Complex modes identification via Hilbert transform S. Gabriele1, D. Spina2, C. Valente3, A. De Leonardis3 1 University “Roma Tre”, Department of Structures Via C. Segre 4/6, 00146, Roma, Italy email:
[email protected] 2
Department of Civil Protection, Seismic Risk Office Via Vitorchiano 4, 00181, Roma, Italy 3
University “G. d’Annunzio”, Department of Engineering V.le Pindaro 42, 65127, Pescara, Italy
Abstract Among the numerous identification methods available for operational modal analysis, the CPR (Complex Plane Representation) method proves efficient and effective in the case of sweep tests where the excitation is not measured. The CPR method has been described in a number of papers by the authors and allows for high precise identification of the modal parameters. In spite of this precision there exist cases where the approximations involved by practical applications need further clarification. The paper is devoted to estimate error bounds due to lack of congruence between the theoretical assumptions and the actual system characteristics or field conditions. The dependence of the amplitude and phase values on the modal density is crucial in this respect. Theoretical and numerical solution are used to validate the CPR results.
1
Introduction
Experimental modal analysis is currently performed to identify the dynamic properties of systems and structures. The modal parameters provide valuable information on the system and its status and their knowledge is exploited in a number of applications that range from system reconstruction to damage assessment to structural health monitoring and so on. The availability of more and more precise and robust methods is therefore a continuous need. Several well grounded methods are presently available for the estimate of the modal parameters [1]. Often, the dynamic response is mapped into the frequency domain to get the spectral properties of the system. The effectiveness of these methods comes from the enlargement of the functional space to comprise both the real and the imaginary part of the system dynamics. Such interesting feature can be exploited also in the time domain by using an appropriate transformation of the dynamic response of the system. The mathematical operator that allows to get the imaginary counterpart of the real measured system motion is the Hilbert transform [2]. This latter operator has been used to set up an identification method, termed Complex Plane Representation CPR, capable to identify the modal parameters of generally damped systems. The method was first proposed in [3] and then refined in [4]. The formulation is specialized to deal with the problem of complex modes identification from the analysis of the dynamic response close to resonance. It has been shown that the CPR method is a natural framework to deal with non classical damped system [5] and turns out to be particularly efficient and effective from a computational viewpoint. However in practical applications the deviations from ideal conditions introduce some sources of error that is important to define precisely. The purpose of the paper is to discuss how the approximations affect the identification. To this end the CPR method is first presented in a rigorous theoretical framework, then possible deviations from the ideal conditions are considered. These are consequences of three aspects: (i) non ideal dynamic response, i.e.
noise corrupted signals, (ii) non ideal experimental conditions, i.e. approximation of the theoretical conditions, (iii) non ideal systems, i.e. coupling of modal contributions. The first two aspects are easy to deal with and a quantification of the errors induced is given. The third aspect deserves more attention and is deeply discussed. Depending on the modal density the CPR method can identify fictitious phase differences among the different degrees of freedom and hence can fail in the identification of real or complex modes. In order to evaluate the errors due to the modal density, the results of the CPR method are compared against two alternative solutions: the theoretical solution obtained through standard modal analysis and the numerical solution provided by a frequency domain analysis. The results are illustrated according to a 2dof mechanical system that allows to control in a simple way the phase changes as a consequence of the modal density in both cases of proportional (real modes) or non proportional (complex modes) damping.
2
The CPR method
The identification of the complex modes is carried out through the Complex Plane Representation method [3,4]. The CPR method analyzes the system response to (quasi)harmonic excitations. These types of excitations are common in the experimental modal analysis field of civil structures [6]. Apart from the above, no further assumption is required for the forcing function so that the method belongs to the family of the so called output only methods. The rationale of the method is summarized below. The response of a N dofs linear system endowed with general non proportional damping, when subjected to a sinusoidal forcing function close to resonant conditions, can be given in the form:
" N % " N % qh (t) = $ ! Ahp ' sin(2(fk t) + $ ! Bhp ' cos(2(fk t) = # p=1 & # p=1 &
(1)
= ( Ahk + ) hk ) sin(2(fk t) + (Bhk + * hk ) cos(2(fk t), h = 1...N Equation (1) states that the dynamic response (time history) q of degree of freedom h at resonance (frequency k) is given by the sum of N modal contributions. The main contribution to mode k is provided by Ahk and Bhk and the contributions due to the other modes are summed up in αhk and βhk. These latter terms are generally small at resonance and can generally be neglected. A more comprehensive representation of qh can be obtained if equation (1) is mapped into the complex plane. This step is accomplished via the Hilbert transform HT [2] that allows to compute the imaginary counterpart of the real (measured) motion qh. The complex signal zh is obtained: +$
z h (t) = q h (t) + iq˜ h (t), where : q˜ h (t) = HT[q h (t)] =
% q (! )[" (t # ! )] h
#1
d!
