10.2. Impulse Response Function

Before deriving the Duhamel Integral, we define the impulse response function h(t) as the response of the system to a unit impulse force applied at t=0: F(t)=δ(t)

The impulse response function, denoted as h(t), is a key concept in understanding how a system reacts to a specific type of force known as a unit impulse. This impulse is represented mathematically as δ(t), which is a function that is zero everywhere except at t=0, where it is theoretically infinite but has an area of one under the curve. Essentially, h(t) captures how the system responds over time when it is hit by this instantaneous force.

Think of h(t) like the ripple effect you see when a stone is thrown into a still pond. Just as the water responds to the stone's impact with ripples spreading out, h(t) represents how the system reacts to a sudden 'hit' or force. The responses at various times after the impact can be thought of as the ripples moving outward.

The solution x(t) for such an input is the unit impulse response, which depends on the damping condition of the system:

When we apply a unit impulse force to the system, we can analyze the resulting motion or response, x(t). This response is called the unit impulse response and is crucial for characterizing behavior under dynamic loading conditions. The nature of x(t) will vary based on the damping condition of the system, which is a measure of how quickly the system dissipates energy.

Imagine pushing a swing lightly. If you let it go, it moves back and forth until it gradually slows down due to air resistance and friction, which represents damping. The swing's path could be considered as its response x(t) to your push—similarly, our system's response to the impulse reflects how different amounts of 'push' from damping alter its motion.

10.2.1 Underdamped Case (ζ<1) Let ω =√k/m be the natural circular frequency and ζ= be the damping ratio. The impulse response function is given by: 1 h(t)= e−ζω nsin(ω t) Where ω =ω √1−ζ2 is the damped natural frequency.

An underdamped system has a damping ratio ζ that is less than 1, which means it experiences oscillations before eventually coming to rest. The formula for the impulse response function h(t) for an underdamped case indicates that the response includes an oscillatory component, expressed by the sine function, multiplied by an exponential decay factor. This means the system will oscillate, but these oscillations will gradually decrease in amplitude over time due to the damping effect.

Consider a child on a swing. If they are pushed lightly (underdamped), the swing will go back and forth several times before coming to a stop, gradually slowing down because friction (damping) reduces their speed with every pass. The mathematical description of h(t) captures this 'bouncing' behavior as the excluded energy diminishes over time.

--