Differential equations are a vital tool in engineering for modeling systems that change over time or space. In structural engineering, they help describe how structures respond to forces, such as weight or pressure.
For the beam in this exercise, the Rayleigh beam theory gives rise to a fourth-order differential equation:
- \(EI \frac{d^4 w(x, t)}{d x^4} = m \frac{d^2 w(x, t)}{d t^2} + p(x, t)\)
This equation represents the balance of forces along the beam, emphasizing how the distributed load \(p(x, t)\) and inertia \(m\frac{d^2 w(x, t)}{dt^2}\) influence the deflection \(w(x, t)\).
Often, engineers solve such equations using techniques like separation of variables or employing numerical methods for complex cases. In this example, a particular solution was derived using trigonometric functions to satisfy both time and spatial dependencies. Understanding these equations allows engineers to predict structure performance over time, ensuring designs can withstand various dynamic loads.