Court Judgements Labeling Using Topic Modeling and Syntactic
Parsing–References

# Court Judgements Labeling Using Topic Modeling and Syntactic Parsing

Yuchen Liu

(2022)

###### Abstract

In regions that practice common law, relevant historical cases are essential
references for sentencing. To help legal practitioners find previous judgement
easier, this paper aims to label each court judgement by some tags. These tags
are legally important to summarize the judgement and can guide the user to
similar judgements. We introduce a heuristic system to solve the problem,
which starts from Aspect-driven Topic Modeling and uses Dependency Parsing and
Constituency Parsing for phrase generation. We also construct a legal term
tree for Hong Kong and implemented a sentence simplification module to support
the system. Finally, we propose a similar document recommendation algorithm
based on the generated tags. It enables users to find similar documents based
on a few selected aspects rather than the whole passage. Experiment results
show that this system is the best approach for this specific task. It is
better than simple term extraction method in terms of summarizing the
document, and the recommendation algorithm is more effective than full-text
comparison approaches. We believe that the system has huge potential in law as
well as in other areas.

##  1 Introduction

###  1.1 Problem Statement

In regions that practice common law, the tribunal relies highly on previous
court decisions to resolve disputes. Legal practitioners frequently need to
find relevant legal judgements at work. Indexing each court judgements by
labels can considerably increase the speed of court judgement recommendation,
and can help users find other judgements with similar topics faster.

This paper will focus on labeling every court judgement by some tags. There
are existing paid websites that provide such tags, for example Westlaw Asia.
However, they maintained the tags by recruiting legal experts to label
manually, which is expensive and time-consuming. Plus, users cannot select the
tags to find similar documents. A better solution must be developing an NLP
system that can automatically extract legally important tags from a court
judgement. It leads to the question that how to build such a system and how to
find similar documents using these tags.

###  1.2 Literature Review

Previous work of Wu, et al. wu22 focuses on semantic search and summarization
using Topic Modeling. It proved that the Topic Modeling based on labeled
aspects (background, injury, losses, and compensations for Personal Injury
cases) shows the best summarization ability. Figure 1 shows the general
process of Aspect-driven Topic Modeling.

Figure 1: Aspect-driven Topic Modeling

For each aspect of documents in the training set, a few significant topics
will be generated. Based on the resultant topics, the similarity between
paragraphs and aspects can be calculated using the normalized inner product of
the word2vec embeddings of the paragraphs and topics [Mikolov et al.2013].
This way, paragraphs that are related to each aspect can be found. By simply
changing paragraphs to sentences in this algorithm, we can find the most
relevant sentences to each aspect. These aspect sentences should
comprehensively summarize the court judgement and will serve as the starting
point of this project.

Two of the classical Syntactic Parsing approaches will be used in our model.

Dependency Parsing finds the asymmetric dependency relations between words.
These relations can be represented as arrows pointing from the heads to the
dependents. Usually, the heads are more grammatically important than
dependents. For example, adjectives normally depend on nouns. The result
should form a non-recurrent tree. We can backtrack from one leaf of the tree
and find all its parent words.

Constituency Parsing uses phrase structure grammar to organize words into
nested constituents. Each constituent is labeled by one of the predetermined
grammar units, called POS Tags, for example, VB denotes verb phrase and PP
denotes prepositional phrase.

###  1.3 Objective

The tags given to each judgement should fulfill the below properties.

  1. 1.

The tags should be/contain legally important terms.

  2. 2.

The tags are free of grammatical/content mistakes that will affect
understanding.

  3. 3.

The tags can navigate the user to other similar documents.

  4. 4.

The tags can describe important aspects of the current judgement.

As illustrated in 1.3, if we generate tags based on the aspect sentences,
requirement 4 will be satisfied [Wu et al.2022]. Therefore, we will only focus
on solving requirements 1-3 in the following methodology. Because of the
regional difference of constitution, we will use Hong Kong as the example and
use the HKLII website as the data source for historical judgements.

###  1.4 Report Outline

Chapter 2 will introduce two supporting modules and the main algorithm. The
similar document recommendation algorithm will then be illustrated. Chapter 3
will show the experiment process and discuss the results. Limitations and the
future work for the project will be pointed out as well. Finally, the
conclusion is given in Chapter 4.

##  2 Methodology

This section will start by introducing the preparation work for the system,
including constructing the legal concept tree and developing the sentence
simplification module. Then, it will compare contextual phrases against legal
terms to discuss if phrase generation algorithms are needed. At last, it will
present a novel model for extracting legally important phrases from sentences.

