Analysis of the non linear bending and membrane stresses associated to the fundamental non-linear mode shape of fully clamped skew plates at large vibration amplitudes

2017 ◽  
Vol 9 (1-2) ◽  
pp. 57-67
Author(s):  
Hanane Moulay Abdelali ◽  
Khalid El Bikri ◽  
Rhali Benamar
Keyword(s):  
Mode Shape ◽  
Linear Mode ◽  
Non Linear

A Semi-Analytical Approach for the Geometrically Nonlinear Analysis of Skew Plates

2014 ◽  
Vol 704 ◽  
pp. 118-130
Author(s):  
Hanane Moulay Abdelali ◽  
Mounia El Kadiri ◽  
Rhali Benamar
Keyword(s):  
Mode Shape ◽  
Algebraic Equations ◽  
Skew Angle ◽  
Geometrically Nonlinear Analysis ◽  
Good Convergence ◽  
Linear Mode ◽  
Skew Plate ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Analytical Expressions

The present work concerns the nonlinear dynamic behaviour of fully clamped skew plates at large vibration amplitudes. A model based on Hamilton’s principle and spectral analysis has been used to study the large amplitude free vibration problem, reducing the non linear problem to solution of a set of non-linear algebraic equations. Two methods of solution have been adopted, the first method uses an improved version of the Newton-Raphson method, and the second leads to explicit analytical expressions for the higher mode contribution coefficients to the first non-linear mode shape of the skew plate examined. The amplitude dependent fundamental mode shape and the associated non-linear frequencies have been obtained by the two methods and a good convergence has been found. It was found that the non-linear frequencies increase with increasing the amplitude of vibration, which corresponds to the hardening type effect, expected in similar cases, due to the membrane forces induced by the large vibration amplitudes. The non-linear mode exhibits a higher bending stress near to the clamps at large deflections, compared with that predicted by linear theory. Numerical details are presented and the comparison made between the results obtained and previous ones available in the literature shows a satisfactory agreement. Tables of numerical results are given, corresponding to the linear and nonlinear cases for various values of the skew angle θ and various values of the vibration amplitude. These results, similar to those previously published for other plates, are expected to be useful to designers in the need of accurate estimates of the non-linear frequencies, the non linear strains and stresses induced by large vibration amplitudes of skew plates.

An integrated design technique of advanced linear-mode-shape method and serviceability drift optimization for tall buildings with lateral–torsional modes

2010 ◽  
Vol 32 (8) ◽  
pp. 2146-2156 ◽  
Author(s):  
M.F. Huang ◽  
K.T. Tse ◽  
C.M. Chan ◽  
K.C.S. Kwok ◽  
P.A. Hitchcock ◽  
...  
Keyword(s):  
Mode Shape ◽  
Integrated Design ◽  
Tall Buildings ◽  
Design Technique ◽  
Linear Mode

Vertical shear-induced resonant triads in Keplerian discs

2019 ◽  
Vol 488 (3) ◽  
pp. 4207-4219 ◽  
Author(s):  
Yuri Shtemler ◽  
Michael Mond
Keyword(s):  
Temporal Evolution ◽  
Shear Instability ◽  
Vertical Shear ◽  
Class Iii ◽  
Radial Temperature ◽  
Linear Interaction ◽  
Linear Mode ◽  
Proposed Model ◽  
Non Linear ◽  
Resonant Triads

ABSTRACT The vertical-shear instability (VSI) is studied through weakly non-linear analysis of unmagnetized vertically isothermal thin Keplerian discs under small radial temperature gradients. Vertically global and radially local axisymmetric compressible perturbations are considered. The VSI excites three classes of quasi-resonant triads of non-linearly interacting modes characterized by distinct temporal evolution. Most of the triads belong to the two-mode regime of non-linear interaction. Such triads are comprised of one unstable non-linear mode that grows quasi-exponentially, and two other modes that practically decoupled from the former. The latter two modes perform non-linear oscillations around their either linear prototypes (class I) or respective initial values (class II). The rest of the resonant triads belong to class III where all three modes exhibit non-linear oscillations. The proposed model describes an intermediate non-linear stage of the VSI prior to its saturation.

Quadratic Mode Shape Components From Ground Vibration Testing

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
L. H. van Zyl ◽  
E. H. Mathews
Keyword(s):  
Mode Shape ◽  
Ground Vibration ◽  
Axial Displacement ◽  
Vibration Test ◽  
Transverse Displacement ◽  
Vibration Testing ◽  
Linear Mode ◽  
Quadratic Mode ◽  
Acceleration Signals ◽  
Shape Component

Points on a vibrating structure generally move along curved paths rather than straight lines. For example, the tip of a cantilever beam vibrating in a bending mode experiences axial displacement as well as transverse displacement. The axial displacement is governed by the inextensibility of the neutral axis of the beam and is proportional to the square of the transverse displacement; hence the name “quadratic mode shape component.” Quadratic mode shape components are largely ignored in modal analysis, but there are some applications in the field of modal-basis structural analysis where the curved path of motion cannot be ignored. Examples include vibrations of rotating structures and buckling. Methods employing finite element analysis have been developed to calculate quadratic mode shape components. Ground vibration testing typically only yields the linear mode shape components. This paper explores the possibility of measuring the quadratic mode shape components in a sine-dwell ground vibration test. This is purely an additional measurement and does not affect the measured linear mode shape components or the modal parameters, i.e., modal mass, frequency, and damping ratio. The accelerometer output was modeled in detail taking into account its linear acceleration, its rotation, and gravitational acceleration. The response was correlated with the Fourier series representation of the output signal. The result was a simple expression for the quadratic mode shape component. The method was tested on a simple test piece and satisfactory results were obtained. The method requires that the accelerometers measure down to steady state and that up to the second Fourier coefficients of the output signals are calculated. The proposed method for measuring quadratic mode shape components in a sine-dwell ground vibration test seems feasible. One drawback of the method is that it is based on the measurement and processing of second harmonics in the acceleration signals and is therefore sensitive to any form of structural nonlinearity that may also cause higher harmonics in the acceleration signals. Another drawback is that only the quadratic components of individual modes can be measured, whereas coupled quadratic terms are generally also required to fully describe the motion of a point on a vibrating structure.

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