14.2. Single Degree of Freedom Systems (SDOF)

• Mass-Spring-Damper System: Simplest representation of structural systems. • Differential Equation of Motion: mx¨+cx˙ +kx=0

In this chunk, we explore the mathematical modeling of Single Degree of Freedom (SDOF) systems. An SDOF system can be represented as a Mass-Spring-Damper model, which is one of the simplest forms used in structural analysis. The mass represents the structure's inertia, the spring represents the structure's stiffness, and the damper accounts for the energy dissipation (like how shock absorbers work in cars).

Additionally, the motion of the system can be described by the differential equation of motion given by mx¨ + cx˙ + kx = 0, where:

• m represents the mass of the system,
• c is the damping coefficient,
• k is the stiffness of the spring, and
• x represents displacement. This equation helps us understand how the system responds to dynamic forces over time.

Consider a simple swing at a park: when a child pushes the swing (dynamic force), the mass of the child and the swing, along with the ropes acting like a spring, dictate how high and fast the swing moves. When the swing is pushed and allowed to move freely, it behaves according to the mass-spring-damper model, where it goes up and down until the motion gradually stops due to air drag (damping).

For an undamped system:

o = √(k/m) (rad/s)

This chunk discusses the concept of undamped natural frequency, denoted here as ω (omega). The equation given states that the natural frequency is determined by taking the square root of the stiffness (k) divided by the mass (m). This relationship shows that if stiffness increases (the system is ‘stiffer’), the natural frequency increases; conversely, if mass increases, the natural frequency decreases.

In simpler terms, the natural frequency indicates how quickly the system wants to oscillate back and forth when disturbed, assuming there is no energy loss (damping).

Imagine a child on a trampoline. If the trampoline is very tight (high stiffness), when the child jumps, they will bounce back quickly, indicating a higher natural frequency. If the trampoline is saggy (low stiffness), the same jump will result in a slower bounce, showing a lower natural frequency.

• Natural frequency is expressed in Hz or rad/s. • A higher stiffness or lower mass leads to a higher natural frequency.

In this portion, we cover how natural frequency is quantified. Natural frequency can be expressed in hertz (Hz), which measures cycles per second, or in radians per second (rad/s), which is often used in physics and engineering contexts.

The takeaway here is that an increase in stiffness (the resistance of a system to deformation) tends to increase the natural frequency, while an increase in mass (how much the system weighs) tends to decrease the natural frequency. Understanding these relationships is crucial for engineers designing structures that need to withstand dynamic forces, such as earthquakes.

Think of two musical instruments: a violin string (high stiffness, high frequency) versus a bass guitar string (lower stiffness, lower frequency). The difference in material properties leads to different vibrations and sounds, similar to how structures with differing stiffness and mass will oscillate at different natural frequencies during disturbances.

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