The future works will further study the effects of parameter variation on the vibration of rectangular, circular and composite plates. For rectangular plates, effects of geometric imperfections on frequency-load interaction of biaxially compressed antisymmetric angle ply rectangular plates can be further studied. Besides, the effects of four different boundary conditions can be checked same as the circular plate. As for composite plate, cross-ply composite plate will be the next step. How will hysterical damping and viscous damping change the vibration mode will also be interesting for angle-ply and cross-ply composite plates.
APPENDIX A
MATLAB PROGRAM FOR COMPOSITE PLATE
Throughout this thesis Matlab is utilized to conduct Runge-Kutta method to solve the Modified-Duffing equation from the vibration problem. The most complicated and representative example will be angle-ply composite plate vibration problem. In this Appendix, the Matlab program that is used in thesis is given as another way to understand and solve this problem. This program can be subject to more than 10 parameter variations. And this will be of good help to future learners in this area of research.
The main program is:
%Used in composite rectangular plate calculation
%Use ODE45 to solve the Duffing eqn. Then find the T for the sin curve. %FInd linstedt's result Y and initial imperfection is set to be N
%[k,e,a2]=Cd(E1,E2,G12,r12,v1,theta,h,M,n,bc,imp,E)
N=-4;m=1;
[k,e,a]=Cd(200,5,2.5,0.25,0.5,pi/6,0.005,1,1,1,1.5,5); r=3*e/8-5*a*a*e*e/12;
while(N<4),
options=odeset; options.RelTol=0.00000001; options.AbsTol=0.0000001; options.MaxStep=0.0005; clear t; clear x; [t,x]=ode45('ODE_comp',[0,80],[N,0]); q=1; clear P; for i=10:1000, if x(i,2)*x(i+1,2)*100000000>0,i=i+1;
else P(q)=t(i); q=q+1; i=i+1; end; end; Y2(m)=1+r*N*N-15*e*e/256*N^4; %while(x(i)>x(i+1)),i=i+1; end; T2(m)=(36)*pi/(-P(1)+P(37))/k^0.5; %T(m)=2*pi/t(i); X2(m)=N;
%if x(i)>(x(i+1)-0.0001),T(m)=4*pi/(t(i)+t(i+1));else break; end; m=m+1; N=N+0.1;
end
Since different boundary conditions will give us different parameters for the modified-Duffing equation. Two similar programs have been edited. They will be called upon by the main program for the parameter calculation, which are Cd and Cd2. Cd corresponds to the boundary condition of simply supported while Cd2 corresponds to the boundary condition of clamped.
Cd:
function [k,e,a2]=Cd(E1,E2,G12,r12,v1,theta,h,M,n,bc,imp,E)
% boundary condition 1: inplane movable; BC 2: inplane immovable [A,B,D]=comp(E1,E2,G12,r12,v1,theta,h); AA=inv(A); BB=-A\B; DD=D-B*(A\B); aa=E*h*AA;
bb=BB/h; dd=DD/(E*h^3); Caa=@(M,n) -(2*bb(2,3)-bb(3,1))*M^3*n+(2*bb(1,3)-bb(3,2))*n^3*M; Cbb=@(M,n) aa(2,2)*M^4+(2*aa(1,2)+aa(3,3))*(M*n)^2+aa(1,1)*n^4; Cdd=@(M,n) dd(1,1)*M^4+2*(dd(1,2)+2*dd(3,3))*M*M*n*n+dd(2,2)*n^4; % use double a,b,d to show a*, b*, d*.
c0=-Cbb(M,n)/Caa(M,n); c1=n*n/(32*M*M*aa(2,2)); c2=M*M/(32*n*n*aa(1,1)); a11=A(1,1)/E/h; a12=A(1,2)/E/h; a21=A(2,1)/E/h; a22=A(2,2)/E/h; if bc==1 e1=0; e2=0; end if bc==2
e1=(a21*M*M+a22*n*n)*pi*pi/8; e2=(a11*M*M+a12*n*n)*pi*pi/8; end e0=((M*M*e2+n*n*e1)/pi/pi)+2*M*M*n*n*(c1+c2); k=Cdd(M,n)+Cbb(M,n)*Cbb(M,n)/Caa(M,n)+2*imp*imp*e0; e=e0/k; a2=3*imp*e0/e/k; And Cd2:
%This is for clamped angle-ply
function [k,e,a2]=Cd2(E1,E2,G12,r12,v1,theta,h,M,n,bc,imp,E)
% boundary condition 1: inplane movable; BC 2: inplane immovable [A,B,D]=comp(E1,E2,G12,r12,v1,theta,h); AA=inv(A); BB=-A\B; DD=D-B*(A\B);
aa=E*h*AA; bb=BB/h; dd=DD/(E*h^3); Caa=@(M,n) -(2*bb(2,3)-bb(3,1))*M^3*n+(2*bb(1,3)-bb(3,2))*n^3*M; Cbb=@(M,n) aa(2,2)*M^4+(2*aa(1,2)+aa(3,3))*(M*n)^2+aa(1,1)*n^4; Cdd=@(M,n) dd(1,1)*M^4+2*(dd(1,2)+2*dd(3,3))*M*M*n*n+dd(2,2)*n^4; % use double a,b,d to show a*, b*, d*.