(2)
#$
Upon substitution of (1) in (2) and using the modulation property [2] one gets:
z h (t) = ( Ahk + iBhk )[sin(2!fk t) " i cos(2!fk t)]
(3)
The use of the complex signal zh(t) in place of the system response qh(t) allows to compute the modal parameters of the system: the instantaneous envelope a(t) and frequency f(t) functions and the mode
shapes ψ. The first two parameters are immediately obtained by the relations given in [7]: ah ( t ) = z h ( t ) , 2!fh ( t ) = d ["z h ( t )] dt . During resonant conditions both the amplitude of the motion and the vibrating frequency should remain constant and can be used to check the stationary of the response to be analyzed. More useful is their identification in the free decay range at the end of the stationary phase that allows to estimate the dissipating and vibrating properties of the system [7, 8]. This item is presently well grounded and is not further discussed.
2.1
Mode shapes identification – ideal conditions
What is relatively new is the possibility to identify the mode shapes. To this end, equation (3) allows to institute a relation of simple proportionality among the oscillations of the different N degrees of freedom:
z h (t) = ! hk z r (t) = (" hh + i# hk )z r (t), " hk =
Ahk Ark + Bhk Brk A B $ Bhk Ark , # hk = hk rk2 2 2 Ark + Brk Ark + Brk2
(4)
In equation (4) qr, i.e. zr, is defind as the reference degree of freedom and ξhk and ηhk are respectively the real and the imaginary part of the k-th eigenvector scaled such that ψrk = 1. If zh(t) and zr(t) are interpreted as the dynamic response in the complex domain the identification of the mode shape components can be carried out through the minimization of the following error function (the symbol * stands for complex conjugate): +$
! hr = % [z h (t) " # hk z r (t)][z h (t) " # hk z r (t)]* dt
(5)
"$
The stationary point of εhk is found by taking the derivative of equation (5) with respect to ψhk, i.e. to ξhk and ηhk. Using the position (2) together with the orthogonality property and the energy conservation property of the HT [2], the following expressions are obtained: +&
!" hr = 0 $ ' [2# hk q r2 (t) % 2q h (t)qr (t) + 2# hk q r2 (t) % 2 q˜ h (t) q˜ r (t)]dt = 0 !# hk %& +&
!" hr = 0 $ ' [2( hk q˜ r2 (t) + 2q h (t) q˜ r (t) + 2( hk q r2 (t) + 2 q˜ h (t)q r (t)]dt = 0 !( hk %&
(6)
Equation (6) consists of two linear and independent equations that allow to compute the real and imaginary part of the sought mode shape: +#
$ [q (t)q (t) + q˜ (t) q˜ (t)]dt h
! hk =
"#
r
h
+#
2 $ q (t)dt 2 r
"#
+#
r
$ [q (t) q˜ (t) + q˜ (t)q (t)]dt h
; % hk = "
"#
r
h
+#
2 $ q (t)dt 2 r
"#
r
(7)
2.2
Digitized noise polluted signals
The solution (7) is valid under ideal conditions. In practice the experimental records qh(t) are known only in a discrete set of nt time instant tp and are noise polluted. When the noise can be modeled by a white noise Poisson process the following relations hold: nt
! qs (t p )e j (t p ) = 0, p=1 nt
! e (t s
p
nt
nt
! q˜s (t p )e j (t p ) = 0,
! q (t s
p=1
nt
! e˜ (t
)e j (t p ) = R1" sj ,
s
p=1
p
p
)e˜ j (t p ) = 0,
p=1
(8)
nt
! e˜ (t
)e˜ j (t p ) = R1" sj ,
s
p=1
p
)e j (t p ) = R3" sj
p=1
where ! sj is the Kronecker symbol and R1, R2 and R3 are positive constant. Then the equations (7) take the form:
nt
! hk =
nt
" [qh (t p ) + eh (t p )][qr (t p ) + er (t p )] + " [ q˜ h (t p ) + e˜h (t p )][ q˜r (t p ) + e˜r (t p )] p=1
p=1
nt
2" [qr (t p ) + er (t p )]
=
2
p=1
nt
" [q (t h
=
p
(9a)
)qr (t p ) + q˜ h (t p ) q˜ r (t p )]
p=1
np
np
" [q (t )] + " 2
r
p=1
p=1
nt
! hk =
[er (t p )]2
p
nt
" [qh (t p ) + eh (t p )][ q˜ r (t p ) + e˜r (t p )] + " [ q˜h (t p ) + e˜h (t p )][qr (t p ) + er (t p )] p=1
p=1
nt
2" [q r (t p ) + er (t p )] p=1
nt
" [q (t ) q˜ (t ) + q˜ (t )q (t )] h
=
p
r
p
h
p
=
2
r
(9b)
p
p=1
np
np
p=1
p=1
" [qr (t p )]2 + " [er (t p )]2
The comparison between equations (7) and (9) shows the error in the estimates of the modal components ξhk and ηhk due to the noise. It is found that the error depends solely on the noise that affect the reference dof r regardless the noise level present in the other measured dofs. From the above it is apparent that the quality of the estimates depends on the appropriate selection of the reference dof r. In view of equation (9), the criterion adopted to select r corresponds to minimize the noise
