SKYTRUCK AIRPLANE LANDING GEAR BY MEANS OF THE ... contrary to the
classical nonlinear system identification methods, does not require ... The
research presented in this paper aimed at identification of nonlinear parameters
of.
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IDENTIFICATION OF NONLINEAR PARAMETERS OF THE SKYTRUCK AIRPLANE LANDING GEAR BY MEANS OF THE OPERATIONAL MODAL ANALYSIS OUTPUT-ONLY METHOD Joanna IWANIEC, Tadeusz UHL AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics, Department of Robotics and Mechatronics, Mickiewicz Alley 30, 30-059 Krakow, POLAND The paper concerns output-only method combining restoring force, boundary perturbation and direct parameter estimation techniques. This method, on the contrary to the classical nonlinear system identification methods, does not require the knowledge of excitation nor system linear behaviour around any operating point and, therefore, is a convenient method for identification of nonlinear systems working under non-measurable operational loads. In the paper there are presented results of method application to the parameter identification of the Skytruck airplane landing gear. Keywords: operational nonlinear system identification, output–only method, restoring force, landing gear, M28 Skytruck airplane.
1. INTRODUCTION Mechanical structures encountered in industrial practice are nonlinear to some degree. Although the sources of nonlinear structural behaviour can be different [1,2,12,16,25], nonlinear systems, in general, do not follow the superposition principle and exhibit complex phenomena such as jumps, changes in natural frequencies resulting from changes in amplitudes, self-induced and chaotic vibrations, co-existence of many stable equilibrium positions. Therefore classical identification methods formulated for linear systems can’t be applied to identification of nonlinear systems nor it is possible to elaborate general identification method applicable to all nonlinear systems in all instances. Historically, equivalent [23] and stochastic [27] linearization methods were the first methods used commonly for the purposes of nonlinear system identification. In the following years perturbation theory for weakly nonlinear systems was developed [13,17,19] and the
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concept of nonlinear normal modes was introduced [21, 24, 26, 29-33]. Later works [5,6,20] concerned identification of strongly nonlinear systems. Recently the interest is paid to the possibilities of taking advantage of phenomena characteristic for nonlinear systems and design of machines and mechanical systems working in nonlinear ranges of dynamic characteristics [18, 22, 28]. The research presented in this paper aimed at identification of nonlinear parameters of the M28 Skytruck airplane (Fig. 1) landing gear on the basis of measured vibration accelerations resulting from operational loads acting on the airplane during its ride over the apron after landing (Fig. 4). Therefore application of the classical nonlinear system identification methods, which require an input measurement (or estimate) as well as system linear behaviour around an operating point, was not possible. Instead the method [10] combining the restoring force, boundary perturbation and direct parameter estimation techniques (Fig. 2) was used. This method, on the contrary to the classical nonlinear system identification methods, requires neither the knowledge of excitation nor linear behaviour of the considered system around an operating point and, therefore, is a convenient parameter identification method for strongly nonlinear systems working under operational loads, the measurement of which is difficult or impossible to carry out.
Fig. 1. M28 Skytruck model 05. 2. ALGORITHM OF THE ASSUMED OUTPUT-ONLY IDENTIFICATION METHOD Algorithm of the assumed identification method [10] combines three techniques: restoring force, boundary perturbation and direct parameter estimation (Fig. 2). In the first step of the algorithm, nonlinear system parameters are estimated on the basis of system dynamic motion equations with nonlinear restoring forces eliminated beforehand. In the next step, by the use of the direct parameter estimation method, nominally linear system parameters are computed. Since in case of operational measurements the excitation remains unknown, the number of unknown parameters is greater than the number of system dynamic motion equations that can be formulated. Therefore, for the purposes of providing an additional dynamic motion equation and computing the absolute values of system parameters, the boundary perturbation
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method is used. The method consists in adding given mass (inertia) to a particular degree-offreedom and retaking measurements of accelerations of modified system masses.
Fig. 2. Algorithm of the method assumed for the purposes of identification of nonlinear parameters of the Skytruck airplane landing gear. 3. IDENTIFICATION OF NONLINEAR AIRPLANE LANDING GEAR
PARAMETERS
OF
THE
SKYTRUCK
Identification research was carried out for the M28 Skytruck airplane (Fig. 1, Fig. 3), which is a twin-engine monoplane with twin vertical tails produced in the Polish Air Plant (PZL) in Mielec. The airplane is equipped with a tricycle non-retractable landing gear.
