Boundary conditions for different areas of phi

Dear Forum,

This is my first post, I hope I'm not doing anything wrong. I am not a native English speaker, but I hope to translate the mathematical expressions well.

I want to define a function for a cam gear. I have already defined the cam mechanism linearly, now I want to define the actual function using a sinoid. That means, I set up the laws of motion in general. I have done that (f0,f1,f2,f1f2). Now I would like to determine the special parameters and for this I define the motion function and its derivatives (F0,F1,F2,F1F2) again with boundary conditions. For example, for phi between 0° and 10° the motion function of the path should be -1.

I wanted to try it the way you can see below. But no matter what I do, the error remains. I need your help.

N_Vert = 25.4/11;       %11 needles for 25.4mm = 2.31mm

% define Positions
s_hubweg = N_Vert*[0;-1;-1;0;0;1;1;2;2;1;1;0;0];
winkel = [0;10;22.5;32.5;45;55;90;100;112.5;122.5;135;145;180];

% create the laws of movement
z = linspace(0,1,180);

f0 = 0.5*(1-cos(pi*z));	 %added for the forum: f0=f
f1 = (pi/2)*sin(pi*z);   %added for the forum: f1=f'
f2 = (pi^2/2)*cos(pi*z); %added for the forum: f2=f''
f1f2 = f1.*f2;           %added for the forum: f1f2=f'*f''

% nominal parameters for cam gear
C_vn = 1.57;
C_an = 4.93;
C_Mdynn = 3.88;

% Übertragungsfunktion F von phi
phi = linspace(0,360,360);

% define Function for areas of phi
% function of the way
F0(phi>=0 & phi<10)         = flip(f0-1);
F0(phi>=10 & phi<22.5)      = -1;
F0(phi>=22.5 & phi<32.5)    = f0-1;
% ... and more

% function of speed
F1(phi>=0 & phi<10)         = -f1;
F1(phi>=10 & phi<22.5)      = 0;
F1(phi>=22.5 & phi<32.5)    = f1;
% ... and more

% function of acceleration
F2(phi>=0 & phi<10)         = -f2;
F2(phi>=10 & phi<22.5)      = 0;
F2(phi>=22.5 & phi<32.5)    = f2;
% ... and more

% function of momentum
F1F2(phi>=0 & phi<10)       = -f1f2;
F1F2(phi>=10 & phi<22.5)    = 0;
F1F2(phi>=22.5 & phi<32.5)  = f1f2;
% ... and more

% adjust amplitudes
F0 = F0*N_Vert;
F1 = F1*N_Vert;
F2 = F2*N_Vert;
F1F2 = F1F2*N_Vert.^2;

My error starts to appear with this line:

F0(phi>=0 & phi<10)         = flip(f0-1);

And continues with all similar lines.

I hope I've given you all the relevant information about the code so that you can understand my problem.

Can you tell me, how to setup such a process correctly in MATLAB?

I would be very grateful for an answer! Thanks a lot!

Answer 1

[Alan Stevens]

Firstly, phi is size 1x360, whereas f0 is of size 1x180, but, more importantly perhaps, phi>=0 & phi<10 means you want elements that are both less than and greater than zero!. I suspect you meant phi>=0 | phi<10 (or, rather than and). The result of phi>=0 & phi<10 is to return a vector of zeros, and MATLAB indexing starts at 1 not 0.

[Dave K.]

First of all thanks for your time!

Of course you are absolutely right that f0 and phi are of different lengths. I increased the run variable z for f0 from 180 to 360 values, that fits now.

Also, as you suggested, I also replaced the & with |. This fixed the error, but not the problem. The command still doesn't seem to work properly, because as an answer for F, no matter if F_0,1,2 or _1&2, I get a matrix with only zeros. By the way, the matrix has the form [1,360].

So... error solved, but function F_x still not working.

[Dave K.]

And maybe another short thing:

Why do I need to use OR here? For phi between for example 10° and 22.5° I thought I'd say:

For phi bigger than 10° AND smaller than 22.5° the function shall be for example 1. If I would use OR in that case, wouldn't I get function shall be 1 for all phi smaller than 22.5° including the ones even smaller than the wanted 10°?

I believe this is the main problem.

% define Function for areas of phi
% function of the way
F0(phi>=0 | phi<10)         = flip(f0-1);
F0(phi>=10 | phi<22.5)      = -1;
F0(phi>=22.5 | phi<32.5)    = f0-1;
% ... and more
%last line of this area:
F0(phi>=325 | phi<=360)     = 0; %!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

See? I believe the following happens:

For all phi smaller than 360° function F0 shall be ZERO. I believe the OR creates the mistake.

Are you sure AND is wrong here? Because I believe it is not entirely wrong here.

[Alan Stevens]

My apologies. You are right about the &. I misread the conditions!

