# The trade-off between vibration isolation by dynamic and static stiffness

DFM PULSE, our vibration isolation solution, provides design solutions for vibration isolation devices for a wide range of components. The article will introduce the basic statics and dynamics of why a metamaterial approach to vibration isolation may be necessary instead of an existing component.

Hiroaki Natsume

2022,09,16 2022,09,16

# 1質点系の動的な性能について

## 自由振動

\begin{align}
m \ddot{x} + kx = 0 \tag{1-1} \\
\ddot{x} + \omega_n^2 x = 0 \tag{1-2} \\
\omega_n^2 = \frac{k}{m}　\tag{1-3}
\end{align}

x=Ae^{i\omega_n t + \delta}

Aは最大振幅、\delta は位相差で初期値に依存する値です。一般解からわかるように \omega_n2\pi 秒内で何回振動するかを表す量で固有円振動数と呼ばれます。 固有円振動数 \omega_n は、固有周期 T、固有周波数 f_n とは次の関係があります。

f_n = \frac{1}{T} = \frac{\omega_n}{2 \pi} = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}
\tag{1-4}

## 強制振動

\ddot{x} + \omega_n^2 x = F_0
\tag{1-5}

\omega_n^2 x = F
\tag{1-6}

\begin{align}
T_r &= \left|\frac{F}{F_0} \right|
= \left|\frac{\omega_n^2 x}{\ddot{x} + \omega_n^2 x} \right|
= \left|\frac{1}{1 + \frac{1}{\omega_n^2} \frac{\ddot{x}}{x}} \right| \\
&= \left|\frac{1}{1 + \frac{-\omega^2}{\omega_n^2}} \right|
= \left|\frac{1}{1 - u^2} \right|
\tag{1-7}
\end{align}

u は加振周波数と1質点系の固有周波数の比(u=\frac{\omega}{\omega_n} = \frac{f}{f_n})、x は式(1-5) の形式の微分方程式の一般解 x = A e^{i \omega t} を使用して解いています。

これらより次のことがわかります。

• u = 1 のとき、振動が大きく増幅して伝達される（共振現象）
• u\sqrt{2} 以下の範囲では常に振動が増幅される
• u\sqrt{2} を超えたときはじめて振動伝達率 Tr の値が 1 を下回り防振効果を発揮する

つまり防振したい場合は、周波数の比を \sqrt{2} 以上にする必要があります。

### 具体的な防振設計例

100Hz で加振される 1kg の装置から発生する振動の伝達を \frac{1}{8} となるように防振したい場合、周波数比 u=3 となり目標の固有周波数は式(1-7) より以下になります。

3 = \frac{100}{f_n}, \quad
\tag{1-8}
f_n \fallingdotseq 33 [\rm{Hz}]

\frac{100}{3} = \frac{1}{2 \pi} \sqrt{\frac{k}{1}}, \quad
k = \left( \frac{200}{3} \pi \right)^2 \fallingdotseq 43900 [\textrm{N/m}] = 43.9 [\textrm{N/mm}]
\tag{1-9}

### 質量と周波数の関係

m [kg]

F_0 [Hz]

バネ剛性
k [N/mm]
1 100 33 43.9
0.1 100 33 4.39
0.01 100 33 0.439
1 10 3.3 0.439
0.1 10 3.3 0.0439
0.01 10 3.3 0.00439
1 1 0.33 0.00439
0.1 1 0.33 0.000439
0.01 1 0.33 0.0000439

このように、軽いものをより低い加振周波数でも防振するためには、より低い剛性のバネが求められることがわかります。 このバネ剛性で具体的にはどのような形状の部材があるかは、例えばモノタロウさんやミスミさんなどで、バネ剛性の値から部品を検索することができるので、興味がある方は調べてみてください。

