Half-width method - Q value
As the frequency ω of the external force approaches the natural frequency ωo, ζ≪1
(x/A)max＝1/(2ζ) （32）
This is what it will be.
This value, as shown in Figure 5, is the same as the value obtained by dividing the natural frequency ωo by the frequency width Δf, which corresponds to 1/√2 of the peak value (which is 1/2 in terms of power) (Q value: Quality factor).
Q value=ωo/Δω=fo/Δf=1/(2ζ) (33)
For an explanation of why this happens, please refer to the following section on "Q-value".
In the evaluation of vibration-damping materials, the method of determining ζ using the half-width method from an FFT analyzer is commonly used. To accurately determine ζ, it is necessary to increase the frequency resolution of the response magnification (frequency response function) and improve the accuracy of reading Δf.
To increase frequency resolution, you can use a longer sample length or utilize the zoom function.
Figure 5: Half-width method, Q value, and damping ratio ζ
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Q = 1/(2ζ) = ω0/(ω2-ω1)
If we look up ω2 and ω1 from the calculated values, we get ω2 = approximately 20.9 and ω1 = approximately 18.9. Therefore, Δω = ω2 - ω1 = approximately 20.9 - approximately 18.9
ωo = 20
Q = ωo/Δω = 1/(2ζ) = 20 ÷ 2 = 10ζ = 0.05, thus confirming the half-width method.
Force equilibrium in forced steady-state oscillations
When considering the force equilibrium during steady-state oscillation of a forced vibration, the equation of motion for one degree of freedom and its solution, which I will rewrite with a slightly different expression as a summary, are as follows:
equation of motion
mx''＋cx'＋kx＝Fcosωt
x''＋(c/m)*x'+(k/m)*x＝(F/m)*cosωt
＝(F/k)*(k/m)*cosωt ・・・（34）
Here,
ωo＝√(k/m), ζ＝c/Cc＝c/2√(mk), c＝2ζ√(mk),
c/m = 2ζωo, Xs = F/k (static deflection) ... (35)
If we set this, then equation (34) of the equation of motion is
x''+2ζωox'+ωo^2＝Xs*(ωo^2)*cosωt...(36)
The solution x for the forced stationary response is
Let η = ω/ωo,
x＝Dcos(ωt-θ)
D＝F/k*1/√[{1-η^2}^2＋(2ζη)^2]
＝Xs*1/√[{1-η^2}^2＋(2ζη)^2]
θ＝tan-1(2ζη/(1-η^2） ・・・ （37）
The response magnification is
D/Xs＝1/√[{1-η^2}^2＋(2ζη)^2] ・・・(38)
The forces that balance the external force Fcosωt are the inertial force -mx'', the damping force -cx', and the spring force -kx. The relationship between them is:
external force
Fcosωt
displacement
x = Dcos(ωt-θ)
Inertial force is
-mx''＝＋m(ω^2)Dcos(ωt-θ)
Damping force is
-cx'=+cωDsin(ωt-θ)=cωDcos(ωt-θ-π/2)
Spring force is
-kx＝-kDcos(ωt-θ)＝kDcos(ωt-θ-π)
If we consider the phase relationship with respect to the direction of the inertial force,
- The inertial force is in phase with x.
- The external force F is greater than the inertial force, with a movement of θ.
- Damping force lags the inertial force by π/2.
- The spring force lags behind the inertial force by π.
Note that the D response magnification and θ phase change with ω as shown in Figures 3 and 4 of the previous issue. When expressed as a vector with respect to the inertial force and x, it looks like Figure 6.
Please note that the magnitude of the vectors in Figure 6 is an illustrative representation.
Figure 6: Equilibrium of external force F, inertial force mω²D, damping force cωD, and spring force.
In the case of ω < ωo
When ω is small, the inertial force mω²D is small, and the external force F and the inertial force counteract the spring kD.
In the case where ω = ωo
The inertial force lags behind the external force by 90 degrees, resulting in a relationship where the inertial force and spring force are in opposition, and the external force and damping force are in opposition. x becomes maximum (D is maximum), and both the inertial force and spring force are at their maximum. Also, the velocity is at its maximum, and the damping force is at its maximum.
In the case of ω > ωo
As ω increases, the inertial force gradually lags behind the external force by 180 degrees. D decreases as ω increases, and kD and cωD become small. The inertial force is proportional to ω², but since it balances out to a constant magnitude F, x(D) decreases rapidly.
References: Introduction to Modal Analysis, by Akio Nagamatsu, published by Corona Publishing Co., Ltd.
(Excerpt from the email newsletter issued on July 20, 2006)
