NUMERICAL AND EXPERIMENTAL STUDIES ON VIBRATION ANALYSIS OF AIRBORNE EXTERNAL ANTENNA TOWARDS

STRUCTURAL INTEGRITY ASSESSMENT

Nagesh R. Iyer†, G.S. Palani‡, J. Rajasankar‡, N. Gopalakrishnan‡ and G. Sivagnanam† - Director and Corresponding Author, ‡ - Scientist

CSIR-Structural Engineering Research Centre, CSIR Campus, Taramani, Chennai -600 113.

- Scientist, Centre for Air-Borne Systems, Bangalore-560 037.

ABSTRACT A brief explanation of procedure for numerical and experimental investigation of complex engineering structural components has been presented. Details of the finite element analysis of an antenna panel for conducting vibration analysis to extract natural frequencies have been discussed. The panel is discretized with a combination of hexahedron, tetrahedron and wedge finite elements. The mass of the panel is idealised to be consistent with the stiffness representation. Additional non-structural masses due to that of equipment supported by the panel are lumped at specific locations on the panel. For the experimental investigations, the antenna panel has been subjected to sinusoidal/random vibration tests. Resonance search tests have been performed to estimate the natural frequencies and mode shapes of the panel within the excited range of frequencies. Resonant search test has been conducted with a constant base acceleration of 0.5g or less for a frequency band of 5 – 500 Hz in three orthogonal directions using a uni-axial slip table system after rigidly fixing the model on to the table. Resonant frequency is based on the peaks from the frequency response function of response acceleration with reference to base acceleration. Keywords Finite element method; Stiffness; Mass; Damping; Vibration analysis; Antenna panel INTRODUCTION The structural components in many industrial structures such as offshore platforms, ship hulls, box girder bridges and aerospace structures are often subjected to dynamic loads due to machinery, crane operations, helicopter landing, impact shock, blast, etc. The design of these structures involves detailed analysis for static, free vibration and dynamic responses. Free vibration or dynamic response analysis is an order of magnitude more complex than the static analysis and it involves considerable amount of computational effort. Analysis of such complex and large size problems can only be carried out by using high speed computers employing numerical methods such as the finite element method (FEM). Proper representation of stiffness, mass and damping properties in finite element analysis (FEA) is essential for accuracy and reliability of dynamic responses. This necessitates utmost care in modelling the structural components and its associated attributes. The objective of this paper is to provide brief explanation of procedure for numerical and experimental investigation of complex engineering structural components. Details of the finite element analysis of an antenna panel for conducting vibration analysis to extract natural frequencies have been discussed. The panel is discretized with a combination of hexahedron, tetrahedron and wedge finite elements. The mass of the panel is idealised to be consistent with the stiffness representation. Additional non-structural masses due to that of equipment supported by the panel are lumped at specific locations on the panel. For the experimental investigations, the antenna panel has been subjected to sinusoidal/random vibration tests. Resonance search tests

have been performed to estimate the natural frequencies and mode shapes of the panel within the excited range of frequencies. Resonant search test has been conducted with a constant base acceleration of 0.5g or less for a frequency band of 5 – 500 Hz in three orthogonal directions using a uni-axial slip table system after rigidly fixing the model on to the table. Resonant frequency is based on the peaks from the frequency response function of response acceleration with reference to base acceleration. METHODOLOGIES FOR DYNAMIC ANALYSIS The governing equation of motion for a structure subjected to dynamic forces can be expressed in matrix form as

( )tF = uK + u C + u M &&& (1) where M is the mass matrix, C is the damping matrix and K is the stiffness matrix. , and u are the generalized acceleration, velocity and displacement vectors respectively and F(t) is the time-dependent applied force vector. In FEA overall matrices for the structure are formed by assembling the relevant element matrices (Zienkiewicz and Taylor, 2000). A brief discussion on the formulation of stiffness, mass and damping matrices are presented below followed by the solution procedures for free vibration and dynamic response analysis.

u&& u&

Stiffness Representation Structural components in most of the real life structures are generally made up of beams and plates/shells with and without stiffeners. For the purpose of FEA, it is necessary to identify and use finite elements (beam, plate/shell and stiffener elements) which will be reliable and economical in representing the structural behaviour of the structural components. An exhaustive study on the finite element modelling of beams, plates and stiffened plate/shell panels was reported by Palani et al., 1990. Based on the study, an efficient stiffened plate/shell finite element model with provisions for locating the stiffeners arbitrarily was developed. The model as shown in Fig. 1 was derived by combining the 9-noded Lagrangian isoparametric plate/shell finite element with the 3-noded isoparametric beam element by employing suitable transformations without increasing DOF. Transformation matrices were developed (Palani et al., 1993) to take care of eccentricity of stiffeners with respect to the plate mid-surface and to consider arbitrary location of stiffeners in the plate element.

