Page 34 - Fall 2006

Fig. 21. Finite element models and sample mode shapes for a marine propeller (left) and a metal equipment cabinet (right).
The matrices in the finite element system of equations represent the element masses (mij), damping (bij), and stiff- nesses (kij), and contain many, many terms; equal to the num- ber of degrees of freedom (DOF) in the model (N). The more elements that are used, the larger the matrices become. The system shown in equation (28) is usually written more com- pactly in matrix form as:
. (30)
A finite element computer program will assemble the mass, stiffness, and damping matrices (M, B, and K) based on the element geometries and material properties, and solve for the vibration response (the displacement vector d) based on the loads applied in the force vector F. A separate solution is required for each analysis frequency since the assembled matrix is frequency-dependent. Finite element programs can also extract the eigenvalues (normal modes) of the system when no loads are applied. It is easy to see that models that include many elements will require long computational times, and significant storage space due to the size of the matrices. In spite of these computational requirements, FE modeling is extremely popular, and used routinely to simu- late the vibrations in large models of aircraft, automobiles, ships, and submarines. Finite element models are sometimes also used to simulate acoustic regions, although another numerical method—Boundary Element (BE) analysis—is used more commonly for that purpose. We will learn about BE modeling in part 2 of this tutorial, and also see how BE models of fluids may be coupled to FE models of structures.
Summary
In part 1 of this article on structural acoustics, I have pre- sented basic vibration theories for beams, plates and shells, shown the speeds at which different waves travel through solids, how they can reinforce each other in finite structures to form modes, and how those modes define a structure’s mobility. Structural damping limits the vibration peaks at modal resonance frequencies, and at high frequencies can cause the mobility of a finite structure to converge to that of an infinite structure.
In part 2, we will see how these structural vibrations interact with neighboring acoustic media to radiate sound. Conversely, we will also consider how acoustic waves inci- dent on structures cause structural vibration, and subsequent sound re-radiation. As part of our discussion, we will learn about boundary element numerical modeling methods, which are used widely to compute fluid-structure interaction of complex structures in air and water.
Acknowledgments
I thank the members of the ARL/Penn State Structural Acoustics Department, along with several of the graduate stu- dents in Penn State’s Graduate Program in Acoustics (Andrew Munro, Ben Doty, William Bonness, and Ryan Glotzbecker). I also thank Dr. Courtney Burroughs (retired) who taught Structural Acoustics at Penn State before I did, and provided me with valuable guidance while we worked together.AT
The FE models shown here have been constructed with enough elements to resolve spatially the mode shapes in the examples. At very high frequencies, when the structural wavelengths shorten to the point where they are similar to the element sizes, the FE model becomes inaccurate. A good rule of thumb is to use at least six, and preferably eight ele- ments to model a structural wavelength. The wavelengths for a given analysis frequency can be estimated using the wave speed formulas shown earlier in this article.
The equations used to represent a finite element model are assembled into linear matrices. The matrices, although large and complicated, look similar to simple mass-spring- damper lumped parameter model equations assuming time- harmonic frequency dependence eiωt:
. (28)
Recall from basic vibrations theory that the time-harmonic response of a simple spring-mass-damper system is:
. (29)
32 Acoustics Today, October 2006