Starting this time, we will discuss the fundamentals of vibration measurement, one of the most important application areas for the FFT analyzer, a type of frequency analyzer. Unlike ordinary static physical quantities such as mass, length, temperature, and pressure, sound and vibration require consideration of frequency (or vibration frequency). Frequency is the rate at which sound or vibration repeats as a wave phenomenon, and the intensity of the vibration changes depending on its value. Here, we will mainly explain natural frequency (or resonant frequency) and damping ratio.
Vibrations can be divided into complex types such as free vibration, forced vibration, and self-excited vibration, but complex vibrations will not be discussed here.
Free vibration is a phenomenon in which, after an initial external force such as striking or pushing and releasing is applied, the object continues to vibrate at a specific speed (frequency) even without further external force. In contrast, forced vibration is a phenomenon in which an external force is constantly acting on the object, causing it to continue vibrating at the same frequency as that external force. Vibrations in typical rotating machinery such as engines and motors are examples of forced vibration.
Free vibration usually subsides quickly and doesn't cause practical problems, but it's a fundamental phenomenon that reveals all the dynamic characteristics of a machine and is the basis for forced vibration and other phenomena. Therefore, this time, I will talk about free vibration.
First, let's discuss how to express the magnitude of vibration.
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Figure 1. Motion of a spring and a weight (simple harmonic motion)
In Figure 1, when a weight attached to a spring is pulled and then suddenly released, the weight will repeat its motion infinitely at a constant frequency (or period) if there is no damping. If the position (displacement) trajectory of the weight is plotted with time on the horizontal axis, it becomes a sine wave as shown in Figure 1, and this motion is specifically called simple harmonic motion.
Velocity is a quantity that represents how much the position changes per unit time, and acceleration is a quantity that represents how much the velocity changes per unit time. Therefore, if we let the displacement of the mass be x(t), the velocity be v(t), and the acceleration be a(t), then, as shown in Figure 2, they have a differential and integral relationship.
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Figure 2: Interrelationship between displacement, velocity, and acceleration
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Figure 3. Time relationship between displacement, velocity, and acceleration waveforms.
.................................（1）
.................................（2）
The displacement, velocity, and acceleration waveforms of simple harmonic motion (sine wave) are as shown in Figure 3. The displacement waveform lags the acceleration waveform by 180 degrees in phase; in other words, its phase is inverted.
Next, I will explain how free vibrations occur.
Even for complex machines, we consider a model where a point mass m is connected to a spring with spring constant k and a damper with damping force c, as shown in Figure 4. This model is called a one-degree-of-freedom vibration system.
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Figure 4. Model of a 1-degree-of-freedom vibration system
First, for simplicity, let's consider the case where there is no damper (c = 0), as shown in Figure 5.
In Figure 5, when the point mass is displaced by x from a state of no vibration, it receives a restoring force of kx from the spring (a resistance force in the direction of displacement), so the equation of motion for the point mass m is:
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Figure 5. One-degree-of-freedom undamped vibration system
Alternatively, if we consider the product of mass and acceleration as the inertial force (resistance force), and think of it as a balance between external and internal forces, then in free vibration, the external force = 0;
This can be written as an equation of motion equivalent to equation (3). Equation (4) is a second-order differential equation in terms of displacement (t), which is a function of time, and the only functions whose second derivative is equal to itself multiplied by a coefficient can be trigonometric functions or exponential functions.
I will omit the details, but from equation (4), the displacement x(t) is:
Here, C and α are constants determined by the initial conditions;
Natural angular frequency.................................(6)
Natural frequency.................................(7)
Thus, free vibration occurs at a natural frequency (equation (7)) determined only by the spring constant k and mass m, regardless of initial conditions such as the initial amplitude, and does not occur at any other frequency.
The property of a restoring force that is proportional to the displacement, such as the spring constant k, is called stiffness or rigidity.
Generally, stiffness has the property of making vibrations faster, and mass has the property of making vibrations slower. Natural frequency can be described as the frequency at which the interplay between these two properties is balanced.
