# Fast, large amplitude vibrations of compliant cylindrical shells carrying a
fluid

Pawel Zimoch, Eliott Tixier, Julia Hsu, Amos Winter and Anette Hosoi

Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139, USA

###### Abstract

In this fluid dynamics video, we demonstrate the first three circumferential
modes of fast, large amplitude vibrations of compliant cylindrical shells
carrying a fluid.

##  1 Details

We use shells made of vinylpolysiloxane (a silicone-based elastomer) with
elastic moduli of $0.2$, $0.6$ and $1.0$ MPa, as well as shells made of latex.
The shells range from $5$–$10$ mm in diameter, $5$–$50$ mm in length and
approximately $0.1$–$0.2$ mm in thickness. We clamp one end of these shells
onto a rigid nozzle and pass air through them at flow rates ranging from
$0.2$–$2.5$ liters per second. When the flow rate of air in the shells exceeds
a certain critical value, dependent on the dimensions and material properties
of each shell, the shell becomes unstable and begins to vibrate. The mode of
vibration corresponds to one of the circumferential normal modes of vibration
of cantilevered cylindrical shells. Which mode is observed depends on the
dimensions and material properties of the shell. We observed the first three
modes.

In the first mode, commonly known as the “garden hose mode,” the shells
oscillate side-to-side with the frequency of approximately $15$ Hz.

In the second mode, the surface of the shell bends inwards, obstructing the
fluid flow and causing a large jump in the pressure drop across the nozzle. In
this mode, the shell can vibrate with frequencies from $200$–$700$ Hz,
depending on the volumetric flow rate of air. We observed that the frequency
of oscillation is directly proportional to air flow rate. Additionally, when
the shell vibrates in the second mode with average frequencies ranging from
approximately $350$–$550$ Hz, the vibration is unstable and the oscillation
frequency varies widely between periods. The second mode is the most robust
and can be observed in the largest range of parameters in our shells.

In the third mode of vibration, the circumference of the free end of the shell
is divided into three “flaps” oscillating inwards and outwards. In this mode,
the shells vibrate with frequency of approximately $600$–$1000$ Hz.

The images were captured with a Phantom v5.2 high speed color camera. The
images of the second and third mode were captured using a stroboscopic
technique and the final video is a concatenation of frames taken from
different oscillation periods. Each frame is slightly offset in phase,
yielding a slow-motion effect.

##  2 Acknowledgments

We would like to express our gratitude to Felice Frankel for her help with
cinematography, and to Prof. John Bush who kindly lent us the high speed
camera. We would also like to thank Prof. Pedro Reis and Dr. Arnaud Lazarus
for sharing with us their data on elastic properties of vinylpolysiloxane
elastomers.

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##  Appendix A Details of Metrics

###  A.1. Overall Performance.

Moreover, Chaudhry et al. (Chaudhry et al., 2019) devise a metric known as
Learning Curve Area (LCA), which quantifies the speed of learning in a model.
It first defines an average $b$-shot performance, where $b$ represents the
number of mini-batches, subsequent to the completion of training across all
$T$ tasks as follows:

(7) $Z_{b}=\frac{1}{N}\sum_{i=1}^{N}R_{N,i}$

$LCA$ at $\beta$ is the area of the convergence curve $Z_{b}$ as a function of
$b\in[0,\beta]$:

(8)
$LCA_{\beta}=\frac{1}{\beta+1}\int_{0}^{\beta}Z_{b}\,db=\frac{1}{\beta+1}\sum_{b=0}^{\beta}Z_{b}$

The Learning Curve Area (LCA) provides insights into model learning dynamics.
$LCA_{0}$ measures the average $0$-shot performance, similar to forward
transfer ((Lopez-Paz and Ranzato, 2017)). $LCA_{\beta}$, quantifying the area
under the $Z_{b}$ curve, evaluates both average $0$-shot performance and
learning speed. Although two models may achieve similar $Z_{b}$ or $A_{T}$
values, they can differ significantly in $LCA_{\beta}$ due to variations in
learning rates. This metric is crucial for identifying models that quickly
learn from few examples, particularly when $\beta$ is small.

Qin et al. (Qin et al., 2022) propose two metrics designed to evaluate pre-
trained language models (PLMs) based on their performance within learned
domains: Average Perplexity ($AP$) and Average Increased Perplexity
($AP^{+}$). The aforementioned metrics are utilized to assess key capabilities
of PLMs, such as instruction following and safety, as discussed in Wang et al.
(Wang et al., 2024a).

