Direct Integration Of The Equation Of Motion

In direct integration the equation of motion are integrated using a numerical step-by-step procedure. The term direct means that no transformation of the equation of motion into the different form is carried out.The direct integration of the equations of motion provides the response of the system at discrete intervals of time which are usually spaced. In this procedure three basic parameters of displacement, velocitiy aand acceleration are computed. The integration algorithms are based on appropriately selected expressions that relate the response parameters at a given interval of time to their values at one or more previous time points. In general two independent expressions of this nature must be specified. The equation of motion written for the time interval under consideration provides the third expression necessary to determine the three unknown parameters. If the equation of motion is written at a time step which the three parameters are unknown(t + At), the time integration scheme is called implicite integration method. In explicit integration method, the equilibrium equation is written at the time step which all the three basis parameters are known (t).

In time integration scheme, the basic parameters are known at the be-gining of the integration or any points preceding the integration time. These specified or computed values permit the marching scheme to be begun so that response can be computed at as many subsequent points as desired. The accuracy and stability of the scheme depend on the magnitude of the time interval and marching algorithm.

There are several method of direct integration. Newmark method, Wilson, Houbolt are implicit integration methods while Central Difference Method is the explicit integration method.

The stability of each method depends on the time steps taken and some of the methods are unconditionally stable. In an unconditionaly stable method, solution does not blow up choosing a big time step. However, the accuracy of the solution might be affected. To achieve an accurate solution a small time step must be chosen based on the natural period of the system as well as the characteristics of the exciting force.

All the above integration schemes are for the solution of a single equation of motion. While, in the couples dam-reservoir equation we are dealing with a couple of equilibrium equation. The stability of an integration scheme for a single equation does not guarantee the stability of a coupled equation. Therefore, It is needed to develope an approach for solution of coupled equation using the above methods.

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