Lesson 2 Movement Analysis

One of the simplest and most useful mechanisms is the four-bar linkage. Most of the following description will concentrate on this linkage, but the procedures are also applicable to more complex linkage.

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, velocities and accelerations.

The four-bar 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 of a crank-rocker type can rotate continuously through 360, 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 may rotate entirely around or switch into a crossed configuration called an anti-parallelogram linkage[1]. Grashof's criteria states that the sum of the shortest and longest links of a planar four-bar linkage cannot be greater than 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 fixing different links 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. Hartemberg mentions that “run” is a term that means effectiveness with which motion is imparted to the output link; it implies smooth operation, in which a maximum force component is available to produce a force or torque in an output member. Although the resulting output force or torque is not only a function of the geometry of the linkage, but it is generally the result of dynamic or inertia force, which are 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 of a mechanism, the transmission angle changes in value. A transmission angle of 0 degree may occur at a specific position, on which the output link will not move regardless of how large a force is applied to the input link. In fact, due to friction in the joints, the general rule of thumb is to design mechanisms with transmission angle of larger than a specified value. Matrix-based definitions have been developed which measure the ability of a linkage to transmit motion. The value of a determinant (which contains derivatives of output motion variables with respect to an input motion variable for a given linkage geometry [2]) is a measure of the movability of the linkage in a particular position.

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-bar 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 computation.

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 drawn 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 or instant 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 Knnedy's theorem of three centers is very useful. It states that the three instantaneous centers of three bodies moving relative to one another must be along a straight line.

The acceleration of links of a mechanism is of interest because of its effect on inertia force, which in turn influences the stress in the parts of a machine, bearing loads, vibration, and nose. Since the ultimate objective is inertia-force analysis of mechanism and machines, all acceleration components should be expressed in one and the same coordinate system: the inertia frame of reference of the fixed link of the mechanism.[3]

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 the angular acceleration of this body, and is called the tangential acceleration.[4] 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.

Words and Expressions

Notes

[1] 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 may rotate entirely around or switch into a crossed configuration called an anti-parallelogram linkage.

全句翻译：在平行四边形机构中有一种特殊情形，即机构中的输入杆长度等于输出杆长度，连杆长度也等于机架杆长度。这样，该机构的输入杆和输出杆均可以作圆周转动，也可以转换成一种被称为反平行四边形机构的交叉结构。

[2] …which contains derivatives of output motion variables with respect to an input motion variable for a given linkage geometry…

全句翻译：该行列式中包含某个给定机构的输出运动变量对输入运动变量的导数。

[3] …the inertia frame of reference of the fixed link of the mechanism.

全句翻译：机构固定构件的惯性坐标系。

[4] One component has the direction tangent to the path of this point, pointing in the same sense of the angular acceleration of this body, is called the normal or centripetal acceleration.

全句翻译：其中一个分量与该点轨迹相切，其方向与这个刚体角加速度方向相同，这一分量称为切向加速度。

Theoretical mechanics, is a subject of studying general principles of mechanical movement, belongs to common mechanics of classical mechanics. Theoretical mechanics falls into statics, kinematics and dynamics. Statics mainly studies the conditions of acting forces when the body under acting forces is equilibrium, also studies the analytic methods of forces acting on body, and the methods of simplifying force system,etc. Kinematics just studies the movements of body in view of geometry(e.g. track, velocity and acceleration, etc.), not the physical reasons causing the movements of body. Dynamics deals properly with motions under forces.