Jiwon Han is currently a 5th year PhD student in Astronomy & Astrop...
## TL;DR The paper focuses on understanding why coffee spills wh...
Actually, two Ig Nobel Prizes have honored to research that analyze...
Here's a [great video](https://www.youtube.com/watch?v=WGwU_e3uHBc)...
In 2012, Mayer and Krechetnikov also delved into the coffee spillin...
Jiwon, also gave a great Ted talk on this topic, which you can watc...
There's actually a device you can use to transport beverages withou...
The study of liquid dynamics, particularly in the context of sloshi...
The research on the dynamics of liquid oscillation, like that condu...
A Study on the Coffee Spilling Phenomenon in the Low Impulse Regime
Jiwon Han
Korean Minjok Leadership Academy, Gangwon-do, Rep. Korea
(Dated: November 2, 2015)
When a glass of wine is oscillated horizontally at 4Hz, the liquid surface oscillates calmly. But
when the same amount of liquid is contained in a cylindrical mug and oscillated under the same
conditions, the liquid starts to oscillate aggressively against the container walls and results in sig-
nificant spillage. This is a manifestation of the same principles that also cause coffee spillage when
we walk. In this study, we experimentally investigate the cup motion and liquid oscillation during
locomotion. The frequency spectrum of each motion reveals that the second harmonic mode of
the hand motion corresponds to the resonance frequency of the first antisymmetric mode of coffee
oscillation, resulting in maximum spillage. By applying these experimental findings, a number of
methods to suppress resonance are presented. Then, we construct two mechanical models to ratio-
nalize our experimental findings and gain further insight; both models successfully predict actual
hand behaviors.
Rarely do we manage to carry coffee around without
spilling it once [Fig 1]. In fact, due to the very common-
ness of the phenomenon, we tend to dismiss questioning
it beyond simply exclaiming: “Jenkins! You have too
much coffee in your cup!”
Figure 1: Rarely do we walk without spilling coffee.
However, the coffee spilling phenomenon is deceivingly
simple [1]. As a counter-intuitive example, prepare two
liquid containers with distinct geometrical structures;
here, we consider a wine-glass and a normal sized cylin-
drical mug. Pour the same amount of coffee inside each
glass (this is to ensure that Jenkins does not have “too
much coffee” in his cup). Since the human walking mo-
tion consists of periodic movements on the plane parallel
to the walking direction, we will oscillate each cup at a
fixed frequency in order to simulate such oscillatory mo-
tion. Using the mechanical device shown in Fig 2a, we
impose a horizontal excitation X = X
cos(2π × 2t) to
No institutions were involved in this study
each liquid container. According to common sense, since
the amount of coffee is the same inside each cup, the
amount of coffee that spills from the oscillation should
be fairly similar as well. However, this is not the case.
As it is clearly shown in Fig 3b, the coffee motion inside
the wine glass is aggressive while that of the cylindrical
cup is comparatively steady; consequently, the amount
of coffee spilt is significantly different. When the driving
frequency is changed to 4Hz, we are again surprised. Es-
sentially, the liquid behavior inside each container is com-
pletely reversed: while the coffee inside the wine glass re-
mains close to equilibrium, the coffee inside the cylindri-
cal cup oscillates violently [Fig 3c and Fig 3d]. Although
we yet do not have sufficient knowledge of the human
walking motion, such experiment results are enough to
show that the amount of liquid may not be the sole rea-
son behind spilling coffee.
(a) (b)
Figure 2: (a) A mechanical setup to maintain a fixed
freqeuncy during oscillation. (b) A diagram of the effective
cup height.
Indeed, the spilling of coffee is a manifestation of mul-
tiple interactions, ranging from the body-hand coordina-
tion to the resonance properties of the cup-coffee inter-
action. Thus, in order to gain clearer insight, the cof-
fee spilling phenomenon is divided into two regimes: the
low impulse regime and the high impulse regime. The
term “impulse” indicates the maximum magnitude of the
impulse that the cup experiences. Not surprisingly, the
physical properties of each regime are significantly dis-
(a) (b)
Figure 3: Oscillations at (a), (b) 2Hz and (c), (d) 4Hz.
tinct. In the low impulse regime, the interaction between
the cup and coffee is considered as a periodic function;
thus, the oscillation properties are researched extensively.
