Showing papers in "Journal of Sound and Vibration in 2006"

Abstract: The guided Lamb wave is widely acknowledged as one of the most encouraging tools for quantitative identification of damage in composite structures, and relevant research has been conducted intensively since the 1980s. The main aim of this paper is to provide a comprehensive review on the state of the art of Lamb wave-based damage identification approaches for composite structures, addressing the advances and achievements in these techniques in the past decades. Major emphasis is placed on the unique characteristics and mechanisms of Lamb waves in laminated composites; approaches in wave mode selection, generation and collection; modelling and numerical simulation techniques; signal processing and identification algorithms; and sensor network technology for practical utility. Representative case studies are also briefly described in terms of various experimental validations and applications.

Abstract: De-noising and extraction of the weak signature are crucial to fault prognostics in which case features are often very weak and masked by noise. The wavelet transform has been widely used in signal de-noising due to its extraordinary time-frequency representation capability. In this paper, the performance of wavelet decomposition-based de-noising and wavelet filter-based de-noising methods are compared based on signals from mechanical defects. The comparison result reveals that wavelet filter is more suitable and reliable to detect a weak signature of mechanical impulse-like defect signals, whereas the wavelet decomposition de-noising method can achieve satisfactory results on smooth signal detection. In order to select optimal parameters for the wavelet filter, a two-step optimization process is proposed. Minimal Shannon entropy is used to optimize the Morlet wavelet shape factor. A periodicity detection method based on singular value decomposition (SVD) is used to choose the appropriate scale for the wavelet transform. The signal de-noising results from both simulated signals and experimental data are presented and both support the proposed method.

Abstract: Future MEMS devices will harvest energy from their environment. One can envisage an autonomous condition monitoring vibration sensor being powered by that same vibration, and transmitting data over a wireless link; inaccessible or hostile environments are obvious areas of application. The base excitation of an elastically mounted magnetic seismic mass moving past a coil, considered previously by several authors, is analysed in detail. The amplitude of the seismic mass is limited in any practical device and this, together with the magnitude and frequency of the excitation define the maximum power that can be extracted from the environment. The overall damping coefficient (part of which is mechanical) is associated with the harvesting and dissipation of energy and also the transfer of energy from the vibrating base into the system. It is shown that net energy flow from the base through the damper is positive (negative) for ω > ω n ( ω ω n ) , but is zero when ω = ω n . The mechanical part of the damper cannot contribute more power than it dissipates and is neutral, at best, when ω / ω n → ∞ . Maximum power is delivered to an electrical load when its resistance is equal to the sum of the coil internal resistance and the electrical analogue of the mechanical damping coefficient, which differs from what has been claimed. A highly damped system has the advantage of harvesting energy over a wider band of excitation frequencies on either side of the natural frequency, is smaller, but will harvest marginally less power. One possible strategy for variable amplitude excitation is proposed.

Abstract: Current processing of acoustic array data is burdened with considerable uncertainty. This study reports an original methodology that serves to demystify array results, reduce misinterpretation, and accurately quantify position and strength of acoustic sources. Traditional array results represent noise sources that are convolved with array beamform response functions, which depend on array geometry, size (with respect to source position and distributions), and frequency. The Deconvolution Approach for the Mapping of Acoustic Sources (DAMAS) method removes beamforming characteristics from output presentations. A unique linear system of equations accounts for reciprocal influence at different locations over the array survey region. It makes no assumption beyond the traditional processing assumption of statistically independent noise sources. A new robust iterative method seamlessly introduces a positivity constraint (due to source independence) that makes the equation system sufficiently deterministic. DAMAS is quantitatively validated using archival data from a variety of prior high-lift airframe component noise studies, including flap edge/cove, trailing edge, leading edge, slat, and calibration sources. Presentations are explicit and straightforward, as the noise radiated from a region of interest is determined by simply summing the mean-squared values over that region. DAMAS can fully replace existing array processing and presentations methodology in most applications. It appears to dramatically increase the value of arrays to the field of experimental acoustics.

