Continuing from last time, I will explain the vibration isolation of vibration-damping rubber.
To prevent vibrations from small testing equipment placed on a desk from being transmitted to the desk and making noise, vibration-damping mats are sometimes used as a simple vibration isolation measure. The vulcanized rubber used in these vibration-damping mats is made by uniformly mixing compounding agents into the rubber and applying a constant pressure and temperature to give it high elasticity. It has excellent internal friction damping capabilities, is available in various shapes, is easy to handle, and is often used for vibration isolation in machinery.
The key point we focused on last time was to isolate the vibration so that the natural frequency is less than or equal to one-third of the external force frequency.
Figure 1. Vibration transmission rate of a damped 1-degree-of-freedom drive system
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Frequency ratio η=ω/ω0
When using vibration-damping rubber, pay attention to both the static and dynamic spring constants.
The static spring constant Ks is related to the amount of deflection δ when a machine is placed on vibration-damping rubber, and the following relationship exists:
Ks=W/δ...(1)
W: Load (=mG m: Mass G: Gravity)
δ: amount of deflection
The ratio λ of the dynamic spring constant kd to the static spring constant ks is approximately as shown in the following table.
| kinds | λ＝kd/ks |
| Natural rubber | 1.0 (soft) to 1.6 (hard) |
| Chloroprene-based | 1.4～2.8 |
| Nitrile-based | 1.5～2.5 |
From equation (1), the relationship between the deflection δ and the dynamic spring constant kd can be expressed by substituting ks = kd/λ,
kd/λ＝W/δ
Organize
δ/λ = W/kd ... (2)
The natural frequency fo of the system is
fo＝1/2π*√｛kd*G/W）＝1/2π*√｛Ks*λ*G/W｝
＝1/2π*√｛λ*G/δ｝ ・・・（3）
Substituting the units of Ks (kg/cm), δ (cm), and G = 980 cm/s²,
fo＝4.93√｛ksλ/W｝＝4.93√｛λ/δ｝ ・・・（4）
As a rough calculation formula
fo＝5/√δ （Hz） ・・・（5）
This is used.
If we set λ=1 in equation (3), the relationship between the natural frequency and ks can be graphed as shown in Figure 3.
Figure 3 Relationship between natural frequency, load, and spring constant
Load (kg) 𝑓_0=4⋅93√((𝜆⋅𝐾_𝑆)/𝑊) λ=1
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Spring constant (kg/cm)
Vibration-damping rubber comes in various shapes, such as cylindrical and square. For cylindrical rubber, if it's too long, it will buckle into a V-shape due to deflection, so it must be used within its buckling limit.
The selection of general vibration-damping rubber is
- Select a natural frequency that is less than or equal to 1/3 of the frequency of the external force.
- Deflection due to static load should be kept within approximately 10%.
- The sum of static load, dynamic load, and safety margin must be within the allowable load range.
- Choose a material that is suitable for the operating environment, including operating temperature and oil resistance.
It will be.
External forces and their frequencies include the rotational speed per second of a rotating shaft, the rotational speed per second multiplied by the number of blades of a blower, the rotational speed per second multiplied by the number of teeth of a gear, and the number of explosions per second of an internal combustion engine.
<Example 3>
Four vibration-damping rubber pads are used to isolate a machine weighing 200 kg.
The catalog stated that the allowable load was 80 kg, ks = 35 (kg/mm), and λ = 1.4.
The deflection δ and natural frequency fo are
Convert Ks to units
KS＝350 (kg/cm)
Also,
Load per vibration-damping rubber unit W = 200 / 4 = 50 (kg)
From equation (1)
δ＝W/Ks＝50/350≒0.143(cm)
From equation (4)
fo＝4.93√｛1.4÷0.143｝≒15(Hz)
The natural frequency is approximately 15 Hz.
Example 4
A 100kg internal combustion engine is rotating at 900 revolutions per minute. An unbalanced force of 20kg is generated vertically. We want to reduce the vibration transmission force to 5kg using four vibration-damping rubbers. What should be the spring constant K of each rubber?
The angular frequency ω of the external force is
ω＝2πf＝2π×900÷60＝94.2 （rad/s）
The natural angular frequency ωo is
ωo^2＝K/m＝＝WG/m＝980K/100＝9.8K (rad/s）
The vibration transmission coefficient τ for one degree of freedom can be simply calculated using the following equation without damping from the previous issue.
τ＝｜1÷｛1-（ω/ωo）^2｝｜ ・・・（6）
Furthermore, from the condition ω > ωo that the natural frequency is less than or equal to 1/3 of the external force frequency, equation (6) is
5/20≦1÷｛（ω/ωo）^2-1｝＝1÷｛94.2^2/9.8K-1｝
Therefore
K≦181 （kg/cm）
The dynamic spring constant of one is
181÷4≒45.3（kg/cm）
If the static spring constant of one is λ = 1.4,
Ks＝Kd/λ＝45.3÷1.4≒32.5（kg/cm）
After selecting vibration-damping rubber under conditions like those in Example 4, you can check fo and τ using Example 3 and equation (6).
As shown in the example, determine the static spring constant by taking one-third of the frequency of the external force as the natural frequency, and select a vibration-damping rubber with the corresponding spring constant. After selection, you can also check the natural frequency fo and transmission coefficient τ. When selecting vibration-damping rubber, please refer to materials and catalogs provided by specialized manufacturers.
References: Machine Noise and Countermeasures, published by Kyoritsu Shuppan.
(Excerpt from the email newsletter issued on August 25, 2005)