###  2.1 Preparation Work

We have created two standalone modules that support and improve the main
algorithm. Both of the modules have been open-sourced on GitHub.

Legal Concepts Tree contains more than 13000 common legal terms used in Hong
Kong Judiciary. It is adapted from the legal taxonomy developed by Sweet &
Maxwell. As shown in Figure 2, it has a perfect tree structure where parent
terms are more general than their children. The tree is optimized to store in
a hashed map for faster key search. With this tree, we can use keyword
matching to identify the legal terms in the sentence. If necessary, it is also
useful for substituting the legal terms with more general ones.

Figure 2: Part of the legal concepts tree

Sentence Simplification is the task to split complex or compound sentences
into simple sentences, each of which has only one pair of subject and verb.
There are previous works on this task using constituency parsing, such as
Garain et al. 4 and Das et al. 5. We implement and improve the algorithm by
Garain et al. 4 to fit into the latest Stanford Constituency Parsing rule
[Marneffe et al.2021, Schuster et al.2016]. This implementation is also
intelligent to skip the proper names, such as ”Fish and Chip” or ”Johnson and
Johnson”. This module can help to simplify long sentences, which are common in
court judgements, so that operations afterward can be more accurate.

###  2.2 Tag Format Comparison

One critical question to ask is that since the legal concept tree can help to
find legal terms, is it necessary to take one step further to construct
contextual phrases based on the legal terms? For instance, given the sentence
”She suffered a fracture of the superior and inferior public rami of the right
pelvis.”, is tag ”fracture” enough, or is tag ”fracture of rami of pelvis”
better? While phrase tags can contain more local contexts and information than
word tags, they may sometimes be broken during generation. It is impetuous to
draw any conclusion at this point, so one experiment is set up to test this
question. The result is shown in chapter 3. The remaining sections in chapter
2 will focus on generating phrase tags only.

###  2.3 Main Algorithm

As stated in 1.4, Aspect-driven Topic Modelling will be used and the most
relevant sentences to the aspects will be found first. The main algorithm
solves the problem of extracting legally important phrases from each sentence.

Algorithm 1 The algorithm to transform a sentence into tags (variables are
italic)

procedure Generate Tags(sentence $x$)

$sentences$ $\leftarrow$ split $x$ into simple ones, using split sentence
module

$tags\leftarrow[]$

$N\leftarrow n$

for sentence $s$ in $sentences$ do

$dp\\_tree$ $\leftarrow$ the enhanced dependency tree of $s$ $\triangleright$
A

for word $w$ in $s$ do

if $w$ in legal concepts tree then

$words$ $\leftarrow$ the smallest constituents containing $w$ $\triangleright$
B

for $parent$ of $w$ in $dp\\_tree$ do

$words.append$($parent$, preposition betweeen $parent$ and $w$)
$\triangleright$ C

$w\leftarrow parent$

end for

$tags.append$(join $words$ by ” ” in same order of sentence $x$)

end if

end for

end for

return $tags$

end procedure

In step A, the Stanford Enhanced Dependency [Marneffe et al.2021] should be
used. It differs from the Universal Dependency in containing prepositions
inside relations. It will benefit the reconstruction of extracted words in
step C. It should be noted that the relations starting with ”acl:” should be
eliminated as they will cause dependency cycles. Step B finds the shortest
subset of the sentence that contains the matching legal term. Based on
Constituency Parsing, it is to find the shortest component with a tag that
ends with ”P”, such as ”NP”, ”VP”. This will include the neighbor words of the
legal term to make the result more specific. For example, given the
constituency tree in Figure 2.3, assume that the word ”condition” is found to
be legal term.

Figure 3:  Constituency Parsing Tree and Dependency Parsing Tree of the
example sentence (arrows are pointing from parent to children)

Step B finds the smallest component ”His mental condition”. Without this step,
the word ”condition” itself might be too general and ambiguous.

