Hello, I recently used Matlab to find the natural frequencies of a clamped-clamped beam. It was fairly simple, as I construct the global mass and stiffness matrices. Then it's just a matter of using the eig() function. (For sake of simplicity I put beam properties all to 1.) When I choose how many elements (lets call it N) to use, it seems to give me natural frequencies which, seem to be smaller than usual. I somehow stumbled upon the fact that these frequencies are the literature values I have divided by N^2. Anyone got any ideas as why this is so? Here is my code. Thanks!! Edit: For example when N = 10, my first natural frequency is 0.2237 when the exact value is 22.37. Jeff Code (Text): syms y; syms z; area = 1; L = 1; p = 1; E = 1; Ig = 1; n = 10; %number of elements if n == 0 stop end size = 4+((n-1)*2); %define global matrices Mg = zeros(size); Kg = zeros(size); %beam function A = [1 0 0 0 ; 0 1 0 0 ; 1 L L^2 L^3 ; 0 1 2*L 3*L^2]; Ainv = inv(A); Aitrans = transpose(Ainv); Yh = [1 ; y ; y^2 ; y^3]; Ya = [1 ; y]; Ydotdot = diff(diff(Yh)); %Solving for Mass matrix (kinetic energy) prod1 = Yh*transpose(Yh); M = p*area*Aitrans*int(prod1, y, 0, L)*Ainv; %solving for stiffness matrix (potential energy) K = (E*Ig)*Aitrans*int(Ydotdot*transpose(Ydotdot), y, 0, L)*Ainv; %creating the global mass matrix Mg(1:4, 1:4) = M(1:4, 1:4); Kg(1:4, 1:4) = K(1:4, 1:4); if n > 1 i = 1; j = 1; for i=1:n-1 Mg(j+2:j+5, j+2:j+5) = Mg(j+2:j+5, j+2:j+5) + M(1:4, 1:4); Kg(j+2:j+5, j+2:j+5) = Kg(j+2:j+5, j+2:j+5) + K(1:4, 1:4); j = j+2; end end %delete rows/columns based on clamped ends (boundary conditions) Mg(size,:) = []; Mg(size-1,:) = []; Mg(:,size) = []; Mg(:,size-1) = []; Mg(:,1) = []; Mg(:,1) = []; Mg(1,:) = []; Mg(1,:) = []; Kg(size,:) = []; Kg(size-1,:) = []; Kg(:,size) = []; Kg(:,size-1) = []; Kg(:,1) = []; Kg(:,1) = []; Kg(1,:) = []; Kg(1,:) = []; eigs = eig(Kg,Mg); sqrt(eigs)
I haven't checked every line of the code, but I think you have each element of length 1, so the length of your beam depends on the number of elements in the model. If the length of the beam is [itex]l[/itex], the global mass is proportional to [itex]l[/itex] and the global stiffness proportional to [itex]1/l^3[/itex], so the frequences would be proportional to [itex]\sqrt{k/m} = 1/l^2[/itex].