M. Petyt

A finite element method for determining the acoustic modes of irregular shaped cavities

Abstract A twenty node, isoparametric acoustic finite element model is developed for analysing the acoustic modes of irregular shaped cavities. The element is first used to analyse a rectangular enclosure. This indicates that good accuracy can be obtained if one element is used between each nodal plane. Experiments are described on a model van enclosure. A finite element analysis of the model shows very good agreement with the measured results.

A finite element study of harmonic wave propagation in periodic structures

The vibration behaviour of periodic structures may be described by a characteristic “propagation constant”. The natural frequencies of the finite periodic structure are related to the variation of the purely imaginary part of this propagation constant, known as the phase constant. A method is presented for using the finite element technique to evaluate the phase constant of an arbitrary periodic structure. The method is applied to two types of periodic construction. The first is a periodically supported infinite beam. The second is a skin rib structure, with dimensions typical of a tailplane. Two models of the tailplane are considered, a beam model and a plate model.

Free vibration of a curved beam

Abstract Three finite element models are investigated for determining the radial vibrations of a curved beam. The investigations show that rigid body displacements should be closely represented and also that the normal and tangential representations should lead to the same strain energy convergence. One of the models is used to investigate the variation with subtended angle of the six lowest natural frequencies of beams with simply supported, hinged and clamped ends.

Vibration of curved plates

Abstract The free vibration characteristics of a singly curved rectangular plate have been obtained by four theoretical methods and compared with experimental results. The variations of a non-dimensional frequency parameter with aspect ratio, a thickness parameter, curvature and fuselage pressurization are indicated.

Non-linear free vibration of isotropic plates with internal resonance

The geometrically non-linear free vibration of thin isotropic plates is investigated using the hierarchical finite element method (HFEM). Von Karmans non-linear strain–displacement relationships are employed and the middle plane in-plane displacements are included in the model. The equations of motion are developed by applying the principle of virtual work and the harmonic balance method (HBM), and the solutions are determined using a continuation method. The convergence properties of the HFEM and of the HBM are analyzed. Internal resonances are discovered. The variation of the plates mode shape with the amplitude of vibration and during the period of vibration is demonstrated.

Non-linear vibration of composite laminated plates by the hierarchical finite element method

Abstract The geometrically non-linear vibration of thin, laminated composite plates is studied by the hierarchical finite element and the harmonic balance methods. Free and steady-state forced vibration are analysed. The excitations considered are harmonic plane waves at both normal and grazing incidence. The equations of motion are solved by a continuation procedure and the stability of the steady-state solutions is investigated by applying Floquet’s theory. The convergence properties of the hierarchical finite element method and the influence of the middle plane in-plane displacements are discussed, and results are compared with published results.

Vibration of column-supported floor slabs

Abstract The finite element displacement method of analysis is used to determine the vibration characteristics of floor slabs on four column supports. The results obtained are compared with other theoretical solutions and also experimental measurements. The effect of rigidity and finite area of the column supports are investigated. Finally, the vibration characteristics of various arrangements of slabs on many supports are considered.

An investigation into geometrically nonlinear analysis of rectangular laminated plates using the hierarchical finite element method

Abstract The geometrically nonlinear analysis of laminated composite rectangular plates is studied using the hierarchical finite element method (HFEM). The derivation of the equilibrium equations and tangential stiffness matrix are given. Symbolic computation is used to calculate the high-order polynomial integrals needed to establish the stiffness matrices. The Newton-Raphson method is used in the iterative procedure. The convergence property and the numerical stability of the method are discussed. The influence of in-plane displacements on the geometrically nonlinear deformation is also discussed. The results of static analyses indicate that the extension of HFEM to geometrically nonlinear analysis of laminated rectangular plates is very successful. It is believed that this scheme can be easily applied to geometrically nonlinear dynamic analysis of laminated plates.

A finite element study of the vibration of trapezoidal plates

Two high precision, conforming, plate bending elements, one a quadrilateral and the other a triangle, are used to investigate the free vibration characteristics of triangular and trapezoidal plates. An account of the derivation of the element mass and stiffness matrices is included. These elements are used first to obtain the natural frequencies and mode shapes of rectangular, trapezoidal and triangular, simply supported plates with height to base ratios of 1·5. The results of this analysis are compared with those found by other authors. The natural frequencies and nodal patterns of both simply supported and clamped trapezoidal plates with height to base ratios of 6 are then examined, by using the elements described. The ratio of 6 has been chosen as typical of the dimensions of control surface ribs. Four shapes of rib are considered, a rectangle, two symmetric trapezoids, and an isosceles triangle.

Ground vibration in the vicinity of a moving harmonic rectangular load on a half-space

The transmission of vibrations over the surface of the ground, due to a moving, vertical harmonic rectangular load, is investigated theoretically. The interior of the ground is modelled as an elastic half-space. The transformed solutions are obtained using the double Fourier transform. Numerical results for the displacements on the surface are presented for loads moving with speeds up to the Rayleigh wavespeed of the ground.