Single Source Three Dimensional Capture of Full Field Plate Vibrations

Measurement of the vibrations of plates can offer significant challenges to the experimentalist, particularly when the plates are lightweight, exhibit large amplitude deflections, nonlinear responses or are initially curved. The use of accelerometers adds masses which can change the dynamics of lightweight plates. Large amplitude oscillations and initial curvatures cause complications when using a laser vibrometer, as they make it difficult to get consistent reflections back to the receiver. Furthermore, large or nonlinear oscillations challenge inherent assumptions on which the vibrometer’s algorithms depend. A high speed video camera avoids these issues, but makes it hard to extract numerical data. This paper describes a method that extends the capabilities of a high speed video camera by using a mirror, allowing post-processing software to stereoscopically resolve an array of points on the plate surface to 3D coordinates, capturing the complete shape and position of the plate throughout vibration. This method avoids all the problems mentioned above and gives very clear insight into plate vibration. Some example results of this method are presented, using thermally bistable carbon laminate plates filmed at a 1000 frames per second. These plates pose the challenges described, and also exhibit an unusual oscillatory motion where the plates ‘snap’ between two statically stable states. The method is shown to provide clear insight into the rich dynamics of these plates.     

Natural vibrations of a cylindrical shell reinforced with rectangular plates

This paper outlines a technique of determining the natural frequencies and modes of an elastic structure consisting of a cylindrical shell and noncrossing rectangular plates inserted into it. The influence of the position, number, and thickness of the plates on the natural frequencies and modes is analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 89–96, March 2006.     

Bending analysis of thin functionally graded plates using generalized differential quadrature method

In this paper, the differential quadrature (DQ) method is presented for easy and effective analysis of isotropic functionally graded (FG) and functionally graded coated (FGC) thin plates with constant Poisson’s ratio and varying Young’s modulus in the thickness direction. The bending of FG and FGC plates under transverse loading has been studied using the polynomial differential quadrature (PDQ) and the harmonic differential quadrature (HDQ) methods. A three-dimensional elasticity solution for a moderately thick FG plate with exponential Young’s modulus is used as the benchmark. Two examples, including a thin FG rectangular plate and a thin FGC rectangular plate with sigmoidal Young’s modulus, are investigated. The numerical results of PDQ and HDQ methods reveal good agreement with other solutions. Also, it is shown that the formulations for thin FG plates and homogeneous plates are similar, except that the plane strain components of the middle surface in FG plates are not zero.     

Elastic plates under bending solved by pseudocaustics

The method pf pseudocaustics was applied to the study of out-of-plane bending in elastic plates. It is shown that for bending problems where the loading mode is given, the method determines experimentally the complex potential function at selected points along the boundaries. A conformal mapping of the closed smooth curves of each boundary of the plate on to a unit circle allows the determination of the complex potential ϕ (ζ), expressed in the form of a Laurent series. This in turn yields the complete solution of the bent plate. In order to show the efficiency of the method it was applied to two typical examples of thin infinite plates in cylindrical bending, having either a circular central hole, or a square hole. The experimental results corroborate the theoretical results, thus proving that this combined theoretical and experimental method is suitable for solving elastic problems in applications with high accuracy, where other methods fail to yield satisfactory results.     

A C<Superscript>0</Superscript>-type zig–zag theory and finite element for laminated composite and sandwich plates with general configurations

In order to conveniently develop C⁰ continuous element for the accurate analysis of laminated composite and sandwich plates with general configurations, this paper develops a C⁰-type zig–zag theory in which the interlaminar continuity of transverse shear stresses is a priori satisfied and the number of unknowns is independent of the number of layers. The present theory is applicable not only to the cross-ply but also to the angle-ply laminated composite and sandwich plates. On the premise of retaining the merit of previous zig–zag theories, the derivatives of transverse displacement have been taken out from the displacement fields. Therefore, based on the proposed zig–zag theory, it is very easy to construct the C⁰ continuous element. To assess the performance of the proposed model, the classical quadratic six-node triangular element with seven degrees of freedom at each node is presented for the static analysis of laminated composite and sandwich plates. The typical examples are taken into account to assess the performance of finite element based on the proposed zig–zag theory by comparing the present results with the three-dimensional elasticity solutions. Numerical results show that the present model can produce the more accurate deformations and stresses compared with the previous zig–zag theories.     

