Comparing natural frequency by Rayleigh Ritz method(RRM) and assumed mode method(AMM) for simply supported beam with point masses

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Aniket Bharamgonda 2020-1-19

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https://ww2.mathworks.cn/matlabcentral/answers/501014-comparing-natural-frequency-by-rayleigh-ritz-method-rrm-and-assumed-mode-method-amm-for-simply-sup

编辑： Aniket Bharamgonda 2020-1-19

I am trying to get similar results from RRM and AMM for simply supported beam with point masses. But I am getting totaly different result from both cases. Please help me in pointing out error in codes. Below is image representing loading condition and code.

clc
clear
%Program to compute Natural frequency using AMM
L=5; b=0.05; E=69e9; rho=2500;      % Material Propertise
syms x ld
h=0.2;                                    %Beam height
m1=200; m2=90; m3=50; m4=30; m5=10; %Point Masses
Ar=h*b;                                % Beam Area
I=(1/12)*b*(h)^3;                      %Beam Inertia
phi=[sin(pi*x/L) sin(2*pi*x/L) sin(3*pi*x/L) sin(4*pi*x/L) sin(5*pi*x/L) ...
    sin(6*pi*x/L) sin(7*pi*x/L) sin(8*pi*x/L)];
dphi=diff(phi,x);
ddphi=diff(dphi,x);
for i=1:7
    for j=1:7
        im=rho*Ar*int(phi(i)*phi(j),x,0,L)...
        +m1*subs(phi(i)*phi(j),x,0.2*L)+m2*subs(phi(i)*phi(j),x,0.4*L)...
        +m3*subs(phi(i)*phi(j),x,0.5*L)+m4*subs(phi(i)*phi(j),x,0.6*L)...
        +m5*subs(phi(i)*phi(j),x,0.8*L);
        M(i,j)=im;
        ik=E*I*int(ddphi(i)*ddphi(j),x,0,L);
        K(i,j)=ik;
    end
end
A=K-(ld*M);
p=coeffs(det(A),ld);
l=roots(p);
%End of AMM
% Code for RRM
L=5; N=11; b=0.05; E=69e9; rho=2500;      % Material Propertise as per paper
syms x 
h=0.2; m1=200; m2=90; m3=50; m4=30; m5=10; %Design parameters
Ar=h*b;                                % Beam Area
I=(1/12)*b*(h)^3;                      %Beam Inertia
% Mass and Stiffness matrix
for i=1:N
    for j=1:N
        im=double(int(rho*Ar*sin(i*pi*x/L)*sin(j*pi*x/L), x, 0, L)...
        +m1*sin(i*pi*0.2)*sin(j*pi*0.2)+m2*sin(i*pi*0.4)*sin(j*pi*0.4)...
        +m3*sin(i*pi*0.5)*sin(j*pi*0.5)+m4*sin(i*pi*0.6)*sin(j*pi*0.6)...
        +m5*sin(i*pi*0.8)*sin(j*pi*0.8)); 
        ik=double(int(E*I*sin(i*pi*x/L)*sin(j*pi*x/L)*(pi/L)^4*(i*j)^2, x, 0, L));
        K(i,j)=ik;         %Stiffness matrix
        M(i,j)=im;         %Mass matrix
    end
end
%Natural Frequency and modal vector 
A=M\K;
[v,e]=eig(A);
w=double(sqrt(e));
C=length(v);
%End of RRM