Solving coupled ODE symbolically

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saravanakumar D
saravanakumar D 2021-12-25
评论： saravanakumar D 2021-12-26
Ran in:
I need to solve this coupled ODE through laplace transform.
My algorithm
  1. I have written the 2nd order ODE in matrix form.
  2. Took laplace transform
  3. Substituted the initial conditions. (Help Needed: Could not substitute diff(x1(0)) and diff(x2(0)) as zero.
  4. Solve the linear equation (Help Needed: Don't know what to do)
I have attached my code.
syms t s x1(t) x2(t) m1 m2 c1 c2 k1 k2 k_c c_c Delta_m f1 f2 omega f_0
M=[m1 0;0 m2]
M = 
C=[c1+c_c -c_c;-c_c c2+c_c]
C = 
K=[k1+k_c -k_c;-k_c k2+k_c]
K = 
F=[f1;f2]
F = 
X=[x1;x2]
X(t) = 
equ=M*diff(X,t,2)+C*diff(X,t)+K*X==F
equ(t) = 
equ=subs(equ,[f1,f2],[f_0*cos(omega*t),0])
equ(t) = 
L1=laplace(equ)
L1 = 
L2=subs(L1,[x1(0),x2(0),diff(x1(0),t,1),diff(x2(0),t,1)],[0 0 0 0]) %Substituting Zero Initial Condition
L2 = 
% In above step the time derivative has to be zero, but it is not.
Sol=solve(L2,[laplace(x1) laplace(x2)])
Error using sym.getEqnsVars>checkVariables (line 98)
Second argument must be a vector of symbolic variables.

Error in sym.getEqnsVars (line 62)
checkVariables(vars);

Error in sym/solve>getEqns (line 429)
[eqns, vars] = sym.getEqnsVars(argv{:});

Error in sym/solve (line 226)
[eqns,vars,options] = getEqns(varargin{:});

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采纳的回答

Walter Roberson
Walter Roberson 2021-12-25
Ran in:
syms t s x1(t) x2(t) m1 m2 c1 c2 k1 k2 k_c c_c Delta_m f1 f2 omega f_0
M=[m1 0;0 m2]
M = 
C=[c1+c_c -c_c;-c_c c2+c_c]
C = 
K=[k1+k_c -k_c;-k_c k2+k_c]
K = 
F=[f1;f2]
F = 
X=[x1;x2]
X(t) = 
equ=M*diff(X,t,2)+C*diff(X,t)+K*X==F
equ(t) = 
equ=subs(equ,[f1,f2],[f_0*cos(omega*t),0])
equ(t) = 
L1=laplace(equ)
L1 = 
dx1 = diff(x1)
dx1(t) = 
dx2 = diff(x2)
dx2(t) = 
lap1 = laplace(x1)
lap1 = 
lap2 = laplace(x2)
lap2 = 
syms LAP1 LAP2
L2 = subs(L1, {x1(0), x2(0), dx1(0), dx2(0), lap1, lap2}, {0, 0, 0, 0, LAP1, LAP2}) %Substituting Zero Initial Condition
L2 = 
Sol = solve(L2, [LAP1, LAP2])
Sol = struct with fields:
LAP1: (f_0*s*(k2 + k_c + c2*s + c_c*s + m2*s^2))/((omega^2 + s^2)*(k1*k2 + k1*k_c + k2*k_c + c1*m2*s^3 + c2*m1*s^3 + c_c*m1*s^3 + c_c*m2*s^3 + k1*m2*s^2 + k2*m1*s^2 + k_c*m1*s^2 + k_c*m2*s^2 + m1*m2*s^4 + c1*k2*s + c2*k1*s + c1*k_c*s + c_c*k1*s + … LAP2: (f_0*s*(k_c + c_c*s))/((omega^2 + s^2)*(k1*k2 + k1*k_c + k2*k_c + c1*m2*s^3 + c2*m1*s^3 + c_c*m1*s^3 + c_c*m2*s^3 + k1*m2*s^2 + k2*m1*s^2 + k_c*m1*s^2 + k_c*m2*s^2 + m1*m2*s^4 + c1*k2*s + c2*k1*s + c1*k_c*s + c_c*k1*s + c2*k_c*s + c_c*k2*s +…
LAP1s = simplify(Sol.LAP1)
LAP1s = 
LAP2s = simplify(Sol.LAP2)
LAP2s = 
%iLap1 = ilaplace(LAP1s)
%iLap2 = ilaplace(LAP2s)
The ilaplace() take longer than this demonstration facility allows

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