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Saturday, August 8, 2015

Grashof's Criterion: Crank-Rocker Four-Bar Mechanism

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Grashof's criterion for Double Crank Four-Bar Mechanism is almost the same as Crank-Rocker Mechanism. They're both defined as S+L < P+Q. The difference is on the location of the shortest link (S). For double crank mechanism, S must be on the ground link (frame). But for the crank-rocker, S will be on the side link and it must be the input link.

We then move the shortest link from the frame to the side link on LH. With the help of SAM 7.0 Professional software (by Artas Engineering), it can display the paths of desired nodes as shown in the picture.
Crank-Rocker linkage with SAM 7.0 The Ultimate Mechanism Designer
Grashof's Criterion: Crank-Rocker Mechanism using SAM 7.0 Professional Software
The pink curves represent paths of desired nodes. The input link which is now the shortest link is able to make a full revolution. It's called a crank. And the output (can be either L, P or Q) will only oscillate around its pivoting point. It's called a rocker. But if we change the driver to node 4 and let it drives from the link which is not the shortest link, it can't be reversed. The shortest link can't make a full revolution. What we have to do is to swap as shown in the following picture and it becomes a crank-rocker mechanism again which is now driving from the RH pivoting point.
Grashof's criterion: RH driver with SAM Mechanism Design Software
Grashof's Criterion: Crank-Rocker Mechanism Driver on RH side
How can we see the velocity? From the path display, we can see how they move. But to see the velocity, we can choose to display the hodograph.
Path is the line that a moving point describes in the fixed reference system.
Velocity Hodograph is the locus of the arrowhead of the velocity vectors (rotated 90 degree) of a moving point. 
 The hodograph in SAM 7.0 is shown as follows.
Velocity Hodograph in SAM 7.0 The Ultimate Mechanism Designer
Velocity Hodograph in SAM 7.0 The Ultimate Mechanism Designer
The lines on the outside of the path represent CCW direction of the node. And the lines at the inside of the path represent CW direction of the node. From the above picture, we can see that the input link of the crank-rocker mechanism rotates at a constant speed in CCW direction since the hodograph displays uniform lines at outside of the circular path. However, the output link oscillates back and forth with changes in velocity since there are lines of the hodograph both on the outside and inside of the its curve path. And we can quickly point out the position where the mechanism has highest velocity from longest line on the hodograph. SAM 7.0 is also able to plot various design parameters of the mechanism e.g. velocity, displacement, acceleration, length, etc. In the above picture, LH graph shows the velocity profile of the output link which can be traced manually. The following shows the hodograph of the double crank mechanism for a comparison.
Hodograph of Double Crank Mechanism
Hodograph of Double Crank Mechanism
The following is the video showing how to use SAM 7.0 Software to simulate the double crank and crank-rocker mechanisms.

Wednesday, August 5, 2015

Grashof's Criterion: Four-bar Mechanism Double Crank

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One of the most commonly used linkages is the four-bar linkage. It consists of 3 moving links and 1 ground link (also called a frame). There are 4 pin joints connected between those links. And from Gruebler's Equation, it's the mechanism with 1 degree of freedom (1DOF) which requires only 1 actuator to drive and control position of all linkages. The four-bar mechanism consists of the following components.
4-bar linkages components
Four-bar mechanism components
The link that connects to the driver or power source is called the input link. The other link connected to the fixed pivot is called the output link. The remaining moving link connected between input and output links is called the coupler. It couples the motion of the input link to the output link.

There are different configurations of lengths of the four-bar mechanism and it results in different movements of the mechanism. Grashof's Criterion helps classifies into the following categories:
  1. Double Crank --- also called a drag link mechanism
  2. Crank-Rocker
  3. Double Rocker
  4. Change Point
  5. Triple Rocker
In this post, we're going to explore the case of Double Crank which both input and output links are able to rotate through a full revolution. We use Autodesk ForceEffect app on Google chrome to illustrate and demonstrate how the mechanism moves. The links can be setup easily by just dragging. The exact length dimension can also be specified. Once we setup all required items i.e. links, pivots and drive, we can play and see the animation of the linkages with the path of desired tracing points.
Autodesk FORCEEFFECT for Mechanism Design
Autodesk ForceEffect app on Google chrome
Grashof's criterion states that a four-bar mechanism has at least one revolving link if:

S + L ≤ P + Q

where:
S = length of the shortest link
L = length of the longest link
P = length of one of the intermediate length links
Q = length of the other intermediate length link

For the Double Crank category, the following criteria must be satisfied:

Double Crank:

  • S + L < P + Q
  • S is the length of the frame (ground link)
So we setup the links in ForceEffect app as follows.
Grashof's criteria for double crank four-bar linkage
Grashof's Criteria for Double Crank 4-bar Mechanism
The shortest link (S) is the frame which is 400 mm long. The length of the longest link (L) is 1300 mm. The remaining 2 intermediate links (P and Q) have length 700 mm and 1200 mm.

This satisfies Grashof's criteria since (S)400 + (L)1300 < (P)700 + (Q)1200. And this is how this mechanism moves.
Path and velocity vectors of double crank 4-bar mechanism
Paths of double crank four-bar mechanism
Both input and output links can make a full revolution as desired. Let's find more details of other categories in later post.

The following video shows how to use ForceEffect app to simulate the motion of double crank four-bar linkage.

Reference:

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