6. Equations of Motion of SDOF System for Mass as well as Base Excitation

In the context of Earthquake Engineering, the analysis of the dynamic response of structures is fundamental. Most structural systems can be idealized as a combination of simpler systems, of which the Single Degree of Freedom (SDOF) system is the most fundamental. The SDOF model captures the essence of dynamic behavior and forms the basis for understanding the response under various types of excitations — such as due to ground motion (base excitation) and external force (mass excitation).

In earthquake engineering, we study how buildings react dynamically during events like earthquakes. Many buildings can be simplified into basic models, the simplest being the Single Degree of Freedom (SDOF) system. This model helps us understand the overall behavior of structures when they face different forces, either from the ground shaking during an earthquake (base excitation) or from external forces acting directly on them (mass excitation). Essentially, the SDOF system is a foundational concept that allows engineers to predict how a structure will respond to dynamic loads.

Imagine a swing moving back and forth. The swing can be influenced by pushing it (mass excitation) or by someone sitting on it while it’s on a moving platform (base excitation). Just like the swing, buildings respond to the forces they encounter during seismic activities.

• Definition: A mechanical system with only one coordinate required to describe its motion. • Components: o Mass (m) o Spring (stiffness, k) o Damper (damping coefficient, c) • Equation of motion: derived using Newton's Second Law or D’Alembert’s Principle.

A Single Degree of Freedom (SDOF) system is defined as a system that can be described by a single variable, which simplifies its analysis significantly. The components of an SDOF system include: mass (which represents the weight of the structure), spring (which describes the stiffness or elastic response), and damper (which represents energy dissipation through damping). The equation governing this system can be derived from Newton’s Second Law, which forms the basis for analyzing the dynamic behavior of the system when it is subjected to forces.

Think of a car suspension system. The car's mass is the body, the springs absorb shocks from the road, and dampers prevent excessive bouncing. Just like how these components work together to ensure a smooth ride, in an SDOF system, the mass, spring, and damper work together to determine how the system will move and respond to various forces.

• Governing Equation: mu¨(t)+ku(t)=0 • Solution: u(t)=Acos(ω t)+Bsin(ω t) where ω = √(k/m) is the natural frequency. • Interpretation: Pure harmonic motion.

The governing equation of motion for a free undamped SDOF system can be written as 'mu¨(t) + ku(t) = 0', where 'm' is the mass, 'k' is the stiffness, and 'u(t)' is the displacement. The solution of this equation describes the system's motion, revealing that it follows a harmonic pattern (like a wave). The terms 'A' and 'B' represent the initial conditions, and 'ω', determined by the mass and stiffness, indicates how quickly the system oscillates. This means that when there are no external forces acting on the system, it will oscillate indefinitely at its natural frequency.

Consider a child on a playground swing. As they swing back and forth, they follow a smooth motion, repeating the same pattern over and over. This behavior is analogous to harmonic motion, demonstrating how the SDOF system oscillates around an equilibrium position without any forced disturbances.

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