Goutam Pohit

Nonlinear modelling and dynamic analysis of cracked Timoshenko functionally graded beams based on neutral surface approach

The present work accounts Timoshenko beam theory followed by Ritz approximation and an iterative technique to deal nonlinear free vibration problems of cracked Functionally Graded Material (FGM) beams based on neutral surface location. Using neutral surface as a reference rather than the midsurface reduces the complexity of nonlinear problems. It is assumed that crack always remains open. Analysis is carried out for clamped-clamped and clamped-free boundary conditions. Nonlinear frequencies and mode shapes corresponding to first three mode of vibration are obtained for the first time for different crack parameters, amplitudes of vibration and material indexes. The accuracy of the present solution is verified by comparing some of the obtained results with existing solutions. It can be concluded that present results are not only accurate but the methodology is very simple and easy to perform.

Crack modelling and detection in Timoshenko FGM beam under transverse vibration using frequency contour and response surface model with GA

In the present work, dynamic response of cracked Timoshenko beam with functionally graded material properties are obtained by a numerical technique using Ritz approximation. In order to verify the applicability and performance of the formulation, comparisons of the present numerical method with three-dimensional FEM models are made. Crack is assumed to be transverse and open throughout the vibration cycle. Two different crack detection techniques have been proposed. Results obtained by the numerical technique are used in both of the crack detection techniques. In the first technique, the frequency contours with respect to crack location and size are plotted and the intersection of contours of different modes helps in the prediction of crack location and size. In the second technique, crack is modelled using response surface methodology (RSM). The sum of the squared errors between the numerical and RSM regression model natural frequencies is used as the objective function. This objective function is minimised using genetic algorithm optimisation technique. Both the crack detection techniques and the numerical analysis have shown good agreement with each other.

Stability and Bifurcation Analysis of an Axially Accelerating Beam

The present work deals with nonlinear transverse vibration of a beam moving with a harmonically fluctuating velocity and subjected to parametric excitation at a frequency close to twice the natural frequency in presence of internal resonance. The geometric cubic nonlinearity in the equation of motion is due to stretching effect of the beam. The analysis is carried out using the method of multiple scales (MMS) by directly attacking the governing nonlinear integral-partial-differential equations and the associated boundary conditions. The resulting set of first- order ordinary differential equations governing the modulation of amplitude and phase of the first two modes are analyzed numerically to obtain steady state and dynamic bevaviour along with stability as well as bifurcation of the travelling system. The system exhibits trivial, single mode and two mode solutions with pitchfork, saddle-node and Hopf bifurcations. The sensitivity of the system towards the variation in frequency and amplitude of fluctuating velocity component, variation in damping, flexural stiffness and initial points are studied.

Parametric and internal resonances of an axially moving beam with time-dependent velocity

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beamvelocity is assumed to be comprised of a constantmean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of secondmode is approximately three times that of firstmode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

Stability, Bifurcation and Chaos of a Traveling Viscoelastic Beam Tuned to 3:1 Internal Resonance and Subjected to Parametric Excitation

Analytical-numerical approach has been adopted to investigate the stability, bifurcation and dynamic behavior (including chaotic behavior) of axially moving viscoelastic beam subjected to parametric excitation resulting from speed variation in the presence of 3:1 internal resonance between the first two modes of vibration. The governing equation of transverse vibration is a nonlinear integro-partial-differential equation with time-dependent coefficients. The direct method of multiple scales is employed to analyze the joint influence of the combination of parametric resonance and internal resonance with the focus on steady state responses. Equilibrium solutions along with their stability and bifurcations are determined by continuation algorithm while direct time integration is used for dynamic behavior for various system parameters. The results are compared with the previous works depicting the principal parametric resonances of the first and second modes. Significant comparative analysis results are reported in the stability and bifurcation of frequency response analysis. The dynamic responses show a range of behavior viz. stable periodic, mixed mode, quasiperiodic and unstable chaotic motion of the system. Numerical results illustrate various typical and interesting nonlinear phenomena of the traveling system which are not found in the existent literature.

Virtual manufacturing of gears with chip formation

The manufacturing process of gears is fairly complicated owing to the presence of various simultaneous motions of the cutter and the job. In this paper, an attempt is made to simulate the spur and helical-gear manufacturing process, including the chip formation, in a virtual manufacturing environment. The user has the option to manipulate the various parameters necessary for the manufacturing operation. The integrated process may also help to develop an optimised product. For better understanding of the operational principle, an animation facility in the form of a movie is included in the package.

Crack Detection in Rotating Cantilever Beam by Continuous Wavelet Transform

This paper presents an investigation methodology for detection of size and location of open edge transverse crack on a rotating beams using continuous wavelength transform. Generally, vibration characteristic of abeam like structure changes significantly due to the presence of crack. However when the crack is relatively small, it is difficult to identify the presence of the crack from the vibration response data only. In the present case, vibration parameter, such as, mode shapes of damaged beams are obtained using the finite element method. The CWT is implemented with different scales on mode shape of the rotating beam to evaluate crack size and its location. It is also shown that this method can produce satisfactory results with some limitation based on mode shape profile. In addition to the the profile quality, other determining factors are scale and resolution of the detection signal

Nonlinear Dynamics of a Travelling Beam Subjected to Multi-frequency Parametric Excitation

This paper deals with two frequency parametric excitation in presence of internal resonance. The cubic nonlinearity is inserted into the equation of motion by considering the mid-line stretching of the beam. The perturbation method of multiple scales is applied directly to the governing nonlinear fourth order integro-partial differential equation of motion. This leads to a set of first order differential equations known as the reduced equations or normalized reduced equations, which are utilized to determine the additional instability zones, appeared in the trivial state stability plot, the bifurcation and stability of fixed-points, periodic, quasi-periodic, mixed mode and chaotic responses. The transition of system behaviour from stable periodic to unstable chaotic occurs through intermittency route

Dynamics of nonlinearly damped microcantilevers under electrostatic excitation

Nonlinear dynamic behaviour of a cantilever microbeam actuated by a combination of DC and AC loading are investigated in presence of squeeze-film damping. A reduced order model formulated accounting for the nonlinearities of the system arising out of electrostatic forces and squeeze-film damping is numerically simulated to observe the large amplitude dynamic characteristics near primary and superharmonic resonances. The emphasis is on the significance of nonlinear damping in capturing the true dynamic characteristics of microsystems formulated as distributed parameter model. The damping nonlinearity is found to considerably affect the dynamics with a profound stabilising effect on the microsystem. Under the effect of large DC bias voltage, frequency–response curves obtained for different amplitudes of AC excitation exhibit local and global bifurcations. Response sensitivity to initial conditions is investigated near bifurcation points. Findings in the superharmonic resonance domain are emphasised.

Virtual manufacturing of various types of gears and validation of the technique using rapid prototype

Virtual manufacturing of gears is carried out to illustrate the fairly complicated gear generation process and the associated simultaneous motions being provided to the gear blank and the gear cutter. An attempt has been made to capture the phenomenon of chip formation as well. Generation of various types of gears are shown employing two types of operation, one being use of a gear hobbing cutter having multiple cutting edges. As such validation of the virtual manufacturing technique is always a difficult task. Using rapid prototype technology, a novel approach is proposed to ascertain the efficacy of the virtual manufacturing process being followed. The methodology is quite general and can be used for validation of any other virtual manufacturing process as well.