The special material performance, manufacturing process, machining behavior, and operating condition of composite materials cause uncertainties to inevitably appear in the mechanical properties of composite structures. Therefore, variability in mechanical properties must be considered in a mechanical response analysis of composite structures. A method is proposed in this paper to predict the dynamic performance of composite landing gear with uncertainties using experimental modal analysis data and nonlinear static test data. In this method, the nonlinear dynamic model of the composite landing gear is divided into two parts, the linear and the nonlinear parts. An experimental modal analysis is employed to predict the linear parameters with a frequency response function, and the nonlinear parameters caused by large deflection are identified by a nonlinear static test with the nonlinear least squares method. To check the accuracy and practicability of the method, it is applied to drop impact simulations and tests of composite landing gear. The results of the simulations are in good agreement with the test results, which shows that the proposed method is perfectly suited for the dynamic analysis of composite landing gear.
Landing gear is the undercarriage of an aircraft or spacecraft and may be used for either takeoff or landing. For aircraft, the landing gear supports the craft when it is not flying, allowing it to take off, land, and taxi without damage. The impact loads are extremely high when the aircraft is landing [
Composite materials are widely used in aerospace, aircraft, marine, civil, automotive, sport [
The dynamic characteristic analysis of the composite structures has recently attracted more attention. The finite element method is generally used for the analysis of complex structural behaviors of the composite structures. Many excellent theories are available on the finite element method for composite structural analysis. The classical analysis theory is based on the Kirchhoff plate theory, which is the simplest theory among them. The firstorder shear deformation theory is suitable for the global structural behavior of thin and moderately thick laminated composite. Various higherorder shear deformation theories have overcome limitations in both the classical and firstorder shear deformation theories. Layerwise lamination theory, which can predict the interlaminar stresses accurately, assumes a displacement representation formula in each layer, and the theory based on the 3D continuum can predict a composite laminate’s interlaminar stress [
The theories have made significant progress but are based on accurate composite material parameters. The material properties determined from standard specimens tested in the laboratory may deviate significantly from those of actual laminated composite components manufactured in a factory [
The properties of composites can be obtained accurately via experimental tests [
The mixed numericalexperimental technique makes the finite element method suitable for dynamic analysis of the laminated composites. However, the tests for measuring eigenfrequencies of composite plates are almost all linear, as in the works mentioned above, and do not consider the structural nonlinearities of the laminated composites. There are obvious geometrical nonlinearities in the work process of landing gears, so the nonlinear system identification is needed in the nonlinear structural models of landing gears. The nonlinear system identification in structural dynamics has been studied since the 1970s; many excellent methods are available for identification of nonlinear structural models. Timedomain methods, such as the restoring force surface method, the approach based on nonlinear autoregressive moving average with exogenous input model, and Hilbert transformbased data decomposition [
Our study is aimed at expanding a numericalexperimental method based on linear and nonlinear tests for nonlinear modeling and drop impact analysis of composite landing gear.
The equation of motion of an
Any solution to (
A set of coupled modal equations with reduced degrees of freedom is first obtained by applying the modal coordinate transformation:
Generally, a subset of
Then, (
Equation (
In the numericalexperimental method of this paper, the linear part was identified by experimental eigenvalue matrices, eigenvector matrices, and others from experimental modal analysis, and the nonlinear part was identified by a nonlinear static test. The flowchart of the numericalexperimental method is shown in Figure
The flowchart of the numericalexperimental method.
The linear parameters were identified by experimental modal analysis. The experimental modal analysis is a method to describe a structure in terms of its natural characteristics which are the frequencies, damping, and mode shapes. By using signalanalysis techniques, one can easily measure vibration on operating structures and make a frequency analysis.
The frequency spectrum is the description of how vibration levels vary with frequencies which can then be checked against a specification. The frequency response function measurement removes forces spectrum from the data and describes the inherent structural response between defined points on the structure.
In the time domain, structural properties are given in terms of mass, stiffness, and damping. The dynamic equilibrium of an
The corresponding dynamic equilibrium in frequency domain can be described as
The frequency response vector
The matrix
The nonlinear parameters were identified by a nonlinear static test. This test measures the nonlinear displacements of the composite structure
The equation of an
Muravyov [
The nonlinear force vector may be expressed in the following form:
The nonlinear parameters
There is a model of the following form:
The firstorder necessary condition for a minimum is
Then, the values of
The nonlinear dynamic model is built after the linear parameters
The composite landing gear, as shown in Figure
The composite landing gear.
Three composite materials, the carbon cloth, the glass cloth, and the glass fiber with polymeric matrix, are used in the manufacture of the landing gear, as shown in Figure
Composition of the landing gear.
The linear parameters of the composite landing gear were identified by experimental modal analysis. In general, the database is obtained from direct measurements. The frequency response functions are the measured output that will be used to construct the experimental realizations of the linear parameters identification.
The landing gear was tested in fixed constraint at the junction to the fuselage, as shown in Figure
Setup of experimental modal analysis.
The sensors were located in the center of both sides of the landing gear and separated by 20 mm, as shown in Figure
Locations of sensors.
The FRFs (frequency response functions) were computed by a data acquisition and analysis system LMS SCADAS III, and the experimental natural frequencies and modal shapes were obtained by modal analyses using LMS Test.Lab Structure analysis.
The first six modal shapes are shown in Figure
Modal shapes of the composite landing gear.
Table
Frequency versus mode number.
Mode  Frequency/Hz 

