Mridangam shell-cavity analytical acoustic model for shape sensitivity studies

Harikanth MOHANDAS(1) , Chandramouli PADMANABHAN(2)(1) Indian Institute of Technology Madras, Chennai 600036, India, me17s025@smail.iitm.ac.in(2) Indian Institute of Technology Madras, Chennai 600036, India, mouli@iitm.ac.in

Abstract
Mridangam is a double-sided percussion instrument used in South Indian classical music concerts as an accompaniment. It consists of a smaller, centrally loaded membrane on the right and a larger membrane on theleft. These are assembled on a wooden shell which encloses an approximate double-conical acoustic cavity.
In this paper, the effect of the shape of the cavity on the music generated is of interest. For this purpose, a
one-dimensional analytical model based on a diverging-converging conical horn with short cylindrical ends isdeveloped. The first six axial natural frequencies and modes are computed from this model and validated using
a 3D finite element model. Moderate alterations to the geometry of the horn show a marginal shift for the firstmode and almost no significant change at the higher modes. However, cavity response to boundary excitations
for different shapes reveals a possible impact on the vibro-acoustics.
Keywords: Musical acoustics, Double-horn model, Boundary excitation

1

INTRODUCTION

Indian classical music has developed many unique sounding instruments during its existence from the Vedic age
to the present. The designs of some of these are quite intriguing. Two such instruments are the Mridangam
and the Tabla. The former is used in the Carnatic music tradition of the South, while the latter is used in the
Hindustani music tradition of the North. They both are percussion instruments used as a rhythm accompaniment.
An inherent feature of these is that, unlike a regular stretched circular membrane, their membranes produce
overtones which are integer multiples of the fundamental. This makes them harmonic, a rare attribute for a
percussion instrument. Harmonicity is achieved by carefully placing a central mass loading on the leather. The
mass loading appears as a black patch on the membrane. The detailed description of the membrane vibration
characteristics can be found in the paper by Raman [1]. An area yet unexplored on the Mridangam is the shapeof the air cavity and its role on the sound generated.
Mridangam has a shape similar to a barrel. The wooden shell has a slightly off-centre bulge, and attachedat both its ends are the membranes. The membrane played by the right hand is called the karanai, and isharmonic with a central mass loading. On the opposite end is the thoppi membrane, which is not harmonic as
opposed to the karanai. The membranes and the shell enclose an air cavity. It was found from Sooraj’s research
investigation [2] that the wooden shell does not contribute much to the music produced and that the coupling
happens only between the axial modes of the acoustic cavity and axisymmetric membrane vibration modes.
Thus, a simple one-dimensional model would be sufficient for the cavity to carry out shape sensitivity studies.
An analytical model would be more efficient for varying geometric parameters than a finite element modeland hence is chosen for the investigation. After developing a model, an eigenvalue analysis to extract natural
frequencies and mode shapes is followed by the computation of acoustic response to a boundary excitation; thisis carried out for different configurations of the cavity.

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2b

section 2

section 1

2a

Karanai end

2d

Thoppi end

section 4

section 3

z
0

l1

l2

l3

l4

Figure 1. Mridangam air cavity approximation

2

CAVITY ANALYTICAL MODEL

The cavity has a shape resembling a diverging-converging conical horn, with cylindrical ends. Membranes areattached to these cylindrical portions. An analytical model is built based on this approximation, and figure 1
shows this geometry. The lengths l1 , l2 , l3 and l4 are 56, 337, 544 and 604 mm respectively. The karanai
end radius, a is 66 mm while that at the thoppi end, d is 80 mm. The largest cross-sectional area has aradius b, which is 123 mm. For plane wave propagation, the equations for acoustic pressure variation in thecylindrical sections 1 and 4, is well known. Webster’s horn equation [3] is a one-dimensional approximation
for wave propagation in a variable cross-sectional area. It is solved for area variation of a cone, to get thepressure distributions at sections 2 and 3 [4]. Subscripts 1 and 2, are respectively used in denoting functions
at cylindrical and conical sections at the karanai end while subscripts 3 and 4, respectively denote functions atconical and cylindrical sections in the thoppi end. The pressure distribution p, at each section, is as follows.
p1 (z) = Ae− jkz + Be jkz
h
i
a
a
1
 Ce− jk(z−l1 + m ) + De jk(z−l1 + m )p2 (z) =ak z − l1 + m
h
i
b
b
1
 Ee− jk(z−l2 + n ) + Fe jk(z−l2 + n )p3 (z) =bk z − l2 + n
p4 (z) = Ge− jkz + He jkz

