INTRODUCTIONHydraulic bushings are widely used as compliance joints in automotive suspension because their unique, amplitude and frequency dependent dynamic properties are beneficial for both ride quality and vibration isolation [1-2]. Design features in these devices typically include pumping chambers with working fluid, flow channels, elastomeric structure, and stoppers-all of which may introduce nonlinearity into the system. As a result, hydraulic bushing configurations are often analyzed in terms of a “black box” [1, 2, 3]. The linear system theory could be used to characterize the device as a cross-point stiffness magnitude and loss factor, but only at a given displacement amplitude and preload. The transfer function method captures only the global behavior of the device and may not yield any insight regarding amplitude dependent behavior. The use of reduced-order models [4, 5, 6, 7, 8, 9, 10] to analyze hydraulic bushings offers useful simulation results and improved insight into the physics of the device, though prior work on such models has largely been limited to linear system theory [4, 5, 6, 7, 8]. Limited effort has focused on adding a single nonlinear element to more precisely describe the physics of a design feature [9, 10]. Feature-based linear or nonlinear models are often useful to designers, as the link may be clearer between design parameters and resulting properties than with a “black box” or detailed finite element model.

PROBLEM FORMULATIONThe purpose of this paper is to develop feature-based models (based on reduced order dynamic system theory) which can predict unknown amplitude dependent dynamic stiffness behavior of hydraulic bushings. Model development in this paper assumes steady-state sinusoidal excitation at low frequencies (up to on the order of 50 Hz) and only one

device with a single fluid path (inertia track) is considered. The first objective is to use a reduced order model framework to create linear (I) and quasi-linear (II) models with empirical parameters for sinusoidal loading. Second, a method of measuring fluid compliance is introduced and nonlinear models (III A-C) are developed which include nonlinear fluid resistance and compliance elements. Finally, the effectiveness of these models to predict amplitude dependence in stiffness magnitude and loss angle spectra are evaluated. The schematic for a simplified reduced-order hydraulic bushing model is displayed in Figure 1, showing all of the relevant parameters, and Figure 2 shows measured static stiffness curve. Typical measured production bushing dynamic stiffness magnitude and loss angle curves are shown in Figure 3. Table 1 presents an overview of each model proposed in this paper.

LINEAR MODEL (I)A Laplace domain representation of the system dynamics can be useful to characterize the spectral properties of the device. This assumes a linear time-invariant system and carries many assumptions with it, including the principle of superposition and a steady-state input. The dynamic system shown in Figure 1 is characterized by the following transfer function in the Laplace domain (s), based on a reduced order linear system:

(1-a)

Dynamic Analysis of Hydraulic Bushings with Measured Nonlinear Compliance Parameters

Luke Fredette, Jason Dreyer, and Rajendra SinghOhio State University

ABSTRACTHydraulic bushings with amplitude sensitive and spectrally varying properties are commonly used in automotive suspension. However, scientific investigation of their dynamic properties has been mostly limited to linear system based theory, which cannot capture the significant amplitude dependence exhibited by the devices. This paper extends prior literature by introducing a nonlinear fluid compliance term for reduced-order bushing models. Quasi-linear models developed from step sine tests on an elastomeric test machine can predict amplitude dependence trends, but offer limited insight into the physics of the system. A bench experiment focusing on the compliance parameter isolated from other system properties yields additional understanding and a more precise characterization. Amplitude dependence predictions are improved in numerical simulations of the system while using the experimentally determined nonlinear compliance parameter, expanding the capabilities of reduced-order hydraulic bushing models.

CITATION: Fredette, L., Dreyer, J., and Singh, R., "Dynamic Analysis of Hydraulic Bushings with Measured Nonlinear Compliance Parameters," SAE Int. J. Passeng. Cars - Mech. Syst. 8(3):2015, doi:10.4271/2015-01-2355.

