# Discrete Mechanics A Unified Approach by H. Kardestuncer

By H. Kardestuncer

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Note that while the steady state heat flow is represented by a, an elliptic partial differential equation, and is a pure boundary-value problem, the unsteady case, Eq. 5), is a parabolic-differential equation with boundary and initial values. The difference molecule of Eq. n1 + h2 U n 1 i+ - fit n U. 7) in which nand i identify respectively the particular time and location (space). The repetition of this molecule at every interior point of the domain and the introduction of the boundary conditions yield a set of simulataneous equations which eventaully results in the temperature gradient of the domain.

Higher order differential equations can always be reduced to a set of ftrst order simultaneous equations. Consider, for instance, the following equation which represents forced vibration of an elastic system with damping. MU+CU+ KU =p (t). 5) 54 etc. one will obtain When material and damping non-linearities exist. the corresponding stiffness and damping matrices need to be evaluated at each time step. The Newton-Raphson method which refers only to the fU"st derivative of the load-deflection curve is the most common procedute for this putpose.

Fondamenti di Calcolo delle Variazioni. Vol. I, II, Bologna, 1921-23. 4 Prager, W. and Synge, J. , Approximation in elasticity based on the concept of function space, Quarterly App. , Vol. 5, 1947. 5 Synge, J. , The Hypercircle in Mathematical Physics, Cambridge University Press, London and New York, 1957. , The Variational Principles of Mechanics, University of Toronto, 1949. 7 Schecter, R. , The Variational Method in Engineering, McGraw-Hill Book Company, 1967. , Variational Methods in Elasticity and Plasticity, Pergamon Press, 1968.