3D Force Analysis of a Hydraulic Crawler
3D Force Analysis of a Hydraulic Crawler
(OP)
I'm busy with a force analysis of a crawler, pictured below, using Excel. I've colour-coded the main parts into (static) body, yoke, boom and nozzle. I've quickly drawn in the cylinders, but there are 3 pairs: the body/yoke pair swivels the yoke, the yoke/boom pair lifts and lowers the boom, and the final pair lifts and lowers the nozzle.

Here's a simplified diagram of the forces, where (generally) the cylinder forces are labeled F, and the reaction forces are labeled R. (I'm neglecting the weights and the nozzle suction force.)

All I need to calculate is the resultant force on the nozzle for activation of the cylinders for any arbitrary position. I have the cylinder forces, and all the displacements calculated for whatever angles the respective hinges may be in. So now I just need the final piece of the puzzle: the nozzle force!
I'm running into several issues. I thought I could do this like a mechanism, using vectors. But with so many unknowns and the complexity of vector algebra, solving for multiple unknown vectors is impossible (it seems). I'm just really unsure how to proceed. Can I disregard the reaction forces and consider the cylinder forces acting in isolation, and simply sum their effect on the nozzle? Or do I need to write out the full equations for each "link", summing forces and moments to yield a full set of simultaneous equations, then solve that system?
Everything I've seen online just has simplified 2D analyses, like for backhoe excavators. I assume this is a fairly trivial problem for something like an FEA solver, but even that methodology is not the right tool I don't think.

Here's a simplified diagram of the forces, where (generally) the cylinder forces are labeled F, and the reaction forces are labeled R. (I'm neglecting the weights and the nozzle suction force.)

All I need to calculate is the resultant force on the nozzle for activation of the cylinders for any arbitrary position. I have the cylinder forces, and all the displacements calculated for whatever angles the respective hinges may be in. So now I just need the final piece of the puzzle: the nozzle force!
I'm running into several issues. I thought I could do this like a mechanism, using vectors. But with so many unknowns and the complexity of vector algebra, solving for multiple unknown vectors is impossible (it seems). I'm just really unsure how to proceed. Can I disregard the reaction forces and consider the cylinder forces acting in isolation, and simply sum their effect on the nozzle? Or do I need to write out the full equations for each "link", summing forces and moments to yield a full set of simultaneous equations, then solve that system?
Everything I've seen online just has simplified 2D analyses, like for backhoe excavators. I assume this is a fairly trivial problem for something like an FEA solver, but even that methodology is not the right tool I don't think.





RE: 3D Force Analysis of a Hydraulic Crawler
I might of misunderstood but isn't the force to lift the nozzle simply the forces in Fkl and Fkr balanced by the mass of the nozzle acting at its own centre of gravity?
“Do not worry about your problems with mathematics, I assure you mine are far greater.” Albert Einstein
RE: 3D Force Analysis of a Hydraulic Crawler
Cheers
Greg Locock
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RE: 3D Force Analysis of a Hydraulic Crawler
RE: 3D Force Analysis of a Hydraulic Crawler
RE: 3D Force Analysis of a Hydraulic Crawler
RE: 3D Force Analysis of a Hydraulic Crawler
another day in paradise, or is paradise one day closer ?
RE: 3D Force Analysis of a Hydraulic Crawler
RE: 3D Force Analysis of a Hydraulic Crawler
RE: 3D Force Analysis of a Hydraulic Crawler
if you define it as a force at the nozzle, then you work backwards through the structure, and size the actuators accordingly; if you define it as the capacity of the actuators, then you work forwards to find out what force can be applied at the nozzle and ask "is this enough ?"
another day in paradise, or is paradise one day closer ?
RE: 3D Force Analysis of a Hydraulic Crawler
Is the following a fair assessment of the problem?
There are nine independent input variables (rotational position at each of three pivots and force at each of six cylinders) and three output variables (three mutually orthogonal forces applied to some known point on the nozzle).