to signal ratio in the reference dof. To this end, each measurement point is tested as possible reference dof r and the associated error evaluated. Eventually, the best dof to be used as reference is dof s for which:
Es = min( Er ), with Er = "h=1 ! hr , r = 1,2,...,N. N
(10)
It is noted that the index Er offers also a global evaluation of the accuracy of the estimates for the corresponding mode shape.
2.3
Sweep excitations
Sweep tests are often performed in place of pure resonance tests. In these cases the external excitation is a quasi-harmonic function whose frequency is linearly variable in time. If the time variation is slow then quasi-resonant condition can be assumed when the excitation frequency travels around the system frequency k. In this instance the equation (1) is replaced by (αhk and βhk neglected):
q h (t ) = Ahk (t ) sin ! (t ) + Bhk (t ) cos ! (t ), h = 1,..., N
(11)
where both the instantaneous frequency and envelope are no longer constant in time. As a consequence, the modulation property fails and the results (9) can be retained only at the expense of introducing some errors. It is therefore interesting to evaluate those cases in which the error can be neglected. The modulation property states that for time signals composed by the product between an envelope function Ahk(t) or Bhk(t) and an oscillatory function sinθ(t) or cos θ(t), such as in equation (11), if both functions have non overlapping spectra in the frequency domain then the action of the HT is to leave the envelope unchanged and to transform only the oscillatory part:
q h (t ) = Ahk (t ) HT [sin ! (t )] + Bhk (t ) HT [cos! (t )],
h = 1,..., N
(12)
Since in general the envelope or oscillatory functions do not satisfy the above restriction than equation (12) will be error affected:
q h (t ) = Ahk (t ) HT [sin " (t )] + Bhk (t ) HT [cos" (t )] + ! h (t ),
h = 1,..., N
(15)
The global energy of the error function: +$
+$
#$
#$
! = % [" k (t)]2 dt = % [&k ( f )]2 df
(24)
is shown in Figure 1 [6]. The results shown in the figure refer to a linear viscous damped system with normalized frequency (fk = 1 Hz) and unit maximum amplitude. From the figure it can be appreciated that for conventional damping values the error can be neglected.
Figure 1: Error variation vs. damping (fk=1 Hz)
3
Effects of modal density
The CPR method is capable to distinguish and identify real and complex modes. A real mode is detected if all the degrees of freedom (dofs) of a given mode are in phase; on the contrary, a complex mode is detected if there exists a phase difference among the different dofs. In this respect, the modal density introduces modal coupling into the system response and can induce errors in the identified quantities. The modal density is a serious problem for any identification method and especially for those methods that work mode by mode. In fact the modal response is as high altered as close the modes are. Depending on the modal density the CPR method can identify fictitious phase differences and hence fail in the identification of real modes. This aspect is of some interest if one considers that the level of modal complexity can be related to the amount of damage suffered by the system [5]. In conclusion the parameter to be controlled is the phase shift among the different dofs of the system. This phase difference can originate by actual physical system properties as the lack of proportionality between the mass / stiffness matrices and the damping matrix or it can be displayed as a consequence of the progressively higher modal density. In this second case the phase difference is fictitious and depends on the fact that the identification method operates on the whole dynamic response and not just on the contribution of the single mode. In order to evaluate the errors due to the modal density, the results of the CPR method are compared against two alternative solutions: the theoretical solution obtained through standard modal analysis and the numerical solution provided by the FRF analysis since it is a method capable to deal simultaneously with all the excited modes of the system.
