1,721,016 research outputs found
Propagation in waveguides with slowly changing variability
No full textThis thesis investigates structural wave propagation in waveguides with randomly varying material and geometrical properties along the axis of propagation, specifically when the properties vary slowly enough such that there is no or negligible backscattering due to any changes in the propagation medium. This variability plays a significant role in the so called mid-frequency region, but wave-based methods are typically only applicable to homogeneous and uniform waveguides.An analytical tool, the WKB (after Wentzel, Kramers and Brillouin) approximation, is used in order to find a suitable generalisation of the wave solutions for finite waveguides undergoing longitudinal and flexural motion. An alternative wave formulation approximation with piecewise constant properties is derived so that the internal reflections are taken into account, requiring a discretisation of the waveguide. In addition, a Finite Element approximation using an enriched hierarchical basis or Hierarchical Finite Element (HFE) is created, where the variability in the properties of the waveguide is included within the element formulation, thus not requiring a mesh discretisation as opposed to a standard FE solution.A Fourier like series, the Karhunen-Loeve expansion, is used to represent homogeneous and spatially correlated randomness and statistics of the natural frequencies and forced response are derived. Experimental validation is carried out, using firstly a cantilever beam with small masses attached along its length according to a given random field. In the second experiment, an ensemble of glass-fibre reinforced free-free beams, whose variability is characterised by light transmissibility images, is measured. It has been found that the correlation length of the random fields or the scale of the spatial fluctuation is shown to play an important role in the dynamic response statistics. Moreover, the proposed formulations show good agreement with the standard approaches but at a fraction of the computational cost, providing a good framework for uncertainty quantification
Wave propagation in slowly varying one-dimensional random waveguides using a finite element approach
No full textThis work investigates structural wave propagation in one-dimensional waveguides with randomly varying material and geometric properties along the axis of propagation, specifically when the properties vary slowly enough such that there is negligible backscattering due to any changes in the properties of the medium. This variability plays a significant role in the so-called mid-frequency region, but wave-based methods are typically only applied to homogeneous and uniform waveguides. The WKB (after Wentzel, Kramers and Brillouin) approximation can be used to find a suitable generalisation of the wave solution in terms of the change of phase and amplitude of a wave propagating through a non-uniform waveguide, but it is typically restricted to analytical solutions of the equation of motion. In this paper a Wave and Finite Element (WFE) approach is proposed to extend the applicability of the WKB method to cases where no analytical solution is available. The wavenumber is expressed as a function of the position along the waveguide and a Gauss-Legendre quadrature scheme is used to the numerically integrate the phase. The WFE method is used to evaluate the wavenumbers at each integration point, and these are kept to a minimum to minimise computation cost while being able to capture the non-homogeneity to a given accuracy. The wave amplitude is calculated using conservation of power flow. The numerical example of a straight rod with a single propagating wave mode is considered. Random field properties are expressed in terms of a Karhunen-Loeve expansion. The forced response to a point excitation is calculated and results are compared to a standard Finite Element (FE) approach and to the WKB analytical solution. Results show good agreement and require only a few WFE evaluations, providing a suitable framework to account for spatially correlated randomness in waveguides
Flexural wave propagation in slowly varying random waveguides using a finite element approach
No full textThis work investigates structural wave propagation in waveguides with randomly varying properties along the axis of propagation, specifically when the properties vary slowly enough such that there is negligible backscattering. Wave-based methods are typically applied to homogeneous waveguides but the WKB (after Wentzel, Kramers and Brillouin) approximation can be used to find a suitable generalisation of the wave solution in terms of the change of phase and amplitude, but is restricted to analytical solutions. A wave and finite element (WFE) approach is proposed to extend the applicability of the WKB method to cases where no analytical solution is available. The wavenumber is expressed as a function of the position along the waveguide and a Gauss-Legendre quadrature scheme is used to obtain the phase change while the wave amplitude is calculated using conservation of power. The WFE method is used to evaluate the wavenumbers at each integration point. The flexural vibration example is considered with random field proprieties being expressed by a Karhunen-Loeve expansion. Results are compared to a standard FE approach and to the WKB analytical solution. They show good agreement and require only a few WFE evaluations, providing a suitable framework for spatially correlated randomness in waveguides
