hi there i want to get the values of A1,A2,A2,Q1,Q2,Q3 , i have six equations and six initial conditions as Xo, Vo kindly suggest me solution

for i = 1:1:3 
    x(t) = U(i,1)*A1*cos(w2(1,1)*t + Q1) + U(i,2)*A2*cos(w2(2,2)*t + Q2) +U(i,3)*A3*cos(w2(3,3)*t + Q3);
    x= x(t);
    v= diff(x(t));
   
end

x
v

Xo = [1 0 0]
Vo = [0 0 0]

% eqns = [x(1,1) == Xo(1,1) , x(1,2) == Xo(1,2), x(1,3) == Xo(1,3), v(1,1) == Vo(1,1) , v(1,2) == Vo(1,2), v(1,3) == Vo(1,3)];


% S = solve(eqns,[A1 A2 A3 Q1 Q2 Q3])

Answer 1

[Walter Roberson]

for i = 1:1:3 
  x{i} = U(i,1)*A1*cos(w2(1,1)*t + Q1) + U(i,2)*A2*cos(w2(2,2)*t + Q2) +U(i,3)*A3*cos(w2(3,3)*t + Q3);
  v{i} = diff(x{i}, t);
end

Xo = [1 0 0]
Vo = [0 0 0]

eqns = [x{1} == Xo(1,1) , x{2} == Xo(1,2), x{3} == Xo(1,3), v{1} == Vo(1,1) , v{2} == Vo(1,2), v{3} == Vo(1,3)];

S = solve(eqns,[A1 A2 A3 Q1 Q2 Q3])

[Rahul Jangid]

thanks for that but it is not showing me exact value

it is showing as

S =

struct with fields:

A1: [0×1 sym]

A2: [0×1 sym]

A3: [0×1 sym]

Q1: [0×1 sym]

Q2:

[0×1 sym]

Q3: [0×1 sym]

what should i change ???

[Rahul Jangid]

i want to find these values in numeric terms

[Walter Roberson]

please post sample U and w2

[Rahul Jangid]

U = [
   -0.3280    0.7370   -0.5910
   -0.5910    0.3280    0.7370
   -0.7370   -0.5910   -0.3280] 
w2 = [
    0.1981         0         0
         0    1.5550         0
         0         0    3.2470]

[Walter Roberson]

syms A1 A2 A3 Q1 Q2 Q3 t

U = [
 -0.3280    0.7370   -0.5910
 -0.5910    0.3280    0.7370
 -0.7370   -0.5910   -0.3280] 
w2 = [
  0.1981         0         0
       0    1.5550         0
       0         0    3.2470]
for i = 1:1:3 
x{i} = U(i,1)*A1*cos(w2(1,1)*t + Q1) + U(i,2)*A2*cos(w2(2,2)*t + Q2) +U(i,3)*A3*cos(w2(3,3)*t + Q3);
v{i} = diff(x{i}, t);
end

Xo = [1 0 0]
Vo = [0 0 0]

eqns = [x{1} == Xo(1,1) , x{2} == Xo(1,2), x{3} == Xo(1,3), v{1} == Vo(1,1) , v{2} == Vo(1,2), v{3} == Vo(1,3)];

S = solve(eqns,[A1 A2 A3 Q1 Q2 Q3])

This gets me 8 solutions when I test.

Note that if really your U or w2 have more digits of precision than you posted, then it is possible that due to small differences in the values that there might not be any solutions.

[Rahul Jangid]

bro i need to finally find the response curve for all three masses and plot x1,x2,x3 wrt. t

so here is my code please help

%% Intoroduction of variables 
syms m k F1 F2 F3 A1 A2 A3 Q1 Q2 Q3 t

%% Value submission 
m=1;
k=1;
% w=1;

%% Mass, Stiffness AND Force matrices

M=[m 0 0
    0 m 0
    0 0 m];

K = [2*k -k 0
    -k 2*k -k
    0 -k k] ;


%% Equation of motion
format short
[U,w2] = eig(K,M)  


%% Response of the system (here A and Q shows the amplitude and phase difference)
format short
for i = 1:1:3 
    x(t) = U(i,1)*A1*cos(w2(1,1)*t+Q1) + U(i,2)*A2*cos(w2(2,2)*t+Q2) +U(i,3)*A3*cos(w2(3,3)*t+Q3);
    x= x(t);
    v= diff(x); 
end

x
v

Xo = [1 0 0];
Vo = [0 0 0];



% eqns = [x(1,1) == Xo(1,1) , x(1,2) == Xo(1,2), x(1,3) == Xo(1,3), v(1,1) == Vo(1,1) , v(1,2) == Vo(1,2), v(1,3) == Vo(1,3)];

eqns = [x(1,:) == Xo(1,:) , v(1,:) == Vo(1,:)]'
sol = vpasolve(eqns)

[Rahul Jangid]

here Xo and Vo shows the initial conditions of displacement and velocity

[Rahul Jangid]

in previous code w^2 is there so i have corrected it please use this as correct equation

x(t) = U(i,1)*A1*cos(sqrt(w2(1,1))*t+Q1) + U(i,2)*A2*cos(sqrt(w2(2,2))*t+Q2) +U(i,3)*A3*cos(sqrt(w2(3,3))*t+Q3);

[Sam Chak]

Rahul, I guess you want to formulate the equations of motion, which are a coupled set of differential equations. Please confirm.

Otherwise, the way you described your problem can be confusing.

