7.1. FOUR-BAR MECHANISM
A four-link mechanism with four revolute joints is commonly called a four-bar mechanism.
Application of four-bar mechanisms to machinery is numerous. Some typical applications will involve:
b) Link that has no connection to the fixed link is known as the coupler link. A point on this link (which is known as the coupler point) will describe a path on the fixed link, which is called the coupler-point-curve. 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 below
c) 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 crank)
In a four-bar mechanism we can have the following three different types of motion:
The type of motion is a function of the link lengths. Grashof's theorem (or Grashof’s rule) gives the criteria for these various conditions as follows:
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).
Two different crank-rocker mechanisms
c) One double-crank (drag-link) is possible when the shortest link is the frame.
Only double-rocker mechanisms are possible (four different mechanisms, depending on the fixed link).
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.
4.1.2. Dead-Centre Positions of Crank-Rocker Mechanisms
Extended dead-center position is when the crank and the coupler links are extended (q12=q13) and folded dead-center position is when the crank and the coupler are folded on top of each other (q13=q12+p). The oscillation angle of the rocker between the dead-center positions and measured from the extended dead-center to the folded dead-center position is called the swing angle, y. There is a corresponding crank rotation,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:
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).
4.1.3. Transmission Angle
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 transmission angle as:
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 900. Since the angle will be constantly changing during the motion cycle of the mechanism, there will be a position at which the transmission angle will deviate most from 900. In practice it has been found out that if the maximum deviation of the transmission angle from 900 exceeds 400 or 500 (depending on the type of application), the mechanism will lock. In certain cases this maximum deviation must be kept within 200 (e.g. reciprocating pumps) and in certain other applications maximum deviations of up to 700 may be permissible (e.g. aircraft landing gears). One must consider the practical application of a mechanism in order to give a limit to this deviation (whenever in doubt, try to keep this deviation to less that 400 or 500).
One can express the transmission angle in terms of the crank angle and the link lengths as (by writing the cosine theorem for AB0 using the triangles A0AB0 and ABB0 and equating the length AB0).
The minimum and the maximum of the transmission angle can be determined by taking the derivative of the equation (2) with respect to q12 and equating to zero:
The minimum and the maximum values of the transmission angle will be when sin(q12)=0 or when q12=0 or p (when the crank and the fixed link are collinear in extended or folded positions). The minimum and the maximum value of the transmission angle for the four-bar mechanism will be given by:
The critical transmission angle is either mmin or mmax, whichever deviates most from 900 . Sometimes, for the transmission angles greater than 900, instead of m (1800-m) is used for the value of the transmission angle. In such a case, there are two minimum values of the transmission angle ( mmin1=mmin, mmin2=1800-mmax) The most critical transmission angle is the minimum of mmin1 and mmin2. Note that the deviation of the transmission angle from 900 at the two extreme positions will be equal if:
Such four-bar mechanisms are known as centric four-bar. In centric four-bar mechanisms the time ratio is unity (the crank rotation between dead-centers is 1800) 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 900 for the four-bar mechanism whose link lengths are: a2=4, a3=8, a4=6, a1=7.
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.
The maximum and the minimum transmission angle is:
mmin=18.570 (D1=71.430) and mmax=102.640 (D2=12.640). since mmin deviates most from 900, mmin is the critical transmission angle.