2015-01-2355Published 06/15/2015

Copyright © 2015 SAE Internationaldoi:10.4271/2015-01-2355saepcmech.saejournals.org

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(1-b)

(1-c)

(1-d)

(1-e)

(1-f)

(1-g)

where, Kd is dynamic stiffness, FT is the transmitted force, X is the displacement amplitude, kr is the rubber path stiffness, A1 and A2 are the effective pumping areas of each chamber, C1 and C2 are the fluid compliances of each chamber, Ii is the fluid inertance of the inertial track, Ri is the fluid resistance of the inertial track, and aj and bj represent transfer function coefficients of the numerator and denominator respectively. Damping associated with the rubber path is assumed to be insignificant. A frequency (f) domain representation is generated using the following relation, s=i2πf.

Figure 1. Schematic of generic hydraulic bushing with single inertia track. Here, kr denotes the stiffness of the elastomeric structure, C1 and C2 denote the fluid compliances of the pumping chambers, A1 and A2 represent the effective pumping areas of each chamber, Ii is the fluid inertance of the inertia track, and Ri is associated fluid resistance. The system is excited through a displacement input on the inner sleeve, x(t), and the output is the transmitted force through the outer sleeve, FT. Additional state variables for the fluid system include p1 and p2, which are the pressures in each chamber, and qi which is the volume flow rate in the inertia track.

In previous work [4, 5, 6, 7, 8], linear model parameters have been established through a combination of direct calculation and empirical estimation. The static stiffness curve, constitutive equations, and fluid mechanics principles of fluid resistance (related to viscosity) produce most of the parameters, but others are heavily geometrically dependent, and must therefore be estimated given the specific materials and configuration of a particular bushing design.

Furthermore, since the form of the transfer function has been specified, a curve-fitting procedure may be used to estimate the more difficult parameters, such as fluid compliance, C, fluid resistance, R, and effective pumping area, A. For this analysis, some simplifications are made based on the assumption of symmetry in the bushing, i.e. that C1=C2=C, A1=A2=A. A nonlinear constrained optimization using an interior-point algorithm served to optimize the coefficients of the normalized transfer function to minimize the error between the curve-fit and the measured dynamic stiffness curve. The cost function, E, is defined as the sum of the magnitude of error at each frequency, fj.

Figure 2. Static stiffness curve of hydraulic bushing example. The operating range of the bushing is within the linear region, but a nonlinear stopper region also exists for motion control. Here, the displacement is normalized by the displacement at which the stopper engages, and the applied force is normalized by the linearized static stiffness and the stopper engagement displacement.

(2)

The algorithm adjusts a set of parameters in order to minimize E. Choosing the system parameters, C, R, I, and A yields very inaccurate results, presumably due to the extreme range of their values. Choosing the coefficients of the transfer function has proved to be a more effective method. Each coefficient must be positive to make physical sense, so a constraint is added to the algorithm: aj, bj > 0 for all j. The optimization algorithm requires an initial guess, so one must be iteratively supplied until order-of-magnitude transfer function coefficients are found. When the algorithm has optimized all six coefficients, an excellent approximation of the measured dynamic stiffness results. Figure 4 shows the results of this procedure. However, assuming symmetry in the hydraulic bushings, only four unknowns exist in the theoretical model. This means that two equations must be eliminated to fit the system dynamics to the transfer function fit. This is accomplished by averaging the redundant coefficient equations to produce the model parameters with minimal error introduced.

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Figure 3. Dynamic stiffness spectra for an example hydraulic bushing, including (a) stiffness magnitude and (b) loss angle. Here, the dynamic stiffness magnitude is normalized by the static stiffness, ks, and the frequency is normalized by the frequency at which the loss angle is maximized for a 0.1 mm excitation amplitude, ftune. All excitation amplitudes in this paper are peak-to-peak.

Table 1. Hydraulic bushing models.

Linear system models with empirical parameters produced by this method may offer good approximations of hydraulic bushing dynamic properties at a given amplitude and preload. Nevertheless, the model will not be effective at other amplitudes, and still less accurate for other types of inputs which may have a larger dynamic range [10]. Linear models (say, based on a 0.1 mm amplitude test) should always produce the same dynamic stiffness curve regardless of excitation amplitude. This demonstrates the need for more advanced models which may predict amplitude dependent results.

QUASI-LINEAR MODEL (II)Quasi-linear models are developed by a similar method. However, unlike the linear model, the quasi-linear model employs amplitude dependent parameters. Figure 5 shows the amplitude dependence of a production hydraulic bushing. There are significant differences in

peak loss angle and dynamic stiffness amplification between 0.1 and 1.0 mm (peak to peak) excitation amplitudes, although the tuning frequency at which the loss angle is maximized only varies slightly.