Abstract: This paper deals with a semi-analytical finite element (SAFE) method for modeling wave propagation in waveguides of arbitrary cross-section. The method simply requires the finite element discretization of the cross-section of the waveguide, and assumes harmonic motion along the wave propagation direction. The general SAFE technique is extended to account for viscoelastic material damping by allowing for complex stiffness matrices for the material. The dispersive solutions are obtained in terms of phase velocity, group velocity (for undamped media), energy velocity (for damped media), attenuation, and cross-sectional mode shapes. Knowledge of these properties is important in any structural health monitoring attempt that uses ultrasonic guided waves. The proposed SAFE formulation is applied to several examples, including anisotropic viscoelastic layered plates, composite-to-composite adhesive joints and railroad tracks.

Abstract: According to the non-stationary characteristics of roller bearing fault vibration signals, a roller bearing fault diagnosis method based on empirical mode decomposition (EMD) energy entropy is put forward in this paper. Firstly, original acceleration vibration signals are decomposed into a finite number of stationary intrinsic mode functions (IMFs), then the concept of EMD energy entropy is proposed. The analysis results from EMD energy entropy of different vibration signals show that the energy of vibration signal will change in different frequency bands when bearing fault occurs. Therefore, to identify roller bearing fault patterns, energy feature extracted from a number of IMFs that contained the most dominant fault information could serve as input vectors of artificial neural network. The analysis results from roller bearing signals with inner-race and out-race faults show that the diagnosis approach based on neural network by using EMD to extract the energy of different frequency bands as features can identify roller bearing fault patterns accurately and effectively and is superior to that based on wavelet packet decomposition and reconstruction.

Abstract: In this paper some usual vibration-based damage identification techniques (VBDIT) will be reviewed and used for structural damage evaluation With the help of a simple supported beam with different damage levels the reliability of these techniques will be investigated The techniques reviewed herein are based on measured modal parameters which use only few mode shapes and/or modal frequencies of the structure that can be easily obtained by dynamic tests In other words, by realizing two sets of dynamic measurements, corresponding to two moments of the structure lifetime, the dynamic modal parameters can be obtained In order to assess properly the performance of these techniques different noise levels are randomly introduced to the response signals of a simulated beam which is exited by a random force For different levels of damage and noise, the probabilities of damage detection and the probabilities of false alarm for the total number of simulations is evaluated It can be concluded that among the evaluated techniques the strain energy method presents the best stability regarding noisy signals; however, the detection judgement depends on a threshold level which is discussed in this paper The change in mode shape curvature, change in flexibility and change in flexibility curvature methods are also capable to detect and localise damaged elements but in the case of complex and simultaneous damages these techniques show less efficiency

Abstract: In this paper, a time series algorithm is presented for damage identification and localization. The vibration signals obtained from sensors are modeled as autoregressive moving average (ARMA) time series. A new damage-sensitive feature, DSF, is defined as a function of the first three auto regressive (AR) components. It is found that the mean values of the DSF for the damaged and undamaged signals are different. Thus, a hypothesis test involving the t -test is used to obtain a damage decision. Two damage localization indices LI 1 and LI 2 , are introduced based on the AR coefficients. At the sensor locations where damage is introduced, the values of LI 1 and LI 2 appear to increase from their values obtained at the undamaged baseline state. The damage detection and localization algorithms are valid for stationary signals obtained from linear systems. To test the efficacy of the damage detection and localization methodologies, the algorithm has been tested on the analytical and experimental results of the ASCE benchmark structure. In contrast to prior pattern classification and statistical signal processing algorithms that have been able to identify primarily severe damage and have not been able to localize the damage effectively, the proposed algorithm is able to identify and localize minor to severe damage as defined for the benchmark structure.

Abstract: In this paper a method for estimating the damage location in beam and plate structures is presented. A Plexiglas cantilever beam and a steel plate with four fixed boundary conditions are tested experimentally. The estimated mode shapes of the beam are analysed by the one-dimensional continuous wavelet transform. The formulation of the two-dimensional continuous wavelet transform for plate damage detection is presented. The location of the damage is indicated by a peak in the spatial variation of the transformed response. Applications of Gaussian wavelet for one-dimensional problems and reverse biorthogonal wavelet for two-dimensional structures are presented. The proposed wavelet analysis can effectively identify the defect position without knowledge of neither the structure characteristics nor its mathematical model.

Abstract: In this paper, the design of mechanical band-pass filters to be used in energy scavengers is studied. For such filters an ensemble of cantilever beams is proposed where at the tip of each beam a mass, known as the proof mass, is mounted. It is shown that such an ensemble can be made into a band-pass filter when dimensions of the beams and masses of the proof masses are chosen appropriately. A systematic procedure for designing mechanical band-pass filters is given. It is shown that the maximal frequency band of the band-pass filter is limited and cannot be chosen arbitrarily large: The frequency band is independent of dimensions of the beams and masses of the proof masses.

