To understand the behavior of any linear system, consider the following simple example.
To understand the behavior of any linear system, consider the following simple example.

After all the initial oscillations expire, the output frequency of a linear system is always exactly equal to the input frequency, meaning that the frequency of each yellow sphere is exactly equal to the frequency of its corresponding red sphere.
After all the initial oscillations expire, the output frequency of a linear system is always exactly equal to the input frequency, meaning that the frequency of each yellow sphere is exactly equal to the frequency of its corresponding red sphere.

Although each red sphere has a different frequency, each red sphere moves up and down with the exact same amplitude.
Although each red sphere has a different frequency, each red sphere moves up and down with the exact same amplitude.

The output of our system is the position of each yellow sphere.
The output of our system is the position of each yellow sphere.

The frequency determines the ratio of the output amplitude to the input amplitude.
The frequency determines the ratio of the output amplitude to the input amplitude.

The frequency response can be calculated from the system's transfer function.
The frequency response can be calculated from the system's transfer function.

The input to the transfer function is the variable "s."
The input to the transfer function is the variable "s."

S is a complex number with the real and imaginary components along these two axes.
S is a complex number with the real and imaginary components along these two axes.

The output of the transfer function is also a complex number, with a magnitude given by the height of this graph.
The output of the transfer function is also a complex number, with a magnitude given by the height of this graph.

The color symbolizes the "phase" of the transfer function output.
The color symbolizes the "phase" of the transfer function output.

Let's change the color of each point to match the corresponding color of the transfer function at that location.
Let's change the color of each point to match the corresponding color of the transfer function at that location.

We can use these colors to graph the difference between the output phase and the input phase.
We can use these colors to graph the difference between the output phase and the input phase.

In this particular example, the input and output are "in phase" at low frequencies, and they are "out of phase" at high frequencies.
In this particular example, the input and output are "in phase" at low frequencies, and they are "out of phase" at high frequencies.

By changing the transfer function of a system, we can change its frequency response.
By changing the transfer function of a system, we can change its frequency response.

For some transfer functions, the output amplitude at the resonant frequency can grow to infinity.
For some transfer functions, the output amplitude at the resonant frequency can grow to infinity.

This is why for some systems, applying the resonant frequency can destroy the system.
This is why for some systems, applying the resonant frequency can destroy the system.

The transfer function is the Laplace Transform of the Unit Impulse Response.
The transfer function is the Laplace Transform of the Unit Impulse Response.

Much more information is available in the other videos on this channel.
Much more information is available in the other videos on this channel.

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Thank you.
Thank you.