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On estimating system damping from frequency response bandwidths
Journal of Sound and Vibration 330 (2011) 6088–6097
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On estimating system damping from frequency response bandwidths Peter J. Torvik n Air Force Institute of Technology, 1866 Winchester Road, Xenia, OH 45385, USA
a r t i c l e in f o
abstract
Article history: Received 9 December 2010 Received in revised form 17 May 2011 Accepted 30 June 2011 Handling Editor: I. Trendafilova
As damping determines the maximum vibratory response of a system at resonance, reliable estimates of damping are critical both to the design and qualification of systems to be subjected to a vibratory environment and for the evaluation of the effectiveness of modifications or additions provided to increase damping. The sharpness of the frequency response at resonance is often used for this purpose, quantified by the width of the frequency range (bandwidth) for which the response is above some fraction of the maximum response. Several aspects of the use of bandwidth methods in interpreting test results are considered. It is shown that the use of an excessive rate of change of test frequency in a sine sweep leads to overestimates of system damping. A criterion is offered for the identification of the maximum sweep rate for which an observed frequency response function provides a true indication of system damping, rather than an erroneous value dominated by the sweep rate. Applicability of the criterion is demonstrated through the use of results from actual tests. Excessive sweep rates are shown to inflate estimates of system loss factors above the true values in proportion to the square root of the sweep rate. It is also demonstrated with a specific form for an amplitude-dependent stiffness that the resulting nonlinearity can lead to erroneous observations of bandwidth frequencies, as well as the need for further reductions in the sweep rate. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Estimates of the damping of vibrating systems are often made from the observed response to a constant amplitude sinusoidal excitation with frequency swept through resonant peaks. Although based on the response of a single-degree-offreedom system, the bandwidth method is also applicable to continuous or multiple-degree-of-freedom systems if the resonant peaks are so sufficiently well separated that the observed response is that of a single mode. The resulting amplitude–frequency relationship is a frequency response function or Bode amplitude plot. The measure of damping is taken as being proportional to the bandwidth of the response, i.e. the difference between the two frequencies at which the amplitudes are (most typically) 70.7% of the amplitude at the resonance of interest [1]. The accuracy of measurement of these frequencies (half-power or 3 db) is influenced by several factors, such as fidelity of instrumentation, the influence of an adjacent resonance, test procedures, and system nonlinearities. In particular, results obtained have been found to be strongly influenced by the rate at which the frequency is incremented, with excessively high rates leading to erroneous
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(high) estimates of damping. Systems with very light damping are particularly sensitive to this aspect of test procedure. The challenge of obtaining valid measurements is further exacerbated in the presence of a stiffness nonlinearity that can compromise, or even preclude, the determination of the bandwidth of the response. In the development of results, frequencies are expressed in units of rad/s and the system loss factor, Z (ratio of energy dissipated per radian to peak energy stored during the cycle) is used as the measure of damping. Final results, however, are typically given with frequencies in units of Hz and with the system quality factor, Q¼1/Z, used as the measure of damping as these quantities are generally preferred by practitioners. 2. The bandwidth method The procedure for the estimation of system loss factors from measurements of frequencies at a specified fraction of the maximum amplitude (bandwidth frequencies) results from consideration of the steady-state response, xðtÞ ¼ X expðjotÞ, of a single-degree-of-freedom linear damped system to a harmonic excitation FðtÞ ¼ F0 expðjotÞ at frequency o. With mass m, stiffness k, and structural or hysteretic damping [2,3] introduced through a loss factor Z, the complex-valued response amplitude is X¼
Fo =m
o2n ð1 þ jZÞo2
(1)
pffiffiffiffiffiffiffiffiffiffi where on ¼ k=m is the natural or resonant frequency. If a dimensionless frequency ratio, j ¼ o=on , is introduced, the magnitude, 9X9, of the response at frequency o may be characterized by a dimensionless and real-valued ratio, A ¼ k9X9=F0 . Eq. (1) becomes 9X9 1 ¼ A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F0 =k ð1j2 Þ2 þ Z2
(2)
The maximum value, AMAX, of the dimensionless amplitude is 1/Z, occurring at j ¼1, or at an excitation frequency o ¼ on. This maximum value is the magnitude of displacement at resonance normalized by that at static load. It is defined as the system quality factor, Q, and often used as the measure of damping Q AMAX ¼ 1=Z
(3)
At frequencies other than o ¼ on the dimensionless response has lower values. In particular, there exist two frequencies of excitation, o1 and o2, at which the response is some fraction, r, of the maximum value. Solving Eq. (2) for the frequencies with A¼rAMAX yields, for light damping, that
Z¼
o2 o1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi on 1=r 2 1
(4)
While there is latitude [4] in the choice of r, the most typical value is r ¼1/O2. This choice leads, for linear systems with light damping, to the familiar relationship, Z ¼(o2 o1)/on. The frequency response functions of Eq. (2) corresponding to Z ¼1/Q¼0.01 (solid curve) and Z ¼1/Q ¼0.005 (dashed curve) are shown in Fig. 1, with the half-power bandwidths for A/Q ¼r ¼1/O2 as indicated. 3. Influence of sweep rate on the observed frequency response But the bandwidth method presumes the applicability of Eq. (1), i.e. that the system is in steady-state motion. In consequence, the frequency response function of Eq. (2) is actually the loci of successive states of dynamic equilibrium.
Fig. 1. Estimation of system damping from half-power bandwidths.
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Thus, if such frequency response functions are to be obtained experimentally by driving a test object through resonance with constant excitation amplitude and frequency varying at a constant rate, it is essential that that rate of change be sufficiently low as to enable satisfactory approximations to successive states of steady-state response. An initial attempt [5] to establish a criterion for the necessary sweep rate led to the suggestion that, for a linear system, the sweep rate (Hz/s) should be less than (pfn/3Q)2, where fn is the frequency (Hz) and Q is the system quality factor for the mode of interest. 3.1. A criterion for selecting sweep rates An estimate of the maximum satisfactory sweep rate can be obtained by considering a system in steady-state motion at the resonant frequency, on, with dimensionless amplitude, A(on). If the excitation is removed, the amplitude decays as a transient. For a system with loss factor Z, the envelope of the decay of this transient is AT ðtÞ ¼ Aðon ÞexpðZon t=2Þ
(5)
t ¼ 2=Zon
(6)
The time constant of the system is
The time that would be required for the transient amplitude to decay from a value equal to the steady-state amplitude at resonance, A(on), to the value of the steady-state amplitude at a bandwidth frequency, A(or)¼ r A(on), is tr ¼ t lnð1=rÞ
(7)
But the system is not in dynamic equilibrium at t¼ tr. The time, dt, required for decay of the transient and establishment of steady-state motion at the new excitation frequency, or, and new amplitude, A(or), is better estimated as being some multiple, M, of the time constant of the system. Let dtMIN ¼Mt and require that
dt 4
2M
Zon
(8)
The change in excitation frequency during this time interval is the difference between the frequency at maximum amplitude and the frequency at a fraction r of maximum amplitude. This difference is one-half the bandwidth. From Eq. (4) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi do ¼ 12Zon 1=r 2 1 (9) Thus, a sweep rate in Hz/s, S, formed from the ratio of Eq. (9) and the inequality of Eq. (8) should satisfy the relationship pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi do=2p pZ2 fn2 1=r2 1 (10) o S¼ 2M dt The necessary sweep rate, expressed in Hz/s, should be SoS1, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1=r2 1 fn 2 S1 ¼ 2M Q
(11)
While the choice of the parameter M is quite arbitrary, an exponential decay to 1% of an initial amplitude requires M¼4.6. Then, at dt ¼Mt the decay of a transient of the same magnitude as the steady-state amplitude at resonance is essentially complete. As the magnitude of the transient resulting from a reduction in steady-state amplitude to a fraction r of this value would likely be less, the inequality of Eq. (8) can be expected to be an overestimate of the time required. Using M ¼4.6, the coefficients of (fn/Q)2 in Eq. (11) become 0.591, 0.341, and 0.256, for r ¼0.5, 1/O2, and 0.8, respectively. Thus, an estimate of the required sweep rate for any r r0.8 is that the sweep rate should satisfy S oSM, with 2 fn (12) SM ¼ 2Q When computed with the anticipated frequency and a Q corresponding to the highest anticipated value (lowest level damping), this criterion is suggested as being useful for the selection of an initial sweep rate for a test series. It is more conservative, by a factor of four, than that previously suggested [5]. But as the estimate of the expected Q may be low, good practice dictates that the sweep rate employed should always be validated by a comparison with results obtained at an even slower rate. If the observed values of Q are comparable to those assumed, the sweep rate may still have been excessive. Further, the criterion of Eq. (12) is based on an assumption of the symmetric response function of a linear system. As is shown in a later section of this paper, even small nonlinearities may lead to the requirement for slower sweep rates. On the other hand, if the values of system Q resulting from bandwidth determinations are found to be much lower than the assumed value, the sweep rate can be adjusted upwards accordingly. The sweep rate criteria as given above are based on consideration of the time required for the decay of a transient resulting from an increase in frequency from the value at resonance. For the symmetric frequency response function of a linear system, the same result is obtained for a frequency decreasing from the value at resonance.