Step C is to extract faraway related words with respect to the matching legal
term using Dependency Parsing. Starting from the legal term w, the program
will recursively find the parent of it in the dependency tree until the root
is reached or a verb is met. Since the verb is connected to its subject using
relation ”obj”, the condition therefore becomes meeting ”obj”. During the
backtracking, the prepositions and conjunctions in the dependency relation
should also be recorded. They are stored after ”:” in the relation within the
set ”nmod:”, ”obl:”, ”conj:”, ”advcl:”. Finally, the words in the
reconstructed phrase should have the same order as the original sentence. For
the example of Figure 2.3, starting from the word ”condition”, ”unstable” is
found to be the first parent and ”bad” is the second. There is a conjunction
word ”and” between the ”bad” and ”unstable”, according to the relation
”conj:and”, so ”and” should be inserted in between. As a result, the tag ”His
mental condition bad and unstable” is generated.

Overall, this algorithm is a hybrid approach that utilizes both Constituency
Parsing in extracting neighboring phrase and Dependency Parsing in finding
faraway relational words. To make the tags more fluent and natural to people,
we also added several heuristic operations to the algorithm.

  1. 1.

If the last word of the recursive step C is a link verb, such as ”is”, ”are”,
we should truncate it.

  2. 2.

If the smallest component found in step B does not help specify the matching
term, for example only adding ”the” in front of the matching term, step B
should be skipped. In the implementation, the skipping condition is that the
length of the component is two and the POS tag for the first word belongs to
{’DT’, ’IN’, ’PRP$’, ’CD’}.

  3. 3.

Sometimes, the preposition ”to” before a verb cannot be successfully
recognized by the Dependency Parsing. We should add ”to” in front of the word
with POS Tag ”VB”.

###  2.4 Similar Documents Recommendation

Assume we have obtained the tags, ultimately the user will choose a few of
them to navigate to other judgements which may mention similar legal concepts.
According to the user survey, users often have a few interested legal concepts
in mind during searching. Therefore, this aspect-specific similar document
matching may be more useful than comparing two judgements by the full text. To
solve the problem, the simplest way is to directly invoke the default search
engine and use the selected tags as search input. This way, the selected tags
are compared with the whole passage of other documents. Another way is to
compare the selected tags with tags from other documents. The theoretical
basis of this approach is that the tags have summarized most aspects of a
court judgement (Requirement 4 in 1.4). Narrowing the searching scope to tags
only can help to eliminate interference from other unimportant texts, making
the recommendation more accurate. The quality of this recommendation will be
tested in the following experiment. Algorithm 2 shows the details of the
recommendation algorithm by comparing selected tags with tags from other
documents.

Algorithm 2 Recommendation Algorithm Details

function CAL_SIMILARITY($emb1$, $emb2$)

return $emb1\cdot emb2/(\|emb1\|\cdot\|emb2\|)$

end function

procedure Recommend Judgements(selected tags $x$)

$ret$ $\leftarrow$ an empty max heap

for all judgement $j$ do

$score\leftarrow 0$

for tag $t1$ of $x$ do

$e1\leftarrow$ embedding of $t1$

$each\\_score\leftarrow 0$

for tag $t2$ of $j$ do

$e2\leftarrow$ embedding of $t2$

$each\\_score\leftarrow$ max($each\\_score$, CAL_SIMILARITY($e1$, $e2$))

end for

$score$ += $each\\_score$

end for

add $j$ to $ret$ based on $score$

end for

return $ret$

end procedure

Basically, the similarity between each selected tag and another document is
equal to the maximum normalized inner product of all possible tag
combinations. The similarity score between all selected tag and another
document is the sum of such similarities.

##  3 Evaluation

This section detailly describes the experiment to evaluate the quality of the
tags generated. Also, the experiment results will be revealed and the future
work for the project will be pointed out.

###  3.1 Experiment setup

As mentioned in section 2.3, the main algorithm for generating phrase tags is
a hybrid approach of Dependency Parsing and Constituency Parsing. To test
whether the use of both parsing techniques is necessary, we set up a
comparison experiment. For each court judgement, four tag sets are generated
using 1. only Dependency Parsing 2. only Constituency Parsing 3. the Hybrid
Approach 4. Word tags only. Each tag set starts from three starting sentences
derived from Aspect-driven Topic Modeling. Each document along with all the
resultant tags will be distributed to at least two participants to minimize
personal bias. Participants are asked to select the tags that fulfill all
three requirements in 1.4. After getting the result from this experiment 1, we
continue to test the tags format issue mentioned in section 2.2. The
recommendation abilities of word tags and phrase tags are tested in this
experiment 2. For each judgement, participants will be provided with the top 5
recommendations from word tags and phrase tags respectively in pairs.
Participants are asked to find the better recommendation, which is more
similar to the original judgement in each pair.