(1)
(2)
(3)
(4)

p
Here, z is axial coordinate with the origin at the karanai end, k is the wavenumber and c = κ/ρ is the speedof sound in the medium. The adiabatic bulk modulus, κ and density, ρ for air are taken to be 142 kPa and1.225 kg/m3 , respectively. For the karanai end, C = mk a20 + j a21 , D = mk a20 − j a21 and m = lb−a
. Similarly for the
2 −l1. Here a0 , a1 , b0 and b1 are constants to be determined.
thoppi end, E = mk b20 + j b21 , F = mk b20 − j b21 and n = ld−b3 −l2Euler’s momentum equation, ρ ∂∂tu = − ∂∂ pz is used to get the acoustic particle velocity u at each section. For thelength of the cavity, any function ε is a piecewise function of the form,
ε1 (z)ε (z)
2ε(z) =ε3 (z)
ε4 (z)

0 ≤ z ≤ l1
l1 ≤ z ≤ l2.
l2 ≤ z ≤ l3
l3 ≤ z ≤ l4

(5)

2.1 Natural frequencies and modes
Fixed boundaries are used at the ends while extracting the natural frequencies and the mode shape functionsof the cavity, giving boundary conditions u1 (0) = 0 and u4 (l4 ) = 0. Ensuring continuity of pressure and its

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gradient at all three interfaces give six compatibility conditions. They are p1 (l1 ) = p2 (l1 ) and u1 (l1 ) = u2 (l1 )at the karanai side cylinder-cone interface, p2 (l2 ) = p3 (l2 ) and u2 (l2 ) = u3 (l2 ) at the cone-cone interface, andp3 (l3 ) = p4 (l3 ) and u3 (l3 ) = u4 (l3 ) at the the thoppi side cone-cylinder interface. All of these give a set ofeight equations, and they are solved simultaneously to get the natural frequencies and its corresponding set of
constants A, B, a0 , a1 , b0 , b1 , G and H. Mode shape functions pr are obtained from the constants for the rth mode.
Validation of these results is carried out using finite element simulation with a 3D model. The model has
hexahedral acoustic elements with quadratic interpolation and an average internodal interval of 5 mm. Only theaxial modes from the simulation results are picked out for validation. Natural frequencies of the actual cavity
obtained from experiments are also available for comparison [2]. Table 1 shows the first six natural frequencies
from the analytical and finite element models, along with the first four experimental values. Figure 2 shows thepressure plots for axial modes 1, 3 and 5. Notice that the analytical approximation agrees well with the 3D
finite element model and is a reasonable assumption for the actual cavity of a Mridangam.
Table 1. Natural frequencies (in Hz) of Mridangam air cavity
Axial mode Analytical Finite element Experimental[2]
1
2
3
4
5
6

354.47575.11853.821122.71402.2
1689.9351.34569.63847.031113.51390.51676.6

Analytical

0.0

0.1

0.2

0.3

0.4

0.5

0.6 0.0

0.1

0.2

370
628
893
1152
-

Finite Element

0.3

0.4

0.5

0.6 0.0

0.1

0.2

0.3

z (m)

z (m)

z (m)

Mode 1Mode 3Mode 5

0.4

0.5

0.6

Figure 2. Pressure plots for axial modes of Mridangam cavity

2.2 Boundary excitation
The membranes and the cavity interact with each other. Modelling the harmonic response of cavity to boundary
excitations is essential to model the coupled acoustics of the Mridangam. Ginsberg [5] has formulated a method,using Green’s theorem, for solving cavity-structure interaction by modal superposition. The cavity response isfound out using a similar approach. The mathematical formulation assumes piston like motion of walls at theends and does not consider structural interaction.
Consider an air cavity with a variable cross-sectional area S(z), as shown in figure 3. Its response to uniform
harmonic motion of walls at either/both of the boundaries (z = 0, l) is of interest. The governing differential
equation for one-dimensional acoustic wave propagation in the cavity is given by,


1 d
dp
S
+ k2 p = 0,
S dz
dz

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(6)

un

S(z)

z=0

z=l

Figure 3. Cavity with variable cross-section excited at a boundary
where k = Ω/c, for the driving angular frequency Ω. The rigid cavity modes pr (z) satisfy


1 d
d pr
S
+ kr2 pr = 0,
S dz
dz
d pr
d pr
=
= 0.
dz 0
dz l

(7)

kr is the wave number for the rth mode. Now, pressure p is written as a linear expansion of mode functions
pr ,Rp(z) = ∑ αr pr (z).

