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3D Force Analysis of a Hydraulic Crawler

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.

RE: 3D Force Analysis of a Hydraulic Crawler

Hi

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

Desertfox is taking moments about Rkr_Rkl, that seems fine in the absence of dynamics. I haven't seen a simple tool for 3d force diagrams, the old Working Model 3d is overkill, as are MBD programs like MSC Adams, however http://www.freebyte.com/cad/dynamic.htm throws up some interesting possibilities.

Cheers

Greg Locock


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RE: 3D Force Analysis of a Hydraulic Crawler

(OP)
Sorry, maybe I wasn't clear enough. All 6 cylinders could be activated together, so it's unlike a backhoe excavator, which is first positioned then has a single activation point, which is probably what you were thinking.

RE: 3D Force Analysis of a Hydraulic Crawler

(OP)
Thanks Greg, I'll look into that. I see some are open source, which might be useful.

RE: 3D Force Analysis of a Hydraulic Crawler

(OP)
Just some further information, I just need to provide a quasi-static analysis, and I'm hoping to integrate the solution into an existing spreadsheet. So far I have the below, which I've been generating as I go along (mostly to check that it's all working). I'll probably add a sheet with an array of plots showing general behaviour of the nozzle force as the position of the structure changes.

RE: 3D Force Analysis of a Hydraulic Crawler

the nozzle force in two (or three) components; one component loads up FBU and FBL, the other FYL and FYR. The loads on FKL and FKR can be determined from moments of the nozzle force about RKL and RKR and then the loads on RKR and RKL can be determined from a free body.

another day in paradise, or is paradise one day closer ?

RE: 3D Force Analysis of a Hydraulic Crawler

You can do this with vectors in 3d cad. Isolate each cylinder (force vector) in combination with the nozzle and find the resultant nozzle vector for that cylinder. You can write a script to draw a collection of the resultant nozzle vectors vs. the various cylinder positions.

RE: 3D Force Analysis of a Hydraulic Crawler

(OP)
Okay, so that means the superposition of the reaction force from each cylinder would work. I considered this early on but dismissed the idea after a simple thought experiment, although I can't remember exactly why now. Why then do analyses separate the links and find the reaction forces? Is this because they specifically need those reaction forces, or is this not really a mechanism (or one simple enough with which to do a straightforward analysis)? I read through this paper on a backhoe excavator: 689 kB PDF

RE: 3D Force Analysis of a Hydraulic Crawler

i think it just depends on how you define the device's functioning envelope.

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

ChrisDanger,

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

(OP)
Yeah, I was recently getting stuck on solving the fairly trivial problem, M = r x F, with M and r known, and solving for F. I found a solution where
    F = M x r / rr + 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. neutral

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

ChrisDanger,

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

(OP)
Hi. Yes, that would be what I'll be doing. What is wrong with that approach, and do you have any suggestions? It's effectively like fixing everything and activating each cylinder pair separately (which is practically possible), then saying "okay, now turn them all on!" Surely since the system is linear superposition applies?

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.

Quote (pylfrm)

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.

RE: 3D Force Analysis of a Hydraulic Crawler

Hi.
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

(OP)
Hi jlnsol,

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:

Quote (Introduction to Statics and Dynamics by Rudra Pratap)

Indeterminate structures are mechanisms

An indeterminate structure cannot carry all loads and, if not also redundant, has more
equilibrium equations than unknown reaction or interaction force components. Such
a structure is also called a mechanism. The stamp machine below is a mechanism if
there is assumed to be no contact at D
.



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

Hi Chris,
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

ChrisDanger,

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

(OP)
Hi. Thanks for the detailed reply. I just woke up (don't worry, it's only 5 a.m. here) and will work through this soon and post my findings/results.

RE: 3D Force Analysis of a Hydraulic Crawler

(OP)
Yes, I see what you did there... Effectively the you're applying moments that result in equal and opposite reaction forces. This would lead to the statically indeterminant case like in my reply to jlnsol above, where there is no contact force at D in the example mechanism pictured. And the way I'm calculating things the resultant force simply becomes zero. But around this point it has values that increase in magnitude at you move away. I don't see any infinities.

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

Goodmorning Chris,

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

(OP)
Hi, thanks. I see you're using a different method, using trig instead of matrix algebra. I've managed to get some values for the Yoke contribution, which is fairly trivial since the axis is always vertical so one can work in the horizontal plane. For the other axes I've been trying to get an equation that yields the point on the hinge axis that makes a perpendicular vector to the nozzle (for the yoke it's just <x_nozzle, y_nozzle, 0>), hoping to avoid transforming the coordinate systems.

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 / rr to get the resultant force at the nozzle.



RE: 3D Force Analysis of a Hydraulic Crawler

(OP)
That's awesome, thanks! I just finished the other two. I realised the symmetry meant that the perpendicular vector joined the axis at the mid-plane (to "uncross" M using r to get F). So that also turned out to be fairly trivial too. But it's great to know that you got the same answer for the yoke! Thanks again.

RE: 3D Force Analysis of a Hydraulic Crawler

ChrisDanger,

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 moments
 

I 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

(OP)
Hi,

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

ChrisDanger,

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

(OP)
Yes, the boom and knuckle hinges are always in a horizontal plane, and the resultant force is in the vertical plane that contains the yoke hinge.

RE: 3D Force Analysis of a Hydraulic Crawler

ChrisDanger,

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

I'm just skipping all the other calcs to suggest that the method of virtual work would probably be easier.

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

Hi,
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.

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