Parent element Fig. 1 Arbitrarily Located Stiffened Plate/Shell Model

Mass Representation It is well known from the literature (Bathe, 2000 and Iyer et al., 2003) that different methodologies exist to formulate element mass matrices. The familiar and preferred approach is to represent mass either in "consistent" form or in "lumped" form. The representation of the mass of the structural elements plays an important role in the computation of the natural

frequencies (in the high frequency range) and the dynamic responses accurately with least computational effort. Four different mass lumping schemes are generally in practice to conduct dynamic analysis. These are (i) consistent mass matrix (CMM), (ii) consistent diagonal lumping (CDL), (iii) proportional lumping (PL) and (iv) equal lumping (EL). Among these schemes CMM has been derived by assuming the variation of acceleration field over an element to the same as the variation of displacement field over that element. Hence, the shape functions that describe the displacement field over the element have been used in deriving the kinetic energy and the element mass matrices. In the CDL scheme there are only diagonal elements and they are obtained by scaling the diagonal entries of CMM, and the total mass of the element is conserved. Use of Lobatto integration points for the Lagrangian elements to evaluate CMM also results in CDL scheme (Iyer et al., 2003). Proportional and equal lumping schemes are based on physical lumping of the mass to the element nodes. Hence the mass matrices obtained by PL and EL are also diagonal matrices. In diagonal lumping schemes, the effect of rotary inertia is not considered. It may be noted that the total mass is preserved in all the mass schemes. However, lumping as well as the distribution of the mass of the structural elements influence the free vibration as well as the dynamic responses. The advantage of having diagonal mass matrix in an eigenvalue analysis is the ease and reduction of computational effort as well as requirement of lesser storage space. Damping Representation In general the structural damping can be assumed to be viscous, i.e., velocity proportional. With this assumption, the damping matrix C is generally formulated a combination of fraction of mass and stiffness matrices. Such a damping, known as 'Rayleigh Damping', can be expressed as

C = α M + β K (2) where α and β are scalar constants. Considering the equation of motion (eqn (1)) with null force vector and substituting for C from eqn (2)

M && + [α M + β K] + K u =0 (3) u &uTo determine the damping matrix C, the values of α and β must be found. For this, the set of equations in eqn (3) have to be uncoupled and any two uncoupled equations can be used to evaluate the two unknowns α and β (Bathe, 2000). It is usual to use the lowest two frequencies and damping factors to determine α and β. Free Vibration Analysis The free vibration analysis of structures is generally carried out by solving the generalized eigenvalue problem. A generalised eigenvalue problem for undamped free vibration analysis can be expressed as

K φ = λ M φ (4) Various algorithms have been developed and are still being developed for the extraction of eigenvalues and the corresponding eigenvectors. Depending upon the property of the eigenvector used in these algorithms, they may be broadly classified into three categories viz., (i) vector iteration methods, that is forward iteration, wherein the vector on L.H.S. of eqn (4) is assumed or inverse iteration, wherein the vector on R.H.S of eqn (4) is assumed. (ii) transformation methods using the property of

φT K φ = Λ and φT M φ = I (5) is taken advantage of in trying to find an orthogonal matrix φ which will transform the K matrix into a diagonal matrix whose elements are the eigenvalues of the system. The transformation methods usually adopted are the Jacobi transformation and the Householder-QR-inverse (HQRI) iteration scheme. HQRI scheme is designed to solve standard

eigenproblems. Jacobi method can be used to solve the generalised eigenproblems as well. (iii) polynomial Iteration technique where the property

⎢K - µ M ⎪ = 0 when µ = λ (6) is used to extract the eigenvalues. Combining one or more of these techniques several algorithms were developed. Some of them, generally used (Hughues, 1989) for FEA are determinant search method, subspace iteration method and Lanczos method. The purpose of the analysis is to compute undamped natural fundamental frequencies of the panel structure. The analysis is performed by using a sophisticated finite element model employing 3-D solid elements. The overall stiffness and strength of all the structural members in the structure are suitably accounted for in the analysis. The analysis computes deflections at the nodes, and stresses in the elements/members. The second phase relates to the undamped free vibration analysis of the panel structure to compute its natural frequencies.