Determination of the natural frequencies of vibration of nonuniform slabs

The natural frequencies of vibrations of laminated plates are determined in a three-dimensional formulation by analytical separation of the sought functions for plate thickness. The system of differential equations which describes the natural vibrations of the plates is solved analytically. The solution makes it possible to study plates with a large number of layers, including orthotropic plates with elastic characteristics that vary through the thickness. Numerical experiments show that a step approximation can be used to approximate the variable elastic modulus. Ukrainian Transportation University, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 2, pp. 47–53, February, 1999.     

Marcus Integration Method for Shearable Plates

A method proposed by Marcus [5] to integrate the classical biharmonic equation of simply supported, unshearable plates with polygonal contour is extended to apply to shearable plates as well, provided the supporting device is of the ‘hard’ type.     

The effect of boundary conditions on sound loudness radiated from rectangular plates

This paper describes the effect of boundary conditions on sound loudness radiated from rectangular plates. Based on the Rayleigh integral, the Zwicker’s loudness model and the dynamic response of the plates, the sound loudness radiated from vibrating plates and the effects of simple support, clamped support and free support on sound loudness are separately studied. The effects of boundary conditions on sound intensity level, sound intensity density and critical-band level are also studied, taking the frequency selectivity of human hearing into account. The transformation from sound intensity level to sound loudness radiated from rectangular plates, due to the frequency-selectivity characteristics of human hearing, is also illustrated.     

Numerical stress-strain analysis of a nonthin elastic plate

A numerical solution to elastic-equilibrium problems for nonthin plates is proposed. The solution is obtained by using the curvilinear-mesh method in combination with Vekua’s method. The efficiency (rapid convergence and accuracy) of this approach is demonstrated by solving test problems for thick plates that can also be solved exactly or approximately by other methods. A numerical solution is obtained to the bending problem for orthotropic nonthin plates of constant and varying thickness __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 119–126, March 2006.     

Fully Developed Mixed Convection Flow Between Inclined Parallel Plates Filled with a Porous Medium

A fully developed mixed convection flow between inclined parallel flat plates filled with a porous medium is considered through which there is a constant flow rate and with heat being supplied to the fluid by the same uniform heat flux on each plate. The equations governing this flow are made non-dimensional and are seen to depend on two dimensionless parameters, a mixed convection parameter λ and the Péclet number Pe, as well as the inclination γ of the plates to the horizontal. The velocity and temperature profiles are obtained in terms of λ, Pe and γ when the channel is inclined in an upwards direction as well as for horizontal channels. The limiting cases of small and large λ and small Pe are considered with boundary-layer structures being seen to develop on the plates for large values of λ.     

2-D PIV measurements of oscillatory flow around parallel plates

Oscillatory flow in stacks of parallel plates is essential for the working of “standing wave” thermo-acoustic devices. In this paper, the flow in the transition from stack to open tube is studied experimentally using particle image velocimetry. When the flow is directed outwards of the stack, vortices originate behind the stack plates. The Strouhal to Reynolds ratio determines the vortex pattern behind the stack plates, varying from a single vortex pair to a complete vortex street. The influence of different plate-end shapes and porosities are also studied. The streaming velocity is measured using two different methods.     

On the Justification of Plate Models

In this paper, we will consider the modelling of problems in linear elasticity on thin plates by the models of Kirchhoff–Love and Reissner–Mindlin. A fundamental investigation for the Kirchhoff plate goes back to Morgenstern (Arch. Ration. Mech. Anal. 4:145–152, 1959) and is based on the two-energies principle of Prager and Synge. This was half a century ago.     

Solving the elastic bending problem for a plate with mixed boundary conditions

The bending problem for an arbitrarily outlined thin plane with mixed boundary conditions is solved. A technique based on the methods of potentials and balancing loads is proposed for constructing Green’s function for the Germain-Lagrange equation. This technique ensures high accuracy of approximate solutions, which is checked against Levi’s solution for rectangular plates __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 104–112, May 2006.     

Problem of a randomly oriented crack in a box-shaped shell

This paper deals with the stress state of a box-shaped shell formed by two semi-infinite plates joined at a right angle. The plates are homogeneous but have different thicknesses. The shell is weakened by a finite rectilinear crack of unit length which reaches one edge of the shell. The orientation of the crack and the load on its edges are arbitrarily chosen. The problem is solved with the assumption that the thickness of the plates is small compared to the length of the crack, which allows an asymptotic formulation of the problem. The problem is reduced to a special type of Riemannian vector problem in which the stress-intensity factor allows matrix factorization in accordance with Khrapkov’s scheme. The asymptotes of the resulting solution and the stress-intensity factor are examined in relation to the thickness of the shell and the angle formed by the crack and the edge of the shell. Translated from Prikladnaya Mekalinika, Vol. 34, No. 12, pp. 48–54, December, 1998.     

