Journal of Theoretical and Applied Mechanics, Sofia, 2013, vol. 43, No. 1, pp. 1926

VIBRATION MODAL SOLUTIONS DEVELOPING OF THE

ELASTIC CIRCULAR MEMBRANE IN POLAR

COORDINATES BASED ON THE FOURIER-BESSEL

SERIES

Amir Javidinejad

Zodiac Aerospace,

7330 Lincoln Way, Garden Grove, CA USA,

e-mail:amir.javidinejad@gmail.com

[Received 27 July 2012. Accepted 21 January 2013]

Abstract. This paper is written to show the development of the vibra-tion modal solutions of elastic circular membranes in polar coordinates us-ing the Fourier-Bessel series. The ordinary differential equation approachis utilized and the Laplacian of wave equation in polar coordinates is usedto develop the solution of the membrane vibrations. A Fourier-Bessel so-lution is developed for the vibration of the elastic circular membrane inspecific separation of variables is elaborated and based on the initial andboundary conditions. A numerical example is provided to show the ap-plication of such theory.Key words: Membrane vibration, Fourier-Bessel series, eigenvalues.

1. Introduction

In 1738 Daniel Bernoulli published a number of theories on the oscil-lations of heavy chains and G. N. Watson in 1922 in his book of Treatise onthe Theory of Bessel Functions used this similar approach to investigate useof Bessel functions for development of solutions of the oscillating systems us-ing Euler theories. Later these developments lead into development of modalsolutions of the oscillating rectangular membranes by various authors such asNakhle H. Asmar in 2005 and others earlier, using boundary value differentialequations theories.

The application of circular membranes in systems such as pizeoelectricpressure sensors, drum heads and biological systems such as eardrums, has de-veloped the need to investigate the modal solutions of such circular membranesin more detail. Also, due to circular geometrical properties of such membranes,

20 Amir Javidinejad

the study of the oscillating behavior of such membranes in Polar coordinatebecomes more useful than the study in Cartesian coordinate system. Thus,this paper attempts to investigate modal behavior of the circular membranesin polar coordinates in detail using Bessel-Fourier approach.

According to Newtons second law, the sum of the forces on a smallportion of a body is equal to the mass of that small portion times the accel-

eration

(2u

t2

)of the body in motion. Also, the sum of forces on a vibrating

membrane is equal to the tension on the membrane times the stretch of themembrane during oscillations [1]. This force balance equality gives the equationthat governs the behaviour of the vibrating membrane which mathematically isin the Laplacian form. This concept can be derived from elementary geometri-cal mechanics of vibrating bodies that leads to an equation known as the waveequation describing the motion of a vibrating membrane. The following waveequation according to Kreyszig [1] for circular plates can be utilized to developthe vibration modal solutions of the circular membranes in polar coordinates:

(1)2u

t2= c2

(2u

r2+

u

rr2+

2u

r22

)

where, c2 is the membrane tension divided by the membrane density, r is theradius of the membrane and is the angle around the axis of the membrane.

For this wave equation the following solution is valid,

(2) u = F (r, )G(t).

Substituting (2) and its derivative into (1), one would have:

(3) GF = c2(2F

r2G(t) +

F

rrG(t) +

2F

r22G(t)

).

By separation,

(4)G

c2G=

1

F

(2F

r2+

F

rr+

2F

r22

).

As such the expressions on both sides must equal a constant to derivea solution. Which must be a negative constant (-k2) to satisfy the boundaryconditions without being zero [1].

(5)G

c2G=

1

F

(2F

r2+

F

rr+

2F

r22

)= k2.

Vibration Modal Solutions Developing of the Elastic Circular Membrane . . . 21

Rewriting the equation (5) in terms of G one would have:

(6)G

c2G= k2.

and rewriting equation (5) in terms of F one would have:

(7)1

F

(2F

r2+

F

rr+

2F

r22

)= k2.

Equations (6) and (7) yield the two differential equations following:

(8) G + 2G = 0,

where = ck and,

(9)2F

r2+

F

rr+

2F

r22+ k2F = 0.

Substitute the following expression into equation (9):

(10) F = W (r)Q().

One would have the expression:

(11) QW +1

rQW +

1

r2WQ + k2WQ = 0

which can be simplified as,

(12) Q + Qr2W + rW + r2k2W

W= 0.

Letting,

(13) Q

Q=

r2W + rW + r2k2W

W= n2

the following expressions are derived;

Q + n2Q = 0(14)

r2W + rW + (k2r2 n2)W = 0.(15)

Letting s = kr, then (k/s) = (1/r) and ds/dr = k,

22 Amir Javidinejad

Now,

dW

dr=

dW

dsk(16)

d2W

dr2=

d2W

ds2k2.(17)

Substitute, expression (16) and (17) into equation (15) and the followingexpression can be derived:

(18)d2W

ds2k2 +

1

r

dW

dsk +

(k2

n2

r2

)W = 0.

Substitute r = s/k and one would have:

(19)d2W

ds2+

1

s

dW

ds+

(1

n2

k2r2

)W = 0.

This is a Bessels equation as described in many mathematical literature[1]. Which has the Bessel equation solution of,

(20) Wn(r) = Jn(s) = Jn(kr).