Fig. 3a,b. M28 Skytruck model 05 and c) elements of its landing gear.
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Iwaniec J., Uhl T. Vibration acceleration time histories were measured on the airplane landing gear by
means of two three-axial piezoelectric sensors placed on the spindle (sensor 1, ‘lower’) and above the absorber on the airplane body (sensor 2, ‘upper’). Two measurement sessions in two flights with different airplane masses were taken and recorded by the use of the SCADAS III analyzer. In order to provide mass values necessary for the purposes of verification of analysis results, the airplane was weighed after each flight. Basic measurement conditions for individual flights are gathered in the Table 1. Tab. 1. Measurement conditions in the individual flights.
Flight 1 (M2 – ΔM2) Flight 2 (M2)
Sampling frequency [Hz] 200 200
Number of samples 46336 124160
Airplane mass after flight [kg] 6930 7200
Identification of system parameters was carried out on the basis of parts of vertical vibration acceleration time histories, resulting from operational loads acting on the airplane during ride over the apron after landing (Fig. 4). For such a case it was assumed that each wheel suspension works independently and sprung mass of each wheel suspension (Fig. 5) equals 1/3 of the total mass of the airplane body. ― 1 st flight ― 2 nd flight
― 1 st flight ― 2 nd flight
b)
a cceleration
a cceleration
[ m/s 2 ]
[ m/s 2 ]
a)
Sensor 1, ‘lower’ time [s]
Sensor 2, ‘upper’ time [s]
Fig. 4. Analyzed parts of vibration accelerations recorded in the considered flights.
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For each wheel suspension the model presented in the Fig. 5 [10] was assumed. M 1 : unsprung mass,
K3 x2
M2 N1
K2
N2
K1
}
C2 x1
M1 C1
}
xb
}
}
Sprung mass (1/3 of the total mass of the airplane body)
M 2 : sprung mass, K1 : stiffness in tire, K2 : stiffness in suspension,
Elements of landing gear
C1 : damping in tire,
Unsprung mass (rocker and wheel)
x1 : displacement of the M 1 ,
Tire
Fig. 5. Assumed model of each wheel suspension.
C2 : damping in suspension, x2 : displacement of the M 2 , xb : displacement of the tire patch, N1 :
nonlinear
force
in
For the M1 and M2 masses the following dynamic motion equations were formulated: M 1 { x1 } + ( C1 + C 2 ){ x 1 } − C 2 { x 2 } + ( K 1 + K 2 ){ x1 } − K 2 { x 2 } + N 1 ( { x1 ( t )} , { x 2 ( t )} , { x 1 ( t )} , { x 2 ( t )} ) + + N 2 ( { x1 ( t )} , { xb ( t )} , { x 1 ( t )} , { x b ( t )} ) = C1 { x b } + K 1 { x1 } M { x } − C { x } + C { x } − K { x } + ( K + K ){ x } = N ( { x ( t )} , { x ( t )} , { x ( t )} , { x ( t )} ) 2 1 2 2 2 1 2 3 2 1 1 2 1 2 2 2 (1) Rearranged dynamic motion equation for the sprung mass M2: M 2 { x2 } = −C 2 ( { x 2 } − { x 1 } ) − K 2 ( { x 2 } − { x1 } ) − K 3 { x 2 } + N 1 ( { x1 ( t )} , { x 2 ( t )}, { x 1 ( t )}, { x 2 ( t )} ) (2) expresses relation between acceleration of the sprung mass M2 and relative velocity or relative displacement between masses M1 and M2. On the basis of the graphical representation of these relations it is possible to determine the character of damping and stiffness restoring force acting on the wheel suspension. For the purposes of detection of nonlinear damping and stiffness restoring forces acting on the Skytruck airplane landing gear, measured vibration accelerations were integrated offline in time domain to estimate vibration velocities and displacements in the considered measurement pointes. Each integration was preceded by removal of constant components from analyzed signals. Identified nonlinear damping and stiffness restoring force characteristics are presented in the Fig. 6a and Fig. 7a. Identified nonlinear damping and stiffness restoring forces were modelled by the use of function in the form of [10]:
f n ( { ∆x } , { ∆x} ) = M 2 ⋅ ( { ∆x } − a ⋅ { ∆x} ) , −1 { ∆x } = − arctg ( d ⋅ b ⋅ ( { ∆x } + c{ ∆x} ) )
{ ∆x} = { x1 } − { x 2 }
(3)
Computed parameters of model of nonlinear stiffness characteristic are as follows: a = -100, b = 130, c = 0, d = 12,2. For model of nonlinear damping characteristic the following values
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of parameters were assumed: a = -100, b = 130, c = 120, d = 3,5. Comparison of measured and estimated stiffness restoring force fn1 is presented in the Fig. 6a while the measured and estimated damping restoring force fn2 is shown in the Fig. 7a. Estimation errors defined as a differences between measured and estimated restoring forces are depicted in the Fig. 6b and
b)
a) Estimation error [m /s2]
Acceleration of the mass M2 [m/s2]
Fig. 7b.