FINITE ELEMENT ANALYSIS OF THE ANTENNA PANEL Basic parameters and different views of the antenna panel are shown in Figs. 3.1 (a) to 3.1 (b). Vibration analysis of the structure has been carried out by considering the same finite element model as that used for linear static finite element analysis. The models are generated by using 3-D solid elements.

l

704.01

865.76

The extent and finenescomplexity of the strucexample, local analysis members/regions, since on the other hand, for thpost- processors would busing the pre-processor.correctness and details oimportant to consider adthe region of interest butto minimize the effects structure, identification important role in the acseveral important aspec

Fig. 1 Antenna Pane

s of the finite element model to be considered depend on the ture, area of interest and the accuracy of the results required. For would be able to provide better accuracy of forces in the primary the structure is represented in the idealization with high refinement; e local analysis general purpose finite element software with pre- and e required. The finite element model could be developed with ease

One can view the model in any orientation in order to judge the f modelling and thereby improve the accuracy of results. It is also

equate representation of the structure as well as loads not only within also to sufficient extent on either side of the region of interest so as of assumptions in boundary conditions. Proper idealization of the

of nodal points and suitable order of numbering of nodes play an curacy as well as the computational cost of the solution. There are ts, which must be considered in the idealization of the antenna

structure as a three-dimensional model. The stiffness and strength of the material/attachments used are modeled/represented in a scientific and judicious manner. The standard checks for volume, aspect ratio, etc. are made before finalizing and going ahead with the investigation. 4.1 FE Modelling The accuracy of results of the analyses depends considerably on the expertise and engineering judgement employed in idealizing the structure. The software employed has a special pre-processor that facilitates the necessary input to the analysis program to be generated with minimum of effort. The geometric properties are accounted appropriately to represent the actual position in the structure. A generalized and well-accepted approach is followed for the analysis of the antenna panel. The overall stiffness and strength of all the components/parts are suitably accounted for in the analysis. The solid element is defined by eight nodes with each node having three degrees of freedom (DOF), namely three translations. The geometry, node locations, and the coordinate system for this element are shown in Fig. 2(a). The element is defined by eight nodes and isotropic material properties. The notations for stresses and the stress actions in a solid element are shown in Fig. 2(b). As per this, the panel and its sub-components are discretised by using solid finite elements. The model for the analysis represents the antenna structure. It is ensured that all the geometry has been modelled. The formulation of 8-noded solid element is based on the assumption that the material is elastic and isotropic in this particular investigation. All the non-structural components have been represented by lumped masses at appropriate locations.

(a) Local System of Axes & Notations (b) Notations for Stresses

Fig. 2 Notations for Solid Element

Figs. 3 and 4 give the details of idealization of the antenna panel structure as a 3-D solid model showing the nodes and elements. This model has been arrived at after a number of trials and satisfying the convergence requirements on results. Zoomed views of typical locations have also been shown for clarity. The details of the finite element model, material properties and mass are given below.

Material : Aluminium 2024-T351 Density : 2780 kg/m3

Tensile yield strength : 324 MPa Modulus of elasticity : 73.1 GPa Poisson’s ratio : 0.33 Acceleration : 9.81 m/sec2

Total mass of the component : 8.18 kg Structural mass : 7.38 kg Additional mass : 0.80 kg

Before proceeding with the finite element analyses, quality of the mesh/model generated has been checked. Quality and accuracy of developed FE model has been carried out with respect to face corner angle, aspect ratio and shape factor. The details are furnished in Table 1. The characteristics of finite element mesh used for analyses and details of the quality check are given below:

Front view Zoomed view

Fig. 3 FE Mesh – Front view

Zoomed View

Front View

Rear View Zoomed View

Rear View

Zoomed View Fig. 4 FE Mesh – Rear view FE Mesh Characteristics Hexahedron elements = 139349 (76.84 %) Tetrahedron elements = 34780 (19.18 %) Wedge elements = 7227 (3.98 %) Total number of finite elements = 181356 Number of nodes = 245830 Number of equations = 737490

Table 5.1 Quality check of the finite element mesh

Limit No. of failing elements Criteria Element type

Min Max Number % Hexahedron 10 160 410 0.29

Wedge 10 160 7 0.09 Face Corner Angle (degrees)

Tetrahedron 5 170 26 0.07 Aspect Ratio Max. 10 2112 0.011

Hexahedron Wedge

Not applicable Shape Factor

Tetrahedron 0.0001 2 0.005

Boundary conditions: All the degrees of freedom of the nodes at 6 support points each located at top and bottom

faces are fixed All the degrees of freedom of the nodes at 13 support points in the rear face of the panel

are fixed Undamped free vibration analysis has been carried out for the panel. Table 2 shows the computed frequency values for the first ten modes. Fig. 5 shows the plot of mode shape for modes 1 to 3 respectively.

