A New Method for Optimizing Friction Damping in Randomly Excited Systems

Transcription

1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 4T St., New York, N.Y GT-08 A The Society shall not be responsible for statements or op ions advanced in papers or in dis. cussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal, Papers areavailable from ASME for fifteen months after the meeting. Printed in USA. Copyright 989 by ASME A New Method for Optimizing Friction Damping in Randomly Excited Systems T. M. CAMERON and J. H. GRIFFIN Department of Mechanical Engineering Carnegie Mellon University Pittsburgh, Pennsylvania 523 ABSTRACT A method is developed that can be used to calculate the stationary response of randomly excited nonlinear systems. The method iterates to obtain the fast Fourier transform of the system response, returning to the time domain at each iteration to take advantage of the ease in evaluating nonlinearities there. The updated estimates of the nonlinear terms are transformed back into the frequency domain in order to continue iterating on the frequency spectrum of the staionary response. This approach is used to calculate the response of a one degree of freedom system with friction damping that is subjected to random excitation. The one degree of freedom system provides a single mode approximation of systems (e.g. turbine blades) with friction damping. This study investigates various strategies that can be used to optimize the friction load so as to minimize the response of the system.. INTRODUCTION In this paper we present a new method for analyzing and evaluating friction damper designs. This method provides greater flexibility than many other methods thereby allowing new design strategies to be investigated. We then present several strategies for friction damper design and demonstrate that the optimal friction slip load depends on the selected objective. Finally, we investigate the effect of excitation bandwidth on the optimal slip load and develop guidelines for design when the bandwidth of the excitation is unknown. First, however, we attempt to motivate this work and explain the procedures followed for the subsequent investigations. The general problem of how to design a friction constraint so as to minimize the response of a system to random excitation is of interest in a variety of applications. In gas turbine engines we are interested in designing friction dampers so as to minimize buffet stresses in turbine blades. Buffet stresses are induced by random fluctuations in the combustion processes and tend to be most significant at maximum power. In such an application the input excitation could be measured and then expressed in terms of its frequency content by taking its fast Fourier transform (FFT). Consequently, one goal of the current work is to develop a method that can be used to analyze the response of a nonlinear system when the input excitation is defined in terms of its FFT. Limitations in design analysis methods also place limitations on design objectives that may be pursued. Classical analysis techniques for friction damper design are restricted in the situations to which they apply and their use is frequently unwarranted. In the case of random excitation, for example, equivalent linearization (EL), a popular analysis technique (cf. Roberts, 98), can only be used to analyze Gaussian white noise (GWN) excitation and second order response statistics, such as the root mean square (rms) response. Even time integration techniques, besides their computational cost, are difficult to employ if the design objective is to reduce the system response over a specified frequency range. The approach used here calculates the response of the system in both the frequency and time domains. This method, The Alternating FrequencylTime (AFT) domain method (Cameron, 988; Cameron and Griffin, 989) may be used for Monte Carlo simulations as an alternative to numerical integration for stochastically excited systems. The AFT method is introduced and used to solve for the response of nonlinear systems with multiple frequency excitations in Cameron and Griffin, 989. Readers are referred to that paper for a review of spectral methods as applied to vibrating systems. For conciseness, here we discuss only those papers related to random excitation. In the cases considered here the AFT method offers appreciable computational savings over numerical integration as well as the ability to calculate a wide variety of statistics in either the time or frequency domain. The AFT method complements EL in situations with either sinusoidal or GWN excitation (situations where EL is applicable). The AFT method converges more easily at lower slip loads, where EL is inaccurate, and EL is accurate at higher slip loads, where the AFT method may have difficulty converging. Taking advantage of this flexibility we investigate two factors which influence the optimal slip load of a friction damper. The first factor is the output measure or response statistic used to evaluate the effectiveness of the friction damper. In addition to the rms displacement, the most common response measure for stochastically excited systems, we calculate an average maximum response and a narrow band frequency domain reponse. For this investigation we use a GWN excitation and perform a sequence of simulations in which the friction slip load is varied. All three output measures are calculated for each simulation and the slip load which minimizes each output measure is found. These optimal slip loads differ by almost a factor of two. Reasons for the differences are discussed. Presented at the Gas Turbine and Aeroengine Congress and Exposition June 4-8, 989 Toronto, Ontario, Canada