###  A.2. Memory Stability.

Chaudhry et al. (Chaudhry et al., 2018) introduce the Forgetting Measure (FM),
a metric designed to quantify the extent of forgetting a model experiences for
a specific task. The forgetting for a given task $T_{j}$ after sequential
training on tasks up to $T_{N}$ is quantified as the difference between the
highest proficiency ($max(R_{l,j})$) achieved on task $T_{j}$ during initial
training and its proficiency ($R_{N,j}$) after subsequent learning phases:

(9) $f_{j}=\max_{l\in\\{1,\ldots,N-1\\}}(R_{l,j}-R_{N,j}),\quad\forall j<N.$

For the purpose of quantifying forgetting in previous tasks, the function
$f_{j}$ is defined within the interval $[-1,1]$ for $j<N$.

Furthermore, to account for the number of tasks previously encountered, the
Forgetting Measure (FM) at the $N$-th task represents the mean level of
forgetting across all preceding tasks:

(10) $FM=\frac{1}{N-1}\sum_{j=1}^{N-1}f_{j}$

A lower FM indicates better retention of previous tasks. Here, the expansion
or $R_{j,j}$ serves as a more effective quantifier of retained knowledge
concerning past tasks, as opposed to using max. Nonetheless, max remains a
valuable estimator for assessing the extent of forgetting that occurs
throughout the learning process.

Davari et al. (Davari et al., 2022) propose a method named linear probes (LP)
to assess representation forgetting. This approach measures the effectiveness
of learned representations via an optimal linear classifier trained on the
frozen activations of a base network. Representation forgetting is quantified
by evaluating the change in Language Processing (LP) performance before and
after the introduction of a new task. Formally, for each model
($f_{\theta_{i}}$) at time step $i$ of a task sequence, the classifier
($W^{*}_{i}$) is optimized as:
$W^{*}_{i}=\arg\min_{W_{i}}\mathcal{L}(W_{i};f_{\theta_{i}}(X_{i}),Y_{i})$,
where $\mathcal{L}$, $X_{i}$, and $Y_{i}$ represent the objective function,
input data, and labels for task $i$, respectively. The degree of
representational forgetting between two model states, $\theta_{a}$ and
$\theta_{b}$, where $\theta_{b}$ is derived later in the sequence, is
evaluated by calculating the difference in scores:
$Score(W_{a}f_{\theta_{a}}(X_{a}),Y_{a})-Score(W_{b}f_{\theta_{b}}(X_{a}),Y_{a})$,
where $Score$ represents the performance metric, such as accuracy, on the
task.

Kemker et al. (Kemker et al., 2018) introduce three metrics designed to CF:
$\Omega_{\text{base}}$, $\Omega_{\text{new}}$, and $\Omega_{\text{all}}$.
$\Omega_{\text{base}}$ assesses retention of initial learning,
$\Omega_{\text{new}}$ measures recall of new tasks, and $\Omega_{\text{all}}$
evaluates overall proficiency in maintaining old knowledge and acquiring new
information.

(11)
$\Omega_{\text{base}}=\frac{1}{N-1}\sum_{i=2}^{N}\frac{\alpha_{\text{base},i}}{\alpha_{\text{ideal}}}$
(12)
$\Omega_{\text{new}}=\frac{1}{N-1}\sum_{i=2}^{N}\frac{\alpha_{\text{new},i}}{\alpha_{\text{ideal}}}$
(13)
$\Omega_{\text{all}}=\frac{1}{N-1}\sum_{i=2}^{N}\frac{\alpha_{\text{all},i}}{\alpha_{\text{ideal}}}$

where $N$ represents the total number of sessions, $\alpha_{\text{new},i}$ is
the test accuracy after learning session $i$, $\alpha_{\text{base},i}$ denotes
the accuracy on the initial session after $i$ sessions, and
$\alpha_{\text{all},i}$ refers to the test accuracy across all test data for
classes encountered up to point $i$. The ideal performance
($\alpha_{\text{ideal}}$) is defined as the offline MLP accuracy on the base
set. To facilitate comparative analysis across different datasets,
$\Omega_{\text{base}}$ and $\Omega_{\text{all}}$ are normalized by
$\alpha_{\text{ideal}}$. Consequently, unless a model surpasses
$\alpha_{\text{ideal}}$, normalized results will range from 0 to 1, enabling
consistent cross-dataset comparisons.

Additionally, researchers (Koh et al., 2022) devise a novel metric, termed the
Knowledge Loss Ratio (KLR), quantifies knowledge degradation using principles
from information theory.