However, in the high impulse regime, the interaction be-
tween the cup and coffee is momentary and aggressive.
Oscillation properties carry less importance in such a
regime. Spilling from casual walking falls under the for-
mer regime; spilling after tripping on a stone falls under
the latter. In the present paper, the low impulse regime
is set to be the main focus of study.
Also, the effective cup height (which is defined to be
the height of the cup subtracted by the liquid equilib-
rium level [Fig 2b]) is not considered as a variable in
this study for two reasons. First, the role of the effective
height of the cup in spilling is rather straight forward. If
the effective height of the cup is large enough, the coffee
is unlikely to spill unless it is flipped over. On the other
hand, if the effective height of the cup is close to zero,
that is, if the cup is filled to its brim, the liquid is much
more likely to spill. Thus, the taller the cup and lesser
the coffee, the less you spill. Such a relationship is not
investigated to further extent in this study [2]. Second,
as much as it is simple, the role of effective cup height
is also absolute; thus, it should be considered more as a
classification than a variable. Again, such an extra classi-
fication is not included in this study, for it will complicate
the research more than deepen our understanding of the
Thus, in this paper, we study the conditions that max-
imize the amplitude of coffee oscillation under the low
impulse regime. In the Experiment Studies section, the
liquid oscillation properties and the cup’s motion prop-
erties are investigated. Here, a surprising feature of the
cup (hand) movement during walking is observed from its
frequency spectrum. Then, combining the results from
each investigation, it will be revealed how the interplay
between the cup and coffee leads to spilling. By apply-
ing this knowledge, a number of methods to reduce coffee
spilling are presented as well. Next, in the Model Stud-
ies section, two mechanical models of the “normal hand”
posture and the “claw-hand” posture are proposed. They
are each the oscillating-pivot single pendulum and the
oscillating-pivot double pendulum; both models are con-
structed upon the bold assumption that coffee, at least in
this study, can be treated as a simple pendulum. Surpris-
ingly, simulation studies reveal that both models success-
fully predict the important physical properties discovered
through experiment. We then conclude the paper with a
summary of our discoveries.
From experience, we know that our carrying hand is
usually strong enough to be essentially unaffected by the
coffee’s impact on the cup. This subtle insight can im-
mensely simplify the situation: instead of analyzing both
directions of influence, we can limit ourselves to one.
Therefore, it is physically sound to interpret the coffee-
cup system as a forced oscillator. The driving force,
which is synchronized with the carrying hand’s motion,
is directly exerted on the liquid from the inner walls of
the cup. Since we are in the low impulse regime, this
driving force is considered periodic; if the driving fre-
quency corresponds to the resonance frequency of coffee,
the sloshing amplitude reaches its maximum and results
in spilling. Thus, the question that we must investigate
is clear: what are the resonance conditions of this forced
A. Liquid Oscillation Properties
In order to determine the resonance conditions, the
first and foremost information that must be acquired is
the resonance frequency of the oscillator. Here, the os-
cillator is coffee. From the assumption that our liquid in
consideration is incompressible, irrotational, and invis-
cid, the following equation predicts the natural frequen-
cies of the various modes of fluid oscillation in an upright
cylindrical container [[3], [4]].
)[1 +
] (1)
Of the various modes of oscillation, our main interest is
the first antisymmetric mode. This is because of two rea-
sons: the first antisymmetric mode involves the largest
amount of liquid mass movement, and as we will see in
the following section, its frequency corresponds to the
driving frequency (at least partially). Thus, by substi-
tuting the parameters in equation 1 with the generic cup
dimensions [5] of 82mm diameter and 95mm height, the
generic σ and ρ values for coffee [6], and
= 1.841, we
calculate the first antisymmetric mode frequency to be
approximately 3.95Hz. Indeed, this value is dependent
on the specific dimensions of the cup, and it is helpful to
have a sense of how much the natural frequency would
change according to the radius of the cup. Such a rela-
tionship is illustrated in Fig 4. Interestingly, the equation
predicts a different amount of change in the natural fre-
quency when the radius is either increased or decreased:
an increase in the radius will not cause the natural fre-
quency to change as much as it would if it were decreased.