(8)

r=1

The following integral equation, based on Green’s second identity, is constructed to evaluate the coefficients in
Eq. 8.
Z l
0pr

1 d dp1 d d prS−p
SS dz dz
S dz dz
dz = pr



Z l
Z l
d pr ldp ld pr d p
d p d pr
−p
−
)S
dz +
)S
dz
(
(
dz 0
dz 0
dz S dz
dz S dz
00(9)From Euler’s momentum equation and Eqs. 6 and 7, this reduces to(kr2 − k2 )

Z l




 Zl
d pr d p
d p d prppr dz = − jρΩ pr (l)u(l) − pr (0)u(0) −
(
)S
−(
)S
dz.
dz S dz
dz S dz
00(10)Further, introducing modal damping with damping ratio ζ , Eq. 10 becomes,(kr2 − k2 + j2ζ kr k)

Z l




 Zl
d p d prd pr d p
)S
−(
)S
dz.
ppr dz = − jρΩ pr (l)u(l) − pr (0)u(0) −(dz S dz
dz S dz
00(11)

A system of equations is formed using R modes. These are solved to get the coefficients. Ensuring velocity
continuity at the moving wall-fluid interfaces renders the boundary conditions u(0) and u(l).
Eq. 11 is extended for the Mridangam cavity to find its response to boundary excitations. Note that pressure p,
mode shape functions pr and area S are piecewise in the domain of the cavity (0 ≤ z ≤ l4 ). Hence, the integrals
are to be evaluated for each sub-domain and added together. For a vibration magnitude un = 1 mm, driving
frequency is swept up to 1000 Hz. First six modes are taken to reduce the truncation error in this frequency
range. The damping ratio is assumed to be 3% for each mode. For excitation at one end, the other is kept
fixed. When karanai end is excited keeping thoppi fixed, boundary conditions are u(0) = un and u(l4 ) = 0. For
thoppi side excitation, this changes to u(0) = 0 and u(l4 ) = un . Response at a point for each of these excitations
is shown in fig 4. Validation is done using the finite element model from the previous section. At the walls, a
uniform velocity is given to each node such that they add up to the value of un in the analytical model. Note,there are non-axial modes present in the 3D finite element model for the frequency range considered up to the6493

p (Pa)p (Pa)
Analytical

Analytical

Finite element

Finite element

1.2
3

1.0
0.8

2
0.6
0.4

1

0.2

200

400

600

800

1000

ν (Hz)

200(a) Excitation at karanai, response at z = l2

400

600

800

1000

ν (Hz)(b) Excitation at thoppi, response at z = l1

Figure 4. Cavity response to boundary excitations
sixth axial mode. Influence of these modes might be the reason for slight mismatch seen at the peaks of thecurves.

3

SHAPE SENSITIVITY STUDIES

2aKaranai end

2b

2a

2d

Thoppi end

2d

Thoppi end

Karanai end

For different geometric variations of the cavity, acoustic characteristics are studied to understand the influenceof shape on the sound generated by the Mridangam. Three cases are considered for this purpose, as follows. In
one case, length of karanai side cone is increased by 0.1 m (about 20% length of the conical sections), while
the length of thoppi side cone is decreased by the same amount (figure 5a, λ = +0.1 m). Other parameters are
kept unaltered. For the next, the length of the thoppi side cone is increased (λ = −0.1 m), while the karanaiside cone length is decreased to keep the overall length same. In the final case, a single cone connecting thecylindrical ends replaces the two cone design (see figure 5b).

z
z
0

l1

l2+

l3

l4

0

l1

l3

l4

(b) Single cone cavity(a) Figure corresponding to λ = +0.1 m

Figure 5. Geometries for shape sensitivity studies

3.1 Natural frequencies
The first step in shape sensitivity study is to extract the natural frequencies and modes for the geometric configurations discussed above. It is done by following the procedure described for the original case in the previous
section. Table 2 shows the first six natural frequencies for the three cases studied. The numbers in brackets
indicate the percentage deviation from the original model. Apart from fundamental mode of the single conecase, every mode is within 6% of the original. The single cone first mode shows a 20% deviation, which isconsiderable. Further investigation is required to understand the effect of such a shift.

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Table 2. Natural frequencies of geometries for shape sensitivity study
Mode
1
2
3
4
5
6

Frequency (Hz)λ = −0.1 m
λ = +0.1 m
349.64 (-1.4) 335.97 (-5.2)
591.50 (+2.8) 607.02 (+5.5)
846.21 (-0.9) 845.10 (-1.0)
1130.9 (+0.7) 1114.6 (-0.7)
1396.8 (-0.4) 1407.4 (+0.4)
1689.8 (-0.0) 1689.5 (-0.0)

Original
354.47575.11853.821122.71402.2
1689.9Single cone
283.38 (-20)
564.35 (-1.9)
845.86 (-0.9)
1127.5 (+0.4)
1409.2 (+0.5)
1691.0 (+0.1)

Brackets indicate percentage deviation from original.