Abstract: A sensitivity-based finite element (FE) model updating is carried out for damage detection in this paper. The objective function consisting of the modal flexibility residual is formulated and its gradient is derived. The optimization algorithm used to minimize the objective function and damage detection procedures are explained. The proposed procedure is firstly illustrated with a simulated example of the simply supported beam. The effect of noise on the updating algorithm is studied. It is demonstrated that the behavior of proposed algorithm on noise is satisfactory and the identified damage patterns are good. Afterwards, the procedure is applied for the tested reinforced concrete beam, which is damaged in the laboratory. Despite all the elements in the FE model are used as updating parameters which is considered as the extreme adverse condition in FE model updating, the identified damage pattern is comparable with those obtained from the tests. It is verified that the modal flexibility is sensitive to damage and the proposed procedure of FE updating using the modal flexibility residual is promising for the detection of damaged elements.

Abstract: Socio–acoustic surveys were carried out as part of the Soundscape Support to Health research programme to assess the health effects of various soundscapes in residential areas. The study was designed to test whether having access to a quiet side of one's dwelling enhances opportunities for relaxation and reduces noise annoyance and other adverse health effects related to noise. The dwellings chosen were exposed to sound levels from road traffic ranging from about L Aeq , 24 h = 45 – 68 dB at the most-exposed side. The study involved 956 individuals aged 18–75 years. The results demonstrate that access to quiet indoor and outdoor sections of one's dwelling supports health; it produces a lower degree and extent of annoyance and disturbed daytime relaxation, improves sleep and contributes to physiological and psychological well-being. Having access to a quiet side of one's dwelling reduces disturbances by an average of 30–50% for the various critical effects, and corresponds to a reduction in sound levels of (LAeq,24h) 5 dB at the most-exposed side. To protect most people (80%) from annoyance and other adverse effects, sound levels from road traffic should not exceed (LAeq,24h) 60 dB at the most-exposed side, even if there is access to a quiet side of one's dwelling (LAeq,24h⩽45 dB).

Abstract: Ground vibration is an important aspect of the environmental impact of rail traffic. Vibration from about 2–200 Hz is caused by trains moving on the ground surface or in tunnels. The wave field thus created must be modelled in three dimensions because of the excitation under each axle and the movement of the train. For arbitrary geometry of structures and ground surface to be allowed in the analysis, numerical models are required. In most practical situations, the ground and built structures, such as tunnels and tracks, can be considered to be homogeneous in the track direction and may be modelled using the wavenumber finite/boundary element method which is formulated in terms of the wavenumber in that direction. Compared with a conventional, three-dimensional finite/boundary element model, this model is more computationally efficient and requires far less memory since discretization is only made over the vertical–transverse section of the ground and/or built structures. With this model it is possible to predict complete vibration spectra. In this paper, the wavenumber-based modelling approach is outlined and then the applicability of the method to surface vibration and tunnel vibration analyses is demonstrated.

Abstract: Many structural components can be regarded as waveguides. They are uniform in one direction so that the cross section of the waveguide has the same physical and geometric properties at all points along the axis of the waveguide. In this paper a method is presented to calculate the forced response of such a structure using a combination of wave and finite element (FE) approaches. The method involves post-processing a conventional, but low order, FE model in which the mass and stiffness matrices are typically found using a conventional FE package. A section of the waveguide is meshed and the eigenvalues and eigenvectors of the resulting transfer matrix found. The eigenvectors form a set of basis functions for the analysis of the structure as a whole, allowing the global dynamic stiffness matrix to be built easily and then the forced response to be calculated very efficiently. The main advantage of the approach over the alternative waveguide/FE approach often termed the spectral FE method, is that conventional FE packages can be used to form the stiffness and mass matrices so that structures with complex geometries or material distributions can be analysed with relative ease. To demonstrate the efficacy of the method examples of the forced response for a finite beam and plate-strip are presented.

