(3) Natural frequency and damping constant
Once the natural frequency and attenuation constant are known, the form of the solution equation is known, and the solution can be obtained by substituting these values into the respective coefficients.
This time, let's go back in the reverse direction from the previous issue and see what's happening.
The basic equations (3), (4), and (4') from the previous issue, and the example equations (5), (13), and (13') are summarized below. (Some additions)
ζ≪1
x''＋2ζωox'＋ωo^2x＝0 ・・・・・・・・・・・・ (3)
x＝e^(-ζωot){Ccos(√1-ζ^2*ωot)+Dsin(√1-ζ^2*ωot)}
＝Ae^(-ζωot)cos(√1-ζ^2*ωot−Φ) ・・・・・・・・・ （4）
ωn = √1 - ζ^2 * ωo ≈ ωo
Φ = tan⁻¹(D/C)
ACD is a constant determined by the initial conditions.
x＝Ae^(-ζωot) cos(ωot-Φ) ・・・・・・ (4')
x''＋2x'＋400x＝0 ・・・・・・・・・・ （5）
x＝e^-t｛cos20t＋1/20*sin20t｝ ・・・・・・・・・・ （13）
＝1.0012e^-tcos（20t-Φ） ・・・・・・・・・・ （13’）
Φ＝tan^-1（1/20/1）＝tan-1（1/20）
Let's compare the coefficients.
The natural frequency ωo and the damping constant ζ are
ωo^2＝400
2ζωo＝2
twist
Natural frequency: ωo=2πfo=20
fo ≈ 3.2 (Hz)
Damping constant: ζ = 1/20 = 0.05
Damped free vibration frequency: ωn ≈ ωo
(3-1) Example
What happens when the damping constant becomes larger in equation (5)?
x''＋2x'＋400x＝0 ・・・・・・・・・・ （5）
Let's keep the natural frequency the same as in equation (5) and set the damping constant to 0.3.
ωo＝20
ζ＝0.3
The equation of motion, using equation (3) as a reference, is as follows:
ωo^2＝400
2ζωo＝12
x‘’＋12x‘＋400＝0 ・・・・・・・・（14）
Taking ζ into consideration, refer to equation (4)
ωn＝√1-ζ^2*ωo≒19.1
ζωo＝6
x＝e^-6t(Ccos19.1t＋Dsin19.1t)
Based on the initial conditions t=0, x=1, x'=0, refer to the previous issue.
x＝e^-6t｛cos19.1t＋6/19.1sin19.1t｝
x＝1.048e^-6tcos（19.1t-θ） ・・・・・・・・・（15）
θ＝tan-1（6/19.1)
Figure 2 shows the solution to equation (14), equation (15), and the solution to equation (5), equation (13'), superimposed on each other.
It can be seen that equation (15) results in a faster decay time compared to equation (13'), and that the decay frequency is slightly lower than the natural frequency due to the influence of the decay constant.
Figure 2 Waveforms with different damping constants (conceptual diagram)
(Excerpt from the email newsletter issued on March 23, 2006)
