Identification of Boundary Conditions Using Natural Frequencies

The present investigation concerns a disc of varying thickness of whose flexural stiffness $D$ varies with the radius $r$ according to the law $D=D_0 r^m$, where $D_0$ and $m$ are constants. The problem of finding boundary conditions for fastening this disc, which are inaccessible to direct observation, from the natural frequencies of its axisymmetric flexural oscillations is considered. The problem in question belongs to the class of inverse problems and is a completely natural problem of identification of boundary conditions. The search for the unknown conditions for fastening the disc is equivalent to finding the span of the vectors of unknown conditions coefficients. It is shown that this inverse problem is well posed. Two theorems on the uniqueness and a theorem on stability of the solution of this problem are proved, and a method for establishing the unknown conditions for fastening the disc to the walls is indicated. An approximate formula for determining the unknown conditions is obtained using first three natural frequencies. The method of approximate calculation of unknown boundary conditions is explained with the help of three examples of different cases for the fastening the disc (rigid clamping, free support, elastic fixing). Keywords: Boundary conditions, a disc of varying thickness,inverse problem, Plucker condition.Comment: 19 page

Linear modal analysis of L-shaped beam structures: parametric studies

Linear modal analysis of L-shaped beam structures indicates that there are two independent motions, these are in-plane bending and out of plane motions including bending and torsion. Natural frequencies of the structure can be determined by finding the roots of two transcendental equations which correspond to in-plane and out-of-plane motions. Due to the complexity of the equations of motion the natural frequencies cannot be determined explicitly. In this article we nondimensionalise the equations of motion in the space and time domains, and then we solve the transcendental equations for selected values of the L-shaped beam parameters in order to determine their natural frequencies. We use a numerical continuation scheme to perform the parametric solutions of the considered transcendental equations. Using plots of the solutions we can determine the natural frequencies for a specific L-shape beam configuration

Influence of the boundary conditions on the natural frequencies of a Francis turbine

Natural frequencies estimation of Francis turbines is of paramount importance in the stage of design in order to avoid vibration and resonance problems especially during transient events. Francis turbine runners are submerged in water and confined with small axial and radial gaps which considerably decrease their natural frequencies in comparison to the same structure in the air. Acoustic-structural FSI simulations have been used to evaluate the influence of these gaps. This model considers an entire prototype of a Francis turbine, including generator, shaft, runner and surrounding water. The radial gap between the runner and the static parts has been changed from the real configuration (about 0.04% the runner diameter) to 1% of the runner diameter to evaluate its influence on the machine natural frequencies. Mode-shapes and natural frequencies of the whole machine are discussed for all the boundary conditions testedPostprint (published version

On the bipenalty method: why is it advantageous to add stiffness and mass

In a recent paper, Askes et al [1] proposed the simultaneous use of stiffness and inertia of large magnitude to model constraints in time domain analysis. From a frequency domain perspective, as stiffness and inertia have opposite effects on the natural frequencies, this seems counter-intuitive. With increasing stiffness, the natural frequencies either increase or remain unchanged, whereas the opposite is true for inertia. However, it can be shown, through very simple illustrative examples, that the natural frequencies and modes of continuous systems can be found in this way, and that there are advantages in using both stiffness and mass simultaneously

Effect of Damping on the Natural Frequencies of Linear Dynamic Systems

An analysis is presented of the effect of weak damping on the natural frequencies of linear dynamic systems. It is shown that the highest natural frequency is always decreased by damping, but the lower natural frequencies may either increase or decrease, depending on the form of the damping matrix

Electromagnetic resonances of cylinders and aircraft model with resistive wires

The natural frequencies of the electromagnetic resonances of conducting bodies with attached wires were determined. The bodies included twp cylinders and an approximate scale model of the NASA F-106B aircraft. All were three feet in length. Time domain waveforms of B-dot and D-dot were obtained from a sampling oscilloscope, and Prony analysis was used to extract the natural frequencies. The first four natural frequencies of the cylinders (and wires) were determined, and a comparison with calculated results of other investigators shows reasonable agreement. Seven natural frequencies were determined for the F-106B model (with wires), and these were compared with results obtained by NASA in 1982 during direct lightning strikes to the aircraft. The agreement between the corresponding natural frequencies of the model and the aircraft is fairly good and is better than that obtained in the previous work using wires with less resistance. The frequencies lie between 6.5 MHz and 41 MHz, and all of the normalized damping rates are between 0.14 and 0.27

Natural Frequency Analysis Of All Edges Clamped Flexible Thin Plate

In this paper, the analysis of natural frequencies for all clamped edges rectangular flexible thin plate is carried out using Finite Difference (FD) and Finite Element (FE) approaches. According to the literatures, the differential equation of plate was obtained by considering the Kirchhoff hypotheses and Newton’s law. The dynamic differential model is developed by using the FD to obtain the natural frequencies of given plate; for this purpose, a displacement model is converted to combination of sine and cosine functions in form of Fast Fourier Series. In second method, modes of vibration are driven by FE method using the ABAQUS software. The obtained natural frequencies of both methods are evaluated and compared with previous literatures; the outcomes can explain that the improved FD method’s results are more accurate in compare with FE method’s

Electromagnetic resonances of cylinders and aircraft model with resistive wires

Laboratory experiments were done to determine the natural frequencies of the electromagnetic resonances of conducting bodies with attached wires. The bodies include two cylinders and an approximate scale model of the NASA F-106B aircraft. All are three feet in length. Time-domain waveforms of B-dot and D-dot were obtained from a sampling oscilloscope, and Prony analysis was used to extract the natural frequencies. This work is an extension of previous work, but smaller, more resistive wires have been used. The first four natural frequencies of the cylinders (and wires) were determined, and a comparison with calculated results of other investigators show reasonable agreement. Seven natural frequencies were determined for the F-106B model (wire wires), and these have been compared with results obtained by NASA in 1982 during direct lightning strikes to the aircraft. The agreement between the corresponding natural frequencies of the model and the aircraft is fairly good and is better than that obtained in the previous work using wires with less resistance. The frequencies lie between 6.5 MHz and 41 MHz, and all of the normalized damping rates are between 0.14 and 0.27