Relative displacement (x1 – x2 ) [m /s]
Relative displacement (x 1 – x 2 ) [m /s]
Fig. 6. a) stiffness force identified in measurement data (···) and estimated (***), b) estimation error. Since in practice value of mass M2 remains unknown, estimates (fn1 and fn2) of nonlinear restoring forces are determined to within a scale factor of M2. Linear system parameters can be computed by means of direct parameter estimation technique on the basis of dynamic motion equation of the mass M2 with estimates of nonlinear restoring forces subtracted:
{ x2 } − f n1 − f n 2
=
C2 M2
({ x 1 } − { x 2 } ) +
K2 M2
({ x1 } − { x 2 } ) −
K3 { x2 } M2
(4)
Acceleration of the mass M2 [m/s2]
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a) Estimation error [m /s2]
b)
Relative velocity () [m /s]
Relative velocity () [m /s]
Fig. 7. a) Damping force reconstructed on the basis of the measurement data (···) and estimated (***), b) estimation error. By the use of this technique, for the considered Skytruck airplane, the following equations can be formulated:
{T21 ( jω )} = { X 2 ( jω )} ⇒ K 2 1 − 1 + K 3 = ωk2 M 2 , { X 1 ( jω ) } T21 ( ωk )
k = 1, 2, , N f
(5)
K2 (6) K 2 + K3 where: {T21(jω)}: transmissibility function between displacements of the sprung M2 and T21 ( 0 ) =
unsprung M1 masses (with nonlinear restoring forces subtracted), X1(jω), X2(jω): Fourier transforms of x1(t) and x2(t), T21(0): transmissibility function evaluated at zero frequency. Since in practice the exact airplane mass M2 remains unknown, direct parameter estimation method provides two equations with three unknowns - M2, K2 and K3, which means that at this stage of the analysis only ratios of linear parameters (with respect to the mass M2) are available. For the purposes of formulating an additional ‘missing’ dynamic motion equation, boundary perturbation method can be used. The method consists in introducing an additional mass ΔM2 altering dynamic behaviour of the considered system and retaking measurements for such a modified system. An additional ‘missing’ equation is as follows: ' ' { T21 ( jω )} = {{ X 2' (( jω))}} ⇒ K 2 1 − ' (1 ) + K 3 = ω p2 M 2 , X 1 jω T21 ω p
{
p = 1, 2, , N 'f
(7)
}
where: T21' ( jω ) : transmissibility function between displacements of the sprung (M2+ΔM2) ' and unsprung M1 masses (with damping ignored), N f : number of spectral lines of useful data.
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At this stage of analysis, on the basis of equations (5), (6), (7), absolute values of demanded linear parameters can be determined. Computations of transmissibility functions {T21} and { T21' } were carried out by the use of the VIOMA (Virtual In-Operation Modal Analysis) toolbox dedicated for Matlab. Assumed signal processing parameters are gathered in the Table 2 while the comparison of
Magnitude of transmissibility function
estimated transmissibility functions is shown in the Fig. 8.
{T21} - for M2
{T21} - for M2 – ΔM 2
Frequency [Hz]
Fig. 8. Comparison of estimated transmissibility functions. There is a clear difference between transmissibility functions estimated in both cases – higher mass of the airplane ({T21}) shifts resonances towards lower frequencies. Tab. 2. Signal processing parameters for estimation of transmissibility functions T21 and T21’.