Natural frequency statistics of waveguides with slowly changing spatially correlated material variability
No full textThis paper concerns wave propagation in random waveguides and, in particular, when there is spatially correlated variability in the material and/or geometric properties. The one dimensional waveguide is modelled using an analytical formulation for wave propagation and a formulation for the wavenumber is given considering slowly varying proprieties, i.e. the change given by the random field is such that there is no backscattering from a propagating wave. When propagating over a finite distance, the total phase change of a wave is given by the integral of the spatially distributed wavenumber. Spatial correlation is then seen to be important in determining the statistics of this phase (mean, variance, etc.). This information can in turn be used to calculate the natural frequencies and forced response of finite structures. Longitudinal motion in a finite length thin rod is then considered and, given a second order homogenous random field with a certain kind of autocorrelation function, an analytical solution for its Karhunen-Loeve expansion is used for deriving an expression for its natural frequencies. The slowly varying proprieties assumption is used in order to find an analytical expression the for probability density function of the natural frequencies. A FE model of the waveguide is also assembled, with the random field discretized on the FE mesh and assumed piecewise constant within the element proprieties. Monte Carlo sampling is used to evaluate statistics of the natural frequencies and the results are compared to those obtained by the wave approach
Structural vibration analysis with random fields using the hierarchical finite element method
No full textElement-based techniques, like the finite element method, are the standard approach in industry for low-frequency applications in structural dynamics. However, mesh requirements can significantly increase the computational cost for increasing frequencies. In addition, randomness in system properties starts to play a significant role and its inclusion in the model further increases the computational cost. In this paper, a hierarchical finite element formulation is presented which incorporates spatially random properties. Polynomial and trigonometric hierarchical functions are used in the element formulation. Material and geometrical spatially correlated randomness are represented by the Karhunen–Loève expansion, a series representation for random fields. It allows the element integration to be performed only once for each term of the series which has benefits for a sampling scheme and can be used for non-Gaussian distributions. Free vibration and forced response statistics are calculated using the proposed approach. Compared to the standard h-version, the hierarchical finite element approach produces smaller mass and stiffness matrices, without changing the number of nodes of the element, and tends to be computationally more efficient. These are key factors not only when considering solutions for higher frequencies but also in the calculation of response statistics using a sampling method such as Monte Carlo simulation
Wave propagation in slowly varying waveguides using a finite element approach
No full textThis work investigates structural wave propagation in one dimensional waveguides with randomly varying properties along the axis of propagation, specifically when the properties vary slowly enough such that there is negligible backscattering, even if the net change is large. Wave-based methods are typically applied to homogeneous waveguides but the WKB (after Wentzel, Kramers and Brillouin) approximation can be used to find a suitable generalisation of the wave solution in terms of the change of phase and amplitude but is restricted to analytical solutions. A wave and finite element (WFE) approach is proposed to extend the applicability of the WKB method to cases where no analytical solution of the equations of motion is available. The wavenumber is expressed as a function of the position along the waveguide. A Gauss-Legendre quadrature scheme is subsequently used to obtain the phase change, while the wave amplitude is calculated using conservation of power. The WFE method is used to evaluate the wavenumbers at each integration point. Moreover, spatially correlated randomness can be included in the formulation by random field properties and in this paper is expressed by a Karhunen-Loève expansion. Numerical examples are compared to a standard FE approach and to available analytical solutions. They show good agreement when compared to either a full FE or analytical solution and require only a few WFE evaluations, providing a suitable framework for efficient stochastic analysis in waveguides
An experimental investigation of the natural frequency statistics of a beam with spatially correlated random masses
