9.7. Numerical Example

Given: • m=1kg, c=1N·s/m, k =4N/m

In this numerical example, we are given three important parameters for a damped system: mass (m), damping coefficient (c), and stiffness (k). Here, the mass of the system is 1 kilogram, the damping coefficient is 1 Newton-second per meter, and the stiffness is 4 Newtons per meter. These parameters are crucial for defining the motion of the system when subjected to external forces, particularly impulsive forces.

Think of a toy car on a smooth surface. The car’s mass is equivalent to the resistance it has to acceleration. The damping coefficient can be compared to friction between the car and the surface, affecting how quickly it slows down after being pushed. Stiffness is akin to how hard the springs in the car’s suspension work to return it to its position after a bump.

Then: • ω_n=2rad/s, ζ=√1/2=0.25 • ω_d=2√(1−0.25²)=1.936rad/s

From the initial parameters, we can calculate the natural frequency (ω_n) and the damped natural frequency (ω_d) of the system. The natural frequency indicates how the system behaves in a free oscillation without any dampening. The damping ratio (ζ) helps determine how the oscillations reduce over time. In this case, with ζ calculated as 0.25, we note that the system is underdamped, which means it will oscillate before eventually coming to rest.

Imagine a swing at a playground. The natural frequency is like how fast the swing moves back and forth when pushed, while the damping ratio shows how quickly it slows down after being pushed. A lower damping ratio means more oscillations before the swing stops.

Impulse response: 1 x(t)= e^{-0.25·2·t} sin(1.936·t) 1·1.936 x(t)=0.5166·e^{-0.5·t} sin(1.936·t)

The impulse response function describes how the system will respond to an impulsive force applied at time t=0. The exponential term, e^{-0.5·t}, represents the decay in amplitude of the oscillation over time due to damping. The sine function, sin(1.936·t), shows that the displacement will oscillate at the damped frequency. This response helps predict the motion of the structure over time following the impulse.

Think of a plucked guitar string. When you pluck it (the impulse), it vibrates and produces sound. The sound starts loud (high displacement) and gradually gets quieter (decaying amplitude) until it stops vibrating. Here, we are mathematically modeling that same behavior for a different system.

This function describes the displacement response of the structure to an impulsive force at t=0.

The derived function provides a complete description of how the structure behaves immediately after the application of an impulse. It allows engineers to understand the maximum displacement, the frequency of oscillations, and the rate at which the system returns to rest. Such understanding is critical in designing structures that can withstand impulsive forces, like those from earthquakes.

Consider a trampoline. When someone jumps on it (the impulse), it bends and oscillates up and down. The shape and speed of that bounce can be predicted using similar mathematical functions, which allow us to ensure that the trampoline can handle repetitive jumps without breaking.

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