9.4. Response of Damped SDOF System to Unit Impulse

For a damped system, the equation of motion is:

mx¨(t)+cx˙(t)+kx(t)=δ(t)

In a damped single degree of freedom (SDOF) system, the equation of motion describes how the system behaves when subjected to forces over time. The equation is adapted from the basic form of motion by including damping effects, where 'm' represents mass, 'c' represents the damping coefficient, 'k' represents stiffness, and δ(t) is the unit impulse applied at time t. This equation tells us how the system will respond to that sudden input, taking into account both its mass and the resistance due to damping.

Think of a car's suspension system. When the car hits a bump (like a unit impulse), the damping in the suspension (like the 'c' in the equation) helps to absorb the shock and prevent the car from bouncing excessively. This is similar to how the damped SDOF system reacts to an impulse.

Let: • ζ = √c • ω n = k/m, natural frequency • ω d = ω n √(1−ζ²), damped natural frequency

In analyzing the response of a damped system, it's essential to introduce critical parameters: the damping ratio (ζ), natural frequency (ω_n), and damped natural frequency (ω_d). These parameters define the behavior of the system under impulsive forces. The damping ratio helps quantify how much the system's oscillations will decay over time, while both natural frequencies are important to determine how quickly the system inherently wants to oscillate versus how quickly it actually oscillates with damping present.

Consider a pendulum. If you swing it gently (no damping), it will swing back and forth at a specific natural frequency. If you add a sponge at the pivot that absorbs energy (like damping), it will lose energy with each swing and take longer to come to rest compared to the undamped case. Here, the sponge introduces damping, which alters both the rate and style of oscillation.

Assuming initial conditions: • x(0−)=0, x˙(0−)=0 • x˙(0+)= 1/m

Then for ζ <1 (underdamped case):

x(t)= e−ζω nt sin(ω d t), t>0

For solving the response of the damped system, we first set initial conditions: both displacement (x) and velocity (x˙) are zero just before t=0, and there's an initial velocity immediately after the impulse given by 1/m. If the system is underdamped (where ζ < 1), the response can be described by a mathematical expression that combines exponential decay and sinusoidal motion, indicating oscillations that decrease in amplitude over time.

Imagine a swing. When someone jumps off (like applying an impulse), the swing starts moving but gradually slows down and eventually stops due to friction (like the damping effect). The mathematical model describes how quickly this happens, with factors like how heavy the person was and how strong the friction is impacting the response.

This describes how the response decays over time due to damping.

The response of a damped system to an impulse is characterized by decreasing oscillations over time. This behavior is due to the energy being dissipated through damping effects, which can be represented in the form of the response equation. Such decay is crucial for engineers to design buildings, bridges, and other structures capable of withstanding impulses like earthquakes, where damped responses can prevent excessive vibrations.

Think of pushing a child on a swing. Initially, they swing high but will gradually swing lower with each push due to air resistance and other losses - this is similar to how the damped system responds. The push is like the impulse, and the swing's decreasing height is akin to the decay in oscillation amplitude.

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