2 The second factor influencing the optimal slip load which we investigate is the frequency bandwidth of the excitation. Analytical results are often developed for either GWN or for a single frequency sine wave excitation and represent the two extreme cases of bandwidth that can be encountered. In practice, actual random excitations tend to be band limited and we wish to establish under what conditions this fact needs to be taken into consideration. For this study we generate a GWN excitation and progressively filter out more frequencies in successive simulations. For each excitation bandwidth we vary the friction slip load to determine which is optimal for that bandwidth. The final set of simulations consists of a frequency sweep with a sinusoidal excitation for each slip load. In the cases considered here the optimal slip load for a GWN excitation is as much as five times higher than the optimum for a sinusoidal excitation. As a result, it is clear that bandwidth can play an important role in establishing the optimum slip load for a frequency limited excitation. In the next section we describe the model for a turbine blade and friction damper used in the studies presented here, explain how the stochastic excitation process is generated, and illustrate how the AFT method is used to perform Monte Carlo simulations of the blade response. q(t*) t (a) Dimensional System E f(t) t x*(t*) *, x *) x(t ) 2. MODELING AND ANALYSIS Rather than present a general formulation of the AFT method (which is available in Cameron, 988, and Cameron and Griffin, 989) we first present the system model and excitation process used in the subsequent studies and then, for the sake of clarity, we demonstrate how the AFT method is applied specifically to this system and used to calculate the response process. ( x) 2. The Model A single degree-of-freedom (sdf) system is shown in Figure (a). There are several advantages to using this model: it has been demonstrated experimentally to be a viable model for many friction damping applications (cf. Griffin, 980), it allows a comparison of results with other research using this model (e.g., solutions obtained by both the AFT method and Equivalent Linearization are compared in Figure 9), it is simpler to use analytically and computationally, and it permits nonlinear friction effects to be studied more easily by minimizing other dynamic effects. The equation of motion for this system is: m z* + c z* + k x* + y*(z*) = q(t*) () and the friction force is y* (z*) = kd - z* (x*, k * ) (2) The *'s designate dimensional variables, as opposed to the nondimensional variables to be introduced shortly. The system parameters are deterministic: mass, m; linear viscous damping coefficient, c; blade stiffness, k; and damper stiffness kd. The stochastic excitation process is q, the response process is x*, and the dots represent differentiation with respect to time, t*. The friction force, which governs the displacement of the damper, z*, is described by a coefficient of friction, p., and a contact force, Cf. The product µcg is the friction slip load. Optimizing the friction damper consists of finding the slip load which minimizes a selected statistic of the response, x*, for the form of excitation the system is expected to encounter. (A more detailed description of design philosophy and procedure may be found in Cameron, 988, 4..) Introducing the nondimensional variables (b) Nondimensional System Figure : Single Degree-of-Freedom Model of Frictionally Damped Blade x* kd x= t=t*(on (3) pcf and notation + kdc cun = m 2 (k + kd) E kkd f=^ (4) d µcg equation () may be nondimensionalized to produce the system in Figure (b). The equation of motion for the nondimensional system is k + 2 is + ( - e) x + y(z) = e f(t) (5) and the nondimensional friction force is y(z) = e z(x, x) (6) The dots in (5) and (6) represent differentiation with respect to the nondimensional time t, defined in equation (3). Equations (5) and (6), with the definitions in (3) and (4), are the equations used here. 2

3 The displacement of the friction element, z(x, z), depends on the history of the response and is difficult to express in explicit analytical form. For small motion, the friction force is strong enough to resist the spring force so no slip occurs. When the displacement x (x = z until the first slip occurs) reaches a magnitude of unity (), the friction force and spring force are equal, with magnitude e, the nondimensional slip load. For displacement x greater than unity, slip occurs since the spring force would exceed the slip load, which it cannot do. While slip occurs, x varies and z remains at a constant magnitude of unity. However, as soon as x changes direction the friction element will stick, z will follow or lead x by a constant displacement, and no slip will occur until the spring force again tries to exceed the slip load, i.e., when z is displaced by an amount unity from its equilibrium position. This hysteretic force can take on an infinity of values within an envelope described by the extremes of motion, x and x+, as illustrated in Figure 2. (This figure applies to the dimensional system, equation (). For the nondimensional system x - and x+ are modified by equation (4) and the output force ranges from - to +.) The AFT method (presented in section 2.3) only requires a