Abstract: This paper is concerned with the use of the Timoshenko beam model for free vibration analysis of multi-walled carbon nanotubes (CNTs). Unlike the Euler beam model, the Timoshenko beam model allows for the effects of transverse shear deformation and rotary inertia. These effects become significant for CNTs with small length-to-diameter ratios that are normally encountered in applications such as nanoprobes. By using the differential quadrature (DQ) method, the governing Timoshenko equations are solved for CNTs of different length-to-diameter ratios and boundary conditions. By comparing results based on the Timoshenko and the Euler beam theories, we show that the frequencies are significantly overpredicted by the Euler beam theory when the length-to-diameter ratios are small and when considering high vibration modes. For such situations, the Timoshenko beam model should be used for a better prediction of the frequencies.

Abstract: This paper presents the experimental validation of a numerical model for the prediction of train induced vibrations. The model fully accounts for the dynamic interaction between the train, the track and the soil. The track geometry is assumed to be invariant with respect to the longitudinal direction, which allows for an efficient solution of the dynamic track–soil interaction problem in the frequency–wavenumber domain. The model is validated by means of several experiments that have been performed at the occasion of the homologation tests of the new HST track on the line L2 between Brussels and Koln. A first set of experiments is used to determine the dynamic soil and track characteristics. In a second set of experiments, the soil transfer functions, the track–soil transfer functions and the track and free field vibrations during the passage of a Thalys high speed train have been measured. These results are used for a step-wise validation of the numerical model that is based on the identified model parameters and allows to study the propagation of errors in the prediction model.

Abstract: This paper presents the application of neural network for the prediction of ground vibration and frequency by all possible influencing parameters of rock mass, explosive characteristics and blast design. To investigate the appropriateness of this approach, the predictions by ANN is also compared with conventional statistical relation. Network is trained by 150 dataset with 458 epochs and tested it by 20 dataset. The correlation coefficient determined by ANN is 0.9994 and 0.9868 for peak particle velocity (PPV) and frequency while correlation coefficient by statistical analysis is 0.4971 and 0.0356.

Abstract: This paper presents a load-dependent controller design approach to solve the problem of multi-objective control for vehicle active suspension systems by using linear matrix inequalities. A quarter-car model with active suspension system is considered. It is assumed that the vehicle body mass resides in an interval and can be measured online. This approach of designing controllers, whose gain matrix depends on the online available information of the body mass, is based on a parameter-dependent Lyapunov function. Since the parameter-dependent idea is fully exploited, the proposed controller design approach can yield much less conservative results compared with previous approaches that design robust constant controllers in the quadratic framework. The usefulness and the advantages of the proposed controller design methodology are demonstrated via numerical simulations.

Abstract: An approximate solution for the free vibration problem of two-dimensional magneto-electro-elastic laminates is presented to determine their fundamental behavior. The laminates are composed of linear homogeneous elastic, piezoelectric, or magnetostrictive layers with perfect bonding between each interface. The solution for the elastic displacements, electric potential, and magnetic potential is obtained by combining a discrete layer approach with the Ritz method. The model developed here is not dependent on specific boundary conditions, and it is presented as an alternative to the exact or analytical approaches which are limited to a very specific set of edge conditions. The natural frequencies and through-thickness modal behavior are computed for simply supported and cantilever laminates. Solutions for the simply supported case are compared with the known exact solution for piezoelectric laminates, and excellent agreement is obtained. The present approach is also validated by comparing the natural frequencies of a two-layer cantilever plate with known analytical solution and with results obtained using commercial finite element software.

Abstract: The frequency range of interest for ground vibration from underground urban railways is approximately 20 to 100 Hz. For typical soils, the wavelengths of ground vibration in this frequency range are of the order of the spacing of train axles, the tunnel diameter and the distance from the tunnel to nearby building foundations. For accurate modelling, the interactions between these entities therefore have to be taken into account. This paper describes an analytical three-dimensional model for the dynamics of a deep underground railway tunnel of circular cross-section. The tunnel is conceptualised as an infinitely long, thin cylindrical shell surrounded by soil of infinite radial extent. The soil is modelled by means of the wave equations for an elastic continuum. The coupled problem is solved in the frequency domain by Fourier decomposition into ring modes circumferentially and a Fourier transform into the wavenumber domain longitudinally. Numerical results for the tunnel and soil responses due to a normal point load applied to the tunnel invert are presented. The tunnel model is suitable for use in combination with track models to calculate the ground vibration due to excitation by running trains and to evaluate different track configurations.

