1. ## Nonlinear system

This is a post by shoulddt:

Originally Posted by shoulddt
Hello all,
I'm new to the forum, so if this post is in the wrong place I appologize.

I am looking for help at deriving a relationship between unknown variables in large-scale system of symbolic nonlinear equations.
I attempted to use Maple's solve command to determine the relationship, but the equations are too complex and the solution is too large to solve for. I was hoping someone could point me towards an alternative method of solving the system.

The equations in my system are below. I need to solve for the relationship between Lt and Ut, as well as deriving an equation for Ssteer.
Known variables are: LMPx, LMPy, LMPz, LCAxOffset, LCALength, La, Lb, UMPx, UMPy, UMPz, UCAxOffset, UCALength, Ua, Ub, SpindleHt, TRLength, LCAWCx, LCAWCy, LCAWCz, Tire_Radius, ITRx, ITRy, ITRz, SAWCx, SAWCy, SAWCz

While those variables are know, their values can change with each scenario, so I have to solve everything sybollically. The system is 20 equations, 20 unknowns. The equations have been tested and are valid.
Any help at all on how to solve it, or any tips or tricks would be much appreciated. The program I need to implement my solution in is Excel, so if anyone knows of any way to do some of it in Excel that would be great.

eq1 := LBJx = LMPx+cos(La)*cos(Lb)*LCAxOffset+(-sin(La)*cos(Lt)+cos(La)*sin(Lb)*sin(Lt))*LCALength ;
eq2 := LBJy = LMPy+sin(La)*cos(Lb)*LCAxOffset+(cos(La)*cos(Lt)+s in(La)*sin(Lb)*sin(Lt))*LCALength;
eq3 := LBJz = LMPz-sin(Lb)*LCAxOffset+cos(Lb)*sin(Lt)*LCALength;
eq4 := UBJx = UMPx+cos(Ua)*cos(Ub)*UCAxOffset+(-sin(Ua)*cos(Ut)+cos(Ua)*sin(Ub)*sin(Ut))*UCALength ;
eq5 := UBJy = UMPy+sin(Ua)*cos(Ub)*UCAxOffset+(cos(Ua)*cos(Ut)+s in(Ua)*sin(Ub)*sin(Ut))*UCALength;
eq6 := UBJz = UMPz-sin(Ub)*UCAxOffset+cos(Ub)*sin(Ut)*UCALength;
eq7 := WCx = LBJx+cos(Scaster)*cos(Ssteer)*LCAWCx-cos(Scaster)*sin(Ssteer)*LCAWCy+sin(Scaster)*LCAWC z;
eq8 := WCy = LBJy+(sin(Scamber)*sin(Scaster)*cos(Ssteer)+cos(Sc amber)*sin(Ssteer))*LCAWCx+(-sin(Scamber)*sin(Scaster)*sin(Ssteer)+cos(Scamber) *cos(Ssteer))*LCAWCy-sin(Scamber)*cos(Scaster)*LCAWCz;
eq9 := WCz = LBJz+(-cos(Scamber)*sin(Scaster)*cos(Ssteer)+sin(Scamber) *sin(Ssteer))*LCAWCx+(cos(Scamber)*sin(Scaster)*si n(Ssteer)+sin(Scamber)*cos(Ssteer))*LCAWCy+cos(Sca mber)*cos(Scaster)*LCAWCz;
eq10 := SpindleHt = sqrt((LBJx-UBJx)^2+(LBJy-UBJy)^2+(LBJz-UBJz)^2);
eq11 := Scamber = arctan((LBJy-UBJy)/(LBJz-UBJz));
eq12 := Scaster = arctan((LBJx-UBJx)/(LBJz-UBJz));
eq13 := TRLength = sqrt((SAx-ITRx)^2+(SAy-ITRy)^2+(SAz-ITRz)^2);
eq14 := CPx = WCx-(sin(Scamber)*sin(Ssteer)+cos(Scamber)*sin(Scaster )*cos(Ssteer))*Tire_Radius;
eq17 := CPz = 0;
eq18 := SAx = WCx+cos(Scaster)*cos(Ssteer)*SAWCx-cos(Scaster)*sin(Ssteer)*SAWCy+sin(Scaster)*SAWCz;
eq19 := SAy = WCy+(sin(Scamber)*sin(Scaster)*cos(Ssteer)+cos(Sca mber)*sin(Ssteer))*SAWCx+(-sin(Scamber)*sin(Scaster)*sin(Ssteer)+cos(Scamber) *cos(Ssteer))*SAWCy-sin(Scamber)*cos(Scaster)*SAWCz;
eq20 := SAz = WCz+(-cos(Scamber)*sin(Scaster)*cos(Ssteer)+sin(Scamber) *sin(Ssteer))*SAWCx+(cos(Scamber)*sin(Scaster)*sin (Ssteer)+sin(Scamber)*cos(Ssteer))*SAWCy+cos(Scamb er)*cos(Scaster)*SAWCz;

2. ## Re: Nonlinear system

One idea that springs to mind would be to linearise the system around the stationary points as a basis for an approximation. But being 20-dimensional renders any visualisation somewhat difficult.

4. ## Re: Nonlinear system

I would question whether the model really needs that much complexity. I'd have thought that some simplification might be achieved.

5. ## Re: Nonlinear system

Originally Posted by Archie
I would question whether the model really needs that much complexity. I'd have thought that some simplification might be achieved.
Due to the nature of what I am modeling it does need to have all of those equations to fully constrain the system. With that being said, those equations are the results of vector based equations that perform rotations and translations to vectors.

Romsek, thank you for the link. I will try to figure it out, but to be honest it looks way over my head. I guess I should've taken more math courses in college haha.