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3.2. Comparisons with results from experiments Experiments have shown that the use of sweep rates significantly above values given by Eq. (12) lead to erroneous (high) apparent values of damping (low Q). The apparent values obtained by bandwidth determinations from the frequency response functions are, in such cases, actually driven by the value of sweep rate, S, rather than by the actual system damping. With excessive sweep rates, the apparent value of Q increases as the sweep rate is reduced. Examples of the consequences of excessive sweep rates are seen in Fig. 2. A titanium (Ti–6Al–4V) beam was excited in the first free–free mode ( 204 Hz) in air at constant excitation voltage, with apparent system quality factors estimated from the half-power bandwidths of the response to sine sweeps with frequencies decreasing from above resonance at three rates [6]. Values obtained are shown as functions of the maximum vibratory amplitude at the mid-span of the beam. For the beam used, a mid-span deflection of 1 mm corresponds to a maximum strain of about 570 me (ppm). At a frequency of 204 Hz, the criterion of Eq. (11) with r¼ 1/O2 suggests that capture of a system Q as high as 2000 necessitates a maximum sweep rate 0.0036 Hz/s. The use of the criterion of Eq. (12) suggests a slightly lower (more conservative) value. Thus, even at the slowest sweep rate (0.0054 Hz/s), the value, Q 1950, obtained at the lowest amplitude of response is likely to have been suppressed by the sweep rate. This sweep rate appears satisfactory for the value at the highest observed response (Q 1400), but may have be marginal for the value obtained the value obtained (Q 1700) at the intermediate response amplitude. The much lower values of Q obtained with the two higher sweep rates indicate the significant influence of excessive sweep rates. A second example of the influence of excessive sweep rates on apparent system damping is seen in the results obtained for several higher modes of a structural component formed from an unspecified titanium alloy. All tests were conducted in air and with the same constant excitation voltage. Response amplitudes were low and, as the responses of all modes were measured at the same location, amplitudes varied significantly between modes. Values of system Q as shown in Fig. 3 were obtained from bandwidth measurements at an amplitude ratio of r¼1/O2 with sine sweeps at rates ranging from 0.023 Hz/s up to 3.33 Hz/s. Observed values are also given in Table 1, together with the value of sweep rate necessary to assure validity of the result for that particular mode, as predicted with Eq. (11) using the Q observed in that mode at the slowest sweep rate. Values obtained at sweep rates greater than S1 for that mode are shown in italic type. Values obtained at sweep rates satisfying the criteria (plain-face type) agree to within 10%, values obtained at excessive sweep rates are notably lower. Application of the criterion of Eq. (11) suggests that any sweep rate below 1.38 Hz/s should have been satisfactory for the highest damping mode (Q 1080, f¼2175 Hz) and that a sweep rate SoS1 ¼0.38 Hz/s should have been adequate for any modes shown. For the three lower damping (high Q) modes (2060, 2103 and 2115 Hz), the experimental results as shown in Fig. 3 suggest that the rate of 0.42 Hz/s, may have been somewhat too high. Use of the criterion of Eq. (12) with estimates of 2060 Hz and Q¼2000 would have suggested an initial testing at a sweep rates So0.26 Hz/s.
Fig. 2. Observed system quality factors for a free–free beam.
Fig. 3. Observed system quality factors for a structural component.
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Table 1 Observed values of system Q at several sweep rates. Frequency (Hz)
2060 2076 2103 2115 2176 2217
Observed Q at S ¼ 3.33 (Hz/s)
S¼ 1.67 (Hz/s)
S¼ 0.83 (Hz/s)
S ¼0.42 (Hz/s)
S ¼0.023 (Hz/s)
S1 (Hz/s)
946 1014 950 1123 1031 989
1171 1172 1205 1381 1142 1217
1453 1308 1395 1610 1096 1361
1664 1382 1517 1705 1074 1400
1950 1401 1647 1828 1082 1331
0.38 0.75 0.56 0.46 1.38 0.95
Possible influences of stiffness nonlinearities on the frequency response functions used to obtain these results were considered. Those used to obtain the results of Fig. 3 were found for all modes to be very nearly symmetric about the frequency of maximum response, as was assumed in the analysis. The test data used to obtain the results in Fig. 2 were taken at much higher strains, and the slight asymmetry of the response functions suggest a small influence for a nonlinear softening at all but the lowest displacements. The data of Fig. 2 for the lowest sweep rate suggests an amplitudedependent damping with energy dissipated proportional to (about) the 2.4 power of strain. As the suppression of Q is greater for lower damping at lower strains, this amplitude dependence is somewhat masked in the results obtained at the higher sweep rates. Whether a consequence material or air damping, this nonlinearity can be expected to suppress the observed value of Q by about 10% [7]. 3.3. Influence of excessive sweep rates on test results When the rate of change of frequency is too rapid, the change in amplitude is regulated by the sweep rate, rather than by system damping. In such cases the apparent frequency response function is erroneous and a bandwidth taken from it cannot provide a credible measure of damping. That this is indeed the case can be verified by evaluating the consequence of a hypothesis that the observed response is driven by the rate of sweep, rather than by system damping. The transient induced by the frequency change would then proceed as in Eq. (5), but with an apparent system loss factor resulting from an excessive sweep rate, ZAPPARENT, replacing the true value, Z ¼ ZTRUE. The resulting frequency response function will then have an apparent bandwidth that is determined by the apparent loss factor, ZAPPARENT. Using Eq. (4), the ‘half-bandwidth’ is ffi Z on qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi do ¼ or or ¼ APPARENT (13) 1=r 2 1 2 An estimate of the time for the response to decay to the bandwidth amplitude may be obtained by taking the initial amplitude of the transient, AT, as being the change in the steady-state amplitude resulting from a frequency change, i.e., AT ffi Aðon ÞAðor Þ ¼ Aðon Þð1rÞ. Let the transient amplitude be divided into n increments, DA ¼ AT =n ¼ eAðon Þ so that e ¼ ð1rÞ=n. The change in amplitude over one time increment is then ð1eÞAT ¼ AT expðZAPPARENT on DtÞ
(14)
After solving for Dt, multiplying by a factor N to allow the reestablishment of dynamic equilibrium in each increment and summing over n increments, the total time is
dt2 ¼ nN
2 lnf1=ð1eÞg
ZAPPARENT on
ffi
2nN e 2Nð1rÞ ¼ ZAPPARENT on ZAPPARENT on
(15)
If the frequency is changed at a constant rate S, expressed in Hz/s, over time dt2, then 2pS ¼ do=dt2 . It follows from evaluating the ratio of Eq. (13) and Eq. (15) that
Z2APPARENT ¼
8 pSNð1rÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2n 1=r2 1
(16)
from which is seen the principal result that, regardless of the choice of N or r, with an excessive sweep rate, S, the apparent loss factor is proportional to the square root of that sweep rate. When written in terms of the apparent system quality factor, QAPPARENT ¼1/ZAPPARENT, the result, after division by the true quality factor, QTRUE, is that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi QAPPARENT fn p 1=r2 1 cfn pffiffiffi pffiffiffi ¼ (17) ¼ QTRUE 2Nð1rÞ QTRUE S QTRUE S or QAPPARENT ffi
rffiffiffiffiffi S0 Q S TRUE
(18)
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where S0 ¼ c2
fn QTRUE
!2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼
p 1=r2 1
fn
2Nð1rÞ
QTRUE
!2 (19)
This quantity, with frequency fn is Hz, is an indication of a threshold sweep rate, S0, in Hz/s, at which observed values of Q are influenced by the sweep rate, giving suppressed values if S 4S0. Conversely, if S 5S0, the transient response is not restricted by an excessive sweep rate. For intermediate values, it may be expected that both the sweep rate and system damping will influence the decay of transients. Some indication of the value of parameter c may be obtained by noting from Eq. (17) that, if an excessive pffiffiffi sweep rate, S, is determining the value of Q, then observed values of Q and frequency should be related by QAPPARENT S=fn ¼ c. Using this, values of c may then be computed from test results obtained with sweep rates believed to be excessive. For all 12 data points of Fig. 2 taken at 0.67 Hz and having displacements less than 0.6 mm, the sweep rate also exceeded the criterion of Eq. (12) by a factor of four or more. For these twelve data points, the average of the computed values of c is found to be 0.90, with standard deviation of 0.14. For all data of Fig. 3 taken at 3.33 Hz/s except for the measurements at 2176 Hz, the sweep rate exceeded the criterion of Eq. (12) by more than a factor of four. For these five data points, the average value of c is found to be 0.87, with a standard deviation of 0.064. As all data of Figs. 2 and 3 were obtained with r ¼1/O2, this apparent value of c ffi 0.9 is strictly applicable only to that case. However, the dependence of the value of c on the value of r can be inferred from the definition of c in Eq. (17). This suggests a variation of less than 10% over the range 0.5 oro0.8. Another estimate of the value of the parameter c can be estimated from Eq. (19) by choosing a factor of N ¼4.6 in Eq. (19) to allow for the decay of 99% of the transient resulting from each incremental change in amplitude. This suggests a somewhat higher value, with c then ranging from 1.07 to 1.14 for 0.5 or o0.8. Thus, for r ¼1/O2, the use of the 99% criterion to determine the value of c leads to S 4 S0 ffi 1:17ðfn =QTRUE Þ2 , whereas the use of test results for the evaluation suggests that a slower sweep rate S 4 S0 ffi 0:81ðfn =QTRUE Þ2 strongly influences the determination of damping. The general consistency of these findings is taken as confirmation that a sweep rate of more than 3–5 times the value from Eq. (12) is excessive, and suppresses the observed value of system quality factor, Q, and further, that that suppression is proportional to the square root of the sweep rate. Such suppressions should be expected for