Abstract: Linear thermal buckling and free vibration analysis are presented for functionally graded cylindrical shells with clamped–clamped boundary condition based on temperature-dependent material properties. The material properties of functionally graded materials (FGM) shell are assumed to vary smoothly and continuously across the thickness. With high-temperature specified on the inner surface of the FGM shell and outer surface at ambient temperature, 1D heat conduction equation along the thickness of the shell is applied to determine the temperature distribution; thereby, the material properties based on temperature distribution are made available for thermal buckling and free vibration analysis. First-order shear deformation theory along with Fourier series expansion of the displacement variables in the circumferential direction are used to model the FGM shell. Numerical studies involved the understanding of the influence of the power-law index, r/h and l/r ratios on the critical buckling temperature. Free vibration studies of FGM shells under elevated temperature show that the fall in natural frequency is very drastic for the mode corresponding to the lowest natural frequency when compared to the lowest buckling temperature mode.

Abstract: A numerical model is presented to predict vibrations in the free field from excitation due to metro trains in tunnels. The three-dimensional dynamic tunnel–soil interaction problem is solved with a subdomain formulation, using a finite element formulation for the tunnel and a boundary element method for the soil. The periodicity of the geometry in the longitudinal direction of the tunnel is exploited using the Floquet transform, limiting the discretization to a single-bounded reference cell. The responses of two different types of tunnel due to a harmonic load on the tunnel invert are compared, both in the frequency–wavenumber and spatial domains. The first tunnel is a shallow cut-and-cover masonry tunnel on the Paris metro network, embedded in layers of sand, while the second tunnel is a deep bored tunnel of London Underground, with a cast iron lining and embedded in the London clay.

Abstract: This paper introduces a simple method for time-varying vibration decomposition based on the Hilbert transform. The non-stationary frequency of the largest component is estimated as an average function of the instantaneous frequency of the composition, and the corresponding envelope is estimated according to synchronous demodulation. The method is demonstrated using computer simulation of different types of non-stationary and nonlinear vibration.

Abstract: Based on the continuum mechanics and a multiple-elastic beam model, the nonlinear free vibration of embedded multi-wall carbon nanotubes considering intertube radial displacement and the related internal degrees of freedom is investigated. By using the incremental harmonic balanced method, the iterative relationship of nonlinear amplitude and frequency for the single-wall nanotube and double-wall nanotube are expressed. In the numerical calculation, the amplitude frequency response curves of the nonlinear free vibration for the single-wall and double-wall nanotubes are obtained. The effects of the surrounding elastic medium, van der Waals forces and aspect ratio of the multi-wall nanotubes on the amplitude frequency response characteristics are discussed.

Abstract: The analysis of vibration from railway tunnels is of growing interest as new and higher-speed railways are built under the ground to address the transport problems of growing modern urban areas. Such analysis can be carried out using numerical methods but models and therefore computing times can be large. There is a need to be able to apply very fast calculations that can be used in tunnel design and studies of environmental impacts. Taking advantage of the fact that tunnels often have a two-dimensional geometry in the sense that the cross section is constant along the tunnel axis, it is useful to evaluate the potential uses of two-dimensional models before committing to much more costly three-dimensional approaches. The vibration forces in the track due to the passage of a train are by nature three-dimensional and a complete analysis undoubtedly requires a model of three-dimensional wave propagation. The aim of this paper is to investigate the quality of the information that can be gained from a two-dimensional model of a railway tunnel. The vibration transmission from the tunnel floor to the ground surface is analysed for the frequency range relevant to the perception of whole body vibration (about 4–80 Hz). A coupled finite element and boundary element scheme is applied in both two and three dimensions. Two tunnel designs are considered: a cut-and-cover tunnel for a double track and a single-track tunnel dug with the New Austrian tunnelling method (NATM).

Abstract: This study compared the effects of road, rail, and aircraft noise and tested the applicability of the equivalent noise level for the evaluation of sleep disturbances. Sixteen women and 16 men (19–28 years) slept during 3 consecutive weeks in the laboratory. Eight persons slept in quiet throughout. Twenty-four persons were exposed to road, rail, or aircraft noise with weekly permuted changes. Each week consisted of a random sequence of a quiet night (32 dBA) and 3 nights with equivalent noise levels of 39, 44, and 50 dBA and maximum levels of 50–62, 56–68, and 62–74 dBA, respectively. The polysomnogram was recorded during all nights, sleep quality was assessed and performance tests were completed in the morning. Subjectively evaluated sleep quality decreased and reaction time increased gradually with noise levels, whereas most physiological variables revealed the same reactions to both the lower and considerably stronger reactions to the highest noise load. Aircraft noise, rail and road traffic noise caused similar after-effects but physiological sleep parameters were most severely affected by rail noise. The equivalent noise level seems to be a suitable predictor for subjectively evaluated sleep quality but not for physiological sleep disturbances.