Curve-fitting the model parameters to each measured spectrum produces amplitude-dependent bushing properties. Table 2 shows the change in each parameter for three excitation amplitudes. In these results, the inertance remains constant. This is because the mass effects of the fluid should not be influenced by frequency dependent behavior up to about 50 Hz, so a theoretical value of inertance is used based on turbulent flow assumption [12]. Physical intuition suggests that the fluid resistance would increases with increased amplitude, since more fluid is pumped between the chambers. Analysis indicates that the resistance parameter is most affected by a variation in amplitude, and this is why previous work focused on this topic [10]. The compliance term varies slightly, but the effective pumping area increases by only five percent. These two parameters work together as there is interaction between the structural and fluid system; accordingly, the results are very sensitive to changes in such parameters. Finally, the rubber path stiffness drops off by about five percent, which is relatively small since the model is less sensitive to this parameter.

Figure 4. Verification of identified parameters for linear model I. (a) Dynamic stiffness magnitude and (b) loss angle. Key: -measured stiffness; . - transfer function coefficient fit; - physical parameter fit. Using the extracted parameters produces a good approximation of the measured stiffness, while just using the form of the transfer function with empirical coefficients can improve the accuracy.

Figure 6 compares the measured dynamic stiffness curves and quasi-linear model for three amplitudes, showing general agreement in both stiffness magnitude and loss angle at each amplitude. Nevertheless, the quasi-linear model is empirical in nature and provides little insight into the physics that would produce such dynamic properties. Additionally, development of quasi-linear models requires complete dynamic stiffness data at every amplitude of interest. Further, they cannot predict responses to non-sinusoidal input since the parameters (by definition) are defined for specific sinusoidal amplitudes. A more detailed investigation is therefore necessary to understand the nature and extent of the nonlinearity.

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Figure 5. Measured amplitude dependent dynamic stiffness spectra. (a) Normalized dynamic stiffness magnitude for a hydraulic bushing and (b) corresponding loss angle for three excitation amplitudes. Key: − 0.1 mm excitation amplitude; − 0.5 mm amplitude; − 1.0 mm amplitude. Dynamic stiffness magnitudes and loss angles decrease with higher excitation amplitude, indicating nonlinearity in the system.

Table 2. Effect of displacement amplitude on quasi-linear model parameters.

NONLINEAR MODELS (III)The prior section suggests that the nonlinear model parameters would depend on excitation amplitudes and mean loads, as well as excitation type (such as sinusoidal or transient). Utilization of nonlinear models has significant potential for accuracy with complicated nonlinear devices like a hydraulic bushing since the physics could be better understood.

The fluid resistance parameter, R, varies the most between the different amplitudes, and it is also the focus of previous work in nonlinear hydraulic bushing models [10]. Since the flow oscillates, under sinusoidal testing, formulations from steady-state fluid mechanics may not necessarily apply. However, the following turbulent flow formulation for pressure drop (Δp) in a tube may be applied to dynamic flow conditions in some cases [11]:

(3)

Here, Li is the length if the inertia track, μ denotes the dynamic viscosity of the fluid, ρ is the fluid density, Di is the hydraulic diameter of the tube, and qi represents the volume flow rate of the fluid. From this equation, a nonlinear resistance function (RNL(qi)) is derived, as follows:

(4)

The nonlinear fluid resistance function varies with flow rate, increasing with higher amplitude, which is consistent with the quasi-linear model results. Other resistance mechanisms may also occur due to the complex flow geometry of the pumping chambers and channels. The pressure drops due to bends, corners, and entrance/exit effects are characterized as minor losses, and are collectively defined by a single resistance function (RML) which is proportional to flow (as shown below), where ξML is an empirical, geometry-based minor loss coefficient:

(5)

Figure 7 compares these two resistance types with a linear resistance term based on the laminar flow formulation [12]. At very low flow rates, only the laminar flow formulation provides significant damping to the fluid system. However, it is unlikely that fully developed flow will occur in a hydraulic bushing under sinusoidal excitation and complex channel geometry, suggesting the nonlinear resistance mechanism. Resistance due to minor losses are not likely to dominate the viscous losses in hydraulic bushings, but their addition should enhance model accuracy in some loading scenarios.