sweep rates above S0 ffiðfn =QTRUE Þ2 . The sweep rate criterion previously suggested [5] is essentially this value. But, as is shown here, such a sweep rate is a better indication of the threshold at which results are influenced by the sweep rate, rather than a value at which they are not. It should also be noted that in addition to introducing error into the determination of damping, an excessive sweep rate also introduces error into the determination of the frequency for maximum amplitude (resonant frequency). When frequencies are swept upwards through resonance from below, the consequence is an apparent resonant frequency that is higher than the true value, and conversely if frequency is swept downwards from above. Cognizance of this possible source of error is especially critical when small errors become highly significant, as in determining the material properties of a ¨ berst beam by comparing resonant frequencies before and after coating [1]. thin coating on an O 4. Influences of nonlinearity in stiffness While the loss factor of a real system may also be amplitude-dependent, we focus here on the consequences of nonlinearities in stiffness and take the loss factor, Z, to be independent of amplitude. The influence of a damping nonlinearity (dissipated energy not proportional to stored energy) on estimates of damping from the bandwidth measurements has been discussed elsewhere [7,8]. If the stiffness of a system varies with response amplitude, the system is nonlinear, and changes in the frequency for maximum response will be observed as amplitudes of excitation (and response) are varied. In such cases, the frequency response function will not be symmetric about a resonant frequency. There are two consequences. If the nonlinearity is of sufficient strength, a bifurcation may occur so that a portion of the response will not be observable [9]. For weaker nonlinearities, accurate capture of the steeper slope of one side of the response function will necessitate the use of reduced sweep rates. 4.1. A model of a stiffness nonlinearity It has been suggested [10] that systems having modest changes in stiffness with amplitude might be modeled by a complex-valued stiffness of the form kn ¼ kð1 þ eFf9X9g þ jZÞ
(20)
Here e is a small parameter and the monotonic function F{9X9} has null value when the magnitude of the response, 9X9, is zero. The inclusion of the term eF{9X9} in the real part of the complex stiffness modifies Eq. (2) for the magnitude of the dimensionless steady-state response. The response, A ¼k9X9/Fo, to a harmonic excitation of amplitude Fo then becomes 1 A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ eFf9X9go2 =o20 Þ2 þ Z2
(21)
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pffiffiffiffiffiffiffiffiffiffi where o0 ¼ k=m is the resonant frequency at infinitesimal response amplitude. As in the linear system, the maximum value, AMAX, of the dimensionless response is determined by the system loss factor, Z, and may be written as a system quality factor, Q, Eq. (3). The maximum value of the response (resonance) is at a frequency, oR, that varies with maximum amplitude as the excitation level is changed according to
o2R ¼ o20 ð1þ eFf9XMAX 9gÞ ¼ o20 ð1 þ eFf9F0 AMAX =k9gÞ
(22)
For such a system, the function eF{9X9} may be constructed from test data by evaluating the frequencies at the maximum amplitudes observed in response to different levels of excitation. The nonlinear stiffness as used here is the ratio of the maximum values of force and deflection over a cycle of vibration (a secant modulus), rather than the instantaneous relationship that is inherent in a nonlinear differential equation such as n x€ þ2zoo x_ þ o2o ð1 þ a9x9 Þx ¼ ðF0 =mÞsin o t
(23)
The method of Krylov and Bogoliubov for non-autonomous systems [11] may be applied to Eq. (23). Application of this method of averaging has been found [10] to give to give amplitudes for the first approximation of Eq. (23) equal to those of n Eq. (21) with Z ¼2z and e Ff9X9g ¼ k9X9 for a ¼ 4 k=3 when n¼2, and for 3 pk=8 ¼ a when n ¼1. This agreement is taken as justification for the use of a complex-modulus with amplitude-dependent stiffness of the form of Eq. (20) as a means of obtaining frequency response functions, Eq. (21), for systems with modest monotonic nonlinearities in stiffness. 4.2. Observability of bandwidth frequencies At a fraction, r, of maximum amplitude, 9X9 ¼ rAMAX F0 =k and the response, Eq. (21) is A ¼ rAMAX ¼
r
Z
1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ eFfrAMAX F0 =kgo2r =o20 Þ2 þ Z2
(24)
After solving for o2r =o20 , it follows that the two bandwidth frequencies, o1 and o2, must satisfy
o22 þ o21 ¼ 2o20 ð1þ eFfrAMAX F0 =kgÞ
(25)
and that the loss factor, Z, is
Z¼
o22 o21 o2 o21 ð1þ eFfrAMAX F0 =kgÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 o2 þ o21 1=r 2 1 2o0 1=r 1
(26)
Eq. (26) is obtained without the assumption of small values of the amplitude-independent damping, as long as the complex stiffness is adequately described by the assumed form, Eq. (20). It may also be applied to the linear case (F ¼0). But for a nonlinearity of sufficient strength, the response function, Eq. (21) is not a single-valued function of frequency. In consequence, the mathematical solution is not stable at all frequencies and one of the bandwidth frequencies is not physically observable due to the bifurcation, the condition for which may be found by differentiating Eq. (21) and examining the sign of dA/do. The instability is found to occur for over a range of amplitudes, g ¼A/Q, such that d Z2 eF 9F0 A=k9 4 1 pffiffiffiffiffiffiffiffiffiffiffi ffi (27) dA 2 ðgÞ 1g2 4.3. An example The potential influences of a stiffness nonlinearity are clarified through consideration of a specific simple model, in which the stiffness change is taken as linear in amplitude, i.e. eF{9X9} ¼ k9X9, where k is determined from the change in stiffness at some reference force. Equivalently kn ¼ kð1 þ pA=Ap þjZÞ
(28)
where Ap is the dimensionless response at which the stiffness differs by a fraction, p, from the value at infinitesimal displacement. It is particularly convenient to take the reference values at the maximum dimensionless amplitude, Q, for a specific force, Fo ¼Fp, so that Ap ¼1/Z. It follows from Eq. (22) that the maximum value of the response amplitude, Q¼1/Z, is at pffiffiffiffiffiffiffiffiffiffiffi oR ¼ o0 1 þp (29) which may be above or below o0, depending on whether p is positive (strain-stiffening system) or negative (strain-softening). Using Eqs. (29) and (22), (25), and (26) become
Z¼
o22 þ o21 ¼ 2o2R
(30)
o22 o21 ð1 þ pÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o22 þ o21 1=r2 1
(31)
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For Z 51, o1 þ o2 ffi 2o0O(1þrp) and Eq. (31) becomes Z ¼ ð1 þrpÞ=ð1 þ pÞðo2 o1 Þ=o0 . Thus, if the stiffness nonlinearity, p, is only a few percent and the damping is light, use of the standard bandwidth formula introduces negligible error. However, the use of Eq. (31) or a small damping approximation presumes that the frequencies o1 and o2 can be determined from the frequency response function. That this may not be the case is readily demonstrated by example. Frequency response functions given by Eq. (21) are shown as Fig. 4 for Z ¼0.01 and three values (p ¼ 0.02, 0.03, and 0.04) of the stiffness parameter for a softening system. With p ¼ 0.04, the lower branch has negative slope for amplitudes between 55 and 95. With p¼ 0.03, the negative slopes occur between amplitudes of 67 and 91. Such points are not physically observable due to the bifurcation instability [9]. Awareness of the possibility of the ‘jump’ phenomenon resulting from a stiffness nonlinearity is essential if credible results are to be obtained with the bandwidth method. Examples may be seen in the computed values of response, Eq. (21), shown in Fig. 5 for a system with nonlinear softening, p ¼ 0.4, and Z ¼0.01 (Q¼ 100). If frequency is increased from below resonance, the observed values will approximate the vertical upward path so that the true bandwidth cannot be observed above AZ55. While the true bandwidth is observable below A¼ 55, the actual maximum cannot be observed and the value of the amplitude ratio, r, necessary for the estimation of damping cannot be determined. If the frequency is swept downward from above, the actual maximum value and the resonant frequency will be observed, but from A¼95 to about A ¼34 the observed values will approximate a vertical downward trajectory. Any bandwidth determination between these values will lead to an invalid (high) estimate of the loss factor. In principle, if frequency is swept downwards, a bandwidth determination could be made above the critical amplitude. However, a credible determination using a value of r approaching unity requires very high accuracy and precision of measurement. A rudimentary error analysis of Eq. (31) shows that the maximum fractional error in loss factor due to frequency discretization is only the inverse of the number of observations in the bandwidth. But while a 2% uncertainty in the measurement and discretization of amplitude gives only a maximum uncertainty of 75.4% in the loss factor when r ¼0.5, at r ¼0.9 it can become as much as 724%. Moreover, the use of such high values of r as would be required introduces significantly greater error if the loss factor is also amplitude dependent [7]. Further, with values of r approaching unity, the necessary sweep rate, Eq. (12), becomes very low. For the special case of the linear variation in stiffness, eF{X} ¼pZA, the instability of Eq. (27) results for 9p9Z2.6Z, with positive p for a nonlinear hardening system and negative p for the case of softening. The first occurrence is at amplitude g ¼ A/Q¼O(2/3). For values of the nonlinearity just above the value leading to the bifurcation, such as the case in Fig. 4 for p ¼ 0.03, a bandwidth taken in the range 67oAo91 will be a small overestimate. As is seen from Eq. (27), the allowable nonlinearity for valid observations diminishes with decreasing damping. For example, if Q¼2000, the bifurcation occurs at with a 0.13% change in stiffness. As the resonant frequency is proportional to the square root of stiffness, this corresponds
Fig. 4. Dimensionless response of a nonlinear softening system for three values of the nonlinearity parameter.