Farther more expression (14) has the solution [1]:

(21) Qn = cos(n) for n = 0, 1, 2, . . .

Thus, following the expression (10) one would have:

(22) F = Jn(kr) cos(n)

and by expression (2):

(23) u(r, t, ) = F (r, )G(t) = Jn(kr) cos(n)G(t)

where, the Eigen-function G(t) can be expressed by the Fourier series as fol-lowing [2],

(24) Gmn(t) = amn cos(ckmnt) + bmn(ckmnt) for m = 0, 1, 2, . . .

Thus, expression (23) can be rewritten as,

(25) umn(r, t, ) = [amn cos(ckmnt) + bmn sin(ckmnt)]Jn(kmnr) cos(n)

Vibration Modal Solutions Developing of the Elastic Circular Membrane . . . 23

(for R being the radius of the circular membrane, kmn = mn/R), where theFourier series coefficients are:

amn =2

R2J21(mn)

R0

rf(r)J0

(mnR

r)dr(26)

bmn =2

cmnRJ2

1(mn)

R0

rg(r)J0

(mnR

r)dr(27)

and initial displacement and initial velocity are as follows, respectively:

(28) f(r) = u(r, 0) and g(r) =u

t

t=0

.

Also, the mn is the mth positive zeros of Jn(s) that have the followingvalues for the first three modes: (The positive zeros can be determined byplotting the zero-th order Bessel function for a range of s; assuming s from0 to 10 or higher):

(29) 10 = 2.4048, 20 = 5.5201, 30 = 8.6537.

2. Boundary conditions

The following boundary conditions are the trivial boundary conditionsthat are applicable to the solution of vibration modes for the circular membranein polar coordinates. For a circular membrane fixed at the outer boundary, thedeflection of the membrane at radius r = R is null thus boundary equation 29following can be derived.

(30) u(R, t, ) = 0.

To develop a numerical solution one can also assume that the initialdisplacement of the membrane, f(r) is zero and there exists an initial velocityof the vibrating membrane, g(r).

(31) f(r) = 0 and g(r) 6= 0.

24 Amir Javidinejad

3. Numerical example

A numerical example for the first three vibrational modes is illustratedhere. The MathCAD software is utilized to carry out the numerical example andto illustrate the determination of the constants of the functions. The followingkmn factors used in equation (25) for a circular membrane with radius R of 1ft (0.3048 m) and c of 2ft/sec (.111 Nm2/kg), are determined for m = 3 andn = 0:

(32)

k1,0 :=alp1,0R

; k1,0 = 2.405;

k2,0 :=alp2,0R

; k2,0 = 5.52;

k3,0 :=alp3,0R

; k3,0 = 8.654.

The following Fourier series for an initial velocity of 8.333E-3 ft/sec(0.002539898 m/sec) and initial deflection of zero coefficients are determined:(note: amn coefficients are all zero since the initial deflection is zero):

b1,0 :=

[2

(c alp1,0 R)(J1(alp1,0))2

]

R0

r

(1

12

) J0

(alp1,0

r

R

)dr;

b1,0 = 2.776 103;

b2,0 :=

[2

(c alp2,0 R)(J1(alp2,0))2

]

R0

r

(1

12

) J0

(alp2,0

r

R

)dr;

b2,0 = 8.037 104;

b3,0 :=

[2

(c alp3,0 R)(J1(alp3,0))2

]

R0

r

(1

12

) J0

(alp3,0

r

R

)dr;

b3,0 = 4.099 104.

By substituting these values into equation (25), one would have thefollowing vibrational deflection modes, for modes 1, 2 and 3, respectively. Thex-axis is the radial distance of the membrane. The y-axis is the deflection of the

Vibration Modal Solutions Developing of the Elastic Circular Membrane . . . 25

Fig. 1. The first three vibrational modes of the circular membrane

26 Amir Javidinejad

membrane. The S1 is mode 1, S2 is mode 2 and S3 is mode 3 for the circularmembrane. Whereas, the figures in the left are the deflection magnitudes andthe figures in the right are the 3D depictions of the mode shapes of the circularmembrane.

4. Conclusion

The Fourier-Bessel solution of the circular membrane vibration modeswas developed utilizing the wave equation in polar coordinates. The developedvibration modes are based on Bessel functions with solution derivatives from theFourier series. The solutions are a purely mathematical approach to vibrationalnormal modes development in Polar coordinates that are an extension of thesolution development in Cartesian coordinate systems.

REFERENCES

[1] Kreyszig, E. Advanced Engineering Mathematics, New York, John Wiley andSons, 1999.

[2] Churchill, R. V., J. W. Brown. Fourier Series and Boundary Value Prob-lems, New York, McGrawhill, 1987.

[3] Jackson, D. Fourier series and orthogonal polynomials, Mathematical Assoc.of America, D.C., Washington, 1941.

[4] Krantz, S. G. A Panorama of Harmonic Analysis, Mathematical Assoc. ofAmerica, D.C., Washington, 1999.

[5] Walker, J. S. Fourier Analysis, Oxford, Oxford Univ. Press, 1988.

[6] Zygmund, A. Trigonometric Series, Cambridge, Cambridge Univ. Press, 1968.