Abstract: This paper investigates the dynamic response of a class of electrostatically driven microelectromechanical (MEM) oscillators. The particular systems of interest are those which feature parametric excitation that arises from forces produced by fluctuating voltages applied across comb drives. These systems are known to exhibit a wide range of behaviors, some of which have escaped explanation or prediction. In this paper we examine a general governing equation of motion for these systems and use it to provide a complete description of the dynamic response and its dependence on the system parameters. The defining feature of this equation is that both the linear and cubic terms feature parametric excitation which, in comparison to the case of purely linear parametric excitation (e.g. the Mathieu equation), significantly complicates the system's dynamics. One consequence is that an effective nonlinearity for the overall system cannot be defined. Instead, the system features separate effective nonlinearities for each branch of its nontrivial response. As such, it can exhibit not only hardening and softening nonlinearities, but also mixed nonlinearities, wherein the response branches in the system's frequency response bend toward or away from one another near resonance. This paper includes some brief background information on the equation of motion under consideration, an outline of the analytical techniques used to reach the aforementioned results, stability results for the responses in question, a numerical example, explored using simulation, of a MEM oscillator which features this nonlinear behavior, and preliminary experimental results, taken from an actual MEM device, which show evidence of the analytically predicted behavior. Practical issues pertaining to the design of parametrically excited MEM devices are also considered.

Abstract: Gears are one of the most common and important machine components in many advanced machines. An improved understanding of vibration signal is required for the early detection of incipient gear failure to achieve high reliability. This paper mainly consists of two parts: in the first part, a 6-degree-of-freedom gear dynamic model including localized tooth defect has been developed. The model consists of a spur gear pair, two shafts, two inertias representing load and prime mover and bearings. The model incorporates the effects of time-varying mesh stiffness and damping, backlash, excitation due to gear errors and profile modifications. The second part consists of signal processing of simulated and experimental signals. Empirical mode decomposition (EMD) is a method of breaking down a signal without leaving a time domain. The process is useful for analysing non-stationary and nonlinear signals. EMD decomposes a signal into some individual, nearly monocomponent signals, named as intrinsic mode function (IMF). Crest factor and kurtosis have been calculated of these IMFs. EMD pre-processed kurtosis and crest factor give early detection of pitting as compared to raw signal.

Abstract: The natural frequencies for bending vibrations of Timoshenko cracked beams with simple boundary conditions have been obtained. The beam is modelled as two segments connected by two massless springs (one extensional and another one rotational). This model promotes discontinuities in both vertical displacement and rotation due to bending, which are proportional to shear force and bending moment transmitted by the cracked section, respectively. The differential equations for the free bending vibrations are established and then solved individually for each segment with the corresponding boundary conditions and the appropriated compatibility conditions at the cracked section. The problem is also solved by the perturbation method and both procedures are applied to the case of a simply supported cracked beam. The results show that the perturbation method provides simple expressions for the natural frequencies of cracked beams and it gives good results for shallow cracks.

Abstract: The free bending vibration of rotating tapered beams is investigated by using the dynamic stiffness method. The range of problems considered includes beams for which the depth and/or width of the cross-section vary linearly along the length. First, the governing differential equation of motion of the rotating tapered beam in free flap bending vibration is derived for the most general case using Hamilton's principle, allowing for the effects of centrifugal stiffening, an arbitrary outboard force and the hub radius term. For harmonic oscillation the differential equation is solved for bending displacement by applying the Frobenius method of series solution. The expressions for bending rotation, shear force and bending moment at any cross-section of the beam are also obtained in explicit analytical form. Then the dynamic stiffness matrix is developed, by relating the amplitudes of forces and moments to those of the displacements and rotations at the ends of the harmonically vibrating tapered beam. Next the Wittrick–Williams algorithm is used as a solution technique to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of some illustrative examples. A parametric study is carried out to demonstrate the effects of rotational speed, taper ratio and hub radius on the results, which are discussed and compared with the published ones. Finally some conclusions are drawn.