SUS304 40 11 0.8 0.44

# 1質点系の静的な性能について

これまで計算してきた振動に対する問題（動的な問題）に対して、以下のように重力加速度 g がかかるのみの部材の静的な状態について考えます。

\delta = \frac{m g}{k}
\tag{2-1}

\delta = \frac{1 \times 9.8}{0.44} = 22.2 \textrm{mm}
\tag{2-2}

# 剛性-質量-周波数のトレードオフについて

これまで示してきた動的な特性と静的な特性をまとめると、以下のトレードオフがあることがわかります。

1. 剛性が小
• 1次固有周波数が低くなるため、より低周波数から防振することができる
• 静的たわみが大きくなり、防振対象物を設置しただけで過大な変形が発生し底付きする可能性がある。
2. 剛性が大
• 1次固有周波数が高い周波数になるため、防振が難しくなる
• 静的たわみが小さくなり、重量物でも安定して支えることができる。

# About the dynamic performance of 1 mass-spring model

Consider the following undamped 1 mass-spring model with mass m and spring stiffness k.

## Free vibration

The relationship between displacement and force with the point mass in free vibration can be expressed as follows

\begin{align}
m \ddot{x} + kx = 0 \tag{1-1} \\
\ddot{x} + \omega_n^2 x = 0 \tag{1-2} \\
\omega_n^2 = \frac{k}{m}　\tag{1-3}
\end{align}

\omega_n is called the natural circular frequency, which has the following relationship with the natural period T and natural frequency f_n.

f_n = \frac{1}{T} = \frac{\omega_n}{2 \pi} = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}
\tag{1-4}

Equation (1-4) shows that for constant mass, the natural frequency increases in proportion to the square root of the stiffness, i.e., the higher the stiffness, the higher the natural frequency.

## Forced Vibration

Next, consider how the response changes as the natural frequency changes in the following forced vibration problem. For the free vibration problem treated in Equation (1-1), consider the situation where a forced vibration F_0 at an circular frequency of \omega is given.

The equation of motion of a vibrated point mass from the balance of forces is as follows

\ddot{x} + \omega_n^2 x = Fe^{i \omega t} = F_0
\tag{1-5}

On the other hand, the vibration is transmitted through this spring, and the reaction force F generated at the support point side is generated by the term affected by the spring in equation (1-5), which is

\omega_n^2 x = F
\tag{1-6}

From the above, the transmission coefficient of vibration from the vibration source to the support, T_r, is obtained by taking the ratio of Equations (1-5) and (1-6) as follows

\begin{align}
T_r &= \left|\frac{F}{F_0} \right|
= \left|\frac{\omega_n^2 x}{\ddot{x} + \omega_n^2 x} \right|
= \left|\frac{1}{1 + \frac{1}{\omega_n^2} \frac{\ddot{x}}{x}} \right| \\
&= \left|\frac{1}{1 + \frac{-\omega^2}{\omega_n^2}} \right|
= \left|\frac{1}{1 - u^2} \right|
\tag{1-7}
\end{align}

u is the ratio of the excitation frequency to the natural frequency of the 1 mass-spring model (u=\frac{\omega}{\omega_n} = \frac{f}{f_n}), and x is solved using the general solution x = A e^{i \omega t} of the differential equation of the form (1-5).

Equation (1-7) can be graphed as follows

From these we can see the following

• When u = 1, vibrations are transmitted with great amplification (resonance phenomenon)
• Vibrations are always amplified in the range u less than \sqrt{2}.
• Only when u exceeds \sqrt{2}, the value of vibration transmission coefficient Tr falls below 1 and vibration isolation is achieved.

In other words, if you want to isolate vibration, the ratio of frequencies must be greater than \sqrt{2}.

### Specific vibration isolation design examples

If we wish to isolate the transmission of vibration generated by a 1 kg device vibrated at 100 Hz so that \frac{1}{8}, the frequency ratio u=3 and the target natural frequency is the following from equation (1-7).

3 = \frac{100}{f_n}, \quad
\tag{1-6}
f_n \fallingdotseq 33 [\rm{Hz}]

From equation (1-4), the spring stiffness k in this case is

\frac{100}{3} = \frac{1}{2 \pi} \sqrt{\frac{k}{1}}, \quad
k = \left( \frac{200}{3} \pi \right)^2 \fallingdotseq 43900 [\textrm{N/m}] = 43.9 [\textrm{N/mm}]
\tag{1-7}

In actual vibration isolation design, it is rare to find off-the-shelf components that exactly match the performance obtained above, so it is necessary to search for components with similar values in catalogs. In many cases, vibration isolation component manufacturers provide a selection flow on their websites or in their catalogs, which describes what type of vibration isolation component to select without detailed calculations, or provide tables and graphs to help you find the right component.