3.1
Theoretical solution
The theoretical modes of a linear N-dofs system are derived using standard techniques, [9]. The procedure is briefly resumed. According to the system considered in equation (1), the equations of motion in homogeneous conditions are:
Mq˙˙ + Cq˙ + Kq = 0
(25)
In the general case of systems endowed with general damping, decoupling is achieved by recast the problem in the state space form. Denoting by x = [ q˙ ,q ] the new variables vector, equation (25) is rewritten in block matrix form:
M * x˙ + K * x = 0,
"M 0 % "C K% M* = $ , K* = $ ' ' # 0 !K& #K 0 &
with
(26)
where assuming x = xe st , the general eigenproblem is given by:
(K
*
+ sM * )x = 0
(27)
whose eigensolutions are provided by the matrix A: !1 "!M !1C !M !1 K% A = !M * K * = $ ' 0 & # I
(28)
As A is a real non-symmetric matrix, the solution leads to a set of 2N complex conjugate eigenpairs {si ,x i } and the complex modes of the system are computed as:
fi =
Re( si ) Im( si ) T ; "i = ; # i = { x n+1 ... x 2 N } i with i = 1,3,5,...,2N $1 2! Im( si )
(29)
where fi, νi and ψi are the i-th frequency, damping ratio and mode shape respectively.
3.2
Numerical solution via FRF
The frequency response function of a general dissipative N-dof system subjected to a harmonic excitation whose frequency ωk = 2πfk coincides with the k-th resonance frequency of the system is given by the following expression:
% ! rq* ! rp* ! rq! rp ' H pq = + + 2 ' r =1 " # + i " $ " 1$ # " r# r + i " k + " r 1$ # r2 r r k r r '& N
(
)
(
)
( * * *)
(30)
The amount of modal complexity is given by the joint contribution of the actual physical properties of the system and by the spurious terms introduced by the modal density. The effects of these latter can be easily quantified using a system endowed with real modes. In the case of a proportionally damped system, the contribution of the conjugate modes in equation (30) vanishes so that it reduces to:
H pq =
(
! kq! kp
" k# k + i " k $ " k 1$ # k2
)
% N ! rq! rp +,' 2 ' r +k " # + i " $ " k r 1$ # r &' r r
(
)
( * * *)
(31)
where the contribution of the interested mode k has been separated with respect to the other N-1 modal contributions that pollute the pure response in k. The fictitious phase shift is exactly the combined contribution of the modes under summation. Each r ≠ k mode will add a different phase contribution and will be weighted by a different ψrp value for each dof. Each spurious contribution depends on νr and ωr and will be progressively smaller as νr decreases and as ωr departs from ωk. Such cases, in fact, magnify the denominator of the second term on the right side of (31) and make vanishing the spurious contributions. As opposed to the theoretical values (29), the equations (30) and (31) will be used as necessary in the following to compute the phase shifts on a numerical basis.
3.3
Mechanical system
In order to discuss the problem of the phase difference between different dofs due to the modal density, it has been devised a simple, yet comprehensive model capable to highlight the main features of the problem, Figure 1.
Figure 2: 2-dofs system used to study the effects of the modal density The 2-dofs system of Figure 1 allows to control the phase changes as a consequence of the modal density in both cases of proportional (real modes) or nonproportional (complex modes) damping. The system has been chosen since it possesses some important properties that make it particularly effective to study the errors induced by the modal density when the CPR method is used. The mass matrix M of the system is diagonal, the stiffness matrix K is banded and the damping matrix C is adjusted as necessary to provide proportionality or not. The only variable of the problem is the stiffness of central spring k2* = (k2 – ∆k2) so that the system preserves its symmetry regardless the values attributed to k2*. In particular, k2 is kept constant and ∆k2 is changed in the interval [0, k2]. In the case in which C is proportional to K, i.e. ci = ηki, the system has two real modes ψ1 = {1, 1} ψ2 = {1, −1} that are invariant with ∆k2. When the system oscillates according to the first mode, the two masses m1 and m2 move in phase so that the central spring stays unstretched and does not participate to the motion. As a consequence, also the first frequency f1 is invariant with ∆k2. On the contrary, when the system oscillates according to the second mode the two masses m1 and m2 move out of phase in opposite sides and hence the central spring deforms and contributes to the motion. As a
consequence, the second frequency f2 depends on the values taken by ∆k2. It is noted that f2 decreases when ∆k2 increases. It is also noted that making ∆k2 progressively increasing f2 moves progressively towards f1.