In order to observe the effect of nonlinear model parameters on the spectral properties, the governing ordinary differential equations must be integrated. While many numerical methods exist to accomplish this goal, a Runge-Kutta algorithm is employed for this study using Matlab [13]. The excitation input, x(t) is specified as a sinusoidal input at frequency f as shown below, with mean value xm, where xa is the peak-to-peak amplitude:

(6)

The output of this simulation is the transmitted force, including both the rubber and fluid paths. Since the simulated equations are nonlinear, the output may be non-harmonic for a sinusoidal input. However, dynamic stiffness from the linear system theory assumes that both the displacement and transmitted force both contain only a single frequency. Therefore, another nonlinear constrained optimization algorithm is applied to produce the sinusoid which most closely matched the simulated output. Figure 8 shows an example of a non-harmonic system response with the following sinusoidal fit which is used to calculate the dynamic stiffness:

(7)

(8)

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where, Ft,fit is the amplitude and θfit is the phase of the sinusoidal fit. Further, Kd is the complex dynamic stiffness approximation for the nonlinear system.

Figure 6. Comparing quasi-linear model (II) of hydraulic bushing with measured data at three excitation amplitudes. (a) 0.1 mm (b) 0.5 mm (c) 1.0 mm excitation amplitude. Key: - measured dynamic stiffness curve; - quasi-linear model. Quasi-linear models offer reasonably accurate approximation of measured dynamic stiffness data.

Figure 7. Comparison of fluid resistance formulations at low and high flow rates. Key: - (L) fully developed laminar flow in a tube; - (T) turbulent flow in a tube; - (ML) minor losses. For the low-flow regime, laminar resistance is much higher than the other formulations. At a certain flow rate, however, turbulent resistance becomes dominant. Minor losses do not become dominant, but they may become sufficiently high to significantly influence the overall resistance.

Nonlinear model (III-A) with a nonlinear resistance element produces similar amplitude dependence trends as the measured data as shown in Figure 9 that shows the effect of nonlinear resistance on the dynamic stiffness at several excitation amplitudes. This model has improved predicting capabilities over the linear and quasi-linear models, but can produce largely inaccurate results if some of the other parameters are out of tune, particularly the fluid compliance and effective pumping area.

While the reduced order model (III-A) with all linear parameters except for a nonlinear resistance element appears to offer reasonable predictions of the measured dynamic data, this type of nonlinear system analysis is only “locally” valid. This means that model results only predict behavior under very similar excitations. Such a model may not produce accurate results under transient loading, or a loading situation with a large dynamic range. Further study is necessary to understand the influence of other nonlinearities on the system behavior.

MEASUREMENT OF COMPLIANCE PARAMETER: CONCEPTThe structure of hydraulic bushings typically involves irregular geometry and a nonlinear elastomeric material. Since the fluid compliance characterizes the elastic behavior of the fluid container, these properties may lead to a nonlinear compliance parameter. Understanding the nature of this nonlinearity requires an experimental investigation. The effective fluid compliance, C1,2 is first calculated by a combination of the compliance of the container, the compliance associated with the compressibility of the fluid, and the compliance associated with any entrained gas in the fluid:

(9)

Figure 8. Nonlinear model results example with harmonic fit. Key: - non-harmonic simulated force; - sinusoidal fit. Using a sinusoidal fit allows for the frequency-domain calculation of dynamic stiffness.

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Figure 9. Results of nonlinear model III-A with only a nonlinear resistance element showing amplitude dependence trends. Key: − 0.1 mm; − 0.5 mm; − 1.0 mm. Amplitude dependence trends are similar to measured data for the step-sine input.