subroutine to describe the nonlinear force with an assumed form of the input -- like the description above. An explicit analytical description of the nonlinearity is not required. Output Force by the AFT method, and the psd of the continuous GWN process used by EL. If the sample deviates for the discrete process have variance 62 then the discrete autocorrelation, R[m], of f[n] is: R[m] = E{ f[n] f[n+m] } = c2 3[m] (7) where E is the expectation operator, S[-] is the discrete dirac delta function, and m is the discrete time difference between the samples of f[.] in the expectation. The power spectral density, S[k], of f[n] is the discrete Fourier transform (DFT) of R[m]: N- S[k] _ I R[m] e- m=0 i2xmk/n = 62 (8) where N is the number of time samples taken of f(t). Since the discrete autocorrelation, R[m], represents samples of the continuous autocorrelation (cf. Papoulis, 984, p. 290), S[k] is the DFT representation of the continuous psd, S(w). In the absence of aliasing and leakage errors, and ignoring periodic continuations implicit in the DFT, the relationship between S(w) and S[k] is: S[k] = S (2. (k-) AT AT N ) k =,..., N (9) x(t) Consequently, for a discrete GWN process with psd 62, the psd of the continuous GWN process is, from equations (8) and (9), S(w) = 62 AT (0) Figure 2: Force Hysteresis of Friction Damper To complete the equation of motion we must still prescribe the excitation process represented by f(t) in equation (5). 2.2 The Excitation Process The kernel excitation process used in the studies presented here is Gaussian white noise (GWN). For the first study -- dependence of the optimal slip load on the response statistic selected according to the design objective -- all the excitations are GWN. In the second study -- the dependence of the optimal slip load on the frequency bandwidth of the excitation -- different bandwidths are produced by generating and then filtering GWN. Therefore, we first examine how GWN is simulated. 2.2(a) Simulating GWN In anticipation of the AFT method, which uses a sampled version of the excitation and response processes, the task is to obtain a digital simulation of Gaussian white noise. Gaussian white noise is simulated digitally by generating a sequence of independent, normally distributed random deviates which are assumed to represent samples of a continuous GWN process at a uniform sampling period, AT. We denote the continuous process "f(t)" and the discrete process "f[n]," where f[n] = f(n AT). The normal deviates may be generated with standard numerical algorithms (we use a modified version of the function "gasdev" in Press, et al., 986). To compare results of the AFT simulations with solutions obtained by EL it is necessary to establish the relationship between the power spectral density (psd) of the discrete GWN process, used Observe that the psd's described by equations (8) and (0) are constant. A process with a constant psd is called "white," and a process with normally distributed sample amplitudes is called "Gaussian." The processes described here are both Gaussian and white. 2.2(b) Filtering GWN For the second study we obtain different bandwidths of random excitation by lowpass and bandpass filtering GWN. In order to implement the filter with a DFT we used the Parks-McClellan finite impulse response, linear phase design program (McClellan et al., 979). Linear phase is not important since the phase process of GWN is random, but the finite impulse response is necessary for a DFT implementation. The Parks- McClellan algorithm can be shown to provide minimum mean square error with respect to the ideal filter for a given number of samples (Parks and McClellan, 972). The important factor in the bandwidth of the excitation is the relative bandwidth with respect to the "natural" frequencies of the system. We use the term natural frequency loosely to refer to the largest frequency component in the system response. The natural frequency of the system in Figure (b) is bounded by the natural frequencies of the linear extremes corresponding to a zero slip load and the stuck slip load, the slip load at which the damper never slipsl. For the nondimensional system the zero slip load natural frequency is - e and the stuck slip load natural frequency is unity. In the studies performed here the value of e is 0., a typical value for gas turbine applications, so the range of natural frequencies is 0.95 to. t These undamped natural frequencies roughly bound the "natural" frequency of the nonlinear system. At the lower extreme, near zero slip load, viscous damping reduces the natural frequency to slightly below the undamped value. This is verified computationally in Cameron, 988,

4 Five bandwiths of excitation are considered in the second study: GWN (frequency is limited to 0Nyquis ]t/at), two lowpass bands, one narrow bandpass band, and sinusoidal excitation (with a frequency sweep). Only the intermediate three cases require filtering. The actual amplitudes of the filter transfer functions, IH(co)I, are shown in Figure 3. The pass and stop bands specified for the Parks-McClellan program are shown in Table. The bandwidth of the bandpass filter was selected to cover the variation in the natural frequency of the system.. IH(w)I O- Narrow BPF Low Cutoff LPF 0.2 -W High Cutoff LPF w Figure 3: Magnitudes of Filter Transfer Functions Table : Pass & Stop Bands for Filtered Excitations (Upper Frequency Limit Determined by onyquist) Name of Filter Pass Band Stop Band(s) Narrow Band Pass Low Cutoff Low Pass High Cutoff Low Pass White Noise (No Filter) 0.9<_w <_w!