Wavelet Based Spectral Finite Elements For Wave Propagation Analysis In Isotropic, Composite And Nano-Composite Structures
Wave propagation is a common phenomenon in aircraft structures resulting from high velocity transient loadings like bird hit, gust etc. Apart from understanding the behavior of structures under such loading, wave propagation analysis is also important to gain knowledge about their high frequency characteristics, which have several applications. The applications include structural health monitoring using diagnostic waves and control of wave transmission for reduction of noise and vibration. Transient loadings with high frequency content are associated with wave propagation. As a result, the higher modes of the structure participate in the response. Finite element (FE) modeling for such problem requires very fine mesh to capture these higher modes. This leads to large system size and hence large computational cost. Wave propagation problems are usually solved in frequency domain using fast Fourier transform (FFT) and spectral finite element method is one such technique which follows FE procedure in the transformed frequency domain. In this thesis, a novel wavelet based spectral finite element (WSFE) is developed for wave propagation analysis in finite dimension structures. In WSFE for 1-D waveguides, the partial differential wave equations are reduced to a set of ODEs using orthogonal compactly supported Daubechies scaling functions for temporal approximation. The localized nature of the Daubechies basis functions allows finite domain analysis and imposition of the boundary conditions. The reduced ODEs are usually solved exactly, the solution of which gives the dynamic shape functions. The interpolating functions used here are exact solution of the governing differential equation and hence, the exact elemental dynamic stiffness matrix is derived. Thus, In the absence of any discontinuities, one element is sufficient to model 1-D waveguide of any length. This elemental stiffness matrix can be assembled to obtain the global matrix as in FE and after solution, the time domain responses are obtained using the inverse wavelet transform. The developed technique circumvents several serious limitations of the conventional FFT based Spectral Finite Element (FSFE). In FSFE, the wave equations are reduced to ODEs using FFT for time approximation. The remaining part of the formulation is quite similar to that of WSFE. The required assumption of periodicity in FSFE, however, does not allow modeling of finite length structures. It results in “wrap around” problem, which distorts the response simulated using FSFE and a semi-infinite (“throw-off”) element is required for imparting artificial damping. This artificial damping occurs as the “throw off” element allows leakage of energy. In some cases, a very high damping can also be considered instead of “throw off” element to remove wrap around effects. In either cases, the damping introduced is much larger than any inherent damping that may be present in the structure. It should also be mentioned that even in presence of the artificial damping, a larger time window is required for removing the distortions completely. The developed WSFE method is completely free from such problems and can efficiently handle undamped finite length structures irrespective of the time window considered. Apart from this, FSFE allows imposition of only zero initial condition and in contrary any initial conditions can be used in WSFE. Though FSFE has problem in modeling finite length undamped structures for time domain analysis, it is well suited for performing frequency domain study of wave characteristics, namely, the determination of spectrum and dispersion relations. WSFE is also capable of extracting these frequency dependent wave properties, however only up to a certain fraction of the Nyquist frequency. This constraint results from the loss in frequency resolution due to the increase in time resolution in wavelet analysis, where the basis functions are bounded both in time and frequency. A price has to be paid in frequency domain in order to obtain a bound in the time domain. The consequence of this analysis is to impose a constraint on the time sampling rate for the simulation with WSFE, to avoid spurious dispersion. WSFE for 2-D waveguides are formulated using Daubechies scaling functions for both temporal and spatial approximations. The initial and boundary conditions, however, are imposed using two different methods, which are wavelet extrapolation technique and periodic extension or restraint matrix respectively. The 2-D WSFE is bounded in both the spatial directions unlike 2-D FSFE, which is essentially unbounded in one spatial direction. Apart from this, 2-D WSFE is also free from “wrap around” problem similar to 1-D WSFE due to the localized nature of the basis functions used for temporal approximation. In this thesis, WSFE is developed for isotropic 1-D and 2-D waveguides for time and frequency domain analysis. These include elementary rod, Euler-Bernoulli and Timoshenko beams in 1-D modeling, and plates and axisymmetric cylinders in 2-D modeling. The wave propagation responses simulated using WSFE for these waveguides are validated using FE results. The advantages of the proposed technique over the corresponding FSFE method are also highlighted all through the numerical examples. Next part of the thesis involves the extension of the developed WSFE technique for modeling composite and nano-composite structures to study their wave propagation behavior. Due to their anisotropic nature, analysis of composite structures, particularly high frequency transient analysis is much more complicated compared to the corresponding metallic structures. This is due to the presence of stiffness coupling in these structures. Superior mechanical properties of composites, however, are making them integral parts of an aircraft and thus they often experience such short duration, high velocity impact Loadings. Very few literatures report the response of composite structures subjected to such high frequency excitations. Here, WSFE is formulated for a higher order composite beam with axial, flexural, shear and contractional degrees of freedom. WSFE is also formulated for composite plates using classical laminated plate theory with axial and flexural degrees of freedom. Simulations performed using these WSFE models are used to study the higher order and elastic coupling effects