457.643 Structural Random Vibrations In-Class Material: Class 19
III-2. Random Vibration Analysis of Linear Structures (contd.)
Spectral representation of nonstationary process
Used PSD Φ𝑋𝑋(ω) for spectral representation of a stationary process 𝑋(𝑡). What kind of model to use for spectral representation if 𝑋(𝑡) is non-stationary?
Main purpose: describe the change of the frequency content over time 1) Generalized PSD Φ̂𝑋𝑋(ω1, ω2): Fourier transform of 𝜙𝑋𝑋(𝑡1, 𝑡2)
Assuming 𝑋̃(ω) =2π1 ∫ 𝑋(𝑡)𝑒−∞∞ −𝑖𝜔𝑡𝑑𝑡 exists in the mean-square sense,
Φ̂𝑋𝑋(ω1, ω2) ≡ E[𝑋̃(ω1)𝑋̃∗(ω2)]
= 1
(2𝜋)2∫ ∫ E[𝑋(𝑡∞ 1)𝑋∗(𝑡2)]𝑒−𝑖(𝜔1𝑡1−𝜔2𝑡2)𝑑𝑡1𝑑𝑡2
−∞
∞
−∞
= 1
(2𝜋)2∫ ∫ 𝑒∞ −𝑖(𝜔1𝑡1−𝜔2𝑡2)𝑑𝑡1𝑑𝑡2
−∞
∞
−∞
Can show (from the formulation for 𝜙𝑋𝑋(𝑡1, 𝑡2) of a linear system) Φ̂𝑋𝑋(𝜔1, 𝜔2) = Φ̂𝐹𝐹(𝜔1, 𝜔2)𝐻(𝜔1)𝐻∗(𝜔2)
It is also noted that
𝜙𝑋𝑋(𝑡1, 𝑡2) = ∫ ∫ 𝑒∞ 𝑖(𝜔1𝑡1−𝜔2𝑡2)𝑑𝜔1𝑑𝜔2
−∞
∞
−∞
Question: Practical? No, because
It is difficult to assign physical meaning (two ω’s?)
The time term does not appear in the PSD although it is important for nonstationary process
2) Instantaneous PSD Φ𝑖(𝜔, 𝑡) (Page 1952)
𝜙𝑖(𝜏, 𝑡) = E [𝑋 (𝑡 −𝜏
2) 𝑋∗(𝑡 +𝜏 2)]
Φ𝑖(𝜔, 𝑡) = 1
2𝜋∫ 𝜙∞ 𝑖(𝜏, 𝑡)𝑒−𝑖𝜔𝜏𝑑𝜏
−∞
3) Physical PSD (Mark 1970)
Φ𝑝(𝜔, 𝑡)𝑤 = E [|∫ 𝑤(𝑡 − 𝜏)𝑋(𝜏)𝑒−𝑖𝜔𝜏𝑑𝜏
∞
−∞
|
2
]
where 𝑤(𝑡 − 𝜏) is the “window” function that captures PSD around the time 𝑡 4) Evolutionary PSD (Pristley 1965, 1967)
Let us consider two different versions of inverse FT
𝑋(𝑡) = ∫−∞∞ 𝑋̃(𝜔)𝑒𝑖𝜔𝑡𝑑𝜔 Riemann integral
𝑋(𝑡) = ∫ 𝑒−∞∞ 𝑖𝜔𝑡𝑑𝑆(𝜔) Stieltjes integral
~ generalization of Riemann integral by use of “increment process” 𝑑𝑆(ω)
※ Increment Process 𝑑𝑆(ω)
(1) Can use Fourier integral even when 𝑋̃(ω) =𝑑𝑆(ω)
𝑑ω does not exist because 𝑑𝑆(ω) is smoother
𝑋(𝑡) = ∫ 𝑒∞ 𝑖𝜔𝑡𝑑𝑆(𝜔)
−∞
-- Fourier-Stieltjes integral
(2) “Orthogonal” increment process 𝑑𝑆(𝜔)
E[𝑑𝑆(𝜔1)𝑑𝑆∗(𝜔2)] = Φ(𝜔1)𝛿(𝜔1− 𝜔2)𝑑𝜔1𝑑𝜔2
It has been proved that (Lin & Cai 1995), for an orthogonal increment process,
a) Priestley’s idea (toward “evolutionary” PSD): How about…
𝑋(𝑡) = ∫ 𝐴(𝜔, 𝑡)𝑒𝑖𝜔𝑡𝑑𝑆(𝜔)