Table 2 Computed Frequencies

Mode No. Frequency (Hz) Mode No. Frequency

(Hz) 1 279.01 6 628.40 2 346.53 7 632.76 3 537.93 8 641.42 4 591.51 9 663.68 5 602.00 10 680.66

Mode1 f=279.01Hz Mode 2 f=346.53Hz

Mode 3 f=537.93Hz Mode 4 f=591.51Hz

Fig. 5 Natural mode shapes of the antenna panel

The mode shape indicates the expected trend for the given geometry and support conditions of the panel. The fundamental natural frequency of the panel is obtained as 279.01 Hz.

EXPERIMENTAL STUDIES The antenna panel has been subjected to sinusoidal/random vibration tests. Resonance search tests have been performed to estimate the natural frequencies and mode shapes of the panel within the excited range of frequencies. Resonant search test has been conducted with a constant base acceleration of 0.5g or less for a frequency band of 5 – 500 Hz in three orthogonal directions using a uni-axial slip table system after rigidly fixing the model on to the table. Resonant frequency is based on the peaks from the frequency response function of response acceleration with reference to base acceleration. Resonant search test Resonant search test is performed through (a) Experimental modal analysis (EMA) technique, under SISO (single input, single output)

type using a roving impact hammer and a fixed acceleration response with a light-weight accelerometer. This is also equivalent to a test of fixed excitation and roving response. This test is conducted only in the out-of-plane direction of the panel (Z-direction).

(b) Base excitation through a slip-table under sweep sine excitation of 0.5g peak acceleration for a varying frequency band of 5-500Hz and a low sweep rate of 0.10Hz/sec frequency build-up. This is to ensure zero frequency distortion in the frequency response curve. These tests are conduced on all three orthogonal directions.

The reasons for conducting resonant search tests using two methods is (i) The out-of-plane resonant frequencies, as obtained by the resonant search tests are likely

to be influenced by the stiffness of the fixture, which will be a finite value. (ii) The frequencies, which cannot be excited by base excitation, are likely to be excited by

EMA tests. Test facilities include

(a) LDS make HBT-900 – SB seismic mass slip table (b) Vibrator and a power amplifier. (c) LDS make random vibration controller and Siemens make Computer. (d) PCB/ B&K make accelerometers and associated conditioners (e) Dual channel AND-3524 fast fourier transform analyser.

The panel was subjected to base excitation of 0.5g or less in a sweep sine excitation mode of frequency range 5.0Hz to 500Hz with a linear sweep rate of 0.05Hz/sec, frequency variation for all the three orthogonal directions of motion. Pre-test standard ambient check Pre-test was conducted to include calibration of measuring equipment and estimation of the dynamic characteristics of the bare shaking table to identify its response to any specified random input, such that these frequencies do not interfere within the frequency range of interest. Experimental modal analysis In this test, the antenna panel was rigidly connected to the side angles, by means of screws and bolts, at specified locations, leaving a 10 – 20mm gap at the bottom. The angles in turn are bolted to the fixture plate, through adjustable oblique holes. The whole arrangement was fixed onto the LDS-make slip table available at the “Advanced Seismic Testing and Research Lab” of CSIR-SERC. B&K accelerometer was fixed at a critical location. The instrumented impact hammer was used to excite the structure at as many points as possible. The resulting frequency response function was collected and plotted. Fig. 6 show the details of the fixture drawings for the integrated PR-SSR antenna array assembly for vibration tests in Z-direction. Similar fixtures were arranged for vibration tests in X- and Y-directions.

Sweep sine base excitation After rigidly fixing the panel onto the slip-table assembly, the panel was subjected to base excitation of 0.5g or less in a swept sine excitation mode of frequency range 5.0Hz to 500Hz with a linear sweep rate of 0.10Hz/sec, frequency variation. This frequency variation has given sufficient time for vibration to build-up and dwell at resonance. The test was conducted with the test item hard-mounted. The purpose of the test was to ensure that the fundamental resonant frequency recorded did not coincide with the platform fundamental frequency. Two piezo-electric based accelerometers were mounted, one for measurement of applied excitation and the other for collecting the response of the antenna panel. The accelerometers were connected to suitable charge-based conditioners and the conditioned output was connected to the dual channel fast fourier transform (FFT) analyser. The spectrum averaging was done on the collected data and the frequency response function of the antenna panel was given as the result. A control accelerometer was used in addition to the above two accelerometers for controlling the base input to the specimen. As the slip-table is uni-axial, test was conducted on all the three orthogonal directions after suitably re-orienting the specimen. Counter-sink Holes at 150 Crs in Both