I think you need something like:

temp = flip(f0-1);
F0(phi>=0 & phi<10)         = temp(phi>=0 & phi<10);
F0(phi>=10 & phi<22.5)      = -1;
temp2 = f0-1;
F0(phi>=22.5 & phi<32.5)    = temp2(phi>=22.5 & phi<32.5);

and similarly for the other F's.

[Dave K.]

No problem :)

Now knowing that do you have an idea how to get the AND function working?

Worst case I have to create a special F0_i for every single line and then at the end add them all together to get the right function. But that would be an enormous pain in the *** and I fear that. :)

[Alan Stevens]

First of all, the following works:

N_Vert = 25.4/11;       %11 needles for 25.4mm = 2.31mm

% define Positions
s_hubweg = N_Vert*[0;-1;-1;0;0;1;1;2;2;1;1;0;0];
winkel = [0;10;22.5;32.5;45;55;90;100;112.5;122.5;135;145;180];

% create the laws of movement
z = linspace(0,1,180);

f0 = 0.5*(1-cos(pi*z));	 %added for the forum: f0=f
f1 = (pi/2)*sin(pi*z);   %added for the forum: f1=f'
f2 = (pi^2/2)*cos(pi*z); %added for the forum: f2=f''
f1f2 = f1.*f2;           %added for the forum: f1f2=f'*f''

% nominal parameters for cam gear
C_vn = 1.57;
C_an = 4.93;
C_Mdynn = 3.88;

% Übertragungsfunktion F von phi
phi = linspace(0,360,180);

% define Function for areas of phi
% function of the way
temp = flip(f0-1);
F0(phi>=0 & phi<10)         = temp(phi>=0 & phi<10);
F0(phi>=10 & phi<22.5)      = -1;
temp2 = f0-1;
F0(phi>=22.5 & phi<32.5)    = temp2(phi>=22.5 & phi<32.5);
% ... and more

% function of speed
F1(phi>=0 & phi<10)         = -f1(phi>=0 & phi<10);
F1(phi>=10 & phi<22.5)      = 0;
F1(phi>=22.5 & phi<32.5)    = f1(phi>=22.5 & phi<32.5);
% ... and more

% function of acceleration
F2(phi>=0 & phi<10)         = -f2(phi>=0 & phi<10);
F2(phi>=10 & phi<22.5)      = 0;
F2(phi>=22.5 & phi<32.5)    = f2(phi>=22.5 & phi<32.5);
% ... and more

% function of momentum
F1F2(phi>=0 & phi<10)       = -f1f2(phi>=0 & phi<10);
F1F2(phi>=10 & phi<22.5)    = 0;
F1F2(phi>=22.5 & phi<32.5)  = f1f2(phi>=22.5 & phi<32.5);
% ... and more

% adjust amplitudes
F0 = F0*N_Vert;
F1 = F1*N_Vert;
F2 = F2*N_Vert;
F1F2 = F1F2*N_Vert.^2;

I'll have a think about your next problem!

[Dave K.]

I will try that in a few minutes! (...I have about 24 lines for every F_i and in total 4 functions, so it seems like I have to use about 96 temp values and that will take its time)

Thank you very much!

[Dave K.]

Unfortunately this solution didn't work either. I now proceeded to create the function manually:

z = linspace(0,1,20);

f0 = 0.5*(1-cos(pi*z));

F0_1 = flip(f0-1);
F0_2 = -1*ones(1,25);
F0_3 = f0-1;
F0_4 = 0*ones(1,25);
F0_5 = f0;
F0_6 = 1*ones(1,70);
F0_7 = f0+1;
F0_8 = 2*ones(1,25);
F0_9 = flip(f0+1);
F0_10 = 1*ones(1,25);
F0_11 = flip(f0);
F0_12 = 0*ones(1,70);
F0_13 = flip(f0-1);
F0_14 = -1*ones(1,25);
F0_15 = f0-1;
F0_16 = 0*ones(1,25);
F0_17 = f0;
F0_18 = 1*ones(1,70);
F0_19 = f0+1;
F0_20 = 2*ones(1,25);
F0_21 = flip(f0+1);
F0_22 = 1*ones(1,25);
F0_23 = flip(f0);
F0_24 = 0*ones(1,70);

F0_T1 = [F0_1 F0_2 F0_3 F0_4 F0_5 F0_6 F0_7 F0_8 F0_9 F0_10 F0_11 F0_12]; 
F0_T2 = [F0_13 F0_14 F0_15 F0_16 F0_17 F0_18 F0_19 F0_20 F0_21 F0_22];
F0_T3 = [F0_23 F0_24];

F0 = [F0_T1 F0_T2 F0_T3];

I chose z as 20, because this way I can get the .5° values. So 20=10°in0.5° steps. This solves it. ^^

Thank you very much anyway!!!