∞
−∞
where
𝐴(ω, 𝑡): frequency-time modulating function
𝑑𝑆(ω): orthogonal increment process representing a stationary “base”
process 𝑋𝑠(𝑡) = ∫ 𝑒−∞∞ 𝑖𝜔𝑡𝑑𝑆(𝜔)
b) In this case, the auto-correlation function is derived as 𝜙𝑋𝑋(𝑡1, 𝑡2) = E[𝑋(𝑡1)𝑋∗(𝑡2)]
= ∫ ∫ 𝐴(𝜔∞ 1, 𝑡1)𝐴∗(𝜔2, 𝑡2)𝑒𝑖(𝜔1𝑡1−𝜔2𝑡2)E[𝑑𝑆(𝜔1)𝑑𝑆(𝜔2)]
−∞
∞
−∞
= ∫∞𝐴(𝜔, 𝑡1)𝐴∗(𝜔, 𝑡2)𝑒𝑖𝜔(𝑡1−𝑡2)Φ𝑆𝑆(𝜔)𝑑𝜔
−∞
Note, for a stationary process:
𝑅𝑋𝑋(𝜏) = ∫ 𝑒∞ 𝑖𝜔𝜏Φ𝑋𝑋(𝜔)𝑑𝜔
−∞
(Note that 𝑋̃(𝜔) does not exist for stationary process) Proof for (→):
𝜙𝑋𝑋(𝑡1, 𝑡2) = E[𝑋(𝑡1)𝑋∗(𝑡2)]
= ∫ ∫ 𝑒∞ 𝑖𝜔1𝑡1−𝑖𝜔2𝑡2E[𝑑𝑆(𝜔1)𝑑𝑆∗(𝜔2)]
−∞
∞
−∞
= ∫ ∫ 𝑒𝑖𝜔1𝑡1−𝑖𝜔2𝑡2Φ(𝜔1)𝛿(𝜔1− 𝜔2)𝑑𝜔1𝑑𝜔2
∞
−∞
∞
−∞
= ∫ 𝑒∞ 𝑖𝜔(𝑡1−𝑡2)Φ(𝜔1)𝑑𝜔1
= −∞
c) For 𝑡1= 𝑡2,
E[𝑋2(𝑡)] = ∫∞|𝐴(𝜔, 𝑡)|2
−∞
Φ𝑆𝑆(𝜔)𝑑𝜔
Note, for a stationary process:
E[𝑋2] = ∫∞Φ𝑋𝑋(𝜔)
−∞
𝑑𝜔
Comparing the two equations, |𝐴(𝜔, 𝑡)|2Φ𝑆𝑆(ω) seems to describe the evolution of the spectral representation of the non-stationary process at time 𝑡, so we can…
d) Define “Evolutionary” PSD (EPSD) as Φ(𝜔, 𝑡) = Φ𝑆𝑆(𝜔)|𝐴(𝜔, 𝑡)|2
to describe the evolution of the frequency content over time using the frequency-time modulating function 𝐴(ω, 𝑡)
e) Special case: uniformly modulated evolutionary process
𝐴(𝜔, 𝑡) = 𝐴(𝑡)
In this case, it is noted
𝑋(𝑡) = ∫ 𝐴(𝑡)𝑒𝑖𝜔𝑡𝑑𝑆(𝜔)
∞
−∞
= 𝐴(𝑡) ∫ 𝑒𝑖𝜔𝑡𝑑𝑆(𝜔)
∞
−∞
= 𝐴(𝑡)𝑋𝑠(𝑡)
Φ𝑋𝑋(ω, 𝑡) = |𝐴(𝑡)|2Φ𝑆𝑆(𝜔)
E[𝑋2(𝑡)] = |𝐴(𝑡)|2𝐸[𝑋𝑠2(𝑡)] = |𝐴(𝑡)|2𝐸[𝑋𝑠2] How to determine 𝐴(𝜔, 𝑡)? Examples:
Kubo & Penzien (1976): Identified 𝐴(𝑡) by statistical analysis of San Fernando earthquake records (Clough & Penzien 1993)
The “envelope function” 𝐴(𝑡) was identified in the form 𝑎1𝑡 ⋅ exp (−𝑎2𝑡)
Jangid (2004, EESD): provided an extensive survey of envelope functions and investigated SDOF nonstationary responses
Other methods for spectral representation of nonstationary processes:
Wavelet transform (Kareem, Spanos, …), Hilbert-Huang transform (with empirical model decomposition; Wen & Gu, 2004, 2009 in JEM), etc.
f) Input-output relationship when evolutionary PSD is used (to be continued)