1  34.09 
2  78.52 
3  84.00 
4  137.81 
5  142.06 
6  214.93 
7  288.72 
8  356.38 
9  433.49 
10  435.64 
11  590.66 
12  658.78 
13  666.52 
14  769.27 
15  824.36 
16  961.63 
17  1068.71 
18  1165.20 
19  1172.93 
20  1303.38 
The nonlinear parameters were identified by nonlinear static test. The test was performed in fixed constraint at the junction to the fuselage, and vertical downward loads were applied on the junction to the wheel, as shown in Figure
Setup of nonlinear static test.
The displacement of the landing gear changed as the load changed. The loads varied from 100 N to 1000 N with increment of 100 N. The displacements were measured by height caliper, and the displacements under different loads are shown in Table
Results of the nonlinear static test.
Load/N  Displacement/mm 

100  2.68 
200  5.36 
300  7.98 
400  10.76 
500  14.12 
600  17.28 
700  19.35 
800  23.55 
900  25.49 
1000  28.61 
The nonlinear parameters of the composite landing gear were identified by (
The displacements of the landing gear under different loads were recalculated after the nonlinear parameters were identified. The results of the identification were compared with the displacementloading curve of the nonlinear static test, as shown in Figure
Results comparison of identification and test.
Figure
The tire modeling of the land gear is classical tire theory, which is based on the work of Mitschke and Wallentowitz [
Tire modeling.
The normal force applied to the tire is proportional to the penetration depth and the penetration rate, described as
The lateral force of the tire is described as a function of slip angle. The slip angle
The parameters of the tire used in our work were measured by experimental method and are shown in Table
Tire parameters.
Parameter  Value 

Radius  0.1 m 
Width  0.075 m 
Vertical stiffness  75000 N/m 
Lateral stiffness  166.55 N/° 
Mass  1.5 kg 

3.45 × 10^{−3} kg·m^{2} 

5.62 × 10^{−3} kg·m^{2} 
Damping  175 N/(m/s) 
The landing gear is designed to absorb and dissipate the kinetic energy of the landing impact, reducing the impact loads transmitted to the airframe. Drop impact analysis is the method most widely used to check performance of the landing gear. Therefore, three drop impact simulations of the composite landing gear were carried out with the nonlinear dynamic model identified in Section
The drop impact simulations were carried out with the identified nonlinear dynamic model. In the simulations, the composite landing gear was dropped from a certain height and the whole process dynamic response of the landing gear was calculated.
To solve the nonlinear dynamic model, (
The nonlinear part of (
Then, the modal space of (
Cacciola and Muscolino put forward a new approach to convert the modal space to iterative equation in discrete time [
Then, (
The solution process of (
In general, we will be searching for one or more solutions to the equation:
Consider the Taylorseries expansion of the function
Using only the first two terms of the expansion, a first approximation to the root of (
Such approximation is given by
The Jacobian matrix of
Then, the solution of (
The convergence criterion of the iterative process was based on the minimum 2norm. In the solution process of (
The drop tests were carried out with the system shown in Figure
System of drop tests.
The wheels were fixed to the bottom of the landing gear, and the equivalent mass of the aircraft was fixed to the junction to fuselage, as shown in Figure
Setup of drop tests.
In drop tests, the composite landing gear was dropped from the same height as the simulations, and impact loads of the ground were applied to the wheels. The whole process dynamic response was collected by sensors, and the data collection system was LMS SCADAS III.
To check the accuracy of the identified nonlinear dynamic model, three groups of drop impact analysis were carried out. Every group contained a drop simulation and a drop test. The drop simulations were based on the identified nonlinear dynamic model, and the experimental method mentioned above was used in the drop tests.
The analysis parameters of drop impact analysis are shown in Table
Parameters of the drop impact analysis.
Number  Equivalent mass/kg  Height/m  AOA/° 

A  120  0.136  0 
B  131  0.46  0 
C  136  0.46  12 
Since the fuselage of the aircraft was replaced by the equivalent mass, the dynamic responses of the equivalent mass were measured in the drop impact analysis. In drop analysis A, the dynamic responses of the equivalent mass after both the drop simulation and drop test were compared, and the results are shown in Figure
Comparison of results for drop analysis A.
Figure
The analysis parameters of the second drop impact analysis are different from those of the first analysis. The dynamic responses of the equivalent mass after both drop simulation and the drop test were compared; the results are shown in Figure
Comparison of results for drop analysis B.
Figure
In the third drop impact analysis, the dynamic responses of the equivalent mass after both the drop simulation and drop test were compared, and the results are shown in Figure
Comparison of results for drop analysis C.
Figure
A method is proposed in this paper to predict the dynamic performance of composite landing gear with uncertainties using experimental modal analysis data and nonlinear static test data. In the method, the nonlinear dynamic model of the composite landing gear is divided into two parts: the linear and the nonlinear parts. Experimental modal analysis is employed to predict the linear parameters with a frequency response function and the geometrically nonlinear parameters are identified by nonlinear static test with the nonlinear least squares method.
To check its accuracy and practicability, the method is applied to drop impact analysis of composite landing gear. Both simulations and tests are conducted for the drop impact analysis, and the errors of the analyses are extremely small. The maximum error of the simulation compared to the test is 4.33% in drop analysis A, the maximum error of drop analysis B is 5.36%, and the maximum error of drop analysis C is 4.55%. The results of the simulations are in good agreement with the test results, which shows that the proposed method can accurately model the dynamic performance of composite landing gear and the method is perfectly suitable for dynamic analysis of composite landing gear.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The project was supported by the Knowledge Innovation Program of the Chinese Academy of Sciences (Grant no. YYYJ1122) and the Innovation Program of UAV funded by the Chang Guang Satellite Technology Co., Ltd. The authors also gratefully acknowledge the support from the Science Fund for Young Scholars of the National Natural Science Foundation of China (Grant no. 51305421).