Chapter 7: The issue of the coupling between shaker and test-item
7.1 S INE SWEEP EXCITATION
Swept sine input differs from a frequency response analysis, useful tool during the S/C verification and design, because it does not satisfy the steady state requirement involved during the calculation of the frequency response (displacement, velocity or acceleration).
In fact, FRA supposed that the analysis is conducted at a given frequency value using a sine wave input with fixed amplitude and frequency. The value contained into frequency response is that corresponds at the response in time domain when the steady state condition is reached (for the considered value of frequency). The full FRA is achieved when the introduced procedure is conducted for all the frequency involved into the considered spectrum.
pag. 68 On the contrary, how introduced in “Chapter 5: Sine vibration”, swept sine input is governed by the “exponential rule” expressed in “Equation (5.4)”. In this case the steady state requirement is not satisfy because every minute the frequency vary with the octave rule. In this case, just one sine wave input is involved.
Particularly, these discrepancies produce several effects in the dynamic response between the theoretical/analytical and the experimental.
7.1.1 The effect of sweep rate
The first cause of discrepancies from FRA to swept sine vibration is the effect of the sweep rate.
In fact, how introduced before the difference between steady state conditions and transient environment produce a transient behaviour, how is shown in Figure 364.
In this case a SDoF is analysed, but the essay is general.
(a) (b)
Figure 36: Peak shift caused by the sweep rate [Ref. 4]
Particularly, “R=0” means the FRA output, “R=±1” means sweep rate up and down.
As can be seen, compared with the steady state conditions, the peak shift it self in the same direction of the sweep rate (i.e. if R>0 the peak moves towards right and vice versa).
In addition, the peak of the response decreases its magnitude in terms of “Q”, quality factor, and reduces its sharpness because a fictitious value of damping ratio appears.
Moreover, equation (7.1) provided by Lalanne, offers a useful model, for exponential sweep, thanks to which quantify the amount of deviation from steady state response.
𝜂 = 𝑄2𝑅 ln(2)
60𝑓𝑘 (7.1)
Particularly, if 𝜂 ≃ 0.1 the swept response is close to the steady state. Else, if 𝜂 grows it deviates from theoretical condition.
In fact, observing the considered equation, it depends with a direct proportionality to the sweep rate “R”, and inverse proportionality to the natural frequency and the square of the damping ratio being 𝑄 = 1/(2𝜁). This suggest that if the damping ratio is low the low frequency modes are greatly influenced for a given sweep rate.
pag. 69 7.1.2 Beating phenomena
Beating phenomena represent one of the most common issues during vibration test, caused, particularly, by the effect of sweep rate introduced in the previous subparagraph. These problems appear around the main resonant frequency. Indeed, it is a transient phenomenon caused by the un-steady state condition typical during the swept sine wave imposed at the base of IUT. In addition, these un-steady conditions produce a severe perturbation of the FRA analysis (steady state analysis).
Moreover, is possible to divide them in three different classes:
• Beating from two close frequency oscillations
• Beating from transient excitation with frequency content close to a resonant frequency
• Beating due to sine sweep rate
Firstly, according to “Beating from two close frequency oscillations” they are modelled in a simplified way using two sinusoidal excitations of unitary amplitude.
That is, the structure undergoes vibrations of frequency equal to the half of the frequencies sum, which are modulated by a function having frequency equal to the half of the frequency difference.
sin(2𝜋𝑓1𝑡) + sin(2𝜋𝑓2𝑡) = 2 cos[𝜋(𝑓1− 𝑓2)𝑡] sin[𝜋(𝑓1+ 𝑓2)𝑡] (7.1)
Where 𝑓𝑖 is the i-th frequency and “t” is the time.
Figure 37: Beating from the superposition of two frequency sinusoidal excitations [Ref. 4]
However, this type of phenomenon rarely appears during vibration test.
Secondly, “Beating from transient excitation with frequency content close to a resonant frequency” is analysed considering different levels of damping ratio and frequency input at given natural frequency.
pag. 70
(a) (b)
Figure 38: Dynamic amplification factor at different damping ratio (a), Response at various damping and input levels (b) [Ref. 4]
Particularly, structure with low damping ratio stressed by a sinusoidal input with frequency content close to that of the natural frequency reach the steady state conditions when the amplitude is constant. Being beating phenomenon is transient and it appear before the steady state and around the initial phase.
Lastly, about “Beating due to sine sweep rate”, Figure 39 and Figure 40 show the typical differences between the effect of the sweep rate, in terms of sweep up and sweep down, with the influence of damping ratio considering the natural frequency coming from FRA about 30 Hz. The ratio 𝑈̈/𝑍̈ in y-axis is the transmissibility (input- output are accelerations).
(a) (b)
Figure 39: The effect of sweep rate (up) and damping [Ref. 4]
pag. 71
(c) (d)
Figure 40: The effect of sweep rate (down) and damping [Ref. 4]
How is possible to see, comparing (a) and (b), a positive and increasing sweep rate produce a positive shift of the frequency at which the peak is reached with a lower value among the natural one coming from FRA. Vice versa, (c) and (d) show the same phenomena with a negative rate of sweep.
However, the main particularity concerned the pattern oscillation after the peak value. They are the beating or the so called “ringing”.
Figure 41: Ringing after the peak [Ref. 4]
This ringing is a result of the system responding at two frequencies of nearly the same value comprising the transient response at the natural frequency and the harmonic response at the swept excitation frequency.
pag. 72 To sum up, the most important remarks concerning the sine sweep excitation:
i. It produces the peak shifting in terms of magnitude, sharpness and frequency at with the peak is reached
ii. Beating phenomena affect the swept sine response, caused by the nearly between the peak at the natural frequency and the transient response caused by the excitation iii. These two effects show a direct direct proportionality to the sweep rate “R”, and inverse
proportionality to the natural frequency and the square of the damping ratio.