###  3.2 Experiment Results

We invited 7 participants to evaluate on tags of 15 independent court
judgements. Table 1 shows the results of experiment 1. The average number of
tags generated by the four approaches per judgement and the percentage of good
tags are provided. In total, 6.8*15 = 102 tags are generated by the hybrid
method, and 1.76*15 = 26.3 of them are good.

Methods | # of tags | # of good tags | Good tags rate
---|---|---|---
Only Dependency Parsing | 853 | 174 | 0204
Only Constituency Parsing | 580 | 156 | 0268
Hybrid approach | 680 | 176 | 0258
Word tags | 1060 | 158 | 0149
Table 1: Statistics of good tags per judgement

From the statistics, we can conclude that the hybrid approach generated the
greatest number of good tags for each judgement (1.76). Also, although the
constituency parsing approach generated the least good tags (1.56), the
percentage of the good tags is the highest (0.268). Considering the purpose of
the model, which is to generate tags for the users to choose, we should prefer
the absolute number of good tags to the rate of it. Therefore, both Dependency
Parsing and Constituency Parsing is necessary. The hybrid approach is the best
phrase generation method. As for the word tags, the number of good tags is
less than phrases, and the rate of good tags is the lowest. As a result, legal
terms are not comparable to contextual legal phrases in terms of fulfilling
the requirements of tags.

Cases | # of pairs | Rate
---|---|---
Phrase tags better | 088 | 0175
Word tags better | 167 | 0336
Both are good | 099 | 0198
Neither is similar | 146 | 0291
Total | 500 | 1000
Table 2: Statistics of better recommendations per document

Table 2 shows the results of the recommendation ability of word tag and phrase
tag in experiment 2. Since 5 pairs of external documents are recommended for
each judgement, there are 15*5=75 pairs of comparison tested. Here, we use the
hybrid approach for the phrase tag generation.

From the table, we can find that the frequency of word tags being better is
higher than the number of phrase tags being better. This indicates that there
is still room for improvement for the recommendation algorithm based on phrase
tags. The possible direction will be discussed in section 3.4.

###  3.3 Limitations

The percentage of good tags from the hybrid approach is around 26%. Although
the purpose of the task does not require tags to have an extremely high
acceptance rate, higher rate can definitely resolve some misunderstandings and
improve user experience. Based on the feedbacks from experiment participants,
the reason of bad tags mostly owes to not fulfilling requirement 2 from
section 1.4. They often present broken or confusing meanings. The root of this
problem is obvious. Both Dependency Parsing and Constituency Parsing are
syntactic parsing, indicating that they do not consider semantic information
enough. For example, the correct punctuation of sentence ”Drive the lorry for
loading and unloading goods” should be ”Drive the lorry for loading goods” and
”… for unloading goods”. However, the constituency parsing splits like ”Drive
the lorry for loading” and ”… for unloading goods”. The parsing simply finds
the result with the highest probability in grammar regardless of the obvious
parallelism of antonyms: ”loading” and ”unloading”. Therefore, to improve the
main algorithm, essentially a new parsing method that maximizes both
grammatical and semantic probability is needed. Furthermore, since the main
algorithm starts from most relevant sentences of each aspect, the result of
AdTM will affect the final performance to large extent. Therefore, it is also
necessary for the algorithm to find better document summarization methods
alternative to AdTM that can elevate the upper limit.

###  3.4 Future Work

One promising way to improve the recommendation based on phrase tags is giving
weights to different tag combinations. As illustrated in 2.4, we are summing
over the maximum of similarities of combinations by considering all tags
equal. However, different tags come from different sentences that belongs to
topics in four different aspects. If two tags come from the same aspect or
even the same topic, their similarity score should be assigned a higher
weight. For example, two tags both describing the losses of the personal
injury cases should be naturally closer than two that describes different
aspects. What’s more, currently training sets and testing sets are all court
judgements of Personal Injury cases. The effectiveness of this system on other
case types is not tested. Future work should focus on applying to model to
various types of cases. Finding more applications for the system is also an
interesting topic. This model is designed for the HKLII website. However,
after a simple modification of the Legal Concept Tree in section 2.1 to fit
into local context, this model can be applied to other law systems.
Furthermore, other professional areas that have similar concept tree
structures can easily adapt the model to their advantage.

