A response spectrum, simply put, represents the response to an external force.
To find the response, we have to go back to the equations of motion.
As we've seen in previous lessons, the equations of motion are fundamental to dealing with vibration phenomena. Many of you are probably wondering how this works in practice and would like to try solving the equations of motion yourselves.
Let's go back to the beginning and try to see how FFT relates to these equations of motion. Please bear with me.
(1) Equation of motion
The equations of motion for a 1-degree-of-freedom system are as follows when no external forces are acting.
mx‘’＋cx‘＋kx＝0 ・・・（1）
When an external force f(t) acts
mx‘’＋cx‘＋kx＝f(t) ・・・（2）
This is how it is expressed. This formula has appeared many times, but why can it be expressed that way? As a reference, I have quoted Newton's three laws of motion from "Spectral Analysis of Seismic Motion" by Yoshihiko Osaki, published by Kajima Publishing Co., Ltd.
first law
An object that is at rest or moving at a constant velocity in a straight line will remain in that state unless a force acts upon it.
second law
Acceleration (change in velocity) is proportional to the applied force and occurs in the direction of that force.
third law
Action is always in the opposite direction to reaction, and their magnitudes are equal.
These are known as the law of inertia, the law of motion, and the law of action and reaction, respectively.
Now, according to the second law, if we let the acceleration be α and the force be F,
α∝F ・・（3）
Let's define the reciprocal of the proportionality constant as the mass m.
α＝（1/m)F 、F＝mα ・・・（4）
It is expressed as follows. Further rewriting
（-mα）＋F＝0 ・・・（5）
This equation can be considered as a balance between -mα and F, where -mα is called the inertial force. This means that even when an object is moving with acceleration, considering the inertial force allows us to think of it as a static balance of forces, and this is known as D'Alembert's principle.
Let's consider this by applying it to Figure 1, which represents a system with one degree of freedom.
Figure 1
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Imagine a mass m on a floor with no resistance, connected by a spring and a damper. If you move this mass slightly to the right and then release it gently, the spring will force it back to the left, but the damper will prevent it from returning to its original position immediately. This is because the damper exhibits resistance (damping force) proportional to the velocity. The mass may return to its original position while oscillating, or it may return without oscillating. This depends on the size of the mass, spring, and damper.
If we "shift the mass slightly to the right and release it from rest," and consider this moment as time t=0, then there are no external forces after that (f(t)=0), and therefore the equation of motion without external forces is:
-mx"-cx'-kx=0
Initial condition: When t = 0, x = displacement (m)
Since acceleration α is the second derivative of displacement and velocity is the first derivative, the balance between the external force f(t), inertial force -mx'', spring force -kx, and damper damping force -cx' is obtained.
-mx''−cx'−kx＋f(t)＝0
Rewriting this yields equation (2).
When no external force is acting, we can set f(t) = 0 and obtain equation (1).
Since m, c, and k are constants, equations (1) and (2) are called linear differential equations with constant coefficients in mathematics. Next time, we will tackle differential equations.
References
"Spectral Analysis of Seismic Motion" by Yoshihiko Osaki, published by Kajima Publishing Co.
"The Story of Complex Numbers" by Hiroyasu Takao, published by the Japan Science and Technology Federation.
"Practical Mechanical Vibration Theory" by Masaharu Kunieda, published by Rikogakusha.
(Excerpt from the email newsletter issued on January 19, 2006)
