A four bar linkage is the pre-eminent mechanism building block. Four bar linkages are common in aerospace mechanical linkages. At times, a mechanism may be a four bar linkage, but the fact that a mechanism is a four bar linkage may not be obvious. An understanding of four bar linkages is essential to understanding mechanisms in a general sense. The mathematical characteristics of a four bar linkage directly apply to any general mechanism. This module provides an overview of four bar linkages. Some examples are provided. The mathematical equations and analysis techniques are provided in Four Bar Linkage – Equations.
A general four bar linkage is shown in Figure 1. A four bar linkages consists of 4 “bodies” which are the three links plus ground (ground is always considered a link when analyzing mechanisms). For analytical purposes, four bar linkages are portrayed and treated as planar mechanisms. However, in practice implementation of a four bar linkage could be spatial (non-planar). Four bar mechanisms are not always obvious in a mechanism and careful inspection may be required to determine is a linkage is indeed a four bar mechanism. For a spatial four bar linkage, an equivalent planar linkage can often be determined and the analytical techniques discussed in Four Bar Linkage – Equations and Four Bar Linkage – Design Methods apply.
Figure 1 General Four Bar Linkage
The parameters in Figure 1 are defined as follows
O1 ground point for link 1
O2 ground point for link 2
l1 length of link 1
l2 length of link 2
l3 length of link 3
d1 horizontal (x) distance between ground points for link 1 and link 3
d2 vertical (y) distance between ground points for link 1 and link 3
φ1 angle of link 1 to x axis (measured as shown in Figure 1)
φ2 angle of link 2 to x axis (measured as shown in Figure 1)
φ3 angle of link 3 to x axis (measured as shown in Figure 1)
μ angle between link 2 and link 3 (as shown in Figure 1)
Typical terminology for a four bar linkage labels link 1 as the input link, link 2 as the coupler and link 3 as the output link. Of course, link 3 could just as easily be the input link. A four bar linkage is described by the 2 constraint equations
These equations use the parameter definitions shown in Figure 1. These constraint equations state that the sum of distances in the x direction and the sum of distances in the y direction around the four bar linkage must be zero. Note that the equations are nonlinear due to the cos and sin functions. The constraint equations hold true for all possible positions of the linkage, hence they form the basis for mathematical analysis of a four bar linkage. For more details on mathematical analysis, see Four Bar Linkage – Equations.
One characteristic of a four bar linkage is that there is a single degree of freedom (1 DOF). If we let link 1 be the input link, then the position of link 2 and link 3 are uniquely determined by the position of link 1. Thus if the li and di, plus the input link angle, 1, are known, then the positions 2 and 3 can be computed as a function of 1. Thus the four bar linkage has 1 DOF. This characteristic is true of all airplane mechanisms. If the input to the mechanism is known, then the position of all components in the mechanism can be computed. Some mechanisms may have 2 DOFs, using separate inputs such as pilot stick input and autopilot actuator input.
Four bar linkages are very versatile and have been designed to do a multitude of tasks. For example, four bar linkages can be used to have the output travel in a straight line or do general path following. Figure 2 shows a four bar linkage that can be used to draw a straight line.
Figure 2 Four Bar Linkage with Straight Line Output
Referring to Figure 1, maximum efficiency of a four bar linkage is obtained when the angle is kept close to 90 degrees. This minimizes the compression load, Fc, acting on link 3. Fc is wasted force as it does not contribute to the output torque and adds friction and additional column loading to link 3..
Another important characteristic to understand about four bar linkages is that the mechanical advantage of a four bar linkage varies with linkage position. Thus the mechanical advantage is not constant throughout the range of movement – a characteristic of all mechanisms. Mechanical advantage is discussed in Mechanisms – Overview and Mechanical Advantage. In this module, mechanical advantage is shown to be equivalent to the velocity ratio. As shown in Four Bar Linkage – Equations, velocity ratios vary with position. The variation in velocity ratio over the range of movement of a mechanism (i.e., the mechanical advantage) is an important feature to understand. Large variations could be a problem in a overall mechanism design. Generally speaking, small movements around input link angles of 90 degrees are good to minimize variations in mechanical advantage.
Examples of four bar linkages are shown in the figures below. Figure 3 shows a leading edge mechanism for the leading edge of an airplane. This mechanism is a four bar linkage that raises the leading edge section of the wing to give the wing more camber at low speed. The four bar linkage is driven by an actuator connected to one of the links.
Figure 3 Leading Edge Mechanism
A second four bar linkage example is shown in Figure 4. This example is from a fighter jet aircraft that utilizes a series connection of four bar linkages. Note that the complete linkage is not shown. Also, in the airplane this linkage is not planar, even though the picture shows a planar mechanism.
Figure 4 Pushrod Based Flight Control System
Another four mechanism is shown in Figure 5. This mechanism shows two four bar linkages connected by a cable system. The first four bar linkage is the connection between the control column and the forward sector. The ground points for this linkage are the rotational axis for the control column and the bearing attachment for the sector. The second four bar is the aft flight control sector to a flight control surface bellcrank using pushrods. The ground points for the four bar linkage is the bearing attachment for the sector and the bearing attachment for the bellcrank. The input to this four bar linkage is the control column and the output is the elevator position.
Figure 5 Generic Elevator Control Linkage
The last four bar mechanism is a helicopter cyclic control mechanism (blade pitch control) shown in Figure 6. This mechanism is four bar linkage where the coupler is short link between the two rotating shafts. There are several four bar linkages in Figure 6. The first four bar linkage consists of link 1, link 2 and the bellcrank. The second four bar linkage consists of the bellcrank, link 3 and the swashplate. The third four bar linkage consists of the swashplate, link 4 and the propeller blade. Note that the top swashplate is always rotating, while the bottom swashplate is stationary. This allows a stationary mechanism to drive a rotating mechanism.
Figure 6 Helicopter Cyclic Control Mechanism
Further analytical information on four bar linkages can be found in Four Bar Linkage – Equations and design information in Four Bar Linkage – Design Methods. These modules provide the background information for analyzing and designing the linkages shown in Figures 3 through 6.