Figure 4: The natural frequencies as the radius changes.
The fist antisymmetric mode can also easily be ob-
served in the lab. Using the same mechanical device that
was utilized in the introduction, we give the cylindrical
cup a short “pump” and record its subsequent surface
waves. It is important to make sure that the given im-
pulse is at a reasonable magnitude; if the impulse is too
large, unnecessary effects such as the liquid surface break-
ing or other modes of oscillation being excited will be
observed as well. Here, the amplitude of the mechanical
vibrator was set to be 2cm and the frequency was fixed
at 2Hz, which are reasonable values that correspond to
actual dimensions of locomotion. By using the color dif-
ference between the coffee and the background, we track
one point on the liquid surface and plot its height rel-
ative to the equilibrium level. The final graph is pre-
sented in Fig 5a. Visually, the damping oscillation seems
to be monochromatic with an exponentially decreasing
envelope. The former observation can be easily verified
from the frequency spectrum [Fig 5b], which reveals that
the damping oscillation indeed has a single dominant fre-
quency of approximately 3.8Hz. This value is slightly be-
low the predicted frequency of 3.95Hz, most likely due to
the viscosity of coffee and other unconsidered frictional
forces that arise from the cup-coffee interactions [7]. The
second speculation is a bit trickier. The decreasing en-
velope is directly related to the damping coefficient γ;
however, without sufficient knowledge of the input en-
ergy and the rate of dissipation, the damping coefficient
defined as the following definition cannot be accurately
Figure 5: (a) The relative height of a point on the liquid
surface while it oscillates. (b) A FFT analysis of (a) reveals
that the oscillation is fairly close to 3.8Hz.
calculated [8].
γ =
< E
Instead, γ is determined by using an exponential curve-
fit of the enveloped curve of the damping oscillation. The
damping coefficient is revealed to be approximately 0.674
rad/s, with r-square value of 0.9774. A parameter that
can greatly increase this value is discussed in the Sup-
pressing Resonance section.
B. Cup Motion Properties
After investigating the oscillator properties, the next
step is to analyze the driving force: the cup. The cup is
synchronized with our hand’s motion, which is directly
influenced by our bodily movements. Such body-hand
coordination properties have been extensively researched
in biomechanical studies [[9], [10], [11], [12]], and it is
revealed that the hand’s swaying motion during loco-
motion is dictated by our lower body’s “up and down”
movements (H. Pontzer and Lieberman [9] coin the term
“passive mass damper” for our hand’s swaying motion).
However, we need to be cautious of the fact that the
specific mechanism of the hand’s control of the cup may
change according to how we hold the cup. While such
deviations will be investigated in the Suppressing Res-
onance section and the Model Studies section, for now,
we stick to the so-called “normal hand” posture, as illus-
trated in Fig 6a.
In this research, two methods were employed in or-
der to measure the acceleration of the cup during loco-
motion. The first method, which turned out to be un-
successful, was to utilize image processing tools. The
idea was to track the center of mass of the cup while
the cup holder casually walked. That way, it would be
possible to extract the time plot of the cup’s position,
and subsequently, the time plot of the acceleration of the
cup (by taking a second order derivative of the position
data). However, this method was unsuccessful due to
two main reasons. First, the image data was not sensi-
tive enough. If the data is obtained over a long distance,
one would inevitably have to zoom-out; this directly re-
duces the number of pixels by which the position data
is recorded, and results in an extremely “smoothed-out”
data plot. On the other hand, if we zoom in as much
as we want to, the data collection time span is greatly
limited. Unfortunately, we are stuck in a Heisenberg
uncertainty principle-like situation in which we cannot
achieve both measurements with desired quality at the
same time. Second, the visual data was limited to only
one plane of oscillation. Although the plane parallel to
the walking direction is indeed where most of the action
occurs, it would be better if data from all three planes
of oscillation could be acquired as well. Such issues were
solved by adopting the second method: utilizing an ac-
celerometer [13].
