Master Thesis - OUFTI-1
Master Thesis - OUFTI-1 Master Thesis - OUFTI-1
• The logarithmic decrement method: This method is a time domain approach. It calculates damping based on the free decay of oscillations in the structure using Equation 4.2.6. ( ) xn ∆ = ln = 2 π ζ x n+1 √ ⇒ ζ = 1 √ 1 − ζ 2 1 + ( 2 π ∆ ) 2 (4.2.6) where ∆ is the logarithmic decrement between times n and n + 1, x i is the response of the structure at a given time i and ζ is the damping factor. If the assumption of low damping can be applied, this equation is greatly simplied. This simplication is presented in Equation 4.2.7. ∆ ≈ 2 π ζ ⇒ ζ = 2 π ∆ (4.2.7) • The peak-amplitude method: This method is a frequency domain approach. It calculates damping based on the Frequency Response Function (FRF) of the structure. Once the FRF is obtained, the response at the half-power points (illustrated in Figure 4.10) is measured and the damping can be calculated as shown in Equation 4.2.8. Figure 4.10: Illustration of the peak-amplitude method [53] 83
Q = ω k = 1 ω b − ω a 2 ζ ⇒ ζ = 1 ω b − ω a (4.2.8) 2 ω k where Q is the quality factor, ω r is the natural frequency, ω a and ω b are the frequencies of half-power points and ζ is the damping factor. • The Steinberg's equation: This method is an analytical one [54]. It states that the transmissibility at resonance of a PCB is equal to several times (depending of the natural frequency considered) the square root of the natural frequency, as shown in Equation 4.2.9. Q = a √ ω r with ⎧ ⎪⎨ ⎪⎩ a = 0.5 ω r ≤ 100 Hz a = 0.75 100 Hz < ω r ≤ 200 Hz (4.2.9) a = 1 200 Hz < ω r ≤ 400 Hz a = 2 400 Hz < ω r where Q is the transmissibility at resonance and a is the tting factor based on ω r , which is the natural frequency. Unfortunately, the data on which the Steinberg's method is based, are unavailable and therefore, the method is unveriable. For this reason, it will not be used in this thesis. • And a lot of other methods: Circle tting, Stochastic Subspace Identication (SSI), Ibrahim Time Domain (ITD), ... It is important to note that, in case of low damping, the frequency domain methods are pretty poor because the curves on which they are based, are dicult to measure accurately due to the high rate of change of the frequency response curve. So, in case of low damping, time domain methods will be preferred. An other important remark is that damping may vary with the excitation level. So, several tests must be performed at several levels of excitation to determine an accurate value of the damping factor. 4.2.9 Dynamic response computation To avoid large errors when computing the dynamic response of a PCB, two points should be considered: • Enough modes should be computed in the modal solution to excite a signicant fraction (at least 90%) of the total mass of the structure. • If the deection of the PCB is comparable to its thickness, a non-linear analysis will be preferred. 84
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Q =<br />
ω k<br />
= 1<br />
ω b − ω a 2 ζ ⇒ ζ = 1 ω b − ω a<br />
(4.2.8)<br />
2 ω k<br />
where Q is the quality factor, ω r is the natural frequency, ω a and ω b are the frequencies<br />
of half-power points and ζ is the damping factor.<br />
• The Steinberg's equation: This method is an analytical one [54]. It states that<br />
the transmissibility at resonance of a PCB is equal to several times (depending of<br />
the natural frequency considered) the square root of the natural frequency, as shown<br />
in Equation 4.2.9.<br />
Q = a √ ω r<br />
with<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
a = 0.5<br />
ω r ≤ 100 Hz<br />
a = 0.75 100 Hz < ω r ≤ 200 Hz<br />
(4.2.9)<br />
a = 1 200 Hz < ω r ≤ 400 Hz<br />
a = 2 400 Hz < ω r<br />
where Q is the transmissibility at resonance and a is the tting factor based on ω r ,<br />
which is the natural frequency. Unfortunately, the data on which the Steinberg's<br />
method is based, are unavailable and therefore, the method is unveriable. For this<br />
reason, it will not be used in this thesis.<br />
• And a lot of other methods: Circle tting, Stochastic Subspace Identication (SSI),<br />
Ibrahim Time Domain (ITD), ...<br />
It is important to note that, in case of low damping, the frequency domain methods are<br />
pretty poor because the curves on which they are based, are dicult to measure accurately<br />
due to the high rate of change of the frequency response curve. So, in case of low damping,<br />
time domain methods will be preferred.<br />
An other important remark is that damping may vary with the excitation level. So,<br />
several tests must be performed at several levels of excitation to determine an accurate<br />
value of the damping factor.<br />
4.2.9 Dynamic response computation<br />
To avoid large errors when computing the dynamic response of a PCB, two points<br />
should be considered:<br />
• Enough modes should be computed in the modal solution to excite a signicant<br />
fraction (at least 90%) of the total mass of the structure.<br />
• If the deection of the PCB is comparable to its thickness, a non-linear analysis will<br />
be preferred.<br />
84
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