14. Natural Frequencies

In earthquake engineering, understanding the concept of natural frequency is fundamental to the dynamic analysis of structures. Every structure, whether a building, bridge, or tower, has a certain frequency at which it naturally vibrates when subjected to a dynamic force. When the frequency of an external force, such as an earthquake, matches the structure’s natural frequency, resonance occurs—leading to amplified vibrations that can cause severe damage or failure.

Natural frequency is a critical concept in earthquake engineering. It refers to the specific frequency at which a structure can vibrate freely. Every structure, like a building or a bridge, vibrates in a particular way when a force, like an earthquake, is applied. If the earthquake’s frequency aligns with the structure’s natural frequency, it can lead to resonance. Resonance significantly amplifies the vibrations, potentially causing destruction or failure of the structure. Understanding how to determine and mitigate these natural frequencies is essential for engineers in designing safe and resilient buildings.

Think of a swing at a park. Every swing has a natural rhythm at which it moves back and forth. If you push the swing at exactly the right moment (matching its natural frequency), it swings higher and higher. However, if you push at the wrong time, it won’t swing effectively. Similarly, in buildings, if the tremors from an earthquake match the building’s natural frequency, it can cause the building to sway dangerously, just like the swing that goes too high.

• Free Vibration: Occurs when a structure is displaced and allowed to vibrate on its own. • Forced Vibration: When a structure is subjected to an external periodic or random force. • Undamped and Damped Systems: Ideal vs. real-world systems that account for energy dissipation.

Vibrations can be classified into two main types: free and forced vibrations. Free vibrations happen when a structure is disturbed and allowed to oscillate on its own—think of a bell being struck. When struck, it rings naturally without any outside influence. On the other hand, forced vibrations occur when an external force is applied repeatedly, like a person pushing a child on a swing. Additionally, vibrations can be damped or undamped. Undamped systems do not lose energy as they vibrate, which is an ideal scenario, whereas damped systems are realistic as they lose energy over time due to friction or other forces that prevent continuous oscillation.

Consider a guitar string. When you pluck it, it vibrates freely—that's free vibration. If someone continuously strums the string, they're applying a force, and the string vibrates with forced vibration, which changes how it sounds. In buildings, free vibrations can occur as a result of wind or minor movements, while forced vibrations may happen during an earthquake or other man-made activities.

Natural frequency is the rate at which a system oscillates in the absence of any driving or damping force. Mathematically, for a single-degree-of-freedom (SDOF) system:

f_n = rac{1}{2 ext{π}} imesrac{ ext{√}k}{m} Where: • f_n = natural frequency (Hz) • k = stiffness (N/m) • m = mass (kg)

Natural frequency can be formally defined as the rate of oscillation of a system when there are no external forces acting upon it. For a simple system, like a mass attached to a spring (referred to as a single-degree-of-freedom system), the natural frequency depends on two main factors: stiffness (how strong the spring is) and mass (how heavy the object is). The formula shows that natural frequency increases with higher stiffness and decreases with larger mass. Understanding this relationship allows engineers to predict how structures will behave during seismic activity.

Imagine a child on a swing. If the swing is made of a strong, thick chain (higher stiffness), the swing will go back and forth quickly (higher natural frequency). If the child grows heavier (increased mass), the swing will move slower, indicating a lower natural frequency. Just like this swing, structures respond to forces based on their mass and stiffness.

--