A formulation is developed to predict the vibration response of a finite length, submerged plate due to a line drive. The formulation starts by describing the fluid in terms of elliptic cylinder coordinates, which allows the fluid loading term to be expressed in terms of Mathieu functions. By moving the fluid loading term to the right-hand side of the equation, it is considered to be a force. The operator that remains on the left-hand side is the same as that of the in vacuo plate: a fourth-order, constant coefficient, ordinary differential equation. Therefore, the problem appears to be an inhomogeneous ordinary differential equation. The solution that results has the same form as that of the in vacuo plate: the sum of a forced solution, and four homogeneous solutions, each of which is multiplied by an arbitrary constant. These constants are then chosen to satisfy the structural boundary conditions on the two ends of the plate. Results for the finite plate are compared to the infinite plate in both the wave number and spatial domains. The theoretical predictions of the plate velocity response are also compared to results from finite element analysis and show reasonable agreement over a large frequency range.
Frequency Response Functions (FRF) haves been used to aid noise and vibration designs in various industries. Those design considerations include resonance avoidance, vibration reduction, etc. Numerical methods have been widely applied to predict Frequency Response Functions (FRF) of structures. However, the computational resources (i.e., CPU time, memory, disk space) needed to solve large and detailed numerical models are getting large. Furthermore, the need to resolve resonant response peaks can drive up the number of FRF calculations required. Lately, advanced numerical techniques based on a Krylov subspace and Galerkin Projection (KGP) and Pade Approximation have been demonstrated that they can significantly accelerate the overall process by approximating the frequency dependent response (calculating the forced response at only a few frequencies, then using KGP or Pade approximates to reconstruct the FRF for the rest of desired frequency points.) This paper will present the latest enhancements to the KGP: modeling of viscoelastic material via its complex modulus representation and adaptive capability (AKGP) in automating the frequency sweep process via calculated tolerance error. To illustrate the accuracy and efficiency of the new enhancements, a numerical example will be exercised and used as a benchmark to compare different numerical tools with and without the AKGP.
An approximate method for predicting the response of a line-driven, fluid-loaded plate will be presented. The method is based on the free space Green’s function using the rational function approximation developed by DiPerna and Feit. The approximte representations of the Green’s function and pressure and velocity fields will be shown to satisfy the Helmholtz integral equation for the infinite plate. The Helmholtz integral equation will then be used in conjunction with these approximate representations to solve the problem of a finit length plate. Results from the approximate solution will be compared to results from finite-element analysis.
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