_0.85,.<_w 0.0 _< w <_ <_m 0.0 <_ tl _< to 0.0_<w (c) Remarks One further comment regarding the excitation is required. In the studies that follow the response of the system is calculated over a range of slip load values. The slip load does not appear "explicitly in equation (5), it is implicit in f as shown by equation (4). Therefore, the slip load is changed by varying f, and the response is displayed as a function of /f since the slip load is in the denominator of f. The procedure we follow for varying f is to decompose it into the form f(t) = 6-f(t) () where f is a fixed, normalized function and c is a variable scale factor. In the case of GWN f is a standard normal process (zero mean, unit variance) so r is the standard deviation of the f process (for GWN this is the same r that appears in equations (7) to (0)). In the case of a sinusoidal excitation f is a unit amplitude sinusoid, so 6 is the amplitude of f. Therefore the slip load is varied by changing 6. 4 Finally, for the problem to be well defined, we also assume that the response is stationary since the excitation is stationary. This is not always the case with nonlinear systems, chaos is a prime example, but it does hold for the cases studied here (the AFT method cannot converge on a chaotic response). The task of solving the problem using the AFT method is discussed in the next section 2.3 ANALYZING THE SYSTEM RESPONSE The Alternating Frequency/Time domain (AFT) method iterates to obtain the frequency domain response of the system using the discrete Fourier transform (DFT). At each iteration the estimate of the solution is transformed into the time domain to evaluate the nonlinearities since they are described in terms of the state history of the system. The nonlinear forces are transformed back into the frequency domain in order to continue iterating on the frequency spectrum of the response. The transformations between time and frequency are implemented with a fast Fourier transform (FFT). To illustrate the AFT method with the present system consider taking the Fourier transform of equation (5): [(-e-o 2) + j 2i;w] X(w) - F(w) + Y((U) = 0 (2) Here w is the transform variable, capital letters represent the frequency domain counterparts to the lower case time domain functions, and j =. In general the linear terms operating on the unknown response x(t) may be analytically transformed into a coefficient multiplying X(w). It may be possible to analytically transform f(t) into F(w), as in the case of a sinusoidal excitation, or it may not be, as in the case of stochastic excitation. If f(t) cannot be transformed analytically it is sampled in the time domain and transformed with an FFT. The problem with transforming the nonlinear function y(t) into Y(w) is addressed shortly. Since a DFT will be used to compute and represent the frequency dependent functions, we require (2) to hold at the discrete frequencies, cok, introduced by the DFT. Consequently, (2) becomes a system of complex nonlinear algebraic equations: where [( -e-wk2) +j 2Cwk X(wk) - F(wk) + Y(wk) = 0 (3) Wk = 2.i.(k - )k =,...,N (4) AT N Here AT is the sampling period and N is the number of samples in the DFT of each time domain function, and N = + N/2 is the number of unique frequency components represented by the DFT. The unknowns in (3) are the complex values of the frequency domain response, X( ). 2 Y(w) is also unknown, but it does not introduce new, independent unknowns since y depends on the time domain states of the system, x and k. Once x, or X(.Q), is known, so is y. However, since y depends on the system states it cannot be directly transformed into Y((I ). (It is this difficulty which the AFT technique resolves.) In general, due to numerical roundoff, the exact solution to (3) will not be found so the right hand side will have residual values, R(wk), which are not identically zero. 2 XO is the vector (X(tut), X(tu2),..., X(aN) ).

5 U The AFT technique begins with an initial estimate of the solution, X O(w), 3 which is inverse transformed into the time domain to provide the state trajectory, x 0(t) (Figure 4). If z0(t) is required, it is given by the inverse transform of - j X (). 4 Knowing the state trajectory of the system allows y to be mapped from state space domain into the time domain: y(x 0, k0) -, y0(t). It is now possible to Fourier transform the nonlinear term from the time to frequency domain: y 0(t) ^ YO(). Since YO is known in terms of its frequency components, equations (3) may be solved for the next estimate of the solution, X (co), using standard methods for solving nonlinear algebraic equations. This procedure forms one complete iteration of the AFT method. The procedure terminates when an appropriate convergence criterion is satisfied. The convergence criterion used in these studies is to require each component of the residual, R(owk), to be less than a specified tolerance, 0-4. As an order of magnitude comparison, the dominant components of X(wk) have values in the range of 000 to Frequency Domain Time Domain To summarize the AFT method: Linear terms in the differential equation (5) are transformed analytically into frequency domain coefficients of the unknowns X(wk). The forcing term is transformed analytically, if possible, or sampled in the time domain and transformed with an FFT. The AFT technique is used to calculate Y'(tak) based on the current estimate of the solution, Xi(wk). Any method for solving systems of nonlinear algebraic equations may then be used