On simultaneous shape and orientational design for eigenfrequency optimization

Plates with an internal hole of fixed area are designed in order to maximize the performance with respect to eigenfrequencies. The optimization is performed by simultaneous shape, material, and orientational design. The shape of the hole is designed, and the material design is the design of an orthotropic material that can be considered as a fiber-net within each finite element. This fiber-net is optimally oriented in the individual elements of the finite element discretization. The optimizations are performed using the finite element method for analysis, and the optimization approach is a two-step method. In the first step, we find the best design on the basis of a recursive optimization procedure based on optimality criteria. In the second step, mathematical programming and sensitivity analysis are applied to find the final optimized design.     

Designing plates for minimum internal resonances

Internal resonance is a nonlinear phenomenon for a structure when the eigenfrequencies of the structure are commensurable or close to being commensurable. Using optimization we have the possibility to control the eigenfrequencies, i.e., move an eigenfrequency, maximize a given eigenfrequency, or maximize the gap between eigenfrequencies. It is therefore also possible to design a structure that is as free as possible of internal resonance up to mode of order n.We consider plates made of two materials. The designs depend on the boundary conditions and on the frequency range within which the plate should be as free of internal resonance as possible. The two materials can either be two physical materials, or one can be a physical material and the other a weakening of the first material. By doing this we are in principle solving three different problems: a reinforcement problem, a problem of where to put holes in the structure, and, finally the more involved case (from a manufacturing point of view), of two different materials. The optimizations are performed using the finite element method for analysis and the topology optimization approach for design. The optimization problem is formulated using a bound formulation where the objective is to maximize a minimum detuning parameter. Special attention is given to the formulation of the conditions for internal resonance. Using the method presented in this paper it is possible to remove an unwanted nonlinear phenomenon without the use of a nonlinear model and without knowledge of the nonlinearities present in the system.     

A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses

A frequency analysis of an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented. The dimensionless frequency equation for classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions associated to the masses. The resulting transcendental equations are numerically solved for the eigenvalue. On the other hand, the eigenvalue can be predicted merely from the individual beam system carrying a single mass, by virtue of the Dunkerley's formula. A parametric study on the effects of the two masses and their locations is presented for the beam with different boundary conditions. It is found that the Dunkerley's expression can generally yield good approximation if compared with the result associated with the original characteristic equation. The computation time saved owing to the modified Dunkerley method is also illustrated in a comparison. The Dunkerley's method is recommended for the beam carrying more than two masses at different positions, owing to its good approximation and the saving in computational time.     

A comparative study of the eigenvalue solutions for mass-loaded beams under classical boundary conditions

A comparative study of the eigenfrequency analysis for an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented in this short note. The dimensionless frequency equation for different combinations of classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions. Two formulation methods have been commonly used for the boundary-value problem. One is to adopt a single frame originated from the beam's left-end, while another is by dual frames associated with the concentrated mass. It is found that the forms derived by dual frames are more compact than the corresponding expressions by using the single frame. Nevertheless, the comparison for all the cases shows that the dual-frame expressions need more time to obtain the same set of eigenvalues if compared with the time by using the single-frame expressions.     

The radial basis integral equation method for solving the Helmholtz equation

A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=⁴Rln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results.     

Acoustic resonant response of smooth or rough 3D small objects

This paper presents a new method in order to appreciate the roughness of elliptical or spherical small bodies by using ultrasound. This thing is possible by knowing about 2-4 acoustic resonance frequencies. The maximum diameter and medium roughness of the bodies are evaluated. Small bodies such as tumors can be detected, when other medical devices like ultrasound or tomography computerized cannot evaluate the surface shape because of limited spatial resolution. The method is intended to be used in oncological medicine for early non-invasive diagnostic for cancer, because tumor tissue has outer surface more or less rough, depending on whether or not it is malignant.     

Optimal shapes of PZT actuators for laminated structures subjected to displacement or eigenfrequency constraints

In this paper, dynamics, electromechanical couplings, and control of piezoelectric laminated cylindrical shells and rectangular plates are investigated. It is assumed that the piezoelectric layers are distributed on the top and bottom surfaces of the structures. First of all the governing equations and boundary conditions including elastic and piezoelectric couplings are formulated and solutions are derived. Then control of the plate/shells deflections and natural frequencies using high control voltages are studied in order to optimize the structural response. The present formulation of optimal design introduces boundaries of piezoelectric patches as new class of design variables. In addition, classical design variables in the form of ply orientation angles of orthotropic layers are also taken into account. For the actuator/actuator configuration, it was shown that the piezoelectric actuators can significantly reduce deformations/eigenfrequencies of the composite plate. Those effects were dependent on the value of the applied voltage. It was demonstrated that the proper choice of the actuator area is more efficient in reducing deflections/eigenfrequencies. The accuracy of optimal design are verified both with the aid of the FE package ABAQUS and using the standard Rayleigh-Ritz method. The results concerning active vibration control for axisymmetric cylindrical shells are also discussed.     

Computation of Single Eigenfrequencies and Eigenfunctions of Plate and Shell Structures Using an H-adaptive FE-method

The accuracy of the computation of eigenfrequencies and eigenfunctions with FE-methods can be substantially improved with efficient adaptive procedures. For such an adaptive analysis of plate and shell structures a simple a-posteriori error estimator or indicator for the error in the energy norm and L ₂-norm of the eigenfunction u _h for shell and plate structures is proposed.Both indicators hite shall represent the correct convergence of the estimated error. The estimator for the error in the energy norm is used to enlarge adaptively the dimension of the finite element subspace. On the basis of numerical examples the efficiency and the quality of the improved solution is discussed. In order to validate the quantity of the estimated error Aitken’s extrapolation technique is applied.     

The Method of Fundamental Solutions Applied to the Calculation of Eigenfrequencies and Eigenmodes of 2D Simply Connected Shapes

In this work we show the application of the Method of Fundamental Solutions(MFS) in the determination of eigenfrequencies and eigenmodes associated to wave scattering problems. This meshless method was already applied to simple geometry domains with Dirichlet boundary conditions (cf. Karageorghis (2001)) and to multiply connected domains (cf. Chen, Chang, Chen, and Chen (2005)). Here we show that a particular choice of point-sourcescan lead to very good results for a fairly general type of domains. Simulations with Neumann boundary conditionare also considered.     

On the eigenfrequencies of cantilevered beams carrying a tip mass and spring-mass in-span

The present study is concerned with the derivation of the eigenfrequencies and their sensitivity of a cantilevered Bernoulli-Euler beam carrying a tip mass (primary system) to which a spring-mass (secondary system) is attached in-span. After establishing the exact frequency equation of the combined system, a Dunkerley-based approximate formula is given for the fundamental frequency. Using the normal mode method, a second approximate frequency equation is established which is then used for the derivation of a sensitivity formula for the eigenfrequencies. The frequency equations of some simpler systems are obtained from the general equation as special cases. These frequency equations are then numerically solved for various combinations of physical parameters. The comparison of the numerical results with those from exact frequency equations indicate clearly that the eigenfrequencies of the combined system described above can be accurately determined by the present method.