Solving a system of Second Order Equations

Trying to solve but not able to get an output

Need the equations of x1 to x5.

m1 = 0.17; m2 = 0.17; m3 = 0.114; m4 = 0.114; m5=0.114; 
k = 50; k1 = k; k2 = 0.8*k; k3 = 0.7*k; k4 = 0.6*k; k5=0.6*k; k6=0.9*k;
M = [m1 0 0 0 0; 0 m2 0 0 0; 0 0 m3 0 0; 0 0 0 m4 0; 0 0 0 0 m5];
K = [k1+k2 -k2 0 0 0; -k2 k2+k3 -k3 0 0; 0 -k3 k3+k4 -k4 0; 0 0 -k4 k4+k5 -k5; 0 0 0 -k5 k5+k6];

w = 2*pi*40;

syms x1(t) x2(t) x3(t) x4(t) x5(t)

eqn1 = m1*diff(x1,t,2) + (k1+k2)*x1 - k2*x2 == 5*sin(w*t);
eqn2 = m2*diff(x2,t,2) - k2*x1 + (k2+k3)*x2 - k3*x3 == 5*sin(w*t);
eqn3 = m3*diff(x3,t,2) - k3*x2 + (k3+k4)*x3 - k4*x4 == 5*sin(w*t);
eqn4 = m4*diff(x4,t,2) - k4*x3 + (k4+k5)*x4 - k5*x5 == 5*sin(w*t);
eqn5 = m5*diff(x5,t,2) - k5*x4 + (k5+k6)*x5 == 5*sin(w*t);

eqns = [eqn1; eqn2; eqn3; eqn4; eqn5];
S = dsolve(eqns)

Answer 1

[Alan Stevens]

Not sure there is a symbolic solution, but it's easy enough to get a numerical one:

tspan = [0 1];  % Start and end times
% Initial conditions
% x0 = [x1(0), x2(0), x3(0), x4(0), x5(0), v1(0), v2(0), v3(0), v4(0), v5(0)]
x0 = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0];

[t, x] = ode45(@rates, tspan, x0);

plot(t,x(:,1:5)), grid
xlabel('t'), ylabel('x')
legend('x1','x2','x3','x4','x5')

function dxdt = rates(t, x)

m1 = 0.17; m2 = 0.17; m3 = 0.114; m4 = 0.114; m5=0.114; 
k = 50; k1 = k; k2 = 0.8*k; k3 = 0.7*k; k4 = 0.6*k; k5=0.6*k; k6=0.9*k;
w = 2*pi*40;

x1 = x(1); x2 = x(2); x3 = x(3); x4 = x(4); x5 = x(5);
v1 = x(6); v2 = x(7); v3 = x(8); v4 = x(9); v5 = x(10);
f = 5*sin(w*t);

dxdt = [v1; v2; v3; v4; v5;
        (f - ((k1+k2)*x1 - k2*x2))/m1;
        (f - (- k2*x1 + (k2+k3)*x2 - k3*x3))/m2;
        (f - (- k3*x2 + (k3+k4)*x3 - k4*x4))/m3;
        (f - (- k4*x3 + (k4+k5)*x4 - k5*x5))/m4;
        (f - (- k5*x4 + (k5+k6)*x5))/m5];

end

[John D'Errico]

(+1) I would guess it is unlikely a symbolic solution exists at all, so a numerical solution is the only choice. At the same time, I'd bet that a numerical solve is not what the OP was looking to find. Such is life.

[Sam Chak]

I didn't attempt to solve this problem because I'm unsure if the same force, really acts on each mass point in the Mass–Spring System. Hope @Eshwar D can clarify on this matter sooner.

[Eshwar D]

@Sam Chak, the value if 5sin(wt) isn't acting on each m. As the data is not available for each mass yet, it was just assumed to be that for the moment. The actual values will be different.