[Rahul Jangid]

No i have already formulated the equ. of motion thats why i m able to write mass and stiffness matrix

than i found natural frequencies and modes also

then as we know that response of system is summation of natural modes so i have derived x(t) from this i can find displacement of all the masses with respect to time.

so finally i want the displacements of all the masses (to find out the unknowns i have also given initial conditions )

if u have any other idea to get the response curve please suggest me or else help me to find out those displacement equations.

[Walter Roberson]

%% Intoroduction of variables 
syms m k F1 F2 F3 A1 A2 A3 Q1 Q2 Q3 t

%% Value submission 
m=1;
k=1;
% w=1;

%% Mass, Stiffness AND Force matrices

M=[m 0 0
    0 m 0
    0 0 m];

K = [2*k -k 0
    -k 2*k -k
    0 -k k] ;


%% Equation of motion
format short
[U,w2] = eig(K,M)  

U = 3×3

-0.3280 0.7370 -0.5910 -0.5910 0.3280 0.7370 -0.7370 -0.5910 -0.3280

w2 = 3×3

0.1981 0 0 0 1.5550 0 0 0 3.2470

%% Response of the system (here A and Q shows the amplitude and phase difference)
format short
for i = 1:1:3 
    x{i} = U(i,1)*A1*cos(w2(1,1)*t+Q1) + U(i,2)*A2*cos(w2(2,2)*t+Q2) +U(i,3)*A3*cos(w2(3,3)*t+Q3);
    v{i} = diff(x{i}, t); 
end

x

x = 1×3 cell array
    {[(6638091741507547*A2*cos(Q2 + (7002908864245409*t)/4503599627370496))/9007199254740992 - (2661668130624679*A3*cos(Q3 + (913943508336349*t)/281474976710656))/4503599627370496 - (5908457496031831*A1*cos(Q1 + (3567972556901945*t)/18014398509481984))/18014398509481984]}    {[(2954228748015915*A2*cos(Q2 + (7002908864245409*t)/4503599627370496))/9007199254740992 + (1659522935376887*A3*cos(Q3 + (913943508336349*t)/281474976710656))/2251799813685248 - (332708516328085*A1*cos(Q1 + (3567972556901945*t)/18014398509481984))/562949953421312]}    {[- (5323336261249361*A2*cos(Q2 + (7002908864245409*t)/4503599627370496))/9007199254740992 - (2954228748015915*A3*cos(Q3 + (913943508336349*t)/281474976710656))/9007199254740992 - (6638091741507549*A1*cos(Q1 + (3567972556901945*t)/18014398509481984))/9007199254740992]}

v

v = 1×3 cell array
    {[(2432614309330170770701412156971*A3*sin(Q3 + (913943508336349*t)/281474976710656))/1267650600228229401496703205376 - (46485951498277445065388233601723*A2*sin(Q2 + (7002908864245409*t)/4503599627370496))/40564819207303340847894502572032 + (21081214199463155606688465811295*A1*sin(Q1 + (3567972556901945*t)/18014398509481984))/324518553658426726783156020576256]}    {[(1187094855706169954794794625325*A1*sin(Q1 + (3567972556901945*t)/18014398509481984))/10141204801825835211973625643008 - (1516710213722988286690676565563*A3*sin(Q3 + (913943508336349*t)/281474976710656))/633825300114114700748351602688 - (20688194686489267889392397684235*A2*sin(Q2 + (7002908864245409*t)/4503599627370496))/40564819207303340847894502572032]}    {[(37278838691262164489772848433649*A2*sin(Q2 + (7002908864245409*t)/4503599627370496))/40564819207303340847894502572032 + (2699998186389765280096224994335*A3*sin(Q3 + (913943508336349*t)/281474976710656))/2535301200456458802993406410752 + (23684529163896374554619270282805*A1*sin(Q1 + (3567972556901945*t)/18014398509481984))/162259276829213363391578010288128]}

Xo = [1 0 0];
Vo = [0 0 0];



% eqns = [x(1,1) == Xo(1,1) , x(1,2) == Xo(1,2), x(1,3) == Xo(1,3), v(1,1) == Vo(1,1) , v(1,2) == Vo(1,2), v(1,3) == Vo(1,3)];

eqns = [x{1} == Xo(1,1) , x{2} == Xo(1,2), x{3} == Xo(1,3), v{1} == Vo(1,1) , v{2} == Vo(1,2), v{3} == Vo(1,3)];
sol = solve(eqns, [A1 A2 A3 Q1 Q2 Q3])

sol = struct with fields:
    A1: [8×1 sym]
    A2: [8×1 sym]
    A3: [8×1 sym]
    Q1: [8×1 sym]
    Q2: [8×1 sym]
    Q3: [8×1 sym]

cross_check = subs(lhs(eqns) - rhs(eqns), sol)

cross_check = 

structfun(@(S)disp(vpa(S,10)), sol)

Remember that when you use diff() and you do not specify which variable to take the derivative, that MATLAB chooses a variable for you.

Also remember that you are trying to vpasolve() 8 equations in 9 variables (with t being the extra variable)

[Sam Chak]

Okay, @Rahul Jangid. Thanks for your clarification. You want to find the solutions for so that you can plot . What do the 8 sets of tell you? How would you choose a unique set of solution?

[Rahul Jangid]

x1(t) telss me the displacement of mass 1 with respect to time.

[Rahul Jangid]

thanks for your efforts , i have reached to the sol. from your suggestions.