If the properties of the working fluid are known, then the Cliquid component can easily be calculated. The precise amount of entrained gas is very difficult to measure, but this contribution is well known [12]. This leaves the container compliance, Ccontainer, to be experimentally determined. The nonlinear container compliance function is calculated using the slope of the volume vs. pressure curve, so the compliance experiment must measure Vc(p), where Vc is the volume of the chamber at pressure p [12]:

(10)

Measuring the volume of a deformable chamber of irregular geometry is not trivial. Various methods may exist to measure the volume, but given the pressure dependency and the relative ease of pressure measurement, a constitutive relationship between volume and pressure should yield the needed curve. Such a relationship exists in the ideal gas law, where p is pressure, V is volume, n is the molar amount of gas, R is the gas constant, and T is temperature:

(11)

Here, an experiment is proposed that measure volume based on this relationship. Figure 10 shows a schematic of the apparatus, while Figure 11 provides an overview of the method. This particular experiment is validated by measuring the compliance of a spring-return pneumatic cylinder, the compliance of which is known based on the spring properties. With the cylinder as the compliant chamber, measured compliance values agrees well with predictions based on the spring constant, validating the experimental method.

MEASUREMENT OF BUSHING COMPLIANCE FUNCTIONHaving validated the experimental procedure of Figures 10 and 11, the method may be used on the pumping chambers of a production hydraulic bushing. The complexity of design in many production devices make direct measurement challenging. The experimental setup to measure the compliance function of a production hydraulic bushing is shown in Figure 10, where the compliant chamber is the bushing sans metal outer-sleeve inserted into a removable, transparent acrylic sleeve with pressure ports in each chamber. This allows for the manipulation of path configurations, such as blocking off the inertia track.

The results of the experiment suggest a hardening nonlinearity, so a least-squares regression algorithm is applied to fit the data with a linear and two nonlinear approximations, cubic and logarithmic, defined by the following equations:

(12)

(13)

(14)

Here, αj, βj, and γj are numerically determined coefficients and pc is the chamber pressure. The compliance functions are calculated by taking the derivative of the V(p) curves, as defined in equation 10:

(15)

(16)

(17)

Each of these forms has a unique shape, illustrated in Figure 12. The cubic fit produces a hardening compliance, but at high pressures it begins to soften again. This matches the general trend of the quasi-linear compliance parameter, though it may not fully model the physics. The logarithmic fit produces a hardening curve which continues to stiffen asymptotically. The effects of each fit are not directly obvious, and are likely to be sensitive to other model parameters as well as excitation amplitudes and time histories. As is often the case with nonlinear systems, there are several possible solutions and alternate ways to arrive at them; the models used in this paper are by no means exclusive.

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Figure 10. Compliance experiment schematic. An air tank which is assumed to be rigid is fitted with a pressure gage. A needle valve isolates this pressure vessel from a hose leading to a compliant chamber, which is also instrumented. The compliant chamber represents the pumping chambers of a hydraulic bushing.

Figure 11. Flow chart of experimental procedure.

AMPLITUDE DEPENDENT STIFFNESS RESULTS USING NONLINEAR MODEL IIIThe nonlinear compliance functions extracted from the experimental results may be inserted into the model. A correction factor is applied to the measured compliance to account for the elastomeric material's dynamic stiffness amplification and establish proper frequency tuning for the model. Figure 13 shows the effect of the cubic compliance element, as given by equation 16, on system performance. Figure 14 shows the effect of the logarithmic fit given by equation 17. The two curves are nearly indistinguishable, indicating that the functional form used for the nonlinear compliance is of minor importance as

long as the compliance values accurately approximate the compliance of the device. The functional form may be selected to facilitate analysis and solution approximation of the nonlinear system.

Figure 12. Compliance parameters based on nonlinear curve fits and normalized by the linear fit. Key: - linear fit; - cubic fit; - logarithmic fit. The two nonlinear fits to the measured volume curve produce a hardening effect, although the cubic fit does reach a local minimum and begin softening at high pressures.

Figure 13. Results of nonlinear model III-B with nonlinear resistance and cubic-fit compliance elements. Key: − 0.1 mm; − 0.5 mm; −1.0 mm.

When added to the model with a nonlinear resistance, both functional forms proposed in this paper for compliance formulation present a marginal improvement on results. Amplitude dependence trends are largely the result of the nonlinear resistance element; however, the amplitude dependent compliance element improves the deviation in tuned frequency at higher amplitudes. Figure 15 compares the measured dynamic stiffness with two models, one having only a nonlinear resistance element (III-A) and one with both nonlinear resistance and compliance elements (III-C). The tuned frequency (at which the loss angle is at a maximum) shifts down about five percent with a linear compliance. Introducing the nonlinear compliance element brings the shift closer into agreement with measurement.

