When I was in junior high school, I learned about the properties of waves, and the basic concept of acoustic vibration is the same; it's just that it's more complicated. I think you'll remember doing vibration experiments in junior high school by holding a weight and a spring and shaking it. The vibrations of actual cars and buildings are based on the same concept. It's true that acoustic vibration is often seen as difficult. The reason for this is that the phenomena that occur are many simple physical phenomena occurring in the same way, which makes it seem complicated.
1 degree of freedom system
Let's start with a simple mass and a spring. Imagine a kitchen scale. It can be easily represented as "Model 1" as shown in Figure 1.
<Figure 1: 1-degree-of-freedom system model = Mass: M, Spring constant (stiffness): k>
Moving a heavy object requires considerable force, right? Similarly, focusing on mass, a larger weight M makes it harder to move, resulting in slower movement. Lighter objects move more easily, resulting in faster movement. Let's also consider springs. You've probably experienced how a soft spring stretches well and produces large vibrations, while a stiff spring doesn't compress much when pressed with a finger, yet the rebound force is strong, making you feel like you need a lot of force. A stiffer spring will return to its original position faster, even with a small displacement, due to its greater rebound force. The spring constant k is used to express the stiffness (strength) of a spring. Since a spring displaces in proportion to the force, the spring constant is expressed as the ratio of displacement to force. In the case of a spring, if the spring constant is k, the force is F, and the displacement is x, then the following relationship exists.
F = kx
k = x/F (units: m/kg, m/N)
The term "spring constant" is used in the case of springs, but generally the term "stiffness" is used. If you apply force to a scale with your finger only at the beginning and then release it, the scale will vibrate freely up and down repeatedly and gradually stop. This free vibration is called the natural vibration of the scale, and the value expressed as the number of times it vibrates up and down per second, also known as the frequency (Hz), is called the natural frequency. The natural frequency can be calculated. In the model in Figure 1, if the mass is M (kg), the stiffness is k (m/kg), and the natural frequency is f (Hz), then
f = 1/2π × √k/M (Hz), (k/M goes inside the square root)
It will be.
The vibrations of the scale gradually dampen and eventually stop, but in the model in Figure 1, there is no damping element, so the vibrations continue indefinitely.
Let's consider "Model 2," as shown in Figure 2, which adds a damping coefficient C that represents the properties of the damping element.
<Figure 2: Damping 1-degree-of-freedom system model = Mass: M, Spring constant (stiffness): k, Damping coefficient: C>
Damping can be categorized into structural damping, which causes energy loss due to the displacement of atoms and molecules within a material, and external damping, such as air resistance or the viscosity of gaseous fluids like those found in car shock absorbers. While mass can be measured using a scale and stiffness k can be measured from material property tables or the relationship between added mass and displacement, damping cannot be measured separately as structural damping and other types of damping, so it is measured as a total sum. Damping will be discussed in more detail on another occasion.
Now, a model consisting of one mass, one spring, and one damper has only one specific frequency, so it is said to be a one-degree-of-freedom system.
Multi-degree-of-freedom system
While many things in the world are represented by a single-degree-of-freedom system, taking a tall tower as an example reveals that it has multiple natural frequencies for vibration. In reality, it's more common to think of things as multi-degree-of-freedom systems rather than single-degree-of-freedom systems. What does a multi-degree-of-freedom system represent?
If we consider a model in which we add another mass and spring to the previous 1-degree-of-freedom system model, we get the model shown in Figure 3, which is called a 2-degree-of-freedom system.
When we measure this vibration with an FFT analyzer, we can analyze two power spectral peaks: one at the natural frequency where M1 and M2 move up and down in the same direction, and another at the natural frequency where M1 and M2 move up and down in opposite directions.
<Figure 3: Two-degree-of-freedom system model = Masses: M1, M2, Spring constants (stiffness): k1, k2>
If we further increase the mass and springs in a 2-degree-of-freedom system, we get something like Figure 4, which is called a multi-degree-of-freedom model. Extending this idea, it seems possible to simulate vibration phenomena by creating a model composed of many masses and many springs. In fact, simulation is possible if there are no nonlinear elements such as friction or backlash.
<Figure 4: Multi-degree-of-freedom system model = Masses: M1, M2, ..., Spring constants (stiffness): k1, k2, ...>
Generally, the frequencies that cause problems in vibration issues are often low. This is because a certain amount of displacement is required to damage machinery or to radiate sound. For example, if you consider how earthquakes shake and damage buildings, you can understand that large displacements occur at low frequencies. Knowing the vibration characteristics of a building up to the problematic frequency provides sufficient data to consider whether countermeasures are necessary.
From this, we can see that simulation is possible by considering an N-degree-of-freedom system model in which we have N units of mass M, stiffness k, and damping coefficient C that have the necessary and sufficient performance.
(Excerpt from the email newsletter issued on January 22, 2004)
