6.4.1. Equation of Motion for External Force

The applied force is denoted as F(t).

In this section, we first introduce the concept of the applied force F(t). The 'F(t)' notation indicates that the force can change over time, which is an important aspect in dynamic systems. This means that the force might not be constant and may vary based on factors such as external conditions or the specific scenario under consideration. Understanding that external forces can change with time is crucial when analyzing the motion of structures during events like earthquakes or other dynamic loads.

Consider how the force of the wind changes throughout a storm. Sometimes it's gentle, and sometimes it can be extremely strong, affecting how a structure behaves. Just like the wind, external forces acting on a structure can vary over time.

The equation of motion is expressed as: mu¨(t) + cu˙(t) + ku(t) = F(t)

This equation represents the fundamental relationship between the forces acting on a mass (m), the damping (c), and the stiffness (k) of a system governed by external force F(t). Here, mu¨(t) represents the inertia force (mass times acceleration), cu˙(t) represents the damping force (dependent on velocity), and ku(t) signifies the restoring force (dependent on position). By setting this equation equal to the external force, we can analyze how the system responds to various external influences over time.

Imagine a car on a bumpy road. The car's weight (mass) resists sudden changes in motion (inertia). The shock absorbers (damping) help smooth out the ride when the road is rough. The springs in the suspension (stiffness) push the car back to its original position when it goes over a bump. This equation captures exactly how those forces work together when the car encounters bumps in the road, similar to a structure experiencing dynamic loads.

The methods of solution include classical method (Duhamel’s integral), Laplace transform, Fourier Transform, and numerical methods (e.g., Newmark-beta).

To solve the equation of motion, several methods can be employed. The classical method, such as Duhamel’s integral, is typically used for linear systems subjected to time-varying forces. The Laplace transform and Fourier transform are mathematical techniques that convert time-domain equations into frequency-domain equations, which can simplify the analysis and solution process. Numerical methods, like the Newmark-beta method, use computational techniques to approximate solutions, especially for complex systems where analytical solutions may be difficult to obtain.

Think of these methods as different tools in a toolbox. If you need to fix a car, sometimes a wrench (analytical methods) works best, while other times you'll need a computer diagnostic tool (numerical methods). Each tool provides a different way of solving a problem based on its complexity and requirements.

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