This time, continuing from last time, I will talk about the basics of vibration measurement.
The technology of preventing vibrations from being transmitted from vibrating machinery or structures to the outside, or conversely, preventing vibrations from being transmitted from the outside to machinery or structures, is generally called "vibration isolation." The latter is sometimes specifically called "vibration damping." This time, we will discuss vibration transmittance, an important quantitative parameter of vibration isolation technology.
We will consider how vibrations are transmitted when an elastic material such as rubber or a spring is inserted between a vibrating machine and the foundation or floor that supports it. In this case, we will consider it as a one-degree-of-freedom damping system model, as shown in Figure 1.
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Figure 1: Force Transmission to the Foundation
As before, we apply a harmonic excitation force f(t) (an excitation force that can be expressed as a sinusoidal function), which is a forced external force, to a point mass and investigate its response displacement x(t). The harmonic excitation force has amplitude F and angular frequency:
f(t)= Fcos(ωt)
Therefore, the resultant force of the inertial force, viscous resistance, and restoring force present in this vibrating system balances the external force, and the equation of motion is given by equation (2).
At this time, the response displacement in the steady state is:
.................................（3）
Therefore, the amplitude multiplier at this time is:
Xst = F /k (static displacement), let X be the displacement amplitude.
.................................（4）
(Natural angular frequency)
.................................（5）
(Damping ratio)
.................................（6）
This can be expressed as follows (a review from last time).
Next, let fT(t) be the force transmitted from the vibrating mass m to the fixed foundation. We will investigate what happens to this force. The force transmitted to the foundation is the sum of the force through the spring k and the force through the damper c;
Let FT be the amplitude of this resultant force. The component of velocity x(t) is the derivative of the displacement component, so it is clearly a force that advances by 90°. Therefore, the sum of the force vectors is, with X as the displacement amplitude;
.................................（8）
It will be.
The ratio of this transmitted force to the excitation force F is called the vibration transmissibility Tr, and using equation (4), we get:
.................................（9）
.................................（10）
As can be seen from equation (10), the vibration transmission coefficient is a function of the excitation angular frequency and the damping ratio ζ, so when we graph it with the damping ratio as a parameter, we get Figure 2.
The horizontal axis represents the angular frequency normalized by the natural angular frequency ω, and the vertical axis represents the vibration transmittance; both axes are logarithmic.
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Figure 2 Vibration Transmittance
From this figure, we can see that when the excitation angular frequency ω = √2ωn, the vibration transmittance is 1 regardless of the damping ratio ζ, it is greater than 1 when ω < √2ωn, and less than 1 when ω > √2ωn.
Therefore, to reduce the transmission of the excitation force of a vibrating object to the foundation, it is necessary to minimize the natural angular frequency ωn and the damping ratio ζ. However, care must be taken, as making the damping ratio too small will result in an extremely large amplitude at the resonant frequency.
Next, we consider the vibration of machinery installed on a vibrating foundation or floor (Figure 3).
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Figure 3 Transmission of foundation displacement to machinery
Currently, the foundation is vibrating with a displacement u, and we will determine the vibrational displacement of the machine due to displacement excitation.
Vibrations in the foundation;
u(t)= Ucos(ωt) .................................（11）
Assuming the displacement of the machine is x(t), the displacements of the spring and damper are equal to the relative displacement x(t) - u(t), and no forces other than the restoring force and viscous resistance act on the machine, the equation of motion for this machine is:
Rewriting equation (12):
Substituting equation (11) into the right-hand side of equation (13), the right-hand side becomes:
.................................（14）
Substituting this into the right-hand side of equation (13):
.................................（15）
Therefore, equation (15) takes the same form as equation (2), and the steady-state vibration of the machine is equivalent to the case when an excitation force of amplitude U√k² + (cω) ² is applied, so the displacement amplitude X is:
.................................（16）
Therefore, the ratio of the displacement amplitude of the machine to the displacement amplitude of the foundation is:
.................................（17）
This is called the vibration transmission coefficient of displacement, and its formula is exactly the same as that of the vibration transmission coefficient of force.
From these results, in order to prevent foundation vibrations from being transmitted to the machinery as much as possible, just like with force transmission,
It needs to be made to be like this.
Finally, here's a summary.
(1) The technology used to prevent vibrations from being transmitted from vibrating machinery or structures to the outside, or conversely, to prevent vibrations from being transmitted from the outside to machinery or structures, is generally called "vibration isolation."
(2) In the above (1), the latter case is sometimes referred to as "vibration isolation."
(3) The ratio of the excitation force of a vibrating machine to the force transmitted to the foundation is called the vibration transmittance coefficient, and it is an important parameter for evaluating vibration isolation by elastic support.
(4) In order to reduce the excitation force of the vibrating object from being transmitted to the foundation, the natural angular frequency ωn should be made as small as possible and the damping ratio ζ should be made small.
(5) The coefficient of displacement transmission from a vibrating foundation to the machinery mounted on it is given by the same formula as the coefficient of force vibration transmission in (3) above.
【keyword】
Vibration isolation, vibration damping, elastic body, 1-degree-of-freedom damping system, harmonic excitation force, inertial force, viscous resistance force, restoring force, amplitude magnification, natural angular frequency, damping ratio, vibration transmittance, vibration transmittance of displacement
【reference】
- "Introduction to Modal Analysis," by Akio Nagamatsu, Corona Publishing Co., Ltd. (1994)
- "Technologies and Laws for Pollution Prevention," edited by the Editorial Committee for Technologies and Laws for Pollution Prevention, Japan Industrial Pollution Prevention Association (1990).
(Excerpt from the email newsletter issued on November 19, 2015)
