In the previous lesson, we calculated the excitation force in the case of an imbalance. This time, let's consider the displacement caused by the forced vibration resulting from this excitation force.
The equations of motion for a single-degree-of-freedom system without damping are:
mx“＋kx＝Pcosωt ・・・（1）
m: mass
k: Spring constant
P: External force
Since this assumes forced oscillation, we ignore transient states and use this solution.
X＝Acosωt ・・・（2）
Assuming this, substituting into equation (1)
-mAω^2cosωt+kAcosωt＝Pcosωt
A(k-mω^2）＝P
From here,
A＝Xo/｜1-（ω/ωo）^2｜＝Xo/｜1-（f/fo）^2｜・・・（3）
However, Xo = P/k
ωo^2 = (2πfo)^2 = k/m
This is the result.
(3) If we know m, k, or the natural frequency fo and the excitation force P from equation (3), we can determine the vibration amplitude A of the machine.
<Example>
A vibrating screen machine, supported by four springs with a spring constant of 1 kN/mm, and including the exciter, weigh a total of 10 tons, are vibrated using an unbalanced exciter. When the exciter's rotation speed is 750 r/min, the mass of the rotating body is 500 kg, and the eccentricity is 50 mm, what is the amplitude of the vibration displacement of the vibrating screen machine in mm, and what is the force transmitted to the floor?
The excitation force P is greater than last time.
Frequency f=750÷60=12.5 (Hz)
Excitation force P=mrω^2=500×0.05×(2π×12.5)^2=154×10^3(N)
From equation (3), the vibration amplitude A is
Xo＝P/K＝154×10^3÷4000＝38.5(mm)
K = 4 × 1000 × 1000 (N/m)
Natural frequency fo=1/2π*√{4×1000×1000/10000}=3.18(Hz)
A=Xo/｜1-(f/fo)^2｜=38.5÷{12.5^2/3.18^2-1}=2.66(mm)
From the previous article, "Vibration Analysis -19 "Vibration Insulation -2"", the force transmission coefficient τ is
τ = 1/|1-(f/fo)^2|
The force Q transmitted to the floor is
Q=P÷{(f/fo)^2-1}=10.7×10^3(N)
The transfer efficiency (frequency response function) can be measured with an FFT analyzer, and the following calculations can also be performed using it.
For example, when installing precision equipment, it is sometimes necessary to accurately implement vibration isolation measures by predicting the power spectrum of how much vibration the equipment will vibrate as a result of floor vibrations. This can be calculated by multiplying the vibration transmission coefficient (frequency response function) transmitted from the floor to the precision equipment by the vibration power spectrum of the floor. The frequency response function is measured by performing a sine sweep on the precision equipment using a vibration exciter. The vibration of the floor where the equipment is to be installed is measured using Accelerometer to determine its power spectrum. For an example of this multiplication operation, please refer to "Multiplication of Power Spectrum and Frequency Response Function (Equalization Function)" below.
Multiplication of power spectrum and frequency response function
References: Practical Mechanical Vibration Engineering, by Masaharu Kunieda, Rikogakusha
(Excerpt from the email newsletter issued on September 26, 2005)
