9.3. Free Vibration Response of Undamped SDOF System to Unit Impulse

When damping c=0, the equation simplifies to:

mx¨(t)+kx(t)=δ(t)

This chunk introduces the scenario where damping in the system is zero (c=0). When we have no damping, the equation that governs the motion of the system (which is a Single Degree of Freedom or SDOF system) reduces to a simpler form. This indicates that only the mass (m) and stiffness (k) are active in defining the vibrational characteristics of the system when it responds to a unit impulse input, represented as δ(t).

Imagine a swing that is pushed (the impulse) without any frictions or forces slowing it down (no damping). The swing will oscillate back and forth indefinitely following the push as long as it is not subjected to any other forces, similar to how a system behaves under the conditions described in this equation.

Initial conditions just before impulse:

• x(0−)=0
• x˙(0−)=0

Before the impulse is applied at time t=0, we define the initial conditions of the system. Both the displacement x(0−) and the velocity x˙(0−) are set to zero. This means the system is at rest and not displaced prior to the application of the impulse, which helps us understand how the system starts to move in response to the impulse at t=0.

Think of pushing a stationary child on a swing. Before pushing, the child is not moving at all, and the swing is perfectly vertical (x=0). This setup creates a clear starting point for the swing's motion once the push (impulse) is given.

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

Z 0+ Z 0+ Z 0+ mx¨(t)dt+ kx(t)dt= δ(t)dt 0− 0− 0−

To analyze how the impulse affects the system, we integrate the governing equation over a small time interval just before and after the impulse. This integration helps us determine how the velocity of the system changes immediately after the impulse is applied. The Dirac delta function allows us to focus on the instant effect of the impulse on the system.

This is like measuring how much a basketball bounces after being hit by a hand. By observing a tiny moment right before and after the hit, we can directly understand the impact of the push.

m[x˙(0+)−x˙(0−)]=1⇒x˙(0+)=

m

This expression comes from the results of the integration and tells us how the velocity just after the impulse (x˙(0+)) relates to the mass (m). Solving this gives the initial velocity of the system immediately after the impulse is applied. This is crucial for understanding the starting conditions for the free vibration response.

Imagine the moment after you hit a volleyball with your hand. The force (impulse) applied by your hand gives the ball its velocity. If you know how hard you hit it (mass), you can predict how fast the volleyball will be traveling immediately after the hit.

Then, the free vibration response for t>0:

x(t)= sin(ω t)

mω n

After calculating the initial conditions and impacts of the impulse, we derive the equation for the system's free vibration response for any time after t=0. The presence of the sine function indicates that the system will oscillate indefinitely at its natural frequency (ωn), which is determined by the system's mass and stiffness. This formula embodies how the system behaves under free vibrations after being disturbed by the impulse.

This resembles how a perfectly elastic band that is stretched and then released would oscillate back and forth. No matter how it's disturbed, it keeps returning to a steady rhythm defined by its material properties.

This is the impulse response of an undamped SDOF system.

The final statement summarizes that the derived equation represents the complete response of an undamped SDOF system subjected to an impulse. This is foundational in understanding how various structures would react to sudden forces in practical situations, particularly in engineering contexts.

In terms of construction, understanding this response is similar to knowing how a bridge reacts when a heavy truck suddenly drives over it. Engineers must be able to predict the vibrations to ensure safety and stability.

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