8.16 Rayleigh–Ritz Method for Determining Natural …
359
The frequency equation becomes
p
4
− 1224 p
2
E I
ρ A L 4
+ 15.120 p
4
E I
ρ A L 4
2
= 0
The solution of the above equation yields
p 1 = 3.52
E I
ρ A L 4 and p 2 = 22.03
E I
ρ A L 4
8.17 Vibration of Membranes
A membrane is a plate subjected to tensile loading. Entire loading is assumed to be
inplane. There are plenty of application areas of the membranes, such as inflatable
structures, parachutes, drumheads and biomedical components such as the valve of
an artificial heart.
The freebody diagram of the rectangular membrane element is shown in Fig. 8.20.
Let f (x, y, t) be the pressure loading acting in the z-direction. Let T be the intensity
of tension at a point. T remains constant throughout the membrane. On an elemental
area dx dy, forces T dx and T dy act on the sides parallel to y- and x-axes, respectively
(Fig. 8.20).
Fig. 8.20 Freebody diagram of a rectangular membrane
359
The frequency equation becomes
p
4
− 1224 p
2
E I
ρ A L 4
+ 15.120 p
4
E I
ρ A L 4
2
= 0
The solution of the above equation yields
p 1 = 3.52
E I
ρ A L 4 and p 2 = 22.03
E I
ρ A L 4
8.17 Vibration of Membranes
A membrane is a plate subjected to tensile loading. Entire loading is assumed to be
inplane. There are plenty of application areas of the membranes, such as inflatable
structures, parachutes, drumheads and biomedical components such as the valve of
an artificial heart.
The freebody diagram of the rectangular membrane element is shown in Fig. 8.20.
Let f (x, y, t) be the pressure loading acting in the z-direction. Let T be the intensity
of tension at a point. T remains constant throughout the membrane. On an elemental
area dx dy, forces T dx and T dy act on the sides parallel to y- and x-axes, respectively
(Fig. 8.20).
Fig. 8.20 Freebody diagram of a rectangular membrane