Dynamic modelling of thin plate made of certain functionally graded materials

The aim of the contribution is to formulate a macroscopic mathematical model describing the dynamic behaviour of a certain composite thin plates. The plates are made of two-phase stratified composites with a smooth and a slow gradation of macroscopic properties along the stratification. The formulation of mathematical model of these plates is based on a tolerance averaging approach (Woźniak, Michalak, Jędrysiak in Thermomechanics of microheterogeneous solids and structures, 2008). The presented general results are illustrated by analysis of the natural frequencies for two cases of plates: a plate band and an annular plate. The spatial volume fractions of the two different isotropic homogeneous components are optimized so as to maximize or minimize the first natural frequency of the plate under consideration.     

Statics and dynamics of simply supported polygonal Reissner-Mindlin plates by analogy

Summary Free and forced vibrations of moderately thick, transversely isotropic plates loaded by lateral forces and hydrostatic (isotropic) in-plane forces are analyzed in the frequency domain. Influences of shear, rotatory inertia, transverse normal stress and of a two-parameter Pasternak foundation are taken into account. First-order shear-deformation theories of the Reissner–Mindlin type are considered. These theories are written in a unifying manner using tracers to account for the various influencing parameters. In the case of a general polygonal shape of the plate and hard-hinged support conditions, the Reissner-Mindlin deflections are shown to coincide with the results of the classical Kirchhoff theory of thin plates. The background Kirchhoff plate, which has effective (frequency-dependent) stiffness and mass, is loaded by effective lateral and in-plane forces and by imposed fictitious “thermal” curvatures. These deflections are further split into deflections of linear elastic prestressed membranes with effective stiffness, mass and load. This analogy for the deflections is confirmed by utilizing D'Alembert's dynamic principle in the formulation of Lagrange, which yields an integral equation. Furthermore, the analogy is extended in order to include shear forces and bending moments. It is shown that in the static case, with no in-plane prestress taken into account, the stress resultants for certain groups of Reissner-type shear-deformable plates are identical with those resulting from the Kirchhoff theory of the background. Finally, results taken from the literature for simply supported rectangular and polygonal Mindlin plates are yielded and verified by analogy in a quick and simple manner. Received 29 September 1998; accepted for publication 22 June 1999     

Nonlinear vibrations of FGM rectangular plates in thermal environments

Geometrically nonlinear vibrations of FGM rectangular plates in thermal environments are investigated via multi-modal energy approach. Both nonlinear first-order shear deformation theory and von Karman theory are used to model simply supported FGM plates with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities. A pseudo-arclength continuation and collocation scheme is used and it is revealed that, in order to obtain the accurate natural frequency in thermal environments, an analysis based on the full nonlinear model is unavoidable since the plate loses its original flat configuration due to thermal loads. The effect of temperature variations as well as volume fraction exponent is discussed and it is illustrated that thermally deformed FGM plates have stronger hardening behaviour; on the other hand, the effect of volume fraction exponent is not significant, but modal interactions may rise in thermally deformed FGM plates that could not be seen in their undeformed isotropic counterparts. Moreover, a bifurcation analysis is carried out using Gear’s backward differentiation formula (BDF); bifurcation diagrams of Poincaré maps and maximum Lyapunov exponents are obtained in order to detect and classify bifurcations and complex nonlinear dynamics.     

On the Range of Applicability of the Reissner–Mindlin and Kirchhoff–Love Plate Bending Models

We show that the Reissner–Mindlin plate bending model has a wider range of applicability than the Kirchhoff–Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner–Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear effect, the Kirchhoff–Love model fails. This revised version was published online in June 2006 with corrections to the Cover Date.     

Two time scale solution for an unsteady MHD duct flow

The investigation deals with unsteady laminar flow of a viscous, incompressible, electrically conducting fluid between conducting or nonconducting flat plates. A constant magnetic field is suddenly applied perpendicular to the plates and the created electromagnetic effects modify the motion. An approximate solution is obtained using time scales t and t/ε, where ε is the small magnetic Prandtl number.