##  4 Conclusion

This report introduced the model that labels court judgement. The system uses
Aspect-driven Topic Modeling as the starting point. The main algorithm in the
system that extracts phrase tags from a sentence is a hybrid method of
Dependency Parsing and Constituency Parsing. Experiment results show that
these tags outperform general legal terms. Although the aspect-specific
recommendation is more useful than full passage comparison, it sometimes did
not recommend the most similar documents compared with the benchmark
algorithm. Future work should focus on the improvement of the recommendation
algorithm for phrase tags, the seeking of alternative system components and
the discovery of more use cases for the system.

## References

  * [Zhong et al.2020] H. Zhong, C. Xiao, C. Tu, et al., ”How does NLP benefit legal system: A summary of legal artificial intelligence,” arXiv preprint arXiv, 2020.
  * [Wu et al.2022] T. H. Wu, B. Kao, et al., ”Semantic Search and Summarization of Judgments Using Topic Modeling,” in Legal Knowledge and Information Systems, Erich Schweighofer, 2022, pp.100-106.
  * [Mikolov et al.2013] T. Mikolov, I. Sutskever, K. Chen, G. Corrado, and J. Dean, ”Distributed representations of words and phrases and their compositionality,” Proceedings of the 26th International Conference on Neural Information Processing Systems - Volume 2 (NIPS’13), 2013, pp. 3111–3119.
  * [Garain et al.2019] A. Garain, A. Basu, R. Dawn and S. K. Naskar, ”Sentence Simplification using Syntactic Parse trees,” 2019 4th International Conference on Information Systems and Computer Networks (ISCON), 2019, pp. 672-676.
  * [Das et al.2018] B. Das, M. Majumder and S. Phadikar, ”A Novel System for Generating Simple Sentences from Complex and Compound Sentences,” International Journal of Modern Education and Computer Science, 2018, pp. 57.
  * [Marneffe et al.2021] M. C. Marneffe, C. D. Manning, J. Nivre, D. Zeman, ”Universal Dependencies,” Computational Linguistics, 2021, pp. 255–308.
  * [Schuster et al.2016] S. Schuster, C. D. Manning, ”Enhanced English Universal Dependencies: An Improved Representation for Natural Language Understanding Tasks,” Proceedings of the Tenth International Conference on Language Resources and Evaluation (LREC’16), 2016, pp. 2371-2378.


#  Experimental validation of a model for a self-adaptive beam-slider system

Florian Müller1 Maximilian W. Beck1 Malte Krack1 1 University of Stuttgart,
GERMANY

###### Abstract

A system consisting of a doubly clamped beam with an attached body (slider)
free to move along the beam has been studied recently by multiple research
groups. Under harmonic base excitation, the system has the capacity to
passively adapt itself (by slowly changing the slider position) to yield
either high or low vibrations. The central contributions of this work are the
refinement of the recently developed system model with regard to the finite
stiffness of the beam’s clamping, followed by a thorough validation of this
model against experimental results. With the intent to achieve repeatable and
robust self-adaption, a new prototype of the system was designed, featuring,
in particular, a continuously adjustable gap size and a concave inner contact
geometry. The initial beam model is updated based on the results of an
Experimental Nonlinear Modal Analysis of the system (without slider). By
varying the excitation level and frequency in a wide range, all known types of
behavior were reproduced in the experiment. The simulation results of the
updated model with slider are in excellent agreement with the measurements,
both qualitatively (type of behavior) and quantitatively. Minor deviations are
attributed to the system’s sensitivity to inevitable uncertainties, in
particular with regard to the friction coefficient and the linear natural
frequency. It is thus concluded that the proposed model is well-suited for
further analysis of its intriguing dynamics and for model-based optimization
for technical applications such as energy harvesting.

###### keywords:

Non-smooth dynamics, geometric nonlinearity, self-adaption, model updating,
multiple scales

††journal: MSSP

##  1 Introduction

Figure 1: Schematic of self-adaptive system: (a) two-dimensional model of
clamped-clamped beam with attached slider (b) slider detail

