Natural Vibration Mode Shapes Research Articles

Crack identification through scan-tuning of vibration characteristics using piezoelectric materials

This research develops a frequency-based methodology with a scan vibration tuning process for crack identification in beam-type structures coupled with piezoelectric materials. Piezoelectric sensor and actuator patches are mounted on the surface of the host beam synchronously to generate feedback excitations for a tuning process by applying a feedback voltage output from the piezoelectric sensors. The feedback excitations can adjust the stiffness at local section of the beam covered by piezoelectric patches so as to tune its natural vibration mode shapes to amplify the natural frequency change due to the existence of the crack. Piezoelectric patches located at different positions of the beam are activated one by one to realize the scan-tuning process. The crack is identified since the natural frequency change is magnified by the piezoelectric sensor and actuator located at the crack position. Theoretical and finite element models of the scan-tuned beam structures coupled with piezoelectric materials are established. From simulation results, the crack existence and location can be effectively detected through the scan-tuning process with 25% natural frequency change due to a crack located at the middle of the beam. Further parameter studies are conducted to study the effects of the crack location and size on the detection sensitivity.

Nonlinear dynamic stability analysis of the coupled axial-torsional motion of the rotary drilling considering the effect of axial rigid-body dynamics

Dynamic stability of the coupled axial-torsional motion of a drill string is studied in this paper. Considering the drill string as a continuous system, governing equations of axial and torsional motions are derived. The drill string motion is divided into the rigid-body motion and elastic vibrations. Dynamic behavior of axial rigid-body motion is included through an independent equation. Natural vibration mode shapes of a free-free bar and a fixed-free shaft is introduced as trial functions for discretizing axial and torsional equations of elastic vibration, respectively. Due to the presence of state-dependent delay in equations of motion, semi-discretization method is implemented for stability analysis of the delay differential equations. The effect of nominal rotational speed of the drill string (Ω) and nominal weight on bit (W0) are investigated on drilling stability. Stability charts are presented and maximum allowable value of W0 is depicted with respect to Ω. Comparison of the results reveals that considering a constant axial velocity for the top of the drill string which is a common simplifying assumption in the previous researches, affects stability analysis of the drill string dynamics. It is observed that drilling operation is unstable for speeds lower than a threshold value. Based on the stability analysis, the maximum allowable value of W0 shows an increasing and then decreasing behavior with respect to the nominal rotary speed of the drill string. However, by increasing the damping of the system, its behavior changes into a fully decreasing profile with respect to Ω. It can be recognized from the results that both velocity-weakening friction and regenerative cutting have great effect on stability of the system and should be considered in dynamic modeling of the drill string. In this paper effects of several parameters of rock formation and drill string on the stable region are also presented. Finally, variation of steady-state axial speed is depicted with respect to the rotary table angular speed and nominal WOB which can be useful for choosing optimal and safe parameters for drilling.

An asymptotic solution to transverse free vibrations of variable-section beams

The transverse free vibration of a class of variable-cross-section beams is investigated using the Wentzel, Kramers, Brillouin (WKB) approximation. Here the governing equation of motion of the Euler–Bernoulli beam including axial force distribution is utilized to obtain a singular differential equation in terms of the natural frequency of vibration and a WKB expansion series is applied to find the solution. Based on this formulation, a closed form solution is obtained for determination of natural vibration mode shapes and the corresponding frequencies. The first four terms of this asymptotic solution are simplified for homogenous beams to give a compact third-order WKB approximation. Next, the resulting solution is employed to determine the natural frequencies and mode shapes of some examples with and without axial force distribution. The results are then been compared with those in the literature and very good agreement is achieved.

Solution to boundary shape optimization problem of linear elastic continua with prescribed natural vibration mode shapes

This paper presents a practical method of numerical analysis for boundary shape optimization problems of linear elastic continua in which natural vibration modes approach prescribed modes on specified sub-boundaries. The shape gradient for the boundary shape optimization problem is evaluated with optimality conditions obtained by the adjoint variable method, the Lagrange multiplier method, and the formula for the material derivative. Reshaping is accomplished by the traction method, which has been proposed as a solution to boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The validity of the presented method is confirmed by numerical results of three-dimensional beam-like and plate-like continua.

A finite element analysis of the natural frequencies of vibration of the human tympanic membrane. Part I

The present work examines the natural frequencies of vibration of the decoupled human tympanic membrane using the finite element method. A model comprising 49 isoparametric semi-loof thin shell elements was built, based on shape measurements of Kirikae (1960) and our own measurements on several local cadaver ears. A range of material data input was used similar to that assessed earlier by Funnell and Laszlo (1978). The present results indicate the natural vibration mode shapes are simple at low frequencies, but become more complex at higher values as found earlier (Tonndorf and Khanna, 1972; Funnell and Laszlo, 1978), but at reduced frequencies. Data input of the real drum thickenesses and dimensions considerably increases the natural frequencies by about 50% above those for a uniformly thick membrane. It is suggested that realistic eardrum natural frequencies may result from the use of a membrane internal stress parameter in the data file in order to simulate the interplay between radial and circular fibres resulting in the observed drum curvature.