k0=Cbb(2*M,2*n)/(-4*Caa(2*M,2*n)); k1=n*n/(32*M*M*aa(2,2)); k2=M*M/(32*n*n*aa(1,1)); k3=-n*n/(16*32*M*M*aa(2,2)); k4=-M*M/(16*32*n*n*aa(1,1)); k5=-M*M*n*n/(Caa(2*M,2*n)); k6=M*M*n*n/(2*Caa(2*M,4*n)); k7=M*M*n*n/(2*Caa(4*M,2*n)); k8=1/2*((k2-k4-k5+k6+k7/2)+(k1-k3-k5+k6/2+k7/2)+k5-k6-k7); a11=A(1,1)/E/h; a12=A(1,2)/E/h; a21=A(2,1)/E/h;
a22=A(2,2)/E/h; if bc==1 e3=0; e4=0; end if bc==2 e3=(a21*M*M+a22*n*n)*pi*pi/32*3; e4=(a11*M*M+a12*n*n)*pi*pi/32*3; end R=16/9/(pi^4)*(4*M*M*n*n*pi^4*k8+3*M*M*pi*pi*e4/4+3*n*n*pi*pi*e3/4) ; k=2*imp*imp*R+4/9*(-k0*Cbb(2*M,2*n)+Cdd(2*M,2*n)/4+8*M^4*dd(1,1)... +8*n^4*dd(2,2)); e=R/k; a2=3*imp;
Both Cd and Cd2 cited the last program "comp". This is used for the parameter calculation to support upper level program.
Program Comp: function [A,B,D]=comp(E1,E2,G12,r12,v1,theta,h) v2=1-v1; E11=E1*v1+E2*v2; E22=E1*E2/(E1*v2+E2*v1); r21=r12*E2/E1; q11=E11/(1-r12*r21); q22=E22/(1-r12*r21); q12=r12*E22/(1-r12*r21); q21=q12; q66=G12; U1=(3*q11+3*q22+2*q12+4*q66)/8; U2=(q11-q22)/2; U3=(q11+q22-2*q12-4*q66)/8; U4=(q11+q22+6*q12-4*q66)/8; U5=(U1-U4)/2; Q11=U1+U2*cos(2*theta)+U3*cos(4*theta);
Q12=U4-U3*cos(4*theta); Q21=Q12; Q22=U1-U2*cos(2*theta)+U3*cos(4*theta); Q16=U2/2*sin(2*theta)+U3*sin(4*theta); Q26=U2/2*sin(2*theta)-U3*sin(4*theta); Q66=U5-U3*cos(4*theta); A11=h*Q11*2; A12=2*h*Q12; A22=2*h*Q22; A66=2*h*Q66; B16=-h*h*Q16; B26=-h*h*Q26; D11=2/3*h^3*Q11; D12=2/3*h^3*Q12; D22=2/3*h^3*Q22; D66=2/3*h^3*Q66; A=[A11,A12,0;A12,A22,0;0,0,A66]; B=[0,0,B16;0,0,B26;B16,B26,0]; D=[D11,D12,0;D12,D22,0;0,0,D66]; end
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VITA
He Huang was born in DaQing, Heilongjiang, China, on March 15th, 1988. He is the only child in his family. In 2010, He graduated his bachelor degree in Mechanical Engineering in Harbin Institute of Technology. At the same year, he attended University of New Orleans as a Ph.D. student. Supervised by Dr. David Hui, he has been a research assistant and teaching assistant for several courses. During the past four years, he conducted research on the mechanical and thermal properties of composite material, the vibration of rectangular, circular and laminated plates and shells, buckling and the dynamic failure. Currently, he is the teaching assistant of the Material Science and Material Lab.