; it implies smooth operation, in which a maximum force component is available to produce a force or torque in an output member. The resulting output force or torque is not only a function of the geometry of the linkage, it is generally the result of dynamic or inertia force which is often several times as large as the static force. For the analysis of low-speed operations or for an easily obtainable index of how any mechanism might run, the concept of the transmission angle is extremely useful. During the motion
If a mechanism has one degree of freedom (e.g. a four-bar linkage),then prescribing one position parameter, such as the angle of the input link, will completely specify the position of the rest of the mechanism(discounting the branching possibility). We can develop an analytical expression relating the absolute angular positions of the links of a four-bat linkage. This will be much more useful than a graphical analysis procedure when analyzing a number of positions and/or a number of different mechanisms, because the expressions will be easily programmed for automatic compotation.
The relative velocity or velocity polygon method of performing a velocity analysis of a mechanism is one of several methods available. The pole represents all points on the mechanism having zero velocity. Lines drown from the pole to points on the velocity polygon represent the absolute velocities of the corresponding points on the mechanism. A line connecting any two points on the velocity polygon represents the relative velocity for the two corresponding points on the mechanism.
Another method is the instantaneous center or instant center method which is a very useful and often quicker in complex linkage analysis. An instantaneous center is a point at which there is no relative velocity between two links of a mechanism at that instant. In order to locate the locations of some instant centers of a given mechanism, the Kennedy’s theorem of there centers is very useful. It states
Notice that in general there are two components of acceleration of a point on a rigid body rotating about a ground pivot. One component has the direction tangent to the path of this point, pointed in the same sense of angular acceleration of this body, and is called the tangential acceleration. Its presence is due solely to the rate of change of the angular velocity. The other component, which always points toward the center of rotation of the body, is called the normal or centripetal acceleration. This component is present due to the changing direction of the velocity vector.
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