Analytical And Numerical Approaches

Numerical Solution to the Duhamel Integral

Analytical and Numerical Approaches

In the realm of structural analysis, there are two primary approaches to solving the Duhamel integral: the analytical solution and the numerical solution.

Analytical Solution

The analytical approach involves obtaining an exact solution to the integral using mathematical techniques. In this case, the Duhamel integral is expressed as an integral of the equation of motion.

Numerical Solution

When an analytical solution is not feasible, a numerical solution is employed. In part three of our ongoing discussion, we will implement a numerical solution to the Duhamel integral using an appropriate computational method.

Derivation of the General Expression

By substituting the expression of h(t-τ) into the Duhamel integral, we arrive at the general expression:

x(t) = ∫₀^t p(τ)e^-ςωn(t-τ)dτ

where p(τ) represents the applied load, ς is the damping ratio, ω_n is the natural frequency, and t represents time.