Equation (4) can also be derived from an energy perspective. Free vibration is a phenomenon in which the elastic energy (strain energy) stored in a spring is converted into kinetic energy and then the reverse process is repeated, so by the law of conservation of energy:
Constant................................(8)
Differentiating both sides with respect to time gives:
.................................（9）
In other words;
This is equivalent to equation (4).
Next, let's consider damped free oscillation. That is, the case where the damper c is not zero in Figure 4. Generally, a damper has a viscous resistive force proportional to the velocity v(t), so by adding this internal force, which involves energy loss, the equation of motion for viscous damped free oscillation is:
This is how it works. c is called the viscous damping coefficient.
Transforming equation (11):
...............................（12）
Let's assume that. Here;
...............................（13）
...............................（14）
Now, we need to solve the differential equation in equation (12), but whether or not it oscillates depends on the value of the damping ratio ζ in equation (14). When ζ is 1 or greater, it does not oscillate, and when ζ is less than 1 (ζ < 1), it oscillates freely with damping.
Here;
...............................（16）
Damped Natural Angular Frequency
Damping rate
Thus, the role of the damping ratio ζ is not only to stop vibrations (convert vibrational energy into thermal energy), but also to lower the frequency of free vibration.
When the damping ratio ζ is 1, it is determined whether or not oscillation occurs. At that time, from equation (14), the viscous damping coefficient c is equal to m√k², and this value is called the critical damping coefficient.
The ζ, which indicates damping performance, is commonly used not only in machinery but also in electrical control and construction/civil engineering fields, but it's important to note that its terminology differs slightly.
| Electrical and control | Damping coefficient |
| architecture civil engineering | Damping constant |
| machine | Damping ratio |
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Figure 6. Time evolution of displacement waveform due to differences in damping ratio (ζ).
In mechanical engineering, it's common to call it the damping ratio because it's easily confused with the viscous damping coefficient c.
Finally, here's a summary.
(1) Unlike static physical quantities, sound and vibration require consideration of frequency (or vibration frequency) as an axis.
(2) Vibrations can be classified into free vibrations, forced vibrations, and complex vibrations.
(3) When a weight attached to a spring is pulled and then released, its trajectory traces a sine wave, and this motion is called simple harmonic motion.
(4) The displacement, velocity, and acceleration of vibrations are related by differential and integral calculus.
(5) The displacement waveform of simple harmonic motion lags behind the acceleration waveform by 180 degrees in phase, meaning its phase is reversed.
(6) Undamped free vibrations occur at natural frequencies determined solely by the stiffness of the mass and spring constant.
(7) The behavior of viscous damped free vibration is determined by the value of the damping ratio ζ, and when ζ is less than 1, it dampens while vibrating at the damped natural frequency.
(8) When the damping ratio ζ is less than 1, it not only stops the vibration but also lowers the frequency at which free vibration occurs.
(9) The damping ratio ζ has different terminology in fields such as electrical control and civil engineering, so it is important to note that it is called differently in mechanical systems.
【keyword】
Natural frequency, resonant frequency, damping ratio, free vibration, forced vibration, self-excited vibration, complex vibration, simple harmonic motion, displacement, velocity, acceleration, differential and integral calculus, mass, spring constant, damper, one-degree-of-freedom vibration system, restoring force, inertial force, natural angular frequency, natural frequency, stiffness, rigidity, elastic energy, strain energy, kinetic energy, energy conservation law, viscous damped free vibration, viscous damping coefficient, damped natural angular frequency, critical damping coefficient, damping coefficient, damping constant
【reference】
- "Introduction to Modal Analysis," by Akio Nagamatsu, Corona Publishing Co., Ltd. (1994)
- "Introduction to Practical Vibration Analysis," by Yasuhide Takahashi, Naohiro Okutsu, and Takayuki Koizumi, Nikkan Kogyo Shimbun (1983).
(Excerpt from the email newsletter issued on July 16, 2015)