3.2 Boundary excitation response
Boundary excitation response is the next step towards building coupled vibro-acoustic models for shape sensitivity analysis. Eq. 11 is used for the three geometries to get their responses. Karanai and thoppi ends are thepoints of interest, for both excitation and response. When one boundary is excited, the other is kept fixed, as
done previously, i.e., u(0) = un and u(l4 ) = 0 for karanai excitations, and u(0) = 0 and u(l4 ) = un for thoppi
excitations. The magnitude of oscillation un , maximum frequency swept and modal damping ratio ζ remain thesame as used for the analysis of the original shape. The earlier step provides wavenumbers, kr and mode shapefunctions, pr of the three geometries. Note the single cone geometry has only three sub-domains over its lengthand hence, p, pr and S are three sub-function piecewise functions in this case, unlike others.
p Pa

p Pa
4

Original

54

Original

λ- 0.1 mλ

- 0.1 m

λ

0.1 m

λ

0.1 m

3

Single Cone

Single Cone

3
2
2
1
1

200

400

600

800

1000

ν Hz

200

(a) Excitation and response at karanai end
p Pa
5

600

800

λ

- 0.1 m

λ

0.1 m

4

3

2

2

1

1

600

800

1000

(c) Excitation at thoppi, response at karanai

λ- 0.1 mλ

0.1 mSingle Cone

3

400

ν Hz

Original

Single Cone

200

1000

(b) Excitation at karanai, response at thoppi
p Pa
5

Original

4

400

ν Hz

200

400

600

800

1000

ν Hz

(d) Excitation and response at thoppi end

Figure 6. Effect of cavity shape on the response to boundary excitation

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Figure 6 shows the response curves calculated for the three geometries in the study, along with the original.
Near the first natural frequency region, the single cone geometry exhibits higher peak pressure when comparedto the baseline geometry as well as the other two modifications. This would suggest that the vibro-acoustic
coupling could be enhanced with a single cone geometry. Moving the largest cross-section area towards karanai(λ = −0.1 m) reduces the first mode peak pressure amplitude at the karanai when it is excited (see figure6a), while there is an increase in the pressure at the thoppi side for excitation of the thoppi membrane (seefigure 6d). For the other case of moving the section towards thoppi (λ = +0.1 m), the trends are reversed
from the λ = −0.1 m case. In the second mode region, generally, it is a weak response except for the second
curve in figure 6a. The third curve shows a spike near the third mode in figure 6d. Predominant membrane
vibration modes are below 500 Hz. Thus, the first cavity mode in each geometry is likely to couple morewith the membranes. It looks like the single cone will alter the membrane vibrations and hence the soundgenerated. For the other two geometries, there is a distinct increase in the pressure on the membrane opposite
to the direction of change. The increased pressure would possibly lead to more coupling at the first mode whencompared to the baseline, and this needs to be investigated in the near future. Also, the coupling between themembrane vibrations reduces as the location of the largest diameter is moved on either side of the baseline
location.

4

CONCLUSIONS

The analytical model developed is successfully able to capture the acoustics of a Mridangam cavity to a considerable degree. Natural frequencies from this model are in good agreement with experimental values. Moreover,finite element results validate the accuracy. Initial inference from shape sensitivity study is that the single conecavity might impact more on the vibrations of the membranes; the fundamental natural frequency shifts to thelower side and the response near this frequency is highest, at either end, for both excitations. Another observation made is that moving the location of the maximum cavity diameter on the Mridangam shell towards either
end may affect vibrations of the membrane which is opposite to the direction of movement of the largest diameter. The effect of one end excitation on the other end seems to be less sensitive to geometry change, except
for the single cone cavity. To arrive at a definite conclusion requires further analysis by coupling the membrane
and cavity vibrations. This work is in progress.

REFERENCES[1] Raman, C. V. The Indian musical drums, Proceedings of the Indian Academy of Sciences-Section A.
Springer, vol. 1, 1934, pp. 179-188.
[2] G, Sooraj. Vibro-acoustics of a Mridangam. Master’s thesis, Indian Institute of Technology, Madras, 2018.
[3] Eisner, E. Complete solutions of the Webster horn equation. The Journal of the Acoustical Society of America, vol. 41 (no. 4B), 1967, pp. 1126-1146.
[4] Beranek, L. L.; Mellow, T. J. Acoustics: Sound Fields and Transducers, Academic Press, 2012.
[5] Ginsberg, J. H. On Dowell’s simplification for acoustic cavity-structure interaction and consistent alternatives. The Journal of the Acoustical Society of America, vol. 127 (no. 1), 2010, pp. 22-32.
[6] Ramakrishna, B. S.; Sondhi, M. M. Vibrations of Indian musical drums regarded as composite membranes.
The Journal of the Acoustical Society of America, vol. 26 (no. 4), 1954, pp. 523-529.

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