No full textExperimental investigations into the dynamic response of structures with material or geometrical random fields usually depend upon an initial characterization of this variability, with very little control over the statistics at its early manufacturing stage. This provides the need of a minimal number of samples to generate an ensemble of dynamic responses, making such experimental data scarcely found in the literature. In this work, a cantilever beam with small masses attached along its length according to a given discrete random field has an ensemble of natural frequencies measured for a number of correlation lengths. The results can be used to investigate the effects of the correlation length on the subsequent natural frequency statistics. The experimental results are compared with a wave approximation for flexural waves using a continuous random field for the mass density, in order to approximate the mass distribution. Issues concerning this approximation are discussed. In addition, results are also compared with a simple added mass approximation with assumed modes from a FE solution
Wavenumber and natural frequency statistics of waveguides with spatially correlated material variability from finite element analysis
No full textThis paper concerns wave propagation in random waveguides and, in particular, when there is spatially correlated variability in the material and/or geometric properties. The one dimensional waveguide is modelled using a wave and finite element method to evaluate the wave-number. In the numerical examples presented the case of longitudinal motion in a thin rod is considered. The random field is discretized on the FE mesh and the element properties assumed piecewise constant. Analytical solutions for the wavenumbers are then found for each element of the random field mesh. When propagating over a finite distance, the total phase change of a wave is given by the summation of the phase changing in each element. Its statistics can be found by propagating the variability through an eigenvalue problem, by a sensitivity analysis or by Monte Carlo simulation. Spatial correlation is then seen to be important in determining the statistics of this phase (mean, variance, etc). This information can in turn be used to calculate the natural frequencies and forced response of finite structures. The statistics of the natural frequencies of rods of finite length are found by this wave approach are compared to those obtained by solving the full finite element problem
A Bayesian approach for wavenumber identification of metamaterial beams possessing variability
No full textRecent developments in additive manufacturing have allowed for a number of innovative designs in elastic metamaterials and phononic crystals used in several applications, including vibration attenuation. Complex geometric patterns that were otherwise very expensive or unpractical to produce are currently feasible. However, the 3D printing also introduces variability, which can greatly affect the dynamic performance of the metastructure. This work investigates the effects of manufacturing variability on the wavenumber identification of beams with evenly attached resonators, produced from Selective Laser Sintering. A combination of a correlation-based technique and a Bayes framework is proposed to identify the effective wavenumber and the most probable values of some of the design parameters. Typically of interest, for vibration attenuation using metamaterials, are the mass ratio and the resonator natural frequency. For this purpose an analytical model is derived, assuming an infinite number of resonators tuned to the same frequency. These parameters can be highly affected by the manufacturing variability because they are dependent on complex geometrical features of the metastructure. It is shown that the proposed approach can estimate the most likely values of the parameters with less than 4% difference when compared to a benchmark approach; the latter is not only more complex and time demanding, but also based on indirect measurements. Understanding the effects of this variability on the wave propagation represents an important step towards proposing robust designs with respect to the attenuation performance
Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability
No full textThis paper investigates structural wave propagation in waveguides with randomly varying material and geometrical properties along the axis of propagation. More specifically, it is assumed that the properties vary slowly enough such that there is no or negligible backscattering due to any changes in the propagation medium. This variability plays a significant role in the so called mid-frequency region for dynamics and vibration, but wave-based methods are typically only applicable to homogeneous and uniform waveguides. The WKB approximation is used to find a suitable generalization of the wave solutions for finite waveguides undergoing longitudinal and flexural motion. An alternative wave formulation approximation with piecewise constant properties is also derived and included, so that the internal reflections are taken into account, but this requires a discretization of the waveguide. Moreover, a Fourier like series, the Karhunen–Loeve expansion, is used to represent homogeneous and spatially correlated randomness and subsequently the wave propagation approach allows the statistics of the natural frequencies and the forced response to be derived. Experimental validation is presented using a cantilever beam whose mass per unit length is randomized by adding small discrete masses to an otherwise uniform beam. It is shown how the correlation length of the random material properties affects the natural frequency statistics and comparison with the predictions using the WKB approach is given
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