Figure 3: Effect of k2* variation on Amplitude and Phase of H12
Figure 4: Effect of k2* variation on Amplitude and Phase of H22
The modal density is then governed by ∆k2 that leaves the first mode (shape, frequency, damping) unchanged and modifies only the second mode. In order to study the same phenomenon in the presence of complex modes it is necessary to make C non proportional, assuming e.g. c1 = 2ηk1. In these cases one has to abandon the symmetry of the system. When this happens the first mode is no longer invariant. Yet the changes caused by the nonproportionality
are small with respect to those of the second mode that are induced by the stiffness modification so that the first mode can be assumed approximately invariant. The numerical FRF are computed by inserting the theoretical values (29) into the equations (30) and (31). The figures 3 and 4 show the changes of the FRFs H22 and H12 for the case C proportional and ∆k2 varying in [0, 0.9k2]. As ∆k2 increases the peak of the second mode gets close to the peak of the first mode and increases in amplitude until when ∆k2 = 0.9k2 the two peaks become undistinguishable. The FRF as those reported in figure 3 and 4 are used, through their expressions (30) and (31), to evaluate the phase difference between dof q1 and dof q2 as the modal density increases according to changes in k2*. These values will be compared in the next chapter with those identified by means of CPR. As a matter of reference, a typical phase difference is computed as:
!"12 = ! H12 ( f1 ) # ! H 22 ( f1 )
(32)
being ! H12 ( f1 ) and ! H 22 ( f1 ) the FRF phase angles at the first modal frequency f1. It is worth to recall that the existence of a phase difference is the same to say that the relevant mode is complex (not real). In the present case of proportional damping the existence of a phase angle difference can be interpreted as an error. This error is small and negligible for well spaced modes and becomes larger for closely spaced modes. This error would vanish if the system would be excited with a force profile identical to the mode shape. Of course this is unfeasible in real structures and errors due to the modal superposition should be expected.
4
Results
The results are discussed by the help of some examples derived by the system of figure 2. The comparison with the results obtained by the numerical FRF will serve to validate the CPR outcomes. In order to simulate real testing conditions the dynamic force has a sweep-sine variation given by:
[
]
[
]
Fo ( t ) = 4 ! 2 fA + ( fB " fA )t ; # ( t ) = 2! fA t + 0.5( fB " fA )t 2 ; 2
(33)
where [fA, f B] is the frequency range to be tested and the pure sine case corresponds to fA = fB. Two main cases are selected to analyze the CPR effectiveness. For each case the identification of the first mode is reported. It is initially assumed k1 = k2 = k3. In case 1, k2* is varied in [0, 0.9k2] and the damping matrix C is constrained to preserve proportionality. Therefore any detected phase difference depends only on the modal density. Mode shape values are provided for sub-cases: (1a) Δk2 = 0 and (1b) Δk2 = 0.7k2. On the contrary, in case 2 non-proportionality is introduced by imposing that coefficient C11 is two times that of the initial proportional matrix C. For the purpose of comparison, here again the two sub-cases: (2a) Δk2 = 0 and (2b) Δk2 = 0.7k2 are considered. To simulate field conditions a 10% noise is added to the pseudo-experimental signals. In all cases the first frequency f1 = 8.48 Hz is identified with great precision by the CPR method.