to solve (3) for the next estimate of the solution, Xi+l ( wk) The previous two steps are repeated until convergence is achieved. One advantage of the AFT method over numerical integration is that the AFT method directly calculates the steady-state or stationary response of a system and avoids calculating transients, thereby offering computational savings. Since the AFT method has an absolute measure for the accuracy of the solution, the residuals R(cOk), it is not troubled by stiff equations as numerical integration may be. And the AFT method does not face the issue of determining when steady-state has been achieved since it iterates directly on the steady-state solution. IX (w)i X (t) (FFT) Solve Algebraic Equations co (FFT)y (t) w y(x, X) t The weakness of the AFT method is that the DFT can only represent a finite time solution, which the DFT "thinks" is periodic. This poses no problem if the excitation is periodic since the response generally will be periodic as well. However, a random excitation and random response are not periodic. This is not as severe a limitation as it seems; Fourier analyzers and vibration test equipment face it too. Fourier vibration analyzers solve the problem by generating a random excitation signal over a long time. The analyzer chops up the time into shorter pieces and looks at the response of the system in each segment of time. The statistics of the response are calculated in each time segment and averaged together. No single segment represents the entire response of the system, but after the response has been averaged over many segments the statistics do not change appreciably. Assuming the excitation and response are stationary, this average provides a good estimate of the statistics of the response over all time. The AFT method uses the same approach, except that the response is calculated numerically for each time segment rather than experimentally measured. Conceptually, then, for a randomly excited system, the AFT method may be viewed as providing a simulation of vibration experiments with a Fourier analyzer. This approach is called Monte Carlo simulation. (FFT) w Figure 4: Alternating Frequency/Time Domain Technique 3 The superscript denotes the iteration count in the solution procedure. The initial estimate is the zero' iteration, X (y2), the next estimate is count one, X'(Ial), etc. 4 -J 42_X(4) _ -j (wx(w), w2x(tu2),..., o X(wN) ) 3. DESIGN STUDIES Here we present the results of two design studies with a randomly excited, frictionally damped turbine blade. We also compare results of the AFT method and Equivalent Linearization to verify the analysis procedure. In the first study we consider how the optimal slip load with GWN excitation is affected by the statistical output measure -- selected on the basis of the design objective -- used to determine the response of the system. We present three possible measures, explain why a design engineer may be interested in each of them, and show how the optimal slip load differs for each measure. In the second study we consider the effect of excitation bandwidth -- from sinusoidal to GWN excitation -- on the optimal slip load. We conclude from this study that if there is no prior knowledge of the bandwidth of the excitation then it is more conservative to design for a sinusoidal rather than a random excitation. This conclusion applies to the design of blade dampers and should not be generalized to the analysis of disks where multiple modes and superharmonics excited by a random excitation may cause a worse response than a harmonic excitation. 5

6 C The parameters in equation (5) used for both studies are: e = 0. = 0., and 0.0 (value of C is specified in plots of results) and the excitation is given by equation (). Each data point shown in the results consists of an average of between 50 and 03 simulations with the values of e, C, and 6 held constant. For each simulation a different normalized GWN process, f(t), is used. The solution of the Monte Carlo simulation settled down more quickly with the larger value of viscous damping (C = 0.) so fewer simulations were required (= 50). With less damping more simulations (=00) were required for the statistics to converge. A neighboring data point corresponds to another set of simulations with an incremental change in r, but the same values of e and C. The FFT parameters used to discretize the system are: N = 52 tnyquist = 6.4 Other FFT parameters may be deduced from these, e.g., N = 257, AT = , Acu = Effect of Response Measure on Optimal Slip Load In this section we present three blade response statistics which correspond to different design objectives. We define each response statistic and explain the design objectives it measures. Then we present simulation results which compare optimal slip loads for the different response measures. RMS Response The most common response measure used in random vibrations is the root mean square (rms) response defined as: Xrms = E{x (t)} (5) The discrete approximation to the rms response used here is xnns = nn ^x?