Fig. 5. Observability of response with nonlinearity parameter¼ 0.04.
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to a shift in resonant frequency of only 0.065% from the value at infinitesimal amplitude. Thus, the likelihood of invalid measurements due to the nonlinear ‘jump’ phenomenon is very high in lightly damped systems. In such cases, it may be necessary to determine loss factors by other means. Some have been suggested [10]. 4.4. Influence of stiffness nonlinearities on allowable sweep rates The challenge of selecting a sufficiently slow sweep rate is further complicated in the presence of a stiffness nonlinearity. In such cases, even when a bifurcation does not preclude determination of both r-amplitude frequencies, the slope of the response function, dA/do, becomes steeper on one side of the response function (see Fig. 4, p¼0.02), necessitating a further reduction in the sweep rate to accurately capture that portion of the response. Some indication of the potential influence of an asymmetric response function on the necessary sweep rate may be obtained by considering the example used above, i.e. a stiffness reduction proportional to amplitude. For the symmetric response function of a linear system, the necessary frequency changes in Eq. (9) and Eq. (13) were taken as one-half of the bandwidth, do. For the system with stiffness varying linearly with amplitude, the bandwidth frequencies are, from Eq. (24) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2r ¼ o20 ð1 þprÞ 8 o20 Z 1=r2 1 (32) After subtraction from Eq. (22) and some manipulation
oR or ¼ o0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½pð1rÞ 7 Z 1=r 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ pÞ þ or =o0
(33)
This ‘half-bandwidth’ for the nonlinear system may be compared to that for the linear system, Eq. (9), by forming the ratio pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oR or pð1rÞ 7 Z 1=r 2 1 ð2 8 Z 1=r 2 1=2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ o0 (34) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi do ð1 þ pÞ þ or =o0 7 Z 1=r 2 1 For a lightly damped system, the last factor is approximately 1 p/4 and, if 9p9 is only a few percent, may be taken as unity. The ratio of ‘half-bandwidths’ then becomes pffiffiffiffiffiffiffiffiffi 9p9r 1r ðoR or Þ pffiffiffiffiffiffiffiffiffiffi ffi1 (35) do Z 1þr As the bifurcation occurs if 9p9Z2.598Z, the value of this ratio is then in the range of 0.22 to 0.31 for 0.5 rr r0.8. In consequence, the frequency range over which the amplitude must change may be reduced, for the nonlinear system, by as much as this fraction. Thus, sweep rates may have to be lowered by as much as a factor of four as from values determined with Eqs. (11) or (12) in order to assure capture of the steeper slopes in the response functions of systems with nonlinear stiffness. 5. Summary and conclusions The methodology for the estimation of system damping (system loss factor or system quality factor) from the bandwidth of the amplitude response to a sine sweep at constant excitation level has been considered. It is demonstrated by an analysis of a linear system with amplitude-independent damping and through the use of data from tests that an excessive the rate of change of frequency (sweep rate) gives rise to estimates of system loss factors that are higher than the true values, and that these overestimates of loss factors (underestimates of quality factors) are proportional to the square root of the sweep rate. The critical system parameters have been identified. The thresholds for acceptable, and for excessive, sweep rates are found to be proportional to the square of the resonant frequency and inversely proportional to the square of the system quality factor. A criterion for the selection of a sweep rate suitable for initial testing is offered. However, good practice dictates that the sweep rate employed should always be validated by a comparison with results obtained at a slower rate. A criterion for the identification of excessive rates is also provided. While the specific coefficients used in both of these criteria are somewhat arbitrary, the values chosen appear to lead to satisfactory agreement with test results. The consequences of the asymmetry of the response function resulting from an amplitude-dependent stiffness are also addressed. By assuming a specific functional form for a monotonic change is stiffness with response amplitude, a criterion for the appearance of the bifurcation that precludes observation of one of the bandwidth amplitudes is developed. As the degree of nonlinearity at which the bifurcation occurs is found to be proportional to system loss factor, it is shown that very small nonlinearities in stiffness may preclude application of the bandwidth method to lightly damped systems. It is also demonstrated that, with the asymmetric response function, the more rapid variation in response amplitude with change in excitation frequency may necessitate the reduction of sweep rates by as much as a factor of four from values that would be appropriate for a linear system with the same resonant frequency and level of damping.