If so, the following seems like it should work:
Set up an equation for each pivot where the moments of the five forces (two cylinder forces and three nozzle forces) about the axis of rotation sum to zero. There will be three equations and three unknowns, so I'd expect the system to be solvable.
pylfrm
RE: 3D Force Analysis of a Hydraulic Crawler
F = M x r / r • r + k * r, with k = any scalar.
Hence there are an infinite number of solutions, I imagine in a half-plane, since any force can give an equivalent moment depending on the angle. But after much reading and playing around it finally dawned on me that the force generated by the moment will always be orthogonal to r, which is when k = 0. So finally I can solve this thing with simple vector algebra!
Wow. That took a while. I think I'm a little rusty.
Thanks for everyone's help though. I think it edged me closer to the solution and helped define the strategy more clearly.
RE: 3D Force Analysis of a Hydraulic Crawler
So are you independently determining a force vector for each pivot axis that is applied to the point near the end of the nozzle, is orthogonal to the plane containing the pivot axis and the force application point, and balances the moment of the two cylinder forces? Are you then summing these three vectors to get a total resultant force vector?
If so, I don't think that will work. Have I misunderstood?
pylfrm
RE: 3D Force Analysis of a Hydraulic Crawler
I found your previous statement (quoted below) a bit confusing. You mentioned 5 forces (2 cylinder and 3 nozzle). If you consider the nozzle as 3 forces (really 1 force with 3 components) then surely you have to consider the cylinder-pair as 6 forces (2 forces with 3 components each)? Or what did you mean?
Previously I calculated the moment equations from M = r x F but this system is unsolvable (3 simultaneous equations with 3 unknowns, yet they yield a zero-determinant matrix) until you introduce a "reality-check" condition. The way I see it a simple moment will generate an orthogonal force, unless you mean that the force will be dependent on the relative angle of the contact surface and maybe the friction forces, but then we'd introduce all kinds of complications and for this analysis I just need to get an idea of the relative maximal forces depending on system pressure and cylinder sizing.
Please set me straight if anything I say doesn't make sense.
RE: 3D Force Analysis of a Hydraulic Crawler
To answer your question about whether it is a mechanism or not: Yes it is a mechanism because it has freedom to move. It is only holded in position by hydraulic pressure in the cilinders.
Talking about hydraulics: I suggest looking into the working principle of the cilinders first before solving the equations in excel.
as I see it: the two cilinders FKL and FKR work as a team in lifting or lowering the nozzle. If these cilinders are not properly controlled by hydraulic valves they can easily work against each other (action in one gives reaction in the other) and giving additional load on the pivot points and torsion in the nozzle structure.
So I recommend to use one valve and a T splitted piping that feeds these two cilinders simultaneously.
That way FKL will be equal to FKR all the time and in your calculations these two cilinders can be treated as just one cilinder.
This principle is also valid for the cilinder pair FBU and FBL: they can also work against each other easily if not properly laid out hydraulically.
I think you do not want to put unneeded load on yoke/boom pivot points and in the yoke/boom structure.
For the pair FYL and FYR applies the same in my view.
How about this?
RE: 3D Force Analysis of a Hydraulic Crawler
I think the way I'm looking at it, where the nozzle is constrained by contact with the earth to yield the applied force, this stucture can be viewed as not-a-mechanism. At least this is the view from a source I encountered during my recent research. In particular:
You make some fair points about the hydraulic system, and thank you for that, but that's unfortunately outside my scope, as I'm just providing a tool for resizing the cylinders. However, his vehicle has been in service for some time, and as it's been modified with extra weight and capacity, the system pressure has slowly been increased, with the result that cylinder seals are now regularly being blown. This has been passed onto me, just to find a way for someone to see the effects of making changes to the pressure and cylinder sizes to get whatever force they think they need at the nozzle. But it's safe to say that other than the above problem the system is perfectly operational.
RE: 3D Force Analysis of a Hydraulic Crawler
Impressive picture. I think were on the same page here regarding the mechanism/structure once fixated by hydraulics and acting on the ground. Shall I try to develop a formula for this case in Excel tomorrow? It would then start with the given cilinder forces and the angles variation and resulting in nozzle force achievable.
