The tool SD-04 obtains the natural frequencies and associate mode shapes of a tensioned circular membrane with either fixed edge or free edge.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.

**Example: Vibration of a Circular Membrane**

The input in the tool is

After clicking "Run", the results will be shown as follows in another page.

Natural Frequencies (w_{mn} - rad/s)

m = 0

n = 0: w_{00} = 24.04825558

n = 1: w_{01} = 38.3170597

n = 2: w_{02} = 51.35622302

m = 1

n = 0: w_{10} = 55.2007811

n = 1: w_{11} = 70.1558667

n = 2: w_{12} = 84.1724414

m = 2

n = 0: w_{20} = 86.53727913

n = 1: w_{21} = 101.73468135

n = 2: w_{22} = 116.19841172

Analytical Expression of Mode Shape

W_{mn}(r, θ) = J_{n}(γ_{mn} r) cos(nθ)

where γ_{mn} = ω_{mn} / sqrt(T/ρ)

m = 0

n = 0: γ_{00} = 1.20241278

n = 1: γ_{01} = 1.91585299

n = 2: γ_{02} = 2.56781115

m = 1

n = 0: γ_{10} = 2.76003906

n = 1: γ_{11} = 3.50779333

n = 2: γ_{12} = 4.20862207

m = 2

n = 0: γ_{20} = 4.32686396

n = 1: γ_{21} = 5.08673407

n = 2: γ_{22} = 5.80992059

The membrane in consideration is a thin, circular and unstretchable continuum that lies in a plane when in equilibrium, and is subject to uniform in-plane tension force along its boundary. The membrane has lateral or transverse vibration due to external and initial disturbances.

**Figure 1.** A circular membrane in transverse vibration

For the circular membrane in Fig. 1, with radius a, tension T and density ρ, its transverse displacement w(r,θ,t) in free vibration is governed by the differential equation

where r and θ are the polar coordinates, and the Laplacian operator

Two typical types of boundary conditions of the membrane at edge r = a are as follows.

Assume that

Substitute Eq. (5) into Eq. (1), to obtain the eigen-equation for the membrane

where ω is an eigenvalue, and W(r, θ) is the eigenfunction associated with ω .

By separation of variables, write

where n is an integer. This reduces Eq. (6) to

with

By introducing the non-dimensional variable

Eq. (8) is reduced to Besselâ€™s equation of order n

A general solution of the Besselâ€™s equation is

where J_{n} is Bessel function of the first kind, and A can be any non-zero constant.
Thus, the eigenfunction of the circular membrane can be expressed as

Substitute Eq. (13) into boundary condition (3) or (4) yields the characteristic equation, from which the eigenvalues of the membrane can be determined.

For a circular membrane with fixed edge, its characteristic equation is

which has an infinite number of roots, γ _{mn}, m,n = 0, 1, 2...
The non-dimensional roots γ _{mn}a of Eq. (14) are tabulated in Table 1.
Here m is the number of nodal circles and n is the number of nodal diameters, which shall be explained subsequently.

**Table 1.** Roots γ _{mn}a of characteristic equation (14)

It follows from Eq. (9) that the natural frequencies are given by

By Eq. (13), the mode shapes corresponding to Roots γ _{mn} are given by

For example, for a circular membrane with fixed edge and parameters ρ = 0.07, T = 20, a = 1, the mode (0, 1) is plotted in Fig. 2.

**Figure 2.** The spatial distribution of mode (0, 1)

For a circular membrane with free edge, its characteristic equation is

which has an infinite number of roots, γ _{mn}, m,n = 0, 1, 2...
The non-dimensional roots γ _{mn}a of Eq. (17) are tabulated in Table 2.
Here m is the number of nodal circles and n is the number of nodal diameters.
The natural frequencies and mode shapes are also given by Eqs. (15) and (16), respectively.

**Table 2.** Roots γ _{mn}a of characteristic equation (17)

For example, for a circular membrane with fixed edge and parameters ρ = 0.07, T = 20, a = 1, the mode (0, 1) is plotted in Fig. 2.

**Figure 3.** The spatial distribution of mode (1, 0)

1. Den Hartog, J. P., 1985, Mechanical Vibrations, Dover Publications, New York.

2. Inman, D. J., 2000, Engineering Vibrations, 2nd edition, Prentice Hall, Inc., Upper Saddle River, New Jersey.

3. Meirovitch, L., 1967, Analytical Methods in Vibrations, Macmillan, New York.

4. Rao, S. S., 2003, Mechanical Vibrations, 4th edition, Prentice Hall, Inc., Upper Saddle River, New Jersey.

5. Timoshenko, S., 1990, Vibration Problems in Engineering, John Wiley & Sons; 5th edition.

6. Yang, B., 2005, Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes, Elsevier Science, Boston.