"Matrix is close to singular or badly scaled " in line 47 of below code (Newmark integration method for mDOF)

clc

clear all

%%

T_n=50; % natural period

dt=0.05; % time step size

t=0:dt:T_n; % total length of time

n=length(t)-1; % number of time steps

Tol2=1e-8;

Tol1=1e-8; % convergence criteria

%% Determine which special case to use: constant avg. vs. linear

if dt/T_n<=0.551 % Use linear accel. method - closest to theoretical

gamma=0.5;

beta=1/6;

else % Use constant avg. accel. method - unconditionally stable

gamma=0.5;

beta=0.25;

end

%% Establish system properties

omega1=0.5; % natural angular frequency

omega2=0.8;

M=[1, 0; 0 , 2]; % mass

K1=3;

K2=2;

C1=0.1;

C2=0.2; % damping constant

%% Input excitation function

P=zeros(2,n);

P(1,(t<=T_n))=((11*sin(t(t<=T_n)/2))/4 - (2*cos((4*t(t<=T_n))/5))/25 + cos(t(t<=T_n)/2).^2/40 - 3*sin((4*t(t<=T_n))/5).^3);

P(2,(t<=T_n))=(5*sin((4*t(t<=T_n))/5).^3 - 3*sin(t(t<=T_n)/2) - (32*sin((4*t(t<=T_n))/5))/25 - cos(t(t<=T_n)/2).^2/40 - (2*cos((4*t(t<=T_n))/5))/25);

%% Establish initial conditions @ i=1

U(:,1)=[0; 0]; % Initial displacement

V(:,1)=[0.5 ;0.8]; % Intial velocity

N(:,1)=@ (U, V, i) [C1*V(1,i)^2-C1*V(2,i)+K1*U(1,i)-K1*U(2,i)^3; -C1*V(1,i).^2+C1*V(2,i)-K1*U(1,i)+K1*U(2,i)^3+K2*U(2,i)^3+C2*V(2,i)];

A(:,1)=inv(M)*(P(:,1)-N(U, V, 1)); % initaial acceleration

%% Calculations for each time step, i=0,1,2,...,n

Gs=@ (u1, v1, ud1, i) u1(:,i)-ud1(:,i)+dt^2*beta*(M^(-1)*(P(:,i)-N(U, V, i)));

Hs=@ (u1, v1, vd1, i) v1(:,i)-vd1(:,i)+dt*gamma*(M^(-1)*(P(:,i)-N(U, V, i)));

Kt=@ (U, V, i) [K1, -3*K1*U(2,i)^2 ; -K1, 3*(K2+K1)*U(2,i)^2];

Ct=@ (U, V, i) [2*C1*V(1,i), -C1; -2*C1*V(1,i), (C1+C2)];

I=eye(2);

for i=1:1

Ud(:,i+1)=U(:, i)+V(:, i)*dt+dt^2*(1/2-beta)*A(:,i);

Vd(:,i+1)=V(:, i)+dt*(1-gamma)*A(:, i);

U(:,i+1)=Ud(:,i+1);

V(:,i+1)=Vd(:, i+1);

while ((abs(U(1,i+1)-U(1,i))>Tol1) || (abs(U(1,i+1)-U(1,i))>Tol1)||(abs(V(1,i+1)-V(1,i))>=Tol2)||(abs(V(2,i+1)-V(2,i))>=Tol2));

d(:,i+1)=[U(:,i+1);V(:,i+1)]+[(I-dt^2*beta*inv(M)*Kt(U(:,i+1), V(:,i+1), i)), (-dt^2*beta*inv(M)*Ct(U(:,i+1), V(:,i+1), i)); (-dt*gamma*inv(M)*Kt(U(:,i+1), V(:,i+1), i)), (I-dt*gamma*inv(M)*Ct(U(:,i+1), V(:,i+1), i))]\[Gs(U(:,i+1), V(:,i+1), Ud(:,i+1),1);Hs(U(:, i+1), V(:,i+1), Vd(:, i+1), 1)] ;

U(:,i+1)=[d(1,i+1) ;d(2,i+1)];

V(:,i+1)=[d(3,i+1) ;d(4,i+1)];

end

A(:,i+1)=M^-1*(P(:,i+1)-N(U(:,i+1), V(:,i+1), 1));

end

Answer 1

[Rajil Kansal]

Hey,

These warning messages can occur when solving a linear system using the backslash operator (\). They indicate that the system being solved may be poorly conditioned or rank deficient; this can produce incorrect answers. In general the solutions you'll receive are correct, but if you receive these messages it's a good idea to check your results.

You can also refer to https://www.mathworks.com/help/matlab/ref/mldivide.html . In this you can find in the description that MATLAB displays a warning message if A is badly scaled or nearly singular, but performs the calculation regardless.

Also, if you wish to ignore these warning messages, you can turn the warning indicator by typing 'warning off' at the MATLAB command prompt. Alternatively, if you do not want to turn these warnings off entirely, you can use the command 'warning on', which will give shorter warning messages.