SOME EXACT SOLUTIONS OF THE VIBRATION OF NON-HOMOGENEOUS MEMBRANES

✍ Scribed by C.Y. Wang

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 123 KB
• Volume: 210
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.1997.1270

✦ Synopsis

A membrane is non-homogeneous if it has density or thickness variations. Literature on the vibrations of non-homogeneous membranes are few. A composite membrane composed of joining many homogeneous strips was considered by Sato [1] and Kalotas and Lee [2], while membranes composed of two distinct pieces were studied by several authors [3][4][5][6]. Recently, Masad [7] investigated a continuously non-homogeneous rectangular membrane where the density function varies linearly with respect to an edge. Masad used numerical integration and a perturbation method to solve for the natural frequencies.

The purpose of this note is to show that the linear density variation case studied by Masad [6] has a closed form exact solution. Also, we shall present another exact solution for the vibration of a continuously non-homogeneous annular membrane. These exact solutions are important not only in their own right as specific vibration problems, but can also serve as error standards for approximate methods, whether analytic or numerical.

The equation of motion is