RE: 3D Force Analysis of a Hydraulic Crawler
I suppose the "five forces" bit wasn't as clear as it could have been. I think of each piston as providing a single force of known magnitude and known direction. For the reaction force at the nozzle, I think of is as being broken up into the three forces of unknown magnitudes and arbitrarily assigned directions. I said mutually orthogonal because that is often the most convenient form for the result, but it probably isn't actually necessary. Perhaps I am making things more complicated than necessary here, but that's what came to mind first.
To illustrate the problem I see with your latest approach, consider the following simplified example:
- body/yoke pivot will be ignored, and yoke assumed static.
- yoke/boom pivot axis is parallel to the z-axis and located at x = 0, y = 0.
- boom/nozzle pivot axis is parallel to the z-axis and located at x = 1, y = 0.
- nozzle reaction force is applied at x = 2, y = 0, z = 0.
- yoke/boom cylinders apply forces resulting in a moment of (0, 0, 2) about the yoke/boom pivot axis.
- boom/nozzle cylinders apply forces resulting in a moment of (0, 0, -1) about the boom/nozzle pivot axis.
As I understand it, your method would calculate a reaction force vector of (0, -1, 0) to balance moments at the yoke/boom pivot axis, and (0, 1, 0) to balance moments at the boom/nozzle pivot axis. Add these up and you get a total of (0, 0, 0). However, the x-axis component of the reaction force is actually undefined (positive or negative infinity in the limit depending on direction of approach).I picked a special case for this example just so I wouldn't have to do any significant calculations. For a better example not involving infinity, perhaps move the boom/nozzle pivot to x = 1, y = 0.1 or similar. There should be a very large component of the reaction force in the positive x-axis direction, which I believe your method will greatly underestimate.
As for the unsolvable system of equations, unfortunately I have no great insights at the moment. Can you get the method to work for a simplified planar version of the problem with two pivots?
pylfrm
RE: 3D Force Analysis of a Hydraulic Crawler
RE: 3D Force Analysis of a Hydraulic Crawler
Here's what I did (similar to your example but reoriented so as not to confuse myself: I used x longitudinal and z vertical).
Example case:
Changing the Boom moment:
The question is, would this be a problem in reality? I assume if moments did cancel then it would be a transitory case, although if the operator simultaneously lowered the boom and raised the knuckle* he would hardly be trying to maximise any force and would likley be repositioning the nozzle, where there would be no force on the nozzle in any case..
* - Sorry to have introduced the knuckle terminology so late. I was starting to confuse myself with the ambiguity of referring to both the final structure/axis and the resultant force as the nozzle
I had a day off yesterday but will be working on this throughout the day and am quite confident I can get a meaningful answer now. I think the most complicated thing will be doing the coordinate transformations between hinge axes to get the nozzle forces, but I'll probably either post back with more problems or hopefully just some results.
Of course if anything I posted here looks odd/wrong please let me know.
RE: 3D Force Analysis of a Hydraulic Crawler
Here is a calculation for the yoke part only. The green cells can be filled in. Please modify the values as I do not know machine dimensions only angles and cilinder forces.
With the given values in the green cells Cell C34 gives the total horizontal moment that the yoke transfers onto the beam.
(for the YL cilinder the same goniometrics are used as for the YR cilinder however alfa is negative in that case)
If you want me to continu with the beam and nozzle please fill in the green fields.
RE: 3D Force Analysis of a Hydraulic Crawler
Anyway, this is the geometry I used as well as the values I got. (The first set of values are in mm and kN, but I had to change units for the final calculation because I was getting an odd (1000x too small) answer and confusing myself.)
I just used M = r x F to get the individual moments from each cylinder, summed them, then F = M x r / r • r to get the resultant force at the nozzle.
RE: 3D Force Analysis of a Hydraulic Crawler
Just that you know: Total moment on Yoke axis is the same as yours, see revised spreadsheet attached.
RE: 3D Force Analysis of a Hydraulic Crawler
RE: 3D Force Analysis of a Hydraulic Crawler
I have hand-calculated another example, this time avoiding the special case with undefined forces due to the pivots and nozzle being coplanar. I have also changed coordinate systems and terminology to better match what you used.
