## 7.1. FOUR-BAR MECHANISMA four-link mechanism with four revolute joints is commonly called a
Application of four-bar mechanisms to machinery is numerous. Some typical applications will involve: function generation). In such applications we would like to have a certain functional relation such as q_{14} = f(q_{12}) to be realised by the four-bar mechanism. A simple example will be to convert a linear scale to a logarithmic scale within a certain range.
b) Link that has no connection to the fixed link is known as the ) will describe a path on the fixed link, which is called the coupler point. By proper choice of link dimensions useful curves, such as a straight-line or a circular arc, may be found. This coupler point curve can be used as the output of the four-bar mechanism (such as the intermittent film drive shown belowcoupler-point-curvec) The positions of the coupler-link may be used as the output of the four-bar mechanism. As shown in figure below, the four-bar mechanism used for the dump truck requires that the center of gravity of the dumper to move on an inclined straight line while it is being tilted (why?).
The motion characteristics of a-four-bar mechanism will depend on the ratio of the link length dimensions. The links that are connected to the fixed link can possibly have two different types of motion: i) The link may have a full rotation about the fixed axis (we call this type of link - The link may oscillate (swing) between two limiting angles (we call this type of link
).*rocker*
In a four-bar mechanism we can have the following three different types of motion: ii) Both of the links connected to the fixed link can only oscillate. This type of four-bar is called “double-rocker."
The type of motion is a function of the link lengths. Grashof’s rule) gives the criteria for these various conditions as follows: Let us identify the link lengths in a four-bar chain as: l= length of the longest links= length of the shortest link p,q = length of the two intermediate links The following statements are valid (stated without proof. One can prove these statements by using the input-output equation of a four-bar See Appendix AIII for the proof of the theorem).
Then:
Two different crank-rocker mechanisms c) One double-crank (drag-link) is possible when the shortest link is the frame.
- If
*l*+ s > p + q (if the sum of the longest and the shortest link lengths is greater than the sum of the lengths of the two intermediate links).
Only double-rocker mechanisms are possible (four different mechanisms, depending on the fixed link).
- If
*l*+ s = p + q the four possible mechanisms in (1) will result. However these mechanisms will suffer from a condition known as the change point. The center lines of all the links are collinear at this position. The follower linkage may change the direction of rotation. This is an undetermined position.
Galloway mechanism which was patented in 1844) (Fig. 7.6.).
Note that if we multiply or divide all the link lengths by a constant, the ratio of the length of the links, hence the type of four-bar or the angular rotations of the links will not be effected. Therefore it is the ratio of the link lengths, not the link lengths as a whole, which determines the type of four-bar. If our interest is the rotation of the links only, the mechanisms with the same link length ratios will have the same motion characteristics no matter how big or small the mechanism is constructed (this scaling is like multiplying the loop equation by some constant). Out of these types of four-bar mechanisms crank-rocker mechanism has a particular importance in machine design since a continuous rotation may be converted to an oscillation through this type of a four-bar (this statement does not necessarily mean that the other four-bar proportions are not used). We shall now discuss the four-bar mechanism with crank-rocker proportions and important problem related to it.
_{12}, the angular velocity of the rocker is (refer to the velocity analysis of a four-bar mechanism):
Extended dead-center position is when the crank and the coupler links are extended (q f. Sometimes, rather than the corresponding crank angle, time-ratio between the forward and reverse oscillations (strokes) is used. If we assume that the crank is rotating at a constant speed, we define the time ratio as:corresponding crank rotation,Forward stroke of the rocker is when the rocker is moving from extended to folded dead-center position in counterclockwise direction (In machinery forward stroke is the direction of motion during which the rocker is doing work. This definition need not correspond to the kinematic definition given above).
It is rather important to understand how the mechanism will function under loaded conditions in practice while the kinematic characteristics of the mechanism is being considered. By the performance of the mechanism we mean the effective transmission of motion (and force) from the input link to the output link. This also means that for a constant torque input, in a well performing mechanism we must obtain the maximum torque output that is possible and the bearing forces must be a minimum. Of course, torque and force are not the quantities that has been in the kinematics and whatever kinematic quantity we use to define the performance of the mechanism, this quantity will only approximate the static force characteristics of the mechanism. The dynamic characteristics, which is a function of mass and moment of inertia of the rigid bodies, may be several times more than the static forces and the behaviour of the mechanism under the dynamic forces cannot be predicted by kinematics. Still, some rule-of-thumb of the behaviour of the mechanism under load is better then none. Alt defined the
or, the transmission angle can be defined as:
Below the transmission angle for a four-bar mechanism and for a slider-crank mechanisms are shown. It is a simple parameter in which neither the forces nor the velocities are taken into consideration. However, one can judge the performance of the mechanism in the kinematic design stage.
Clearly, the optimum value of the transmission angle is 90 One can express the transmission angle in terms of the crank angle and the link lengths as (by writing the cosine theorem for AB (1) (2) The minimum and the maximum of the transmission angle can be determined by taking the derivative of the equation (2) with respect to q (3) The minimum and the maximum values of the transmission angle will be when sin(q (4)
The critical transmission angle is either mmin or mmax, whichever deviates most from 90 (5) Such four-bar mechanisms are known as ^{0}) and they will have a better force transmission characteristics as compared with the other crank-rocker proportions.
Determine the swing angle, corresponding crank rotation and the maximum deviation of the transmission angle from 90 Since the sum of the longest and the shortest link lengths (4+8=12) is less then the lengths of the two intermediate links (6+7=13) the mechanism is of crank-rocker type and link 2 is the crank. At the dead center positions since the crank and the coupler links are collinear, the four-bar mechanism is of triangular form. And The maximum and the minimum transmission angle is:
m The Classical Transmission Angle Problem Design of Drag-link Mechanisms with Optimum Transmission Angle |