Consider the beam-slider system schematically illustrated in Fig. 1a. Under
harmonic base excitation in the range around the beam’s lowest-frequency
bending mode, this system has shown self-adaptive behavior in experiments
carried out independently by different research groups [1, 2, 3]. The system
was observed to adapt itself in such a way that it achieves and maintains high
vibrations in a wide band of excitation frequencies. After initially small
vibrations, the slider moved to a certain position and the vibration level
increased substantially. Hereby a _signature move_ was observed [1, 4] (which
is also shown later in Fig. 6) as described in the following. The readers are
strongly encouraged to watch the video
https://www.youtube.com/watch?v=qSy8ccbOgn8. While the system vibrates
initially at small level, the slider moves towards the clamping. At a certain
point, the vibrations jump to a higher level and the slider turns back towards
the beam’s center. This movement goes along with a further increase of the
vibration level. At a certain position, which can be different from the beam’s
center, the slider stops and large vibrations are maintained at steady state.
The above described passive re-adjustment to achieve and maintain large
vibrations was achieved for a number of different material pairings and
geometrical dimensions of the basic setup [1, 4, 3]. As this behavior occurs
over a broad band of excitation frequencies in the vicinity of the lowest
natural frequency, the described beam-slider system received considerable
attention in the vibration energy harvesting community [5, 4]. In addition to
that, similar beam-slider systems with various boundary conditions of the beam
were investigated in the context of energy harvesting [6, 7, 8] or vibration
suppression [9]. An overview on energy harvesting systems that are based on
multistable mechanical systems is presented in [10].

#### Vibration-induced sliding

Although the setup of the beam-slider system is seemingly simple, and it has
already been used in first energy harvesting applications, the self-adaptive
behavior is not well-understood. Apparently, the movement of the slider along
the beam is caused by the beam’s vibrations. Consider first the case where the
clearance between slider and beam is negligible compared to the beam’s
transverse vibration amplitude. Then the slider’s transverse displacement and
rotation are constrained to those of the beam, $w(s,t)$ and $w^{\prime}(s,t)$,
respectively, where $w^{\prime}=\partial w/\partial x$, $t$ is time and $s$ is
the slider’s axial position along the beam (Fig. 1a-b). Assuming small
displacements, slow movement of the slider and that the slider can be
reasonably described as a point mass $m$, the equation governing its axial
position reads [11]

$\displaystyle m\ddot{s}+m\left(\ddot{w}+\ddot{w}_{0}\right)w^{\prime}=0\,.$
(1)

Herein, overdot denotes derivative with respect to time, $\ddot{w}$ and
$w^{\prime}$ are evaluated at $x=s$. $\ddot{w}_{0}$ is the imposed base
acceleration, $w$ is the elastic displacement. The second term on the left
hand side of Eq. (1) drives the slider along the beam and can be interpreted
as transverse inertia force, $m\left(\ddot{w}+\ddot{w}_{0}\right)$, projected
onto the rotated beam axis [12]. This driving force is kinematically nonlinear
in the beam deformation. The driving force in accordance with Eq. (1) was used
to explain the movement of the slider both in the case where the flexible host
structure was a beam (e. g. in [13, 14]), and the case where it was a taut
string (e. g. in [11, 12, 15]). An immediate consequence of the above
described theory is that equilibrium points are characterized by
$w^{\prime}=0$ (zero slope), e. g., vibration anti-nodes [15]. Even though the
slider’s mass was larger than that of the beam, and thus non-negligible, the
vibrational deflection shape was clearly dominated by the lowest-frequency
bending mode of the system without slider, regardless of the slider position,
and thus had its anti-node approximately at the beam’s center ($x=L/2$) [1,
16]. The above theory can thus only explain the last part of the signature
move, where the slider indeed moves towards the center. In contrast, the
movement away from the anti-node in the beginning of the signature move cannot
be explained by the above theory.

#### Relevance of clearance and geometric nonlinearity

The movement of the slider away from the anti-node was first reproduced via
simulation in [17]. In the beginning of the signature move, the vibrations are
relatively small, such that the clearance between beam and slider cannot be
neglected. Hence, this clearance was explicitly taken into account in the
model proposed in [17]. The clearance gives rise to dynamical contact
interactions at the four contact points indicated in Fig. 1b, which apparently
leads to an intricate _locomotion_ process. The locomotion process depends
sensitively on the clearance. If the clearance is too small, locomotion is
hampered, as observed in both numerical investigations [17] and experiments
[18, 5]. Besides unilateral contact interactions, dry friction is known to be
crucial: the slider would keep cycling back and forth along the beam in the
frictionless case [17].
Due to the clamping present on both sides of the beam, bending deformation
causes longitudinal stresses, which in turn increase the bending stiffness
(string effect). As can be easily seen e. g. in the video, the beam’s
transverse deformation is not small compared to its thickness. This causes a
considerable geometric hardening effect. Similar to the Duffing oscillator,
the amplitude-frequency curve is strongly bent towards the right, so that the
high-amplitude branch extends over a broad band of excitation frequencies.
This is viewed as cause for the broad-band efficacy intended for energy
harvesting [2]. Moreover, the geometric hardening nonlinearity leads to the
co-existence of low- and high-level vibration states, and is required to
explain the experimentally observed amplitude jumps e. g. during the signature
move [16] (compare also [17], where the beam model was linear and no amplitude
jumps were observed).

