Numerical Solution of MDOF Equations

Numerical Solution of MDOF Equations

The equations of motion for a MDOF system have been solved analytically using the Modal Analysis. Although Modal Analysis is helpful in formulating and understanding some basic concepts of dynamic analysis, it has several limitations of convenience and applicability. In fact, it has even more limitations than the analytical methods used to solve SDOF systems.

In addition to the considerable mathematical effort needed to solve eigenvalue problems and uncouple the simultaneous equations (i.e., make the system matrices diagonal), its formulation requires several assumptions. For example, the method is valid for linear systems only. The orthogonality condition that makes the Modal Analysis convenient, is not guaranteed to be valid for the damping matrix. The practical loading situations can be more complicated and not convenient to solve analytically. Numerical methods must be used in such situations.

As mentioned for SDOF systems, the most widely used numerical approach for solving dynamic problems is the Newmark- method. The method solves the dynamic equation of motion in the (i+1) th

time step based on the results of the i th step.

The dynamic equations of motion for the (i+1) th time step is

Ma i+1 +Cv i+1 +Ku i+1 =f i+1

where the bold small letter ‘a’ stands for the acceleration vector, ‘v’ for velocity vector and ‘u’ for displacement vector. In the Constant Average Acceleration (CAA) method (a special case of Newmark- method where = 0.50 and = 0.25), the velocity and displacement vectors are given by

v i+1 =v i + (a i +a i+1 ) t/2 …………………(11.2)

i+1 =u i +v i t + (a i +a i+1 )t /4 …………………(11.3)

Inserting these values in Eq. (13.1) and rearranging the coefficients, the following equation is obtained,

(M + C t /2 + K t 2 /4)a i+1 =f i+1 – Ku i – (C + K t)v i – (C t/2 + K t /4)a i …………......…..(11.4)

Therefore, if the forcing function f i+1 is known, the only unknown in Eq. (11.4) is the acceleration vector

a i+1 , which can be obtained by matrix inversion (by Gauss Elimination or some other method). Once a i+1 is obtained, Eqs. (11.2) and (11.3) can be used to calculate the velocity vector v i+1 and the displacement vector u i+1 at time t i+1 . These values are used to obtain the results at time t i+2 and subsequent time-steps.

The simulation needs two initial conditions, e.g., the displacement vector u 0 and velocity vector v 0 at time t 0 = 0. Then the initial acceleration vector can be obtained as

Again, any standard method of matrix inversion can be used to solve Eq. (11.5).

Among other methods of numerical solution of the MDOF equations of motion, the Linear Acceleration method and Central Difference method are quite popular. The Linear Acceleration Method is a special case of the Newmark- method with = 0.50 and = 1/6. Instead of assuming constant average acceleration between two time intervals, it assumes the acceleration to vary linearly in between two intervals. Unlike the CAA method, the Linear Acceleration method is not unconditionally stable. However, the time increment needed for its stability is much greater than the interval needed for accurate results, therefore stability is usually not a problem for this method.

Incremental solutions of the equations of motion are also popular, particularly for nonlinear systems. Instead of solving for the total displacement or acceleration at any time, this method solves for the increment (change) in displacement or acceleration. There again, the CAA is widely used.