9. Impulse and Response to Unit Impulse

An impulse is defined as a force of very large magnitude acting over a very short period of time. Theoretically, this is modeled by the Dirac delta function, denoted as δ(t). The mathematical impulse I applied over a time interval t to t1 is:

$I = \int_{t_1}^{t_2} F(t) dt$

For a unit impulse, this simplifies to:

$\int_{-\infty}^{\infty} δ(t) dt = 1$

An impulse is a force applied suddenly and briefly. In mathematics, we often represent this using the Dirac delta function, which is a way to express an infinitely large force over an infinitesimally small duration. The integral of the Dirac delta function over its entire range equals one, which signifies that the total effect of impulse is captured, even though the force acts only momentarily.

Think of an impulse as a quick jab. It’s like tapping a friend on the shoulder really hard for just a moment. Even though the tap is quick, it gets your friend's attention immediately, similar to how an impulse force can cause instant reactions in structures.

For a linear SDOF system subjected to a force F(t), the equation of motion is:

$mx¨(t) + cx˙(t) + kx(t) = F(t)$

For unit impulse input: F(t) = δ(t) So, the governing equation becomes:

$mx¨(t) + cx˙(t) + kx(t) = δ(t)$

The motion of a single degree of freedom (SDOF) system can be described by a second-order differential equation that considers mass, damping, and stiffness. When we apply a unit impulse as the external force, we adjust the equation to reflect this specific force input, allowing us to derive the system's response to the impulse.

Imagine a swing (the SDOF system). If you push it lightly and then suddenly give it a strong push (the impulse), you can analyze how far and how fast it swings based on the swing's weight (mass), how much it wants to swing back and forth (stiffness), and how much it slows down (damping).

When damping c=0, the equation simplifies to:

$mx¨(t) + kx(t) = δ(t)$

Let ω = \sqrt{\frac{k}{m}} be the natural frequency.

Initial conditions just before impulse: • x(0−)=0 • x˙(0−)=0

Integrating the equation over a small interval around t=0:

$\int_{0^-}^{0^+} mx¨(t) dt + \int_{0^-}^{0^+} kx(t) dt = \int_{0^-}^{0^+} δ(t) dt$

Then, the free vibration response for t>0:

$x(t) = \frac{1}{mω_n} sin(ω_n t)$

In an undamped system, we focus simply on the mass and stiffness without considering any damping effects. The resulting response can be modeled as a sine wave that reflects how the system will oscillate in response to the impulse force. The natural frequency determines the speed of oscillation and the amplitude is influenced by the magnitude of the impulse.

Think of a child on a swing. If you give them a push (an impulse) with no one holding them back (no damping), they'll swing back and forth indefinitely until fatigue from air resistance slows them down. Their motion can be described as a smooth back-and-forth swinging motion, akin to the sine wave response.

For a damped system, the equation of motion is:

$mx¨(t) + cx˙(t) + kx(t) = δ(t)$

Let: • ζ = \sqrt{\frac{c}{2 km}} be the damping ratio • ω_n = \sqrt{\frac{k}{m}} be the natural frequency • ω_d = ω_n \sqrt{1−ζ^2}, the damped natural frequency.

Assuming initial conditions: • x(0−)=0, x˙(0−)=0 • x˙(0+)= \frac{1}{m}

Then for ζ <1 (underdamped case):

$x(t) = \frac{1}{mω_d} e^{-ζω_nt}sin(ω_d t), \ t>0$

In systems with damping, the response not only oscillates but also decreases in amplitude over time due to energy loss from effects like friction or air resistance. This equation captures the response of a damped system to an impulse by introducing a decay term that reflects the damping influence. The damped frequency shows how the system's motion slows down more than an undamped system would.

Consider a vibrating guitar string. When plucked (an impulse), it vibrates rapidly but dampens down (loses energy) over time. Each time the string vibrates, it produces sound, but as the energy dissipates through air (like the damping in our equation), the sound fades instead of ringing indefinitely.

The impulse response function, also called Green’s function h(t), is defined as the response of a system to a unit impulse input δ(t).

For a linear time-invariant system, the total response to any arbitrary force F(t) can be obtained using convolution:

$x(t) = \int_{0}^{t} h(t−τ) F(τ) dτ$

Green's function acts as a tool in system analysis, revealing how a system reacts to impulsive forces. Using convolution, we can take the impulse response function and apply it to any force by integrating over time, merging the system's response to many small impulses into one complete response. This principle allows engineers to predict how structures react to more complex loading conditions beyond just simple impulses.

Think of making a smoothie (the response) where you add various ingredients (the forces) one at a time. The final taste is influenced by all the ingredients combined, just like the total response of a system depends on the individual impulses applied over time.

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