on the wave propagation responses. Carbon nanotubes (CNTs) and their composites are attracting a great deal of experimental and theoretical research world-wide. The recent trend in the literature shows a great interest in the dynamic and wave characteristics of CNTs and nano-composites because of their several applications. In most of these applications, CNTs are used in the embedded form as it does not requires precise alignment of the nano-tubes. In addition, the extraordinary mechanical properties of CNTs are being exploited to achieve high strength nano-composite. Apart from the experimental studies and atomistic simulation to study the mechanical properties of CNTs and nano-composites, continuum modeling is also receiving much attention, mainly due to its computational viability. In this thesis, a 1-D WSFE is formulated for multi-wall carbon nanotube (MWNT) embedded composite modeled as beam using higher order layer-wise theory. This theory allows to model partial interfacial shear stress transfer, which normally occurs due to improper dispersion of CNTs in nano-composites. The effects of different matrix materials and fraction of shear stress transfer on the wave characteristics are studied. The responses obtained using other beam theories are also compared. The beam modeling does not allow capturing the radial motions of the CNT, which are important for several applications. These can be effectively captured by modeling the CNT using a 2-D axisymmetric model. Hence, a 2-D WSFE model is constructed to capture the high frequency characteristics of single-walled carbon nanotubes (SWNTs). The response of SWNT simulated using the developed model is validated with experimental and atomistic simulation results reported in the literature. The comparison are done for dispersion relation and also radial breathing mode frequencies. The effects of geometrical parameters, namely the radius and the wall thickness of the SWNT on the higher radial, longitudinal and coupled radial-longitudinal vibrational modes are analyzed. These behaviors are studied in both time and frequency domains. Such time domain analyses of finite length SWNT are not possible with the Fourier transform based techniques reported in literature, although, such analyses are important particularly for sensor applications of SWNT. Spectral finite element method is very much suited for solution of inverse problems like force reconstruction from the measured wave response. This is because the technique is based on the concept of transfer function between the displacements (output) and applied forces (input). In the present work, WSFE is implemented for identification of impact force from the wave propagation responses simulated with FE and used as surrogate experimental results. The results show that WSFE can accurately reconstruct the impulse load applied to 1-D waveguides which include rod, Euler-Bernoulli beam and connected 2-D frame, even with highly truncated response. This is unlike FSFE, where the accuracy of the identified force depends largely on the time window of the measured responses. The detection of damage from the wave propagation analysis is another class of inverse problems considered in this thesis and is of utmost importance in the area of aircraft structural health monitoring. Here, the detection scheme is based on arrival time of the waves reflected from the damage. A novel detection technique based on wavelet filtering is proposed here and it is shown to work efficiently even in the presence of noise in the measured wave responses. Detection of damage requires an efficient damage model to simulate the mode of structural failure. In this regard, two spectrally formulated wavelet elements are proposed, one to model isotropic beam with through-width notch and the second to model composite beam with embedded de-lamination. In the first case, the response of the damaged beam is considered as the perturbation of the undamaged response and the linear perturbation analysis leads to a completely new set of dynamic stiffness matrix. In the second case, the delamination is modeled by subdividing the de-laminated region into separate waveguides and full damage model is established by imposing the kinematics. These models help to simulate wave propagation in such damaged beams to study the effect of damage on the wave response. Noise and vibration are often transmitted from the source to the other parts of the structure in the form of wave propagation. Thus, control of such wave transmission is essential for reduction of noise and vibration, which are the main cause of discomfort and in many cases cause failure of structure. Here, techniques for both passive and active controls of wave are proposed. For active control, a closed loop system is modeled using WSFE with magnetostrictive actuator for control of axial and flexural wave propagations in connected isotropic 1-D waveguides. The feedback is negative velocity and/or acceleration measured at different sensor points. A very new application of CNT reinforced composite for passive control of vibration and wave response is explored in this thesis. For this, a novel concept of nano-composite inserts is proposed. This insert can be made from CNTs dispersed in polymer. The high stiffness of the inserts helps to regulate the power flow in the form of wave propagation from the point of application of the loads to other parts of the structures. The length of the insert, volume fraction of CNTs and position are changed to achieve the required reduction in wave amplitudes. The entire thesis is split up into eight chapters. Chapter 1 presents a brief introduction, the motivation and objective of the thesis. Chapters 2 and 3 give a detail account of wavelet spectral finite element formulation for 1-D and 2-D isotropic waveguides, while Chapter 4 gives the same for composite waveguides. Chapter 5 brings out essential wave characteristics in carbon nanotubes and nano-composite structures, while Chapters 6 and 7 exclusively deal with application of WSFE to some real world problems. The thesis ends with summary and directions of future research. In summary, the thesis has brought out several new aspects of wave propagation in isotropic, composite and nano-composite structures. In addition to establishing wavelet spectral finite element as a useful tool for wave propagation analysis, several new techniques are presented, several new algorithm are proposed and several new concepts are explored.