input-algo-modal-analysis

有限元计算程序

/* [a] : Parameters for Modal Analysis and embedded Newmark Integration */

   no_eigen = 2;
   dt       = 0.03 sec;
   nsteps   = 200;
   beta     = 0.25;
   gamma    = 0.50;

/* [b] : Form Mass and stiffness matrices */

   mass = ColumnUnits( 1500*[ 1, 0, 0, 0;
                              0, 2, 0, 0;
                              0, 0, 2, 0;
                              0, 0, 0, 3], [kg] );

   stiff = ColumnUnits( 800*[ 1, -1,  0,  0;
                             -1,  3, -2,  0;
                              0, -2,  5, -3;
                              0,  0, -3,  7], [kN/m] );

   PrintMatrix(mass, stiff);

/* [c] : First two eigenvalues, periods, and eigenvectors */

   eigen       = Eigen(stiff, mass, [no_eigen]);
   eigenvalue  = Eigenvalue(eigen);
   eigenvector = Eigenvector(eigen);

   for(i = 1; i <= no_eigen; i = i + 1) {
       print "Mode", i ," : w^2 = ", eigenvalue[i][1];
       print " : T = ", 2*PI/sqrt(eigenvalue[i][1]) ,"\n";
   }

   PrintMatrix(eigenvector);

/* [d] : Generalized mass and stiffness matrices */

   EigenTrans = Trans(eigenvector);
   Mstar   = EigenTrans*mass*eigenvector;
   Kstar   = EigenTrans*stiff*eigenvector;

   PrintMatrix( Mstar );
   PrintMatrix( Kstar );

/* 
 * [e] : Generate and print external saw-tooth external loading matrix. First and
 *       second columns contain time (sec), and external force (kN), respectively.
 */ 

   myload = ColumnUnits( Matrix([21,2]), [sec], [1]);
   myload = ColumnUnits( myload,          [kN], [2]);

   for(i = 1; i <= 6; i = i + 1) {
       myload[i][1] = (i-1)*dt;
       myload[i][2] = (2*i-2)*(1 kN);
   }

   for(i = 7; i <= 16; i = i + 1) {
       myload[i][1] = (i-1)*dt;
       myload[i][2] = (22-2*i)*(1 kN);
   }

   for(i = 17; i <= 21; i = i + 1) {
       myload[i][1] = (i-1)*dt;
       myload[i][2] = (2*i-42)*(1 kN);
   }

   PrintMatrix(myload);

/* [f] : Initialize system displacement, velocity, and load vectors */

   displ  = ColumnUnits( Matrix([4,1]), [m]    );
   vel    = ColumnUnits( Matrix([4,1]), [m/sec]);
   eload  = ColumnUnits( Matrix([4,1]), [kN]);

/* [g] : Initialize modal displacement, velocity, and acc'n vectors */

   Mdispl  = ColumnUnits( Matrix([ no_eigen,1 ]), [m]    );
   Mvel    = ColumnUnits( Matrix([ no_eigen,1 ]), [m/sec]);
   Maccel  = ColumnUnits( Matrix([ no_eigen,1 ]), [m/sec/sec]);

/* 
 * [g] : Allocate Matrix to store five response parameters --
 *       Col 1 = time (sec);
 *       Col 2 = 1st mode displacement (cm);
 *       Col 3 = 2nd mode displacement (cm);
 *       Col 4 = 1st + 2nd mode displacement (cm);
 *       Col 5 = Total energy (Joules)
 */ 

   response = ColumnUnits( Matrix([nsteps+1,5]), [sec], [1]);
   response = ColumnUnits( response,  [cm], [2]);
   response = ColumnUnits( response,  [cm], [3]);
   response = ColumnUnits( response,  [cm], [4]);
   response = ColumnUnits( response, [Jou], [5]);

/* [h] : Compute (and compute LU decomposition) effective mass */

   MASS  = Mstar + Kstar*beta*dt*dt;
   lu    = Decompose(MASS);

/* [i] : Mode-Displacement Solution for Response of Undamped MDOF System  */

   for(i = 1; i <= nsteps; i = i + 1) {

    /* [i.1] : Update external load */

       if((i+1) <= 21) then {
          eload[1][1] = myload[i+1][2];
       } else {
          eload[1][1] = 0.0 kN;
       } 

       Pstar = EigenTrans*eload;
       R = Pstar - Kstar*(Mdispl + Mvel*dt + Maccel*(dt*dt/2.0)*(1-2*beta));

    /* [i.2] : Compute new acceleration, velocity and displacement  */

       Maccel_new = Substitution(lu,R); 
       Mvel_new   = Mvel   + dt*(Maccel*(1.0-gamma) + gamma*Maccel_new);
       Mdispl_new = Mdispl + dt*Mvel + ((1 - 2*beta)*Maccel + 2*beta*Maccel_new)*dt*dt/2;

    /* [i.3] : Update and print new response */

       Maccel = Maccel_new;
       Mvel   = Mvel_new;
       Mdispl = Mdispl_new;

    /* [i.4] : Combine Modes */

       displ = eigenvector*Mdispl;
       vel   = eigenvector*Mvel;

    /* [i.5] : Compute Total System Energy */

       e1 = Trans(vel)*mass*vel;
       e2 = Trans(displ)*stiff*displ;
       energy = 0.5*(e1 + e2);

    /* [i.6] : Save components of time-history response */

       response[i+1][1] = i*dt;                            /* Time                  */
       response[i+1][2] = eigenvector[1][1]*Mdispl[1][1];  /* 1st mode displacement */
       response[i+1][3] = eigenvector[1][2]*Mdispl[2][1];  /* 2nd mode displacement */
       response[i+1][4] = displ[1][1];               /* 1st + 2nd mode displacement */
       response[i+1][5] = energy[1][1];                    /* System Energy         */
   }

/* [j] : Print response matrix and quit */

   PrintMatrix(response);
   quit;