Kinematics question

[Ben Coburn]

I'm trying to write a simulation program, mostly as a hobby project, and have what sounds like a simple question that I haven't yet solved:

Is there a closed form solution to the problem of finding the trajectory of one link in a 4-bar linkage, holding one of the links fixed?

I imagine I'll eventually get far enough in Freedom In Machinery that this is obvious, but it seems like an interesting question to ask regardless.

[Tilman]

If you are talking about an analytical solution like a = f(b, c, d, ...) where f does not depend on x, I do not think so. You have to solve this numerically.

I have done a similar project (not yet completed though) ... simulating a double a-arm-suspension with wheel travel and steering ...

[Buckingham]

For a typical double a-arm configuration:

For a defined lower ball joint position, there is (hopefully) only one upper ball joint position that will satisfy the condition of 'the distance between the two ball joints must be equal to the length of the kingpin/upright/spindle'.

[Z]

Originally posted by Ben Coburn:
Is there a closed form solution to the problem of finding the trajectory of one link in a 4-bar linkage, holding one of the links fixed?
Ben,

It depends on what you mean by "closed form solution". If you supply enough inputs, then you can always get all of the required outputs.

I don't think you will find any off-the-shelf equations in Jack's "FiM". Plenty of words, drawings, and deep geometrical stuff, but not much algebra.

However, you might try C. H. Suh, "Suspension Analysis with Instant Screw Theory", SAE paper # 910017. This, and its references, gives a reasonable matrix based approach to the problem. (But note that the section "Screw Axis to replace Roll Axis" draws a wrong conclusion.)

Z

[Sormaz]

Originally posted by Buckingham:
For a typical double a-arm configuration:

For a defined lower ball joint position, there is (hopefully) only one upper ball joint position that will satisfy the condition of 'the distance between the two ball joints must be equal to the length of the kingpin/upright/spindle'.
More often than not there will be two locations that satisfy this condition.

Think about what happens when you reach a toggle point


It is not terribly difficult to solve though, back at the OP. say you label your four points ABCD. You are holding point C and D fixed in space. Point B is constrained to follow a circle around point C and point A follows a circle around D. People love to over-complicate these things. They are just circles.

[BillCobb]

Umm, yes there is, would be the simple answer. Look up "Kinematic Synthesis of Coupler Curves" and other material mentioning 3D linkage simulation using complex number mathematics. Dr. George Sandor was an R.P.I. professor of mine and he and a PhD. candidate (whose name escapes me at the moment) did a thesis on this. You get velocities and accelerations as well as displacements from the solution.

Perfect application to wishbone suspension systems where ride steer, ride caster, and ride camber gradients are prescribed and the mechanism endpoints are solved for. Its the basis for a lot of home grown kinematic only suspension analysis programs. What kills the solution is the departure from reality when you add compliant members (like rubber bushings and steering gear mounts, etc).

[Edward M. Kasprzak]

Back in my undergraduate days we were required to take a course in 4-bar linkage design (and cam profiles, too). The book used was Mechanism Design: Analysis and Synthesis by Erdman & Sandor.

[EPMPaul]

http://www.softintegration.com...rbar/fourbarpos.html (http://www.softintegration.com/chhtml/toolkit/mechanism/fourbar/fourbarpos.html)

That's a good place to start.

The entire site would has a lot of information. If you follow the mathematics reasoning it's easy to derive speeds and acceleration directly from that.