^] (6) i=t [=t where x i is the response for one simulation, each xi is represented by N (= 52) time samples, and there are n (= 50-03) simulations performed for a fixed set of the parameters e,, 6. Individual time samples are identified with the dummy index j. Some methods of analysis are restricted in the statistics which they can calculate, and the rms response may be the most useful measure they offer. Equivalent linearization is an example. Consequently, it may be useful to calculate the rms response numerically in order to verify analytical calculations made with EL. It is also useful to compare the simulated rms response with an experimental rms response in order to assess the validity of the simulation model. The claim is sometimes made (cf. Sogliero and Srinivasan, 980) that the larger the rms or mean square response, the greater the probability of having a larger peak amplitude of vibration. While this may be so, it may be taken to suggest that optimizing the slip load for the rms response is equivalent to optimizing the slip load for the peak response, which we will show is false. Nevertheless, the rms response is still a fair indicator of fatigue. "Average Maximum" Response The only way to determine the maximum response of a blade is to measure its response over all time. As an alternative to this we observe the maximum response of the blade in each finite time AFT simulation. Since each datum is an average of between 50 and 00 AFT simulations we average the maximum response in each simulation to obtain an "average maximum" response, x max. In equation form: n xmax = n Y, max Ix l[j]i j =,..., N (7) where the indices are the same as in equation (6). In other words, xmax is the expected value of the peak response for the time interval of a simulation, N AT. In principle, xmax converges to the average absolute value of the response as NAT - 0 (holding N constant), and Xmax converges to the true peak response as NAT, oo (holding AT constant). Consequently, in practice, the optimal slip load for xmax is a better indicator of the optimal slip load for the peak response than is the optimal slip load for xrms. Calculating X,, ax also provides a better measure of blade fatigue. The difficulty with xmax is that few methods of analysis are capable of calculating it. Narrow Band Freauencv Averaee Response In some applications a designer may be concerned about the response near a particular resonant frequency. In this case the design goal may be to minimize the blade response over a band of frequencies near the expected resonance. In another application the designer may know that the predominant excitation will lie in a certain frequency range, and wants to select the slip load to minimize the frequency response over that range. For these situations we introduce the narrow band average frequency response X(w)ave: [k" X(w)ave = (k- k l+ l) L IX((Ak)I (8) k=k where the lower frequency limit in the band of frequencies being averaged is wkl = (k -)-Ao), and the upper frequency limit is w ka = (k -) Aw. The natural frequency of the system considered here varies from 0.95 to.0, and At = In order to cover the full variation of resonant frequencies we chose to average the response over 0.90 to.05, so kl = 37 and k = 43. Simulation Results Approximately 3000 simulations were performed to obtain the results in Figure 5. Each curve consists of about 60 points, and each point is an average of about 50 simulations. Only 50 simulations were required to achieve convergence since the higher viscous damping was used. The three curves for the different output measures were obtained from the same simulations. The excitation measure used was F(tr)) ave which is the narrow band average frequency content of the GWN excitation, f(t). F(w) ave is defined similarly to the narrow band frequency response in equation (8) 5. In order to compare the optimal slip loads the curves were normalized so that each has a minimum of unity. 5 This input measure is a relic from the other design study. It is important in making these comparisons that the same input measure be used in each case. Filtering the frequency content of the excitation, which is done in the second study, will change the "max" or "rms" measures of the excitation. Since even the narrowest band filter leaves the frequencies within the averaging band intact, the frequency average measure was used to describe the excitation. 6

7 Gaussian White Noise Excitation Varying Bandwidth Excitations.06 e = 0. i; =0. Oumut 4.6 e=0. = W) x ave rms max 4.5 dtput.03 F(63) ave.02.0 X(w)ave F(w)ave ^.00 Curves Normalized to have a Minimum of Unity F(W) ave Figure 5: Effect of Design Objective on Optimal Slip Load The first and most obvious item to note is that the optimal slip loads vary for different output measures which, in turn, are based on the design objective. The high optimal slip load, corresponding to the xmax measure, is almost twice as high as the low optimal slip load, for the X(w) ave measure. The difference between optimal slip loads for x ntt5 and xmax is not surprising. Griffin (980) showed that, with a harmonic excitation, a damper with optimal slip load will stick for half the cycle of blade motion. Therefore a larger peak motion will require the damper to remain stuck longer, which.8 requires a higher slip load for a given damper stiffness. Increasing the slip.7 load too high, however, will cause the damper to stick longer and will be non-optimal. For example, observe that designing the optimal slip load for x m. raises the rms response of the blade, since for most cycles of motion the damper is stuck longer than is optimal. This is a warning against the tendency to design friction dampers with slip loads far greater than optimum. On the other hand, these results confirm that if you must err it may be better to err on the side of too high a slip load. 3.2 Effect of Excitation Bandwidth on Optimal Slip Load In this section we present results of simulations comparing optimal slip loads for