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Figure 14. Simulation of model III-C including nonlinear resistance and log-fit compliance elements. Key: − 0.1 mm; − 0.5 mm; − 1.0 mm. Results are very similar to the cubic-fit compliance model.

The simulations used to generate the stiffness spectra in Figure 15 rely on an empirically determined effective pumping area, based on the 0.1 mm dynamic stiffness. Since the simulations use a 0.5 mm amplitude excitation, amplitude dependent effects may emerge. Adding the nonlinear compliance element has made some improvement but highlights significant error between model and measurement. Judicious selection of an amplitude dependent effective pumping area (obtained via empirical means) should reduce this error.

Figure 15. Comparison of hydraulic bushing dynamic properties and nonlinear simulation result for a 0.5 mm excitation amplitude. Key: - Measured properties; - model III-A; - model III-C. Simulation results show a marginal improvement with the addition of nonlinear compliance elements, but further work is needed to improve prediction accuracy.

CONCLUSIONDevelopment of a reduced-order dynamic model of a hydraulic bushing is a challenging task because of the inherently nonlinear design features. Precise nonlinear mechanisms are difficult to assess a priori for a production device with unknown properties. The modeling process may be facilitated by employing several models as suggested in this article. Linear and quasi-linear models are proposed and compared to measured stiffness. An experiment which measures the fluid compliance of the pumping chambers is proposed. The experiment is validated by using its method to measure the compliance of a spring-return pneumatic

cylinder, the stiffness of which is confirmed by another method. Performing the experiment on a hydraulic bushing has revealed a hardening nonlinearity in the fluid compliance. This nonlinearity, when combined with a nonlinear resistance, improves model accuracy on amplitude dependent trends in stiffness magnitude and loss angle. Introducing this element into simulations also highlights the significance of the effective pumping area at different excitation amplitudes, suggesting future work on this topic.

ACKNOWLEDGEMENTSThe authors would like to thank the member organizations of the Smart Vehicles Concepts Center (www.SmartVehicleCenter.org), such as Transportation Research Center Inc., Honda R&D, F.Tech, and Tenneco Inc. as well as the National Science Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc) for supporting this work.

CONTACT INFORMATIONProfessor Rajendra SinghAcoustics and Dynamics LaboratoryNSF I/UCRC Smart Vehicle Concepts CenterDept. of Mechanical and Aerospace EngineeringThe Ohio State Universitysingh.3@osu.eduPhone: 614-292-9044www.AutoNVH.orghttp://svc.engineering.osu.edu/

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LIST OF SYMBOLS

VARIABLESA - pumping area

a - dynamic stiffness numerator coefficient

b - dynamic stiffness denominator coefficient

C - fluid compliance

D - hydraulic diameter

d - logarithmic volume estimation coefficient

E - transfer function approximation error

FT - transmitted force

f - frequency

I - fluid inertance

j - index number

k - stiffness

- dynamic stiffness

L - length

NTOT - effective total amount of gas in the system

P - fluid pressure

q - volume flow rate of fluid

R - fluid resistance

s - Laplace variable

t - time

V - volume

x - displacement

α - cubic volume estimation coefficient

β - logarithmic volume estimation coefficient

Δp - pressure drop

γ - linear volume estimation coefficient

θ - phase shift

µ - dynamic viscosity

ξ - minor loss coefficient

ρ - fluid density

ϕ - loss angle

SUBSCRIPTS1, 2 - pumping chambers, also coefficient indices

a - amplitude (peak to peak)

c - deformable chamber

cubic - cubic volume estimation

i - inertia track

lin - linear volume estimation

log - logarithmic volume estimation

m - mean value

ML - minor losses

r - rubber path

s - static

T - rigid tank

NL - nonlinear function

SUPERSCRIPTS∼ - complex valued

ABBREVIATIONSL - laminar

lin - linear

log - logarithmic

ML - minor loss

Nom - nominal value

T - turbulent

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