###  A.3. Learning Plasticity.

Intransigence measure (IM), as defined by Chaudhry et al. (Chaudhry et al.,
2018), quantifies a model’s inability to learn new tasks. This measure is
calculated by comparing the performance difference of a task when trained
jointly with other tasks versus when trained in a continual learning setting.
Then the intransigence for the $N$-th task can be defined as:

(14) $IM=R^{*}_{N}-R_{N,N},$

where $R^{*}_{N}$ represents the accuracy achieved on the held-out dataset of
the $N$-th task, $R_{N,N}$ indicates the accuracy on the $N$-th task upon
completion of training in an incremental sequence up to and including task
$N$. Note, $IM_{N}\in[-1,1]$, and lower values indicate superior performance.

###  A.4. Metrics for Continual Pre-training.

CKL (Jang et al., 2022b) introduces a novel metric, named FUAR (FORGOTTEN /
(UPDATED + ACQUIRED) RATIO), which quantitatively measures the efficiency of
each CKL method. It calculates the number of instances of time-invariant
knowledge that a model forgets in order to learn or update one instance of new
knowledge. When FUAR is equal to 1.0, it signifies an equilibrium where one
time-invariant knowledge instance is forgotten on average to obtain a new or
updated knowledge instance. Formally, FUAR is defined as:

(15)
$Eq_{1}=\sum_{i=0}^{N-1}\max(0,\text{Gap}(T_{i}^{F},D_{i},D_{N}))\mathbbm{1}_{\\{T_{i}^{F}\neq
n.d.\\}}$ (16)
$\begin{split}Eq_{2}=\sum_{i=0}^{N-1}\max(0,\text{Gap}(T_{B}^{U},D_{N},D_{i}))\mathbbm{1}_{\\{T_{i}^{F}\neq
n.d.\\}}\\\
+\max(0,\text{Gap}(T_{N}^{A},D_{N},D_{i}))\mathbbm{1}_{\\{T_{i}^{F}\neq
n.d.\\}}\end{split}$

(17) $FUAR(\mathbbm{T}^{F},T_{N}^{U},T_{N}^{A})=\left\\{\begin{aligned}
&\frac{Eq_{1}}{Eq_{2}}\quad&&\text{if }denominator>0\\\
&no\;gain\quad&&\text{otherwise }\end{aligned}\right.$

where $T$ represents an arbitrary task, and $(D_{i})_{i=0}^{N}$ is a sequence
of corpora for LM pretraining. Gap($T,D_{a},D_{b}$) is $\text{Score}(T)$ of
$LM_{a}$ \- $\text{Score}(T)$ of $LM_{b}$, where $LM_{a}$ is pretrained on
$D_{a}$. $\mathbbm{T}^{F}=(T_{i}^{F})_{i=0}^{N-1}$ measures forgetting of
invariant-knowledge from $(D_{i})_{i=0}^{N-1}$. If no task is from $D_{i}$,
$T_{i}^{F}$ is ”n.d.” (not defined). $T_{N}^{U}$ and $T_{N}^{A}$ from $D_{N}$
measure update and acquisition of new knowledge, respectively.

###  A.5. Online CL-Specific Metrics.

Near-future accuracy (NFA) (Al Kader Hammoud et al., 2023) is introduced as a
novel evaluation metric for OCL problem. Unlike traditional evaluation methods
that assess models on immediately subsequent samples, NFA evaluates models on
samples slightly further into the future, using a minimal shift $S$. Such
operation can mitigate label correlation effects, which can adversely impact
the accuracy of model adaptability assessments. The smallest shift $S$ is
selected to ensure that the test sample aligns closely with the distribution
of recently observed training data. The calculation of NFA involves first
checking if the model correctly predicts the label of a future sample, which
can be expressed as
$a_{t}=\mathbbm{1}\\{f_{\theta_{t}}(x_{t+1+S})=y_{t+1+S}\\}$. Subsequently,
the running average is updated using the formula
$A^{RA}t=\frac{1}{t}(A^{RA}{t-1}\cdot(t-1)+a_{t})$.

Yogatama et al. (Yogatama et al., 2019) proposed a novel online codelength
($\ell(D)$), inspired by prequential encoding (Blier and Ollivier, 2018), to
quantify how quickly an existing model can adapt to a new task.

(18)
$\ell(D)=\log_{2}|y|-\sum_{i=2}^{N}\log_{2}p(y_{i}|x_{i};\theta_{D_{i-1}})$

where $|Y|$ is the number of possible labels (classes), and $\theta_{D_{i}}$
represents a particular subset of the dataset $D$. Similar to the approach in
Latent Contextual Allocation (LCA) (Chaudhry et al., 2019), the concept of
_online codelength_ is associated with the area under the learning curve.