different bandwidths of excitation. Much of the work in friction damper optimization has been done for two extremes of excitation bandwidth, sinusoidal and GWN. In regard to actual applications where the precise nature of the excitation may not be known, two questions naturally arise. First, are the regions of acceptable slip loads for sinusoidal and white excitations sufficiently close that a friction damper may be optimized for both types of excitation simultaneously? The answer to this question is generally no, as is shown below. Second, do these bandwidth extremes define the limits of the optimal slip load? In other words, do the optimal slip loads for finite bandwidth random excitations lie between the optimal slip loads for sinusoidal and white excitations? The answer to this question appears to be yes, which is intuitively satisfying. Figure 6 shows that the low pass filtered (LPF) random excitations (cf. Figure 3) gave responses virtually identical to the GWN response. This says that the system effectively filters out frequencies above a certain level so that the difference in the response due to a finite band low pass excitation and the response due to GWN becomes negligible. Figure 7 compares the optimal slip loads for GWN, narrow band pass filtered (BPF) random, and Gaussian White Noise Low Cutoff LPF + High Cutoff LPF F(o) ave Figure 6: Independence of Optimal Slip Load from High Frequency Excitation Due to System Filtering )ave4.5 F(0) ave Different Bandwidths of Excitation E=0. i;=0. - Gaussian White Noise Narrow Band Pass Random Sine Sweep 4.0 A F(63)ave Figure 7: Effect of Excitation Bandwidth on Optimal Slip Load (E = C = 0.) sinusoidal excitations (with = 0.0). Observe that the optimal slip load for the BPF excitation does lie between the sinusoidal and white noise optimal slip loads, suggesting that the GWN and sinusoidal cases supply upper and lower bounds on the optimal slip load. However, the sinusoidally excited system is already "locked up" at the BPF optimal slip load, which is a factor of three higher than the sinusoidal optimal slip load. The GWN optimal slip load is roughly five times higher than the sinusoidal optimal slip load. A friction damper clearly cannot be optimized for all bandwidths of excitation simultaneously. Figure 8 compares optimal slip loads for sinusoidal and white noise excitations with = 0.0. In this case the optimal slip load for the random excitation is a factor of about two or three larger than the optimal slip load for the sinusoidal excitation, which is less dramatic than the previous case (C = 0.0). Also, the deviation from optimal response due to selecting the slip load on the basis of the wrong form of excitation is smaller in this case. The deviation is still substantial, however. 7

8 0 Sinusoidal and Gaussian White Noise Excitations Gaussian White Noise Excitation 60 e = 0.0 =O c=0.0 2;=0.0 ) w) ave 50 - Sine Sweep Gaussian White Noise 40 F(u)) ave 'y Equivalent Linearization AFT Method F(w) ave Figure 8: Effect of Excitation Bandwidth on Optimal Slip Load (e=0.,c=0.0) 3.3 Confirming Simulation Results Prior to beginning a major simulation effort it is wise to run test problems to confirm that the procedures are correct. A large battery of tests were conducted to certify the accuracy of the AFT method and the validity of the simulation results presented above. See Cameron, 988, for details. We present this material now because it relies on the definitions in section 3.. Of the tests conducted, the most interesting is a comparison between simulation results of the AFT and EL methods. Sinha (987) developed damper optimization curves for the system of Figure (b) with C = 0.0 and several values of e using EL. The same system, with e = 0., was simulated using the AFT method in order to compare results. Before presenting the results it is necessary to define the input/output measures used by Sinha. The input measure, l, uses the inverse of the square root of the psd of the white noise excitation, S f, to represent the slip load: R = f = ( 9) 0T The second equality results from applying equation (0). The output measure, y, is a normalized rms response: Y = E(x2).(3. ( - e) = xrms ( - e) (20) E OT E The results of both AFT and EL methods are shown in Figure 9. It may be shown that in the limiting linear case of zero slip load y takes the form: Yeree = c4.74 _ (for e = 0., = 0.0) (2) I Slip Load(a) Figure 9: Comparison of AFT Method and Equivalent Linearization Solutions The AFT solution agrees well with this analytical result. The optimal slip load obtained by each method is the same, which is the important result for damper design, however the actual response magnitudes differ by approximately 0%. This discrepancy is a reminder that both methods are approximate. The numerical approximations of the AFT method introduced an error of 3.3% in an ideal linear problem (Cameron, 988), for which EL is exact. But the analytical approximation of a nonlinearity by EL is known to introduce errors in excess of 0% in cases where exact solutions of nonlinear problems are known (Atalik and Utku, 976). An interesting and useful point regarding the AFT and EL methods is that the regions in which they provide better results are complementary. Due to the many higher frequencies, the AFT method has difficulty converging at higher slip loads, which is where EL is more reliable. The AFT method converges more easily at lower slip loads, which is where EL is unreliable (it over estimates the damper effectiveness, Sinha, 987). Consequently, these two approaches may complement each other in design applications. 