Here's my setup, result, and verification in the form of MATLAB / Octave code:
boom_origin = [0, 0, 0]; boom_axis = [0, 1, 0]; boom_cylinder_moment = [0, -80, 0]; knuckle_origin = [40, 0, 9]; knuckle_axis = [0, 1, 0]; knuckle_cylinder_moment = [0, 40, 0]; nozzle_origin = [80, 0, 0]; nozzle_force = [80/9, 0, -1]; % my hand-calculated result boom_r = nozzle_origin - boom_origin; boom_reaction_moment = cross(boom_r, nozzle_force); boom_total_moment = boom_cylinder_moment + boom_reaction_moment % should equal [0, 0, 0] for equilibrium of moments knuckle_r = nozzle_origin - knuckle_origin; knuckle_reaction_moment = cross(knuckle_r, nozzle_force); knuckle_total_moment = knuckle_cylinder_moment + knuckle_reaction_moment % should equal [0, 0, 0] for equilibrium of momentsI believe your result would be [0.21416 0.00000 -0.04819], but obviously you should confirm. In any case, I'm pretty sure our answers differ drastically. This means you probably disagree with my verification method above, so I'm curious where you think I've gone wrong.
pylfrm
RE: 3D Force Analysis of a Hydraulic Crawler
Yes, I get the result you quoted at the end of you post. My sign is reversed though, so I get the moment equilibrium when I set boom_total_moment = boom_cylinder_moment - boom_reaction_moment. I think this is because I get the force the nozzle exerts as a result of the moment, not the reaction on the nozzle.
Here are my values with your inputs. I'm not sure how you did your hand calc, but I can't see how you got nozzle_force = [80/9, 0, -1].
RE: 3D Force Analysis of a Hydraulic Crawler
Same method I originally suggested, setting up an equilibrium equation at each pivot and solving the resulting system. I used a Pythagorean triple (9^2 + 40^2 = 41^2) for the geometry to keep the trig simple. Note that this was done with with x+ to the right and y+ up.
boom equilibrium:
0 = 80 Nm + 80 m * Fy * 1 + 80 m * Fx * 0
knuckle equilibrium:
0 = -40 Nm + 41 m * Fy * (40/41) + 41 m * Fx * (9/41)
80 Nm + 80 m * Fy = -40 Nm + 40 m * Fy + 9 m * Fx
120 Nm + 80 m * Fy = 40 m * Fy + 9m * Fx
120 Nm + 40 m * Fy = 9m * Fx
120/9 N + 40/9 * Fy = Fx
Fy = -1 N
Fx = 120/9 N - 40/9 N
Fx = 80/9 N
Now that I think about it, you may actually able to use superposition to combine the yoke result with the boom and knuckle result. This will only work if the geometry is just right, i.e. the boom and knuckle reaction force never creates a moment at the yoke pivot, but I suspect that may be the case here.
pylfrm
RE: 3D Force Analysis of a Hydraulic Crawler
RE: 3D Force Analysis of a Hydraulic Crawler
Okay, so you only need to solve a system with two equations and two unknowns. Does my method and result make sense now? Most importantly, would you agree that that the total reaction force at the nozzle must provide equilibrium of moments at both the boom and knuckle pivots?
pylfrm
RE: 3D Force Analysis of a Hydraulic Crawler
Since work is a force through a distance, allowing each cylinder to move a (very) small distance with a known force will result in the nozzle moving through some distance. Since work in = work out (neglecting all the usual suspects) then all one needs it the ratio of movement as a multiplier for the input forces to get the resulting output reactions.
Matrix wise, I believe this is the Jacobian of the kinematic matrix, but it can be done by simple geometry and a spreadsheet to determine sensitivities.
You can do the same with a CAD system and make small changes to the cylinder lengths and measure the nozzle movement.
RE: 3D Force Analysis of a Hydraulic Crawler
Nice to have different solution methods now.
The moment into the beam i found is max at centre position and turning cw or ccw doesn't make a lot of difference.