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Acknowledgments All test data shown here were obtained in the Turbine Engine Fatigue Facility, Wright Patterson AFB, Ohio. References [1] ASTM Subcommittee e33.03, Standard Test Method for Measuring Vibration Damping Properties of Materials, ASTM E 756-05, ASTM International, West Conshohocken, PA, 2005. [2] R.E. Blake, Basic vibration theory, in: A.G. Piersol, T.L. Paez (Eds.), Harris’ Shock and Vibration Handbook, 6th ed., Mc-Graw Hill, New York, 2010. [3] N. Myklestad, The concept of complex damping, Journal of Applied Mechanics 19 (1952) 284–286. [4] A. Nashif, D. Jones, J. Henderson, Vibration Damping, John Wiley and Sons, New York, 1985. [5] P. Torvik, S. Patsias, G.R. Tomlinson, Characterising the damping behaviour of hard coatings: comparisons from two methodologies, Proceedings of the Seventh National Turbine Engine High Cycle Fatigue (HCF) Conference, Palm Beach Gardens, FL, 2002, pp 558–574. [6] A. DeLeon, A Finite Element Evaluation of an Experiment Related to Coating Damping Properties, MS Thesis, AFIT/GA/ENY/09-M03, Air Force Institute of Technology, 2009, pp. 56–61. [7] P. Torvik, A note on the estimation of non-linear system damping, Journal of Applied Mechanics 70 (May) (2003) 449–450. [8] P. Torvik, Determination of mechanical properties of non-linear coatings from measurements with coated beams, International Journal of Solids and Structures 46 (2009) 1066–1077. [9] C. Nataraj, F. Ehrich, Non-linear vibration, in: A.G. Piersol, T.L. Paez (Eds.), Harris’ Shock and Vibration Handbook, 6th ed., Mc-Graw Hill, New York, 2010. [10] P. Torvik, On evaluating the damping of a non-linear resonant system, Paper AIAA 2002-1306, Proceedings of the 43rd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, Denver, Colorado, 2002. [11] N. Bogoliubov, Y. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillation, Gordon and Breach, New York, 1961.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
On estimating system damping from frequency response bandwidths Peter J. Torvik n Air Force Institute of Technology, 1866 Winchester Road, Xenia, OH 45385, USA
a r t i c l e in f o
abstract
Article history: Received 9 December 2010 Received in revised form 17 May 2011 Accepted 30 June 2011 Handling Editor: I. Trendafilova
As damping determines the maximum vibratory response of a system at resonance, reliable estimates of damping are critical both to the design and qualification of systems to be subjected to a vibratory environment and for the evaluation of the effectiveness of modifications or additions provided to increase damping. The sharpness of the frequency response at resonance is often used for this purpose, quantified by the width of the frequency range (bandwidth) for which the response is above some fraction of the maximum response. Several aspects of the use of bandwidth methods in interpreting test results are considered. It is shown that the use of an excessive rate of change of test frequency in a sine sweep leads to overestimates of system damping. A criterion is offered for the identification of the maximum sweep rate for which an observed frequency response function provides a true indication of system damping, rather than an erroneous value dominated by the sweep rate. Applicability of the criterion is demonstrated through the use of results from actual tests. Excessive sweep rates are shown to inflate estimates of system loss factors above the true values in proportion to the square root of the sweep rate. It is also demonstrated with a specific form for an amplitude-dependent stiffness that the resulting nonlinearity can lead to erroneous observations of bandwidth frequencies, as well as the need for further reductions in the sweep rate. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Estimates of the damping of vibrating systems are often made from the observed response to a constant amplitude sinusoidal excitation with frequency swept through resonant peaks. Although based on the response of a single-degree-offreedom system, the bandwidth method is also applicable to continuous or multiple-degree-of-freedom systems if the resonant peaks are so sufficiently well separated that the observed response is that of a single mode. The resulting amplitude–frequency relationship is a frequency response function or Bode amplitude plot. The measure of damping is taken as being proportional to the bandwidth of the response, i.e. the difference between the two frequencies at which the amplitudes are (most typically) 70.7% of the amplitude at the resonance of interest [1]. The accuracy of measurement of these frequencies (half-power or 3 db) is influenced by several factors, such as fidelity of instrumentation, the influence of an adjacent resonance, test procedures, and system nonlinearities. In particular, results obtained have been found to be strongly influenced by the rate at which the frequency is incremented, with excessively high rates leading to erroneous
n
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(high) estimates of damping. Systems with very light damping are particularly sensitive to this aspect of test procedure. The challenge of obtaining valid measurements is further exacerbated in the presence of a stiffness nonlinearity that can compromise, or even preclude, the determination of the bandwidth of the response. In the development of results, frequencies are expressed in units of rad/s and the system loss factor, Z (ratio of energy dissipated per radian to peak energy stored during the cycle) is used as the measure of damping. Final results, however, are typically given with frequencies in units of Hz and with the system quality factor, Q¼1/Z, used as the measure of damping as these quantities are generally preferred by practitioners. 2. The bandwidth method The procedure for the estimation of system loss factors from measurements of frequencies at a specified fraction of the maximum amplitude (bandwidth frequencies) results from consideration of the steady-state response, xðtÞ ¼ X expðjotÞ, of a single-degree-of-freedom linear damped system to a harmonic excitation FðtÞ ¼ F0 expðjotÞ at frequency o. With mass m, stiffness k, and structural or hysteretic damping [2,3] introduced through a loss factor Z, the complex-valued response amplitude is X¼
Fo =m
o2n ð1 þ jZÞo2
(1)
pffiffiffiffiffiffiffiffiffiffi where on ¼ k=m is the natural or resonant frequency. If a dimensionless frequency ratio, j ¼ o=on , is introduced, the magnitude, 9X9, of the response at frequency o may be characterized by a dimensionless and real-valued ratio, A ¼ k9X9=F0 . Eq. (1) becomes 9X9 1 ¼ A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F0 =k ð1j2 Þ2 þ Z2
(2)
The maximum value, AMAX, of the dimensionless amplitude is 1/Z, occurring at j ¼1, or at an excitation frequency o ¼ on. This maximum value is the magnitude of displacement at resonance normalized by that at static load. It is defined as the system quality factor, Q, and often used as the measure of damping Q AMAX ¼ 1=Z
(3)
At frequencies other than o ¼ on the dimensionless response has lower values. In particular, there exist two frequencies of excitation, o1 and o2, at which the response is some fraction, r, of the maximum value. Solving Eq. (2) for the frequencies with A¼rAMAX yields, for light damping, that
Z¼
o2 o1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi on 1=r 2 1
(4)
While there is latitude [4] in the choice of r, the most typical value is r ¼1/O2. This choice leads, for linear systems with light damping, to the familiar relationship, Z ¼(o2 o1)/on. The frequency response functions of Eq. (2) corresponding to Z ¼1/Q¼0.01 (solid curve) and Z ¼1/Q ¼0.005 (dashed curve) are shown in Fig. 1, with the half-power bandwidths for A/Q ¼r ¼1/O2 as indicated. 3. Influence of sweep rate on the observed frequency response But the bandwidth method presumes the applicability of Eq. (1), i.e. that the system is in steady-state motion. In consequence, the frequency response function of Eq. (2) is actually the loci of successive states of dynamic equilibrium.
Fig. 1. Estimation of system damping from half-power bandwidths.
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Thus, if such frequency response functions are to be obtained experimentally by driving a test object through resonance with constant excitation amplitude and frequency varying at a constant rate, it is essential that that rate of change be sufficiently low as to enable satisfactory approximations to successive states of steady-state response. An initial attempt [5] to establish a criterion for the necessary sweep rate led to the suggestion that, for a linear system, the sweep rate (Hz/s) should be less than (pfn/3Q)2, where fn is the frequency (Hz) and Q is the system quality factor for the mode of interest. 3.1. A criterion for selecting sweep rates An estimate of the maximum satisfactory sweep rate can be obtained by considering a system in steady-state motion at the resonant frequency, on, with dimensionless amplitude, A(on). If the excitation is removed, the amplitude decays as a transient. For a system with loss factor Z, the envelope of the decay of this transient is AT ðtÞ ¼ Aðon ÞexpðZon t=2Þ
(5)
t ¼ 2=Zon
(6)
The time constant of the system is
The time that would be required for the transient amplitude to decay from a value equal to the steady-state amplitude at resonance, A(on), to the value of the steady-state amplitude at a bandwidth frequency, A(or)¼ r A(on), is tr ¼ t lnð1=rÞ
(7)
But the system is not in dynamic equilibrium at t¼ tr. The time, dt, required for decay of the transient and establishment of steady-state motion at the new excitation frequency, or, and new amplitude, A(or), is better estimated as being some multiple, M, of the time constant of the system. Let dtMIN ¼Mt and require that
dt 4
2M
Zon
(8)
The change in excitation frequency during this time interval is the difference between the frequency at maximum amplitude and the frequency at a fraction r of maximum amplitude. This difference is one-half the bandwidth. From Eq. (4) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi do ¼ 12Zon 1=r 2 1 (9) Thus, a sweep rate in Hz/s, S, formed from the ratio of Eq. (9) and the inequality of Eq. (8) should satisfy the relationship pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi do=2p pZ2 fn2 1=r2 1 (10) o S¼ 2M dt The necessary sweep rate, expressed in Hz/s, should be SoS1, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1=r2 1 fn 2 S1 ¼ 2M Q
(11)
While the choice of the parameter M is quite arbitrary, an exponential decay to 1% of an initial amplitude requires M¼4.6. Then, at dt ¼Mt the decay of a transient of the same magnitude as the steady-state amplitude at resonance is essentially complete. As the magnitude of the transient resulting from a reduction in steady-state amplitude to a fraction r of this value would likely be less, the inequality of Eq. (8) can be expected to be an overestimate of the time required. Using M ¼4.6, the coefficients of (fn/Q)2 in Eq. (11) become 0.591, 0.341, and 0.256, for r ¼0.5, 1/O2, and 0.8, respectively. Thus, an estimate of the required sweep rate for any r r0.8 is that the sweep rate should satisfy S oSM, with 2 fn (12) SM ¼ 2Q When computed with the anticipated frequency and a Q corresponding to the highest anticipated value (lowest level damping), this criterion is suggested as being useful for the selection of an initial sweep rate for a test series. It is more conservative, by a factor of four, than that previously suggested [5]. But as the estimate of the expected Q may be low, good practice dictates that the sweep rate employed should always be validated by a comparison with results obtained at an even slower rate. If the observed values of Q are comparable to those assumed, the sweep rate may still have been excessive. Further, the criterion of Eq. (12) is based on an assumption of the symmetric response function of a linear system. As is shown in a later section of this paper, even small nonlinearities may lead to the requirement for slower sweep rates. On the other hand, if the values of system Q resulting from bandwidth determinations are found to be much lower than the assumed value, the sweep rate can be adjusted upwards accordingly. The sweep rate criteria as given above are based on consideration of the time required for the decay of a transient resulting from an increase in frequency from the value at resonance. For the symmetric frequency response function of a linear system, the same result is obtained for a frequency decreasing from the value at resonance.
