Modeling and Simulation of a Four Bar Mechanism and a Crane using ProM - PDF

[Project-4]

[Mechanism Studies]

Sasi Bhushan Beera #35763829 Srikanth Avala #35762927

Project4

Four Bar Mechanism

Introduction:

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

loop.

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.

Objective:

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:

6.00

0.50

0.50

Four Bar in Pro-E

Analysis:

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:

Initial Configuration:

# 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:

W3 vs time

W4 vs time

W3dot vs time

W4dot vs time

Fx

Fy

Moment

Analytical Calculations:

Notations:

cos(th1) : C1

sin(th1) : S1

cos(th2):C2

sin(th2):S2

cos(th4):C4

sin(th4):S4

Closed loop equations: position level

l1*C1+l2*C2 = l0+l3*C4

l1*S1+l2*S2 =l3*S4

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.

Force Calculations:

Rocker:

F = (I03*w4dot)/(l3*cos(th))

Crank:

Rx = -F* cos(th)

Ry = -F*sin(th)

M = -F*cos(th)*l1

Results:

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

Force Analysis:

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

Introduction:

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.

Objective

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.

The Base:

``

The arm:

The Rotor:

The supporter(long arm)

The final rendering of the

assembly:

Conclusion:

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.