#### Limitations of the current knowledge and purpose of the present work

The above described signature move is only one of several types of behavior of
the beam-slider system. The slider may also move monotonously towards the
center (typically when it is initially placed near the clamping). In this
case, no amplitude jump occurs; high vibrations are also reached at steady
state. This type of behavior is referred as _trivial adaption_ in the
following, focussed on e. g. in [19], as opposed to the signature move. Both
the signature move and the trivial adaption yield a steady state of high
vibrations, and are in this sense forms of _successful adaption_. An
intermittent behavior is also possible, where the slider undergoes a
_relaxation oscillation_ along the beam (recurrent back and forth movement),
with alternating high and low vibrations [5, 16]. This type of behavior is
only rarely discussed in the literature. Finally, the slider may show
_inactivity_ (typically for low excitation), or move to a certain position
leading to low steady-state vibrations [5, 16]. The latter type of behavior is
sometimes referred to as _self-damping_ [13, 11, 12], as opposed to the
successful self-adaption yielding high vibrations, which is also referred to
as _self-tuning_ [15, 1, 18].
The conditions leading to a certain type of behavior are not understood yet.
In order to obtain this understanding, it is imperative to have a reliable
mathematical model that is capable of accurately reproducing the experimental
observations. Remarkably, most proposed models of the beam-slider system
ignore the clearance and the resulting contact interactions between beam and
slider, and thus fail to qualitatively reproduce the locomotion of the slider.
The purpose of the present work is the quantitative experimental validation of
a recently proposed model, which accounts both for the contact interactions
and the beam’s geometric nonlinearity [16]. To this end, a new prototype of
the beam-slider system is designed, as described in Section 2, taking into
account the experimental experience of the authors and other groups. As it
turns out, it is necessary to refine the model proposed in [16] to account for
the finite compliance of the beam’s clamping. In Section 3, the clamping
stiffness is experimentally identified using the results of an Experimental
Nonlinear Modal Analysis of the beam only (without slider). The extension of
the beam model by the slider and the modeling of the contact interactions
between slider and beam is then described in Section 4. Finally, simulation
results are confronted with measurements in Section 5 for a large range of
operating conditions. Concluding remarks are given in Section 6.

##  2 Experimental setup

For a thorough experimental validation of the simulation model, a new
prototype of the beam-slider system is developed. In the following, the new
design is described in detail, relevant parameters are specified and the
instrumentation for the experimental investigations is explained.

###  2.1 Design of slider, beam and support structure

Based on the scientific findings that the authors and other research groups
made previously, the following desired properties are specified:

  * (1)

The beam should exhibit a pronounced geometric nonlinearity of hardening type
[19, 20, 16].

  * (2)

The clearance between beam and slider should be small (several percent of the
beam’s thickness) [1, 4, 17, 16]. And it should be adjustable due to the known
high sensitivity to this key parameter.

  * (3)

The contact between slider and beam should be well-defined [17, 16].

  * (4)

The slider’s center of mass should be away from the beam in vertical
direction. This was found to increase robustness and speed of locomotion in
preliminary simulations for this work.

From an engineering perspective, three more desired properties are formulated:

  * (5)

The beam should have high resistance against fatigue.

  * (6)

The contact between slider and beam should be robust against wear.

  * (7)

Uncertainties due to the assembly process should be avoided.