The second method proved to be quite successful. The
apparatus, as shown in Fig 6a, is straightforward. By
fixing an accelerometer (or, equivalently, a smartphone)
to the top of the mug, we record all three directions of
acceleration. Since the mug is a hard body, we expect the
acceleration measured on any part of the mug to be equal;
the accelerometer was also strapped to the bottom of the
mug in order to verify that the experiment results were
indeed independent of the position of the accelerometer
Representative acceleration plots in each orientation
and their respective FFT analysis results are presented
in Fig 6b and Fig 6c. Here, the y-axis is the walking
direction, the z-axis is the direction perpendicular to the
ground, and the x-axis is the remaining sideways direc-
tion. From the acceleration time plot, the difference in
the maximum magnitude of acceleration in each axis is
highlighted. The z-axis acceleration has the biggest mag-
nitude, and the x-axis acceleration is almost negligible in
magnitude compared to the other two. This matches our
expectations, since the up-and-down motion of walking
is visually much larger than that of sideways swaying.
According to the results of H. Pontzer and Lieberman
[9], the frequency of the z-axis oscillation should be syn-
chronized with our lower-body movements. Another in-
teresting observation can be made from the frequency
spectrum in each axis. In the acceleration time plot, the
z-axis oscillation seems to have a smaller frequency than
the y-axis oscillation; this is counter-intuitive, since we
expect the cup motion to be have the same frequency
as our body (up-and-down oscillation) itself. In order
to shine a light on such observation, a FFT analysis is
conducted on each acceleration plot.
Indeed, the FFT results are quite enlightening. Let
us first take note of the y-axis frequency spectrum [Fig
6c]. Evidently, the cup does not oscillate at the same fre-
quency of our body. In fact, the motion is not even close
to being purely sinusoidal: at least five or more distinct
harmonic frequencies are contained in the motion. This
directly goes against the daily assumption that our hand
simply goes up and down when we walk. Instead, the
cup-carrying hand undergoes a complex oscillation that
is less than perfectly synchronized with our bodily mo-
tions. We should note that such intricate oscillations do
not stem from the arm itself, but rather the extra degree
of freedom that the wrist allows in the cup motion. An-
other significant observation is made by examining the
specific values of the frequency components in the y-axis
oscillation. Among the distinct harmonic frequencies, the
second harmonic frequency coincides with 3.5 4Hz, which
is the resonance frequency of coffee in regular sized [5]
cylindrical cups. In other words, as we casually walk,
our hand oscillates in such a way that resonates with the
first antisymmetric mode of coffee oscillation; thus, the
likelihood of coffee spilling is maximized. It is impor-
tant to realize that resonance would not likely occur if
higher-frequency modes did not exist in our hand mo-
tion. For example, would one still spill coffee if the cup
was strapped around one’s waist? The answer is prob-
ably “no”, since, as we saw in the introduction, coffee
does not spill as much when it is simply driven at 2Hz.
Again, the particularity of the cup motion that allows
higher-frequency oscillation is highlighted.
Now we shift our focus to the other two results [Fig
6c]. First, the z-axis oscillation clearly exhibits a dom-
inant frequency close to 1.7Hz. There also exist higher
frequencies, but they are rather insignificant compared
to the dominant frequency. This is reflected in our ex-
perience that the walking motion is largely composed of
up-and-down motions, and that the frequency of such
up-and-down motion is what we normally perceive to
be the walking frequency. Although it cannot initiate
a significant level of coffee sloshing, the z-axis oscillation
at 1.7Hz can still amplify the first antisymmetric mode
in two ways. First, since 1.7Hz is close to half of the
resonance frequency, the z-axis oscillation can increase
the amplitude of the coffee once every two cycles after
the first antisymmetric mode is excited by y-axis oscilla-
tions. Second, there is the possibility of subharmonic res-
onance, as in the parametrically driven pendulum [[15],
[16]]. However, such behavior was neither experimen-
tally nor mathematically investigated thoroughly in this
research. Next, it is notable that the x-axis oscillation
has a dominant frequency of approximately 1Hz, which