4. SUMMARY AND CONCLUSIONS A new method for analyzing nonlinear dynamic systems, the Alternating Frequency/Time (AFT) domain method, is applied here to the design and analysis of frictionally damped systems. Advantages of the AFT method for this application include: Flexibility to calculate a wide variety of response statistics and investigate different design objectives in both frequency and time domains for the same simulations Ability to study effects of different forms of excitation, from sinusoidal to Gaussian white noise Similarity to vibration testing with Fourier analyzers allows computer predictions of experimental test results Accurate and efficient calculation of periodic or stationary response, avoids calculating transients The flexibility of the AFT method is exploited in performing two design studies. In the first study we show that the optimal slip load depends on the feature of the response to be minimized. The optimal slip load which minimizes the peak response is larger than the slip load which minimizes the rms response, which is larger than the slip load that minimizes the average frequency response near B

9 resonance. Which feature of the response should be minimized depends on the design objective. Therefore we discuss typical applications and concerns which lead to selecting a particular objective and a corresponding response measure to minimize. In the second design study we show that the optimal friction slip load depends strongly on the bandwidth of the excitation. In the cases examined here, the optimal slip load for a random excitation is as much as five times the optimal slip load for a sinusoidal excitation. Selecting an optimal slip load on the basis of the wrong form of excitation may allow unacceptable levels of response in the system, so care should be exercised to identify the form of excitation the system is expected to encounter. Intuition and numerical evidence from the simulations performed here indicate that the sinusoidal and Gaussian white noise excitations provide upper and lower bounds, respectively, on the optimal slip load for a given system. A nice result of this is that Equivalent Linearization may be used to estimate both of these bounds. Deterministic EL (or the harmonic balance method, which is equivalent) may be used to approximate the lower bound (cf. Griffin, 980), and EL for stochastic excitation may be used to estimate the upper bound (assuming the rms response is to be minimized; cf. Sinha, 987). The computational efficiency of EL makes it the preferred method of analysis when it is accurate. The evidence here and from Sinha is that EL is reliable for lower values of e (e < 0.4, according to Sinha). Solutions calculated by both the AFT and EL methods show the same optimal slip load, despite differences in the predicted response. The cases studied here suggest that when the form of the excitation is unknown it is safer to optimize the slip load on the basis of a sinusoidal rather than a random excitation. There are two reasons for this: () The relative response of a system to a sinusoidal excitation is greater than the relative response to a random excitation (the relative response is the ratio of the response to the excitation), and (2) the variation in the response due to a deviation from the optimal slip load is much greater for a sinusoidal excitation. The slip load optimization curves for the random excitation cases considered here had broad valleys so that the response was more forgiving of a non-optimal slip load. REFERENCES Atalik, T. S., and Utku, S., 976, "Stochastic Linearization of Multidegree-of-Freedom Nonlinear Systems," Journal of Earthquake Engineering and Structural Dynamics, Vol. 4, pp Cameron, T. M., 988, "A New Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems with Application to Design Optimization of Friction Dampers," Ph.D. Thesis, Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, PA. Cameron, T. M., and Griffin, J. H., 989, "An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems," to be published in ASME Journal of Applied Mechanics, Vol. 56. Griffin, J. H., 980, "Friction Damping of Resonant Stresses in Gas Turbine Engine Airfoils," ASME J. of Engineering for Power, Vol. 02, pp McClellan, J. H., Parks, T. W., and Rabiner, L. R., 979, "FIR Linear Phase Filter Design Program," in Programs for Digital Signal Processing, Digital Signal Processing Committee, IEEE Press: New York, pp. 5.- ff. Papoulis, Athanasios, 984, Probability, Random Variables, and Stochastic Processes, Second Edition, McGraw-Hill Book Co.: New York. Parks, T. W., and McClellan, J. H., 972, "Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase," IEEE Trans. Circuit Theory, Vol. CT-9, March, pp Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T., 986, Numerical Recipes, Cambridge University Press: New York. Roberts, J. B., 98, "Response of Nonlinear Mechanical Systems to Random Vibration: Part, Markov Methods; Part 2, Equivalent Linearization and Other Methods," Shock and Vibration Digest; Vol. 3, No. 4, pp (Part ); Vol. 3, No. 5, pp (Part 2). Sinha, A., 987, "Friction Damping of Random Vibration in Gas Turbine Engine Airfoils," ASME paper 87-GT-44, presented at the ASME Gas Turbine Conference and Exhibition, Anaheim, California, May 3 - June 4, 987. Sogliero, G., and Srinivasan, A. V., 980, "Fatigue Life Estimates of Mistuned Blades via a Stochastic Approach," AIAA Journal, Vol. 8, No. 3, pp