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Hello guys, I am an electrical engineer, trying to solve a mechanical engineering problem. I am stuck at the analysis of the 4 bar cross quadrilateral linkage that allows a box as shown in figure 1 to pitch along z axis.

Some information of symbols in figure -1 :

-- The images are the top view with the z axis for each joint coming out of the paper.

-- Joints (All joints are Revolute): J1 (Actuated by motor) J2 (Passive) J3 (Passive), J4 (Passive)

L23 is the input link/ crank

L41 is the output link / rocker

I have done alot of research over the past week and found freundstein equation. For solving it, it is assumed that the input crank angle is given but my motor is mounted at at joint J1. I have to find the pitch at joint J4 as a function of motor angle at Joint J1.  Could some one please guide me in this matter..

The blue bot in the image is the motor axis of rotation( Joint J1). Link L12 is distance from the blue dot (joint J1) to the joint J2 and only link length L12 is important, you can ignore the link above the blue dot..

The housing of the motor as shown in figure below is fixed on the box and the I expect the whole box to pitch as shown in the following images about joint J4.

After some rotation of the motor in clockwise direction, the box moves as shown below by rotating about joint J4.

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If you will go over to the Mechanics Corner blog, I think you will find the information you need in post #3, early in the series. I don't think I ever posted on a four-bar linkage per se, but the methods described there will fit for any planar linkage.

The only thing special about the four-bar is that the equations can be rather difficult to solve, and a numerical solution is usually preferred. Please read through the early posts at the Mechanics Corner, and let me know if  you need further help.

DrD

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Dear DrD, Thank you for the response. I have gone through the posts in your blog. I can do the displacement analysis based on your methods by using vector loop equations and using the x and y parts separately to get two equations. The displacement methods assume that the given or known angle is the input crank angle. In my case, the motor is mounted at the joint joining the rocker and coupler. I can not measure the crank angle because it is a passive joint. What do you suggest for such a situation?

Thank you.

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The vector loop equations do not care what parameter you choose to assign. Just write them as I've indicated, and then put the knowns on one side of the equation and the unknowns on the other side. It is really simple.

I'm not sure what you mean by a "passive joint." This is not terminology that I use.

I presume you wish to use the motor rotation angle as the independent variable. This will be measured between two moving links, but that should not matter. If the motor rotation is denoted as q and the angle of one of the moving links is A, you may very well find yourself looking at some terms involving sin (A+q) or cos (q-A), but everything should still work in the same way.

If you need more help, send the drawings to me in a private message and I'll look further at the problem.

DrD

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Dear Nerd,

I have assembled the equations you want, at least as well as I understand your problem. Please let me know if this answers your question or not.

DrD

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Dear Drd,

Thank you so much. I solved the problem by the following steps

step-1 ) do vector loop analysis and get two equations(x and y)

step-2) derive freudstein equation and simplify into 2 variables ( squaring and adding equations and removing 1 variable using trignometric identities)

step-3) use half angle identities to get the desired joint angle as a quadratic equation whose coefficients depend on independent variable and link lenghts

Your derivation is better! i will use both and see the results.

Thank you again : )

Best Regards,

Nerd

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Dear Nerd,

Please post your work, both solutions and the final results from each. I think a lot of folks would find it very interesting. My intentions, from the equations I gave you was simply to use Newton-Raphson to get the final numbers.

DrD

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Anyone trying to solve a 4 bar linkage, can look at the file I have uploaded. It contains enough information and an external link to a PDF file that helped me in solving my 4 bar linkage.