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3.2. Comparisons with results from experiments Experiments have shown that the use of sweep rates significantly above values given by Eq. (12) lead to erroneous (high) apparent values of damping (low Q). The apparent values obtained by bandwidth determinations from the frequency response functions are, in such cases, actually driven by the value of sweep rate, S, rather than by the actual system damping. With excessive sweep rates, the apparent value of Q increases as the sweep rate is reduced. Examples of the consequences of excessive sweep rates are seen in Fig. 2. A titanium (Ti–6Al–4V) beam was excited in the first free–free mode ( 204 Hz) in air at constant excitation voltage, with apparent system quality factors estimated from the half-power bandwidths of the response to sine sweeps with frequencies decreasing from above resonance at three rates [6]. Values obtained are shown as functions of the maximum vibratory amplitude at the mid-span of the beam. For the beam used, a mid-span deflection of 1 mm corresponds to a maximum strain of about 570 me (ppm). At a frequency of 204 Hz, the criterion of Eq. (11) with r¼ 1/O2 suggests that capture of a system Q as high as 2000 necessitates a maximum sweep rate 0.0036 Hz/s. The use of the criterion of Eq. (12) suggests a slightly lower (more conservative) value. Thus, even at the slowest sweep rate (0.0054 Hz/s), the value, Q 1950, obtained at the lowest amplitude of response is likely to have been suppressed by the sweep rate. This sweep rate appears satisfactory for the value at the highest observed response (Q 1400), but may have be marginal for the value obtained the value obtained (Q 1700) at the intermediate response amplitude. The much lower values of Q obtained with the two higher sweep rates indicate the significant influence of excessive sweep rates. A second example of the influence of excessive sweep rates on apparent system damping is seen in the results obtained for several higher modes of a structural component formed from an unspecified titanium alloy. All tests were conducted in air and with the same constant excitation voltage. Response amplitudes were low and, as the responses of all modes were measured at the same location, amplitudes varied significantly between modes. Values of system Q as shown in Fig. 3 were obtained from bandwidth measurements at an amplitude ratio of r¼1/O2 with sine sweeps at rates ranging from 0.023 Hz/s up to 3.33 Hz/s. Observed values are also given in Table 1, together with the value of sweep rate necessary to assure validity of the result for that particular mode, as predicted with Eq. (11) using the Q observed in that mode at the slowest sweep rate. Values obtained at sweep rates greater than S1 for that mode are shown in italic type. Values obtained at sweep rates satisfying the criteria (plain-face type) agree to within 10%, values obtained at excessive sweep rates are notably lower. Application of the criterion of Eq. (11) suggests that any sweep rate below 1.38 Hz/s should have been satisfactory for the highest damping mode (Q 1080, f¼2175 Hz) and that a sweep rate SoS1 ¼0.38 Hz/s should have been adequate for any modes shown. For the three lower damping (high Q) modes (2060, 2103 and 2115 Hz), the experimental results as shown in Fig. 3 suggest that the rate of 0.42 Hz/s, may have been somewhat too high. Use of the criterion of Eq. (12) with estimates of 2060 Hz and Q¼2000 would have suggested an initial testing at a sweep rates So0.26 Hz/s.
Fig. 2. Observed system quality factors for a free–free beam.
Fig. 3. Observed system quality factors for a structural component.
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Table 1 Observed values of system Q at several sweep rates. Frequency (Hz)
2060 2076 2103 2115 2176 2217
Observed Q at S ¼ 3.33 (Hz/s)
S¼ 1.67 (Hz/s)
S¼ 0.83 (Hz/s)
S ¼0.42 (Hz/s)
S ¼0.023 (Hz/s)
S1 (Hz/s)
946 1014 950 1123 1031 989
1171 1172 1205 1381 1142 1217
1453 1308 1395 1610 1096 1361
1664 1382 1517 1705 1074 1400
1950 1401 1647 1828 1082 1331
0.38 0.75 0.56 0.46 1.38 0.95
Possible influences of stiffness nonlinearities on the frequency response functions used to obtain these results were considered. Those used to obtain the results of Fig. 3 were found for all modes to be very nearly symmetric about the frequency of maximum response, as was assumed in the analysis. The test data used to obtain the results in Fig. 2 were taken at much higher strains, and the slight asymmetry of the response functions suggest a small influence for a nonlinear softening at all but the lowest displacements. The data of Fig. 2 for the lowest sweep rate suggests an amplitudedependent damping with energy dissipated proportional to (about) the 2.4 power of strain. As the suppression of Q is greater for lower damping at lower strains, this amplitude dependence is somewhat masked in the results obtained at the higher sweep rates. Whether a consequence material or air damping, this nonlinearity can be expected to suppress the observed value of Q by about 10% [7]. 3.3. Influence of excessive sweep rates on test results When the rate of change of frequency is too rapid, the change in amplitude is regulated by the sweep rate, rather than by system damping. In such cases the apparent frequency response function is erroneous and a bandwidth taken from it cannot provide a credible measure of damping. That this is indeed the case can be verified by evaluating the consequence of a hypothesis that the observed response is driven by the rate of sweep, rather than by system damping. The transient induced by the frequency change would then proceed as in Eq. (5), but with an apparent system loss factor resulting from an excessive sweep rate, ZAPPARENT, replacing the true value, Z ¼ ZTRUE. The resulting frequency response function will then have an apparent bandwidth that is determined by the apparent loss factor, ZAPPARENT. Using Eq. (4), the ‘half-bandwidth’ is ffi Z on qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi do ¼ or or ¼ APPARENT (13) 1=r 2 1 2 An estimate of the time for the response to decay to the bandwidth amplitude may be obtained by taking the initial amplitude of the transient, AT, as being the change in the steady-state amplitude resulting from a frequency change, i.e., AT ffi Aðon ÞAðor Þ ¼ Aðon Þð1rÞ. Let the transient amplitude be divided into n increments, DA ¼ AT =n ¼ eAðon Þ so that e ¼ ð1rÞ=n. The change in amplitude over one time increment is then ð1eÞAT ¼ AT expðZAPPARENT on DtÞ
(14)
After solving for Dt, multiplying by a factor N to allow the reestablishment of dynamic equilibrium in each increment and summing over n increments, the total time is
dt2 ¼ nN
2 lnf1=ð1eÞg
ZAPPARENT on
ffi
2nN e 2Nð1rÞ ¼ ZAPPARENT on ZAPPARENT on
(15)
If the frequency is changed at a constant rate S, expressed in Hz/s, over time dt2, then 2pS ¼ do=dt2 . It follows from evaluating the ratio of Eq. (13) and Eq. (15) that
Z2APPARENT ¼
8 pSNð1rÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2n 1=r2 1
(16)
from which is seen the principal result that, regardless of the choice of N or r, with an excessive sweep rate, S, the apparent loss factor is proportional to the square root of that sweep rate. When written in terms of the apparent system quality factor, QAPPARENT ¼1/ZAPPARENT, the result, after division by the true quality factor, QTRUE, is that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi QAPPARENT fn p 1=r2 1 cfn pffiffiffi pffiffiffi ¼ (17) ¼ QTRUE 2Nð1rÞ QTRUE S QTRUE S or QAPPARENT ffi
rffiffiffiffiffi S0 Q S TRUE
(18)
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where S0 ¼ c2
fn QTRUE
!2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼
p 1=r2 1
fn
2Nð1rÞ
QTRUE
!2 (19)
This quantity, with frequency fn is Hz, is an indication of a threshold sweep rate, S0, in Hz/s, at which observed values of Q are influenced by the sweep rate, giving suppressed values if S 4S0. Conversely, if S 5S0, the transient response is not restricted by an excessive sweep rate. For intermediate values, it may be expected that both the sweep rate and system damping will influence the decay of transients. Some indication of the value of parameter c may be obtained by noting from Eq. (17) that, if an excessive pffiffiffi sweep rate, S, is determining the value of Q, then observed values of Q and frequency should be related by QAPPARENT S=fn ¼ c. Using this, values of c may then be computed from test results obtained with sweep rates believed to be excessive. For all 12 data points of Fig. 2 taken at 0.67 Hz and having displacements less than 0.6 mm, the sweep rate also exceeded the criterion of Eq. (12) by a factor of four or more. For these twelve data points, the average of the computed values of c is found to be 0.90, with standard deviation of 0.14. For all data of Fig. 3 taken at 3.33 Hz/s except for the measurements at 2176 Hz, the sweep rate exceeded the criterion of Eq. (12) by more than a factor of four. For these five data points, the average value of c is found to be 0.87, with a standard deviation of 0.064. As all data of Figs. 2 and 3 were obtained with r ¼1/O2, this apparent value of c ffi 0.9 is strictly applicable only to that case. However, the dependence of the value of c on the value of r can be inferred from the definition of c in Eq. (17). This suggests a variation of less than 10% over the range 0.5 oro0.8. Another estimate of the value of the parameter c can be estimated from Eq. (19) by choosing a factor of N ¼4.6 in Eq. (19) to allow for the decay of 99% of the transient resulting from each incremental change in amplitude. This suggests a somewhat higher value, with c then ranging from 1.07 to 1.14 for 0.5 or o0.8. Thus, for r ¼1/O2, the use of the 99% criterion to determine the value of c leads to S 4 S0 ffi 1:17ðfn =QTRUE Þ2 , whereas the use of test results for the evaluation suggests that a slower sweep rate S 4 S0 ffi 0:81ðfn =QTRUE Þ2 strongly influences the determination of damping. The general consistency of these findings is taken as confirmation that a sweep rate of more than 3–5 times the value from Eq. (12) is excessive, and suppresses the observed value of system quality factor, Q, and further, that that suppression is proportional to the square root of the sweep rate. Such suppressions should be expected for