### Mass vs. frequency

The following table summarizes the results of calculations for vibration isolation for several combinations of mass and excitation frequency.

Mass
m [kg]
Excitation frequency
F_0 [Hz]
The transfer coefficient becomes 1/8 (u = 3)
Natural frequency F_n [Hz]
Spring stiffness
k [N/mm]
1 100 33 43.9
0.1 100 33 4.39
0.01 100 33 0.439
1 10 3.3 0.439
0.1 10 3.3 0.0439
0.01 10 3.3 0.00439
1 1 0.33 0.00439
0.1 1 0.33 0.000439
0.01 1 0.33 0.0000439

Thus, we can see that a lower stiffness spring is required to isolate lighter objects even at lower excitation frequencies. If you are interested in finding out what specific shapes of components are available with this spring stiffness, you can search for components by spring stiffness value at, for example, Monotaro or MISUMI, and so on.。

As an example of a member with the stiffness of 0.439 N/mm required to isolate vibration when a 1 kg mass point in the above table is vibrated at 10 Hz, the following ready-made member of a simple coil spring was found.

Material Natural height(mm) Diameter(Φmm) wire diameter(Φmm) spring stiffness (N/mm)
SUS304 40 11 0.8 0.44

# Static performance of 1 mass-spring model

In contrast to the problems so far calculated for vibration (dynamic problems), the following is considered for the static state of a member subjected only to gravitational acceleration g.

The deflection under its own weight \delta is given by the following equation from the balance of forces.

\delta = \frac{m g}{k}
\tag{2-1}

Equation (2-1) shows that for a constant mass, the greater the stiffness, the smaller the deflection. Consider the static displacement of the coil spring in the table above, which is determined to be necessary for vibration isolation of a 1 kg spring against 10 Hz excitation.

\delta = \frac{1 \times 9.8}{0.44} = 22.2 \textrm{mm}
\tag{2-2}

The static deflection is 22.2mm, which means that more than half of the spring's natural length of 40mm has been deformed by the static load. When the spring is used for vibration isolation, deformation during forced vibration is added from this displacement, so if the spring is greatly deformed in the static state, it may bottom out due to deformation during forced vibration (the spring is deformed 40 mm and the spring makes contact), losing its vibration isolation effect, and a large load from a collision may be applied to the object to be isolated. The object may be subjected to a large load due to a collision.

In the above example, the vibration isolation is 1 kg at 10 Hz, but assuming a lower frequency vibration isolation, the spring stiffness k is 1/100 of 1 kg at 1 Hz. The amount of deformation in this case is 22.2 \times 100 = 2220\textrm{[mm]} = 2.22\textrm{[m]}, an unrealistic value that makes it difficult to achieve vibration isolation with the methods introduced so far.

Summarizing the dynamic and static characteristics I have shown so far, we can see that there is a tradeoff between

1. Less stiffness
• Lower first-order natural frequencies allow vibration isolation from lower frequencies
• Static deflection is increased and excessive deformation may occur and bottom out just by installing the vibration isolating object.
2. Greater stiffness
• Vibration isolation becomes more difficult due to the higher first-order natural frequencies
• Static deflection is reduced and even heavy objects can be supported stably.

The following is the result of a previous survey of off-the-shelf anti-vibration components. Because of this trade-off, we find that there are no off-the-shelf vibration isolation components commercially available in the low-frequency range. The member at the lower right point of the graph that achieves low-frequency vibration isolation is a member with special spring stiffness, called an air-spring system.

# Summary

This article introduced dynamic and static stiffness and the tradeoffs for vibration isolation that can be obtained from them.

In the next article, I will introduce the specifics of one of the structural metamaterials, called QZS, that may be able to break through the tradeoffs introduced in the last section.

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