Four Bar Mechanism
A four bar linkage or simply a four-bar mechanism is the simplest movable linkage. It consists of four
rigid bodies (called bars or links), each attached to two others by single joints or pivots to form a closed
If each joint has one rotational degree of freedom (i.e., it is a pivot), then the mechanism is usually
planar, and the four-bar is determinate if the positions of any two bodies are known (although there
may be two solutions). One body typically does not move (called the ground link, fixed link, or the
frame), so the position of only one other body is needed to find all positions. The two links connected to
the ground are called grounded links. The remaining one link, not directly connected to the ground link,
is called coupler link. In terms of mechanical action, one of the grounded links is selected to be the input
link, i.e., the link to which an external force is applied to rotate it. The second grounded link is called the
follower link, since its motion is completely determined by the motion of the input link.
Four Bar mechanism
In the figure shown above the first link (input link) is called Crank, the second link Coupler and the third
link is the Follower.
The objective of this project is to simulate the four-bar mechanism using Pro-E and compare the analysis
results with the analytical calculations.
The dimensions of the Four bar mechanism of interest are shown in the figure below:
Length of the Crank : 6 in
Length of the Coupler: 24.7386 in
Length of the Follower: 12 in
The Crank and the Coupler have to be of negligible mass. So the density is appropriately chosen.
The various parameters are tabulated as below:
Volume = length * width*thickness + pi * r^2*thickness
r: radius of curvature of the ends
Link# Length(in) Width(in) Thickness(in) v1 Density Volume Mass
Crank 6 1 0.5 0.3925 1.00E-07 3.3925 3.3925E-07
Coupler 24.7386 1 0.5 0.3925 1.00E-07 12.7618 1.27618E-06
Follower 12 1 0.5 0.3925 0.0007324 6.3925 0.004681867
The Four bar mechanism built in pro-E is as shown in the figure below:
Four Bar in Pro-E
The Four bar mechanism is simulated in Pro-E and both kinematic and dynamic analysis is done to
measure the angle rates and angular acceleration. The Torque and the reaction forces at the Crank-
Ground joint are also measured and are shown in the figures below:
# Angle(rad) Rate(rad/s) Acceleration(rad.s^2)
Crank pi/2 2*pi 0
Follower pi/2 TBD TBD
The angular rates , accelerations of other joints and torque and reaction forces at the Crank-ground joint
are plotted as shown below:
Differentiating the above set of equations w.r.t time we get equations at velocity level:
-l1*S1*w2-l2*S2*w3 = -l3*S4*w4
l1*C1*w2+l2*C2*w3 = l3*C4*w4
Now given w2 we can determine, w3 and w4 at the initial position.
Differentiating the above equations w.r.t time we get equations at acceleration level:
-l1*S1*α2-l1*C1*(w2^2)-l2*S2* α3-l2*C2*(w3^2) = -l3*S4* α4-l3*C4*(w4^2)
l1*C1*α2-l1*S1*(w2^2)+l2*S2* α3-l2*S2*(w3^2) = l3*C4* α4-l3*S4*(w4^2)
α3 and α4 can be determined from the above set of equations.
F = (I03*w4dot)/(l3*cos(th))
Rx = -F* cos(th)
Ry = -F*sin(th)
M = -F*cos(th)*l1
Since pro-E uses relative angles we need to covert them to absolute angles before comparison
# Pro-E Analytical
Relative Absolute Absolute
w3 -360 0 0
w4 180 180 180
α3 282.665 282.665 282.7473
α4 141.354 141.354 141.3717
The hand calculations for the force analysis are submitted in a hand-written format.
The results are tabulated as shown below:
# Pro-E Analytical
Fx -0.0473429 0.0462
Fy -0.012 0.0115
Torque 0.284036 0.2772
Part B - The Dutch Crane
The crane below is a planar four-bar mechanism mounted on a rotating platform. Its critical dimensions
are shown in the schematic below in meters. The maximum motion of the crane is given by its driven
angle Q which varies from 49 degrees at maximum reach to 132 degrees at minimum reach.
The objective here is to render the crane shown above in ProE using reasonable representations for
its components and create an appropriate assembly. The then rendered assembly is to be animated
using ProE mechanism package.
The rendered components:
The major components are modeled according to the crane shown in the fig above.
Therefore, the four bar mechanism is modeled in Pro-E and kinematic and dynamic analysis is
performed to determine the joint rates, accelerations, reaction forces and torques at the joints. And
we compared the Pro-E analysis results with the analytical calculations and they agreed with a good
degree of precision.
The Dutch crane shown is successfully rendered in ProE using idealized models for the
components. The dimensions for the components are approximated to the original values shown in
the figure. The final rendered assembly is shown above and is animated using ProE mechanism
package. The maximum motion of the crane is given by its driven angle Q which varies from 49
degrees at maximum reach to 132 degrees at minimum reach.