Figure 5: CPR identified phase difference
Case 1 The CPR method is used to identify the phase difference between dofs 1 and 2 (Δϕ12 exp). The results relevant to mode 1 are discussed. The real and imaginary components given by equation (7) are used on purpose. For simplicity, it is supposed to excite the system with a pure sine forcing function applied at dof 2. Since the CPR is a time domain method the identified values can be plotted against time. This is done in figure 5 where each curve refers to a different k2* value. Apart from the initial transient and some oscillations at the end of the record due to the Gibbs phenomenon, the plot allows to check the constancy of the phase angle during the stationary resonant condition. The Δϕ12 values are expressed in π. The response starts out of phase (π /2) and then stabilizes to a non null values that increases along with Δk2. The same data are reported in figure 6 but now adopting different axes. The results are plotted against the stiffness values of the central spring. The abscissa is normalized to 1 to better appreciate the results. A value of 1 means k1 = k2 = k3, whereas 0 means no stiffness between dof 1 and 2. In the figure the results of the CPR method are compared with the “exact” numerical results obtained via equation (32). Both results are closed together even if some loss of precision is observed for very small k2*. This point would deserve better discussion especially when stiffness decreasing is intended as damage, but this is not the topic of the paper and it is deferred to future works. The specific results for cases 1a and 1b are indicated with two circles in the curves of figure 6. The relevant numerical values are reported in table 1 and table 2. These data summarize the main result of the work. The tables are organized as follows. They concern the first mode shape and have as many rows as are the system dofs. The columns list the amplitude and phase angle of each dof. The first couple of columns reports the theoretical values computed using equation (29), whereas the second couple of columns reports the values identified using CPR. In the last column the phase difference computed as in (32) is also given. By the tables it is observed that the theoretical mode shapes ! theor = {1, 1} of a proportionally damped system do not exhibit any phase difference ! ! theor = {0, 0}. However if one considers the system response in resonant condition then a fictitious phase difference is detected as a consequence of modal superposition. This phase difference is correctly identified by the CPR method ( !"12 exp) as demonstrated
by the comparison with the numerical results ( !"12 ). Similar consideration apply also for the mode shape amplitudes, but here the changes are small and can be neglected. The dependence of this phenomenon on the modal density is made clear by the comparison between the values of table 1 and table 2.
Figure 6: Phase difference: identified vs. numerical
mode 1
!
theor
! ! theor
!
exp
!"12 exp
!"12 num
dof 1
1.000
0.000
0.998
0.086
0.084
dof 2
1.000
0.000
1.000
0.000
0.000
!"12 exp
!"12 num
Table 1: case 1a mode 1
!
theor
! ! theor
!
exp
dof 1
1.000
0.000
0.965
0.280
0.270
dof 2
1.000
0.000
1.000
0.000
0.000
Table 2: case 1b
Case 2 This is the case where the system shows non proportional damping as opposed to the previous case. Further a noise to signal ratio of 10% is artificially added to the system response. By inspecting table 3 and table 4 it is noted that the theoretical phase ! ! theor is any longer zero as a consequence of the complexity induced by the loss of damping proportionality. The modal amplitude is slightly more corrupted as compared to the values of table 1 and table 2. On the contrary, the phase difference shows significant changes, nevertheless the identified values obtained by the CPR method are very close to the numerical results. This latter result demonstrates that the identification error in the modal parameters is almost insensitive to the measurement noise.
mode 1
!
theor
! ! theor
!
exp
!"12 exp
!"12 num
dof 1
1.000
-0.069
0.979
0.216
0.214
dof 2
0.998
0.000
1.000
0.000
0.000
!"12 exp
!"12 num
Table 3: case 2a mode 1
!
theor
! ! theor
!
exp
dof 1
1.000
-0.170
0.834
0.558
0.538
dof 2
0.998
0.000
1.000
0.000
0.000
Table 4: case 2b
5
Conclusions
The paper deals with the problem of the errors evaluations of the CPR method when applied in actual conditions. The CPR method is a time domain method that works mode by mode according to sweep tests. Several source of errors are considered. Errors related to digitized and noise polluted signals. Errors due to the approximation of the modulation property. Errors deriving from the modal density and the dissipation property of the system. This latter errors affect both the amplitude and phase of the identified mode shape. When the modal parameters are used to interpret the status and the evolution of the system, phase distortion can reveal a serious problem. The problem of phase distortion is studied in a controlled environment by the help of a simple and specially devised mechanical system. Theoretical solutions and numerical solutions obtained using an frequency based identification method are compared with those of CPR. The results show that the phase changes detect by the CPR method are correct and precise. A relation is provided to help understand the ranges where the CPR method can be applied safely.
Acknowledgements The work has been partially granted by PRIN 2007 (Italian National Research Projects).
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