Supplementary Information
Comprehensive Vibrational Dynamics of Half-Open
Fluid-Filled Shells
Markus Lendermann1 , Jin Ming Koh1 , Joel Shi Quan Tan2 , and Kang Hao Cheong1,3,*
1
Science and Math Cluster, Singapore University of Technology and Design (SUTD), 8 Somapah Road, S487372,
Singapore
2
Yong Loo Lin School of Medicine, National University of Singapore, S119228, Singapore
3
SUTD-MIT International Design Center, Singapore University of Technology and Design
*
Corresponding Author: kanghao_cheong@sutd.edu.sg (K. H. Cheong)
Appendix A: Undamped Numerical Model
The boundary conditions at the shell rim (z = H) are those of a free edge with a forcing term:
M1 = 0, N12 − M12 /R1 = 0, (1a)
Q1 + (M12 /B)θ = 0, N1 = cos nθ cos ωt, (1b)
where the transverse shear force Q1 is solved from the moment equilibrium equation
(AM21 )θ + (BM1 )z − M2 Bz + M12 Aθ − Q1 AB = 0, (2)
and M12 and M21 are twisting moment components in the z and θ directions respectively. The boundary
conditions for the fixed bottom of the shell (z = zmin ) are
u = v = w = 0, θ1 = u/R1 + wz /A = 0, (3)
where θ1 is the angle of rotation of the normal to the middle surface about tangents to the θ coordinate
line. For the fluid flow modelled by the Helmholtz equation, the boundary conditions at the glass wall
and free surface are
∇φ · e3 |r=R(z) = wt , (4a)
φtt
φz + = 0, (4b)
g z=h
respectively, where g is the gravitational acceleration.
The finite difference method adopted is more conveniently implemented with a rectangular grid.
To achieve a rectangular two-dimensional grid of sampling points within the fluid domain, a normal-
ization of the radial coordinate η = r/R(z) was performed. Nη × Nz,f points were uniformly sampled
within the two-dimensional grid given by η ∈ [0, 1] and z ∈ [zmin , h]. For the shell domain, a total of Nz,g
points were sampled within z ∈ [zmin , H], of which Nz,f were uniformly sampled within z ∈ [zmin , h]
while the remainder were uniformly sampled within z ∈ [h, H]. Nz,f and Nz,g were determined by the
initial desired number of sampling points along the z direction Nz as follows:
Nz,f = max (dNz /10e, bhNz /Hc) , (5a)
Nz,g = Nz,f + max (dNz /10e, Nz − Nz,f ) , (5b)
1
10−2 Model
Expt.
A (arb.)
10−3
1 2 3 4
t (s)
Figure 1: Measured and best-fit amplitude decay profiles of an empty wineglass, from which time constant τ =
0.515 s and σ = 3.50 × 10−4 were obtained.
ensuring sufficient grid points in the fluid and shell domains for low and high fluid levels respectively.
In the extension of the model with a solid object inserted into the fluid domain, the radial coordinate
is normalized as η = [r − Ri (z)]/[R(z) − Ri (z)] ∈ [0, 1] such that a rectangular grid is maintained. An
additional boundary condition ∇φ · no |r=Ri (z) = 0, where no is the normal vector to the surface of the
solid object, is introduced.
Appendix B: Damped Numerical Model
For the fluid flow modelled by the linearized Navier-Stokes equations, the boundary conditions at the
shell wall and the free surface are
u0 = ut e1 + vt e2 − wt e3 at r = R(z), (6a)
0
u =0 at z = 0, (6b)
∂p ∂ 2 u0z
= 2µ + ρf gu0z at z = H, (6c)
∂t ∂z∂t
∂u0z ∂u0r ∂u0θ 1 ∂u0z
+ = 0, + =0 at z = H. (6d)
∂r ∂z ∂z r ∂θ
Appendix C: Glass Characterization
The shape R(z) and thickness d(z) profiles of each wineglass were characterized as follows. For the
outer surface profile, a photograph of the wineglass filled with dyed water placed in front of a backlit
screen was taken. Computational edge detection then yielded the outer profile. The thickness profile
was obtained by detecting the vertical change in fluid level δz with a travelling microscope when a
small known volume of dyed ethanol δV was added. Where these measurements were taken, the outer
p
radius Ro was measured with a vernier caliper. Thus, the wineglass thickness is given by Ro − δV /πδz.
The neutral surface is then taken to be equidistant from the measured outer and inner profiles. To
facilitate computation in our models, the measured shape and thickness profiles are converted into
P9 P16
closed-form expressions by fitting them to R(z) = i=1 ai z i/4 and d(z) = i=1 bi z i/4 , where ai and bi
are coefficients determined by the least-squares method.
The complex Young’s modulus Y 0 = Y (1 + σj) of the glass material in the numerical damped
model was also characterized. The real part Re{Y 0 } = Y was determined via nonlinear regression of
model calculations of empty-glass natural frequency ωd against measured values. The imaginary part
Im{Y 0 } = Y σ was characterized via nonlinear regression of computed decay time constants against
experimental values, measured by exciting the wineglass to steady-state through the horn driver, and
2
then recording the amplitude decay profile upon abrupt deactivation of the driver. Decay profiles are
assumed to be of the form A(t) = exp (−(t − a1 )/τ ) + a2 , where τ is the time constant and ai are con-
stants to be determined. σ is then obtained by choosing an appropriate value such that the theoretical
quality factor Q of the empty wineglass is related to τ and the natural frequency ωd by 2Q = τ ωd .
Figure 1 presents a sample measured decay profile and best-fit computed profile as obtained from the
procedure. Characterized Y and σ values are given in Table 1 of the main paper.