sweep rates above S0 ffiðfn =QTRUE Þ2 . The sweep rate criterion previously suggested [5] is essentially this value. But, as is shown here, such a sweep rate is a better indication of the threshold at which results are influenced by the sweep rate, rather than a value at which they are not. It should also be noted that in addition to introducing error into the determination of damping, an excessive sweep rate also introduces error into the determination of the frequency for maximum amplitude (resonant frequency). When frequencies are swept upwards through resonance from below, the consequence is an apparent resonant frequency that is higher than the true value, and conversely if frequency is swept downwards from above. Cognizance of this possible source of error is especially critical when small errors become highly significant, as in determining the material properties of a ¨ berst beam by comparing resonant frequencies before and after coating [1]. thin coating on an O 4. Influences of nonlinearity in stiffness While the loss factor of a real system may also be amplitude-dependent, we focus here on the consequences of nonlinearities in stiffness and take the loss factor, Z, to be independent of amplitude. The influence of a damping nonlinearity (dissipated energy not proportional to stored energy) on estimates of damping from the bandwidth measurements has been discussed elsewhere [7,8]. If the stiffness of a system varies with response amplitude, the system is nonlinear, and changes in the frequency for maximum response will be observed as amplitudes of excitation (and response) are varied. In such cases, the frequency response function will not be symmetric about a resonant frequency. There are two consequences. If the nonlinearity is of sufficient strength, a bifurcation may occur so that a portion of the response will not be observable [9]. For weaker nonlinearities, accurate capture of the steeper slope of one side of the response function will necessitate the use of reduced sweep rates. 4.1. A model of a stiffness nonlinearity It has been suggested [10] that systems having modest changes in stiffness with amplitude might be modeled by a complex-valued stiffness of the form kn ¼ kð1 þ eFf9X9g þ jZÞ
(20)
Here e is a small parameter and the monotonic function F{9X9} has null value when the magnitude of the response, 9X9, is zero. The inclusion of the term eF{9X9} in the real part of the complex stiffness modifies Eq. (2) for the magnitude of the dimensionless steady-state response. The response, A ¼k9X9/Fo, to a harmonic excitation of amplitude Fo then becomes 1 A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ eFf9X9go2 =o20 Þ2 þ Z2
(21)
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pffiffiffiffiffiffiffiffiffiffi where o0 ¼ k=m is the resonant frequency at infinitesimal response amplitude. As in the linear system, the maximum value, AMAX, of the dimensionless response is determined by the system loss factor, Z, and may be written as a system quality factor, Q, Eq. (3). The maximum value of the response (resonance) is at a frequency, oR, that varies with maximum amplitude as the excitation level is changed according to
o2R ¼ o20 ð1þ eFf9XMAX 9gÞ ¼ o20 ð1 þ eFf9F0 AMAX =k9gÞ
(22)
For such a system, the function eF{9X9} may be constructed from test data by evaluating the frequencies at the maximum amplitudes observed in response to different levels of excitation. The nonlinear stiffness as used here is the ratio of the maximum values of force and deflection over a cycle of vibration (a secant modulus), rather than the instantaneous relationship that is inherent in a nonlinear differential equation such as n x€ þ2zoo x_ þ o2o ð1 þ a9x9 Þx ¼ ðF0 =mÞsin o t
(23)
The method of Krylov and Bogoliubov for non-autonomous systems [11] may be applied to Eq. (23). Application of this method of averaging has been found [10] to give to give amplitudes for the first approximation of Eq. (23) equal to those of n Eq. (21) with Z ¼2z and e Ff9X9g ¼ k9X9 for a ¼ 4 k=3 when n¼2, and for 3 pk=8 ¼ a when n ¼1. This agreement is taken as justification for the use of a complex-modulus with amplitude-dependent stiffness of the form of Eq. (20) as a means of obtaining frequency response functions, Eq. (21), for systems with modest monotonic nonlinearities in stiffness. 4.2. Observability of bandwidth frequencies At a fraction, r, of maximum amplitude, 9X9 ¼ rAMAX F0 =k and the response, Eq. (21) is A ¼ rAMAX ¼
r
Z
1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ eFfrAMAX F0 =kgo2r =o20 Þ2 þ Z2
(24)
After solving for o2r =o20 , it follows that the two bandwidth frequencies, o1 and o2, must satisfy
o22 þ o21 ¼ 2o20 ð1þ eFfrAMAX F0 =kgÞ
(25)
and that the loss factor, Z, is
Z¼
o22 o21 o2 o21 ð1þ eFfrAMAX F0 =kgÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 o2 þ o21 1=r 2 1 2o0 1=r 1
(26)
Eq. (26) is obtained without the assumption of small values of the amplitude-independent damping, as long as the complex stiffness is adequately described by the assumed form, Eq. (20). It may also be applied to the linear case (F ¼0). But for a nonlinearity of sufficient strength, the response function, Eq. (21) is not a single-valued function of frequency. In consequence, the mathematical solution is not stable at all frequencies and one of the bandwidth frequencies is not physically observable due to the bifurcation, the condition for which may be found by differentiating Eq. (21) and examining the sign of dA/do. The instability is found to occur for over a range of amplitudes, g ¼A/Q, such that d Z2 eF 9F0 A=k9 4 1 pffiffiffiffiffiffiffiffiffiffiffi ffi (27) dA 2 ðgÞ 1g2 4.3. An example The potential influences of a stiffness nonlinearity are clarified through consideration of a specific simple model, in which the stiffness change is taken as linear in amplitude, i.e. eF{9X9} ¼ k9X9, where k is determined from the change in stiffness at some reference force. Equivalently kn ¼ kð1 þ pA=Ap þjZÞ
(28)
where Ap is the dimensionless response at which the stiffness differs by a fraction, p, from the value at infinitesimal displacement. It is particularly convenient to take the reference values at the maximum dimensionless amplitude, Q, for a specific force, Fo ¼Fp, so that Ap ¼1/Z. It follows from Eq. (22) that the maximum value of the response amplitude, Q¼1/Z, is at pffiffiffiffiffiffiffiffiffiffiffi oR ¼ o0 1 þp (29) which may be above or below o0, depending on whether p is positive (strain-stiffening system) or negative (strain-softening). Using Eqs. (29) and (22), (25), and (26) become
Z¼
o22 þ o21 ¼ 2o2R
(30)
o22 o21 ð1 þ pÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o22 þ o21 1=r2 1
(31)
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For Z 51, o1 þ o2 ffi 2o0O(1þrp) and Eq. (31) becomes Z ¼ ð1 þrpÞ=ð1 þ pÞðo2 o1 Þ=o0 . Thus, if the stiffness nonlinearity, p, is only a few percent and the damping is light, use of the standard bandwidth formula introduces negligible error. However, the use of Eq. (31) or a small damping approximation presumes that the frequencies o1 and o2 can be determined from the frequency response function. That this may not be the case is readily demonstrated by example. Frequency response functions given by Eq. (21) are shown as Fig. 4 for Z ¼0.01 and three values (p ¼ 0.02, 0.03, and 0.04) of the stiffness parameter for a softening system. With p ¼ 0.04, the lower branch has negative slope for amplitudes between 55 and 95. With p¼ 0.03, the negative slopes occur between amplitudes of 67 and 91. Such points are not physically observable due to the bifurcation instability [9]. Awareness of the possibility of the ‘jump’ phenomenon resulting from a stiffness nonlinearity is essential if credible results are to be obtained with the bandwidth method. Examples may be seen in the computed values of response, Eq. (21), shown in Fig. 5 for a system with nonlinear softening, p ¼ 0.4, and Z ¼0.01 (Q¼ 100). If frequency is increased from below resonance, the observed values will approximate the vertical upward path so that the true bandwidth cannot be observed above AZ55. While the true bandwidth is observable below A¼ 55, the actual maximum cannot be observed and the value of the amplitude ratio, r, necessary for the estimation of damping cannot be determined. If the frequency is swept downward from above, the actual maximum value and the resonant frequency will be observed, but from A¼95 to about A ¼34 the observed values will approximate a vertical downward trajectory. Any bandwidth determination between these values will lead to an invalid (high) estimate of the loss factor. In principle, if frequency is swept downwards, a bandwidth determination could be made above the critical amplitude. However, a credible determination using a value of r approaching unity requires very high accuracy and precision of measurement. A rudimentary error analysis of Eq. (31) shows that the maximum fractional error in loss factor due to frequency discretization is only the inverse of the number of observations in the bandwidth. But while a 2% uncertainty in the measurement and discretization of amplitude gives only a maximum uncertainty of 75.4% in the loss factor when r ¼0.5, at r ¼0.9 it can become as much as 724%. Moreover, the use of such high values of r as would be required introduces significantly greater error if the loss factor is also amplitude dependent [7]. Further, with values of r approaching unity, the necessary sweep rate, Eq. (12), becomes very low. For the special case of the linear variation in stiffness, eF{X} ¼pZA, the instability of Eq. (27) results for 9p9Z2.6Z, with positive p for a nonlinear hardening system and negative p for the case of softening. The first occurrence is at amplitude g ¼ A/Q¼O(2/3). For values of the nonlinearity just above the value leading to the bifurcation, such as the case in Fig. 4 for p ¼ 0.03, a bandwidth taken in the range 67oAo91 will be a small overestimate. As is seen from Eq. (27), the allowable nonlinearity for valid observations diminishes with decreasing damping. For example, if Q¼2000, the bifurcation occurs at with a 0.13% change in stiffness. As the resonant frequency is proportional to the square root of stiffness, this corresponds
Fig. 4. Dimensionless response of a nonlinear softening system for three values of the nonlinearity parameter.
