Proving a 2D slider crank mechanism fits the Gruebler's Equation

[bortonj88]

Homework Statement

Need to prove that a slider crank mechanism fits the Grueblers eqaution, already done for a four bar mechanism no problem, however having some serious problems with the slider crank. If l is the number of links and j is the number of joints

Homework Equations

Grueblers equation is
3l-2j-4=0

The Attempt at a Solution

Looking at a slider crank mechanism, i deduced that l=4 and j=3, however it does not fit the grueblers equation

 

[TaiwanCountry]

Slide crank is also a kind of four bar.(You need 4 vector to describe that,if it has offset.)

The block count one. i=4 , j=4 .

 

[Mene]

as 3(4)-2(3)-4=0 is not true. After further research, I realized that the slider crank mechanism is a special case of a four bar mechanism. Therefore, the Grueblers equation for a slider crank mechanism should be 3l-2j-2=0. This is because the slider crank mechanism has one fixed joint, which reduces the number of degrees of freedom by 1 compared to a regular four bar mechanism.

In order to prove this, we can use the fact that a slider crank mechanism has 4 links and 3 joints. Plugging these values into the modified Grueblers equation, we get 3(4)-2(3)-2=0, which is true. This shows that a slider crank mechanism does indeed fit the Grueblers equation.

Additionally, we can also visualize the mechanism and count the number of degrees of freedom. A slider crank mechanism has one degree of freedom, which corresponds to the movement of the slider along the fixed link. This is in line with the modified Grueblers equation, which states that the number of degrees of freedom should be equal to 3l-2j-2=3(4)-2(3)-2=1.

Therefore, it can be concluded that a slider crank mechanism fits the Grueblers equation with the modified coefficient of -2 instead of -4, due to its special case as a four bar mechanism with one fixed joint.