Directions Note: Bottom Plate face resting on slip table

should be truly planar and any bowing due to welding is not allowed

AB Note: Side Plate should be Truly Vertical 1100

890 75

75

260 600

8 8

16

10 Typical Counter-sink to

accommodate M8 Allen Bolts

Bottom Plate : 16 mm Thick 860 X 890 mmBottom

Plate (BP)

Side Plate (SP1)

Bracket (AB)

BP

SP

Fig. 6 Fixture details for vibration test – Z direction excitation

Initial resonant search test – Experimental modal analysis (EMA) The initial resonant search test was conducted for two different boundary conditions, namely, (a) Specified supports in the shorter and longer edges and (b) Specified supports only in the shorter edges. Photographs of test set up are shown in Figs. 7 and 8. For convenience, the positions of impact hammer were varied, but the position of response accelerometer was kept constant. Figs 9 and 10 give the frequency response function in terms of m/sec2/N, for the

antenna panel with boundary conditions (a) and (b) giving both the magnitude and imaginary parts. The imaginary or quadrature part of Frequency Response Function (FRF) is closer to the mode shape of the panel for the fundamental frequency of 286.25 Hz. In the case of all edge support condition, only few connections were made along longitudinal edges and there was no strut to arrest deformation at centre. This was not significantly different from short edge support conditions. The first, second and third frequencies and the corresponding made shapes are nearly the same in both the panels tested.

Final Resonance Search Test The test specifications and conduct of the test was similar to the initial resonance search test and only sweep sine tests were performed.

Fig. 7 Integrated PR-SSR Antenna Array Assembly with Accelerometer (X-direction)

Fig. 8 Integrated PR-SSR Antenna Array Assembly with Accelerometer (Z-direction)

Fig. 9 Response (magnitude and imaginary) of the integrated PR-SSR antenna array assembly with additional long side support (Boundary

condition (a)) during roving hammer test

Fig. 10 Response (magnitude and imaginary) of the integrated PR-SSR antenna array assembly with only short side support (Boundary condition (b)) during

roving hammer test SUMMARY AND CONCLUDING REMARKS Airborne external antenna panel has been modeled using 3-D solid elements for carrying out finite element analyses. The choice of the finite element mesh with 3-D solid elements is made due to the complex variations in the geometry of the panel structure. Proper care is

taken while developing the finite element mesh. In addition, the quality checks provided by the finite element analysis software is carried out to ensure overall performance of the model. Undamped free vibration analysis is carried out to extract the first ten frequencies in the second phase of the analysis. The finite element mesh as employed for static analysis is used for free vibration analysis also. First ten undamped natural frequencies of the panel are estimated to be in the range of 279Hz to 680Hz. The fundamental frequency of 285 Hz, as estimated in the out-of-plane Z-direction for the antenna panel, is in good agreement with the theoretically computed value of 279Hz. The fundamental frequency is widely separated from the frequency of antenna assembly (theoretically estimated to be around 20Hz). Hence, local resonances in individual antenna panels may not happen. Acknowledgements The authors thank the Director, SERC for the support provided to carry out the activities of the project. Authors acknowledge the useful technical discussions with Dr. S. Christopher, Director, Centre for Air-Borne Systems, Bangalore. The authors would like to thank their colleagues Shri A. Rama Chandra Murthy and Ms. Smitha Gopinath for useful technical discussions. References

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Iyer, N.R., G.S. Palani and T.V.S.R. Appa Rao (2003), “Influence of mass representation schemes on vibration characteristics of structures”, Jl. Institution of Engineers (India), 84, pp. 19-26.

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Palani, G.S., N.R. Iyer and T.V.S.R. Appa Rao (1993), An efficient finite element model for static and vibration analysis of plates with arbitrarily located eccentric stiffeners, J. Sound Vibr., 166, 409-427.

Robert D. Blevins, Formulas for natural frequencies and mode shape, Van Nostrand Reinhold Company, 1979.

Zienkiewicz, O.C. and R.L. Taylor, The Finite element method, Vol. I: The basis, Vol. II: Solid mechanics, Butterworth-Hieneman Ltd., 2000.