Fig. 5. Observability of response with nonlinearity parameter¼ 0.04.
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to a shift in resonant frequency of only 0.065% from the value at infinitesimal amplitude. Thus, the likelihood of invalid measurements due to the nonlinear ‘jump’ phenomenon is very high in lightly damped systems. In such cases, it may be necessary to determine loss factors by other means. Some have been suggested [10]. 4.4. Influence of stiffness nonlinearities on allowable sweep rates The challenge of selecting a sufficiently slow sweep rate is further complicated in the presence of a stiffness nonlinearity. In such cases, even when a bifurcation does not preclude determination of both r-amplitude frequencies, the slope of the response function, dA/do, becomes steeper on one side of the response function (see Fig. 4, p¼0.02), necessitating a further reduction in the sweep rate to accurately capture that portion of the response. Some indication of the potential influence of an asymmetric response function on the necessary sweep rate may be obtained by considering the example used above, i.e. a stiffness reduction proportional to amplitude. For the symmetric response function of a linear system, the necessary frequency changes in Eq. (9) and Eq. (13) were taken as one-half of the bandwidth, do. For the system with stiffness varying linearly with amplitude, the bandwidth frequencies are, from Eq. (24) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2r ¼ o20 ð1 þprÞ 8 o20 Z 1=r2 1 (32) After subtraction from Eq. (22) and some manipulation
oR or ¼ o0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½pð1rÞ 7 Z 1=r 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ pÞ þ or =o0
(33)
This ‘half-bandwidth’ for the nonlinear system may be compared to that for the linear system, Eq. (9), by forming the ratio pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oR or pð1rÞ 7 Z 1=r 2 1 ð2 8 Z 1=r 2 1=2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ o0 (34) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi do ð1 þ pÞ þ or =o0 7 Z 1=r 2 1 For a lightly damped system, the last factor is approximately 1 p/4 and, if 9p9 is only a few percent, may be taken as unity. The ratio of ‘half-bandwidths’ then becomes pffiffiffiffiffiffiffiffiffi 9p9r 1r ðoR or Þ pffiffiffiffiffiffiffiffiffiffi ffi1 (35) do Z 1þr As the bifurcation occurs if 9p9Z2.598Z, the value of this ratio is then in the range of 0.22 to 0.31 for 0.5 rr r0.8. In consequence, the frequency range over which the amplitude must change may be reduced, for the nonlinear system, by as much as this fraction. Thus, sweep rates may have to be lowered by as much as a factor of four as from values determined with Eqs. (11) or (12) in order to assure capture of the steeper slopes in the response functions of systems with nonlinear stiffness. 5. Summary and conclusions The methodology for the estimation of system damping (system loss factor or system quality factor) from the bandwidth of the amplitude response to a sine sweep at constant excitation level has been considered. It is demonstrated by an analysis of a linear system with amplitude-independent damping and through the use of data from tests that an excessive the rate of change of frequency (sweep rate) gives rise to estimates of system loss factors that are higher than the true values, and that these overestimates of loss factors (underestimates of quality factors) are proportional to the square root of the sweep rate. The critical system parameters have been identified. The thresholds for acceptable, and for excessive, sweep rates are found to be proportional to the square of the resonant frequency and inversely proportional to the square of the system quality factor. A criterion for the selection of a sweep rate suitable for initial testing is offered. However, good practice dictates that the sweep rate employed should always be validated by a comparison with results obtained at a slower rate. A criterion for the identification of excessive rates is also provided. While the specific coefficients used in both of these criteria are somewhat arbitrary, the values chosen appear to lead to satisfactory agreement with test results. The consequences of the asymmetry of the response function resulting from an amplitude-dependent stiffness are also addressed. By assuming a specific functional form for a monotonic change is stiffness with response amplitude, a criterion for the appearance of the bifurcation that precludes observation of one of the bandwidth amplitudes is developed. As the degree of nonlinearity at which the bifurcation occurs is found to be proportional to system loss factor, it is shown that very small nonlinearities in stiffness may preclude application of the bandwidth method to lightly damped systems. It is also demonstrated that, with the asymmetric response function, the more rapid variation in response amplitude with change in excitation frequency may necessitate the reduction of sweep rates by as much as a factor of four from values that would be appropriate for a linear system with the same resonant frequency and level of damping.
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Acknowledgments All test data shown here were obtained in the Turbine Engine Fatigue Facility, Wright Patterson AFB, Ohio. References [1] ASTM Subcommittee e33.03, Standard Test Method for Measuring Vibration Damping Properties of Materials, ASTM E 756-05, ASTM International, West Conshohocken, PA, 2005. [2] R.E. Blake, Basic vibration theory, in: A.G. Piersol, T.L. Paez (Eds.), Harris’ Shock and Vibration Handbook, 6th ed., Mc-Graw Hill, New York, 2010. [3] N. Myklestad, The concept of complex damping, Journal of Applied Mechanics 19 (1952) 284–286. [4] A. Nashif, D. Jones, J. Henderson, Vibration Damping, John Wiley and Sons, New York, 1985. [5] P. Torvik, S. Patsias, G.R. Tomlinson, Characterising the damping behaviour of hard coatings: comparisons from two methodologies, Proceedings of the Seventh National Turbine Engine High Cycle Fatigue (HCF) Conference, Palm Beach Gardens, FL, 2002, pp 558–574. [6] A. DeLeon, A Finite Element Evaluation of an Experiment Related to Coating Damping Properties, MS Thesis, AFIT/GA/ENY/09-M03, Air Force Institute of Technology, 2009, pp. 56–61. [7] P. Torvik, A note on the estimation of non-linear system damping, Journal of Applied Mechanics 70 (May) (2003) 449–450. [8] P. Torvik, Determination of mechanical properties of non-linear coatings from measurements with coated beams, International Journal of Solids and Structures 46 (2009) 1066–1077. [9] C. Nataraj, F. Ehrich, Non-linear vibration, in: A.G. Piersol, T.L. Paez (Eds.), Harris’ Shock and Vibration Handbook, 6th ed., Mc-Graw Hill, New York, 2010. [10] P. Torvik, On evaluating the damping of a non-linear resonant system, Paper AIAA 2002-1306, Proceedings of the 43rd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, Denver, Colorado, 2002. [11] N. Bogoliubov, Y. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillation, Gordon and Breach, New York, 1961.