Fermat's Library | A Study on the Coffee Spilling Phenomenon in the Low Impulse Regime annotated/explained version.

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.

I. INTRODUCTION

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)

(c)

(d)

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

phenomenon.

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.

II. EXPERIMENT STUDIES

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

oscillator?

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]].

g

tanh(



)[1 +

ρg

(



)

] (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

(a)

(b)

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

(2)

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

[14].

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

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