32.3. Response of SDOF Systems to Earthquake Motion

32.3.1 Free Vibration Response

• Undamped and damped cases.
• Natural frequency (ω n) and damping ratio (ζ).

The free vibration response of a single-degree-of-freedom (SDOF) system refers to how the system reacts when it is displaced from its equilibrium position and then allowed to move without any external forces acting on it (i.e., only its inherent properties are at play). In undamped cases, once disturbed, the system continues to oscillate indefinitely. However, in damped cases, there is an energy dissipation effect due to factors like internal friction, which reduces the amplitude of oscillation over time. The natural frequency, denoted as ωₙ, is a key parameter determining how fast the system will oscillate. The damping ratio, ζ, quantifies how oscillations in a dynamic system decay after a disturbance, providing insight into the energy lost during vibrations.

Think of a swing set. If you push the swing and let it go (like an undamped system), it will swing back and forth forever without stopping. But if the swing has friction at the pivot point (like a damped system), it will gradually come to a halt due to energy loss. The speed of its swinging back and forth is determined by the height of the swing (natural frequency), and how quickly it stops is determined by how much friction there is (damping ratio).

32.3.2 Forced Vibration Under Ground Motion

• Use of Duhamel's integral.
• Importance of initial conditions.

Forced vibration occurs when an external force, such as ground motion during an earthquake, causes the SDOF system to oscillate. Duhamel's integral is a mathematical tool used to compute the response of a system to such dynamic loads. It helps us understand how an SDOF system will react to specific ground motions over time, considering its unique properties and initial conditions (like its starting position and velocity). The initial conditions are crucial because they define the state of the system before the external force is applied, impacting how the structure responds.

Imagine a trampoline. If someone jumps onto it (equivalent to forced vibration), the trampoline will oscillate. How high and how fast it bounces back depends not only on the jump force (the ground motion) but also on how stretched or compressed the trampoline was initially when the person jumped (the initial conditions). Duhamel's integral acts like a calculator that helps us predict the exact bouncing behavior based on all these factors.

32.3.3 Construction of Response Spectra

• Based on responses of SDOF systems for various natural periods.
• Design spectrum in IS 1893.

Response spectra are graphical representations that illustrate the peak response (like acceleration or displacement) of SDOF systems subjected to earthquake motion over a range of natural periods. The construction of these spectra helps engineers understand how different structures will behave under seismic loads. The design spectrum outlined in standards like IS 1893 is specifically tailored for seismic design in a particular region or building use, giving engineers essential data to determine how to construct buildings that can withstand earthquakes effectively.

Think of the response spectrum like a fitness chart that shows how different exercises affect various body types. Just as different workouts produce different results on different individuals, different structures with varying natural frequencies (like different exercises) react differently to earthquake forces. The design spectrum serves as a guideline to help choose the best 'workout plan' for different buildings to ensure they can handle the 'workout' during an earthquake.

32.3.4 Elastic vs Inelastic Response Spectrum

• Reduction factor (R) used in design spectrum.
• Capacity spectrum method.

The elastic response spectrum represents the behavior of structures that remain within their elastic limits when subjected to seismic forces, meaning they do not experience permanent deformation. In contrast, the inelastic response spectrum accounts for structures that undergo permanent changes, typically during stronger earthquakes. The reduction factor (R) is applied to the design spectrum to acknowledge that some inelastic behavior is expected in real-world scenarios, allowing for a more realistic approach to design. The capacity spectrum method helps engineers determine a building's actual behavior by analyzing how it can respond under different loading conditions, providing a bridge between theoretical predictions and practical performance.

Imagine a rubber band (elastic response) that can stretch but returns to its original shape when you let go. Now imagine a piece of clay (inelastic response) that deforms permanently when you shape it. The reduction factor is like deciding how much you can stretch the rubber band without losing its form when you're under pressure (like an earthquake). The capacity spectrum method is similar to evaluating both materials (rubber and clay) to see how much pressure each can withstand before yielding.

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