T21 (M2) T21’ (M2 - ΔM2)
Overlap
Window type
Sampling frequency [Hz]
FRF estimator
30% 30%
Flattop Flattop
200 200
H1 H1
Relation between K2 and K3 was derived on the basis of observation that the magnitude of the transmissibility function {T21} approaches 0,7 as the frequency approaches 0 [Hz]: T21 ( 0 ) =
K2 = 0,7 ⇒ K 3 = 0,428 ⋅ K 2 K2 + K3
(8)
Substituting this relation into equation (5), the relation between K2, sprung mass M2 and transmissibility function {T21} at the frequency ω can be written in the following form: 1 = ω 2 M 2 K 2 1,428 − T21 ( ω )
(9)
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Taking into account that K3 = 0,428K2, equation (7) expressing relation between K2, modified sprung mass (M2 – ΔM2) and transmissibility function { T21' } at the frequency ω’ can be rewritten as follows: 1 K 2 1,428 − ' ' T21 ω
( )
= ω '
( ) (M 2
2
− ∆M 2 )
(10)
On the basis of estimated transmissibility functions {T21} and { T21' } three corresponding ‘peaks’ were identified (Fig. 9).
Magnitude of transmissibility function
1 1: peak 1 2: peak 2 3: peak 3
{T21} - for M2 2
{T21} - for M2 – ΔM2 3
Frequency [Hz]
Fig. 9. Corresponding ‘peaks’ of the estimated transmissibility functions. For the values of frequencies and magnitudes corresponding to these ‘peaks’ computations of parameters of the airplane landing gear were carried out (Table 3). Percentage relative error of the mass M2 estimation: Err =
M 2m − M 2e M 2m
M 2m : measured mass ⋅ 100 ⋅ [ %], M 2e : estimated mass
(11)
was used as a measure of parameter estimation accuracy. Values of percentage relative errors of parameter estimation for consecutive ‘peaks’ of transmissibility functions {T21} and { T21' } are gathered in the Table 3.
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Tab. 3. Parameters of estimated transmissibility functions, estimated model parameters and estimation error. Peaks Peak 1 Peak 2 Peak 3
Parameters of transmissibility functions ω ω' T21(ω) T21' ( ω ) [Hz] [Hz] 10,15 0,827 10,938 0,337 6 13,28 0,490 16,406 0,221 1 15,62 0,472 19,531 0,200 5
Estimated parameters M2 [kg]
K2 [N/m]
1832,32 3 3670,50 7 3040,31 9
1021,49 0 1500,13 6 1242,57 8
Estimation error
K3 [N/m]
Err [%]
417,4829
94
6406,167
11
5027,412
30
Values of parameters estimated for peaks 2 and 3 are reasonable and close to each other, while results obtained for the first peak are burdened with inadmissible errors. Such a situation conforms to expectations - since the 2DOF model of airplane wheel suspension was assumed (Fig. 5), only computations carried out for two peaks should provide valid results. 4. CONCLUSIONS AND FINAL REMARKS In the paper there is presented the output-only method that can be used for the purposes of identification of nonlinear parameters of the Skytruck airplane landing gear in the presence of non-measurable operational loads resulting from airplane ride over the apron after landing. Identification of transfer functions (5), (7) in operational conditions (airplane braking and stop in a finite time) is burdened with errors influencing estimates of transfer functions and their peak frequencies. Inaccuracies in frequency estimates result also from the fact that in case of heavily damped systems, such as cars, airplanes or rail coaches, it is difficult to identify the exact location of the peaks. In the carried out research each airplane wheel suspension, which is a multiple degreeof freedom system, was modelled by a simple 2 degree-of-freedom model. Therefore in a theoretical model only two resonant frequencies are present while on the basis of measured response of a real system it is possible to identify a higher number of natural frequencies corresponding to degrees of freedom that are neglected in a theoretical model. Computations carried out for these frequencies lead to estimation of model parameters burdened with gross numerical errors, which differs significantly from values of parameters estimated for other peak frequencies and, therefore, are easy to detect and eliminate. ACKNOWLEDGEMENTS Scientific research was financed from polish means for science (from 2006 till 2009) as the research project n504 026 31/1907: ‘operational modal analysis of nonlinear structures and its applications’.
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