A photograph of the new prototype of the beam-slider system is shown in Fig.
2a. Both ends of the beam are clamped to a frame. The free length of the beam
is nominally straight with uniform rectangular cross section. The frame is
relatively stiff compared to the thin beam in order to enforce bending-
stretching coupling which results in the desired geometric hardening
nonlinearity (1). To be able to precisely adjust the gap size (2), the slider
is made of two identical sliding parts and two identical couplers (Fig. 2b).
Before tightening the screws, the sliding parts can be moved closer together
or away from each other in a continuous way. The final clearance between
slider and beam is adjusted by the aid of a feeler gauge. To realize a well-
defined contact geometry (3), the sliding part’s side facing the beam is
designed concave. Consequently, the nominal contact interface consists of four
lines (or four points in a planar model). An additional threaded hole in the
sliding parts enables to mount additional weights. By adding weight to the
lower side, the center of mass is moved away from the beam (4). The beam is
made of spring steel 1.8159 which provides a high fatigue strength (5). To
minimize contact wear (6), sharp edges are avoided by designing a rounded
contact geometry of the sliding parts instead. To avoid uncertainties during
assembly (7), the clamping mechanism is as follows: The inner screws 1 and 3
(as labelled in Fig. 2a) are M5x25 hexagon socket screws with the task to
provide the normal force which clamps block, beam and frame together. The
outer screws 2 and 4 are fitting screws ISO 7379 with the task to position the
beam and the block relative to the frame. Screw 2 is forming a fit with the
bores in block, beam and frame aligning the parts in $x$-direction. On the
left side, the bore in the beam has a bigger diameter in order to avoid axial
prestress in the beam, i. e. screw 4 only aligns block and frame. Also to
minimize undesired prestresses, the screws are tightened in the order
$1\rightarrow 2\rightarrow 3\rightarrow 4$. In [21, 22], the same frame and
clamping mechanism were used to analyze the softening-hardening behavior of a
curved beam.

Figure 2: Experimental setup: (a) test specimen with instrumentation (b) CAD
drawing of slider illustrating concave contact geometry, stepless adjustable
gap mechanism and additional weight attachment (second sliding part and second
coupler not shown for clarity)

The beam and slider dimensions are close to the system investigated in [4],
where robust adaption was achieved in a broad frequency range. The dimensions
are large compared to, e. g., two of the setups in [1]. This makes it easier
to adjust the slider’s gap size relative to the beam’s thickness. Throughout
the present work, the gap size was adjusted such that a clearance of 5% of the
beam’s thickness is obtained. This (relative) gap size was shown to enable
successful adaption in [16]. The relevant parameters specifying the new
prototype are collected in Tab. 1. Any interested research group is welcome to
reproduce the experiments and/or use this as point of departure for future
research. Thus, all technical drawings (including material specifications) and
CAD files are provided in a repository: https://doi.org/10.18419/darus-2462
[23].

Table 1: Parameters of the beam-slider setup Quantity | Symbol | Value | Unit
---|---|---|---
Free length (beam) | $L$ | 140 | $\mathrm{mm}$
Thickness (beam) | $h$ | 1 | $\mathrm{mm}$
Density (beam) | $\rho$ | 7683 | $\mathrm{kg/m^{3}}$
Mass (beam, free length) | $\rho AL$ | 15.1 | $\mathrm{g}$
Young’s modulus (beam) | $E$ | 210 | $\mathrm{GPa}$
Mass (slider) | $m$ | 46.2 | $\mathrm{g}$
Rotary inertia w.r.t. center of mass (slider) | $J_{yy}^{(C)}$ | 3.15 | $\mathrm{kg\ mm^{2}}$
Distance between contact points (slider) | $B$ | 10 | $\mathrm{mm}$
Distance from center of gravity (slider) | $d$ | 4.1 | $\mathrm{mm}$
Relative gap size | $R/h$ | 1.05 | –

###  2.2 Instrumentation of experiments with slider

To realize base excitation, the frame was mounted onto an electrodynamic
shaker (Brül & Kjær type 4808, driven by amplifyer 2719). Instead of screwing
the frame directly to the shaker’s armature, an impedance head (PCB model
288D01) was placed in between, see Fig. 2a. An accelerometer (PCB model
352C22) was attached to the frame’s top side. The accelerometer was used in
preliminary tests only to verify the acceleration measured by the impedance
head, which was in very good agreement. The beam’s elastic deformation was
measured by orienting the measurement head of a laser Doppler vibrometer (LDV)
(Polytec OFV-552) toward the beam and the corresponding reference head toward
the frame. The slider is expected not to pass the beam’s center. To ensure
that the optical access to the beam is not blocked by the slider, the laser
was aligned to a position in the other half slightly away from the beam’s
center ($x=4L/7$). The slider’s horizontal ($x$) position was tracked by a
triangulation sensor (Micro-Epsilon ILD1420-100). For data acquisition and
supply of the excitation signal (voltage signal operating the shaker via
amplifier), all sensors and the amplifier are connected to a rapid control
prototyping system (dSPACE microlab box). This system was also used to control
the amplitude of excitation to a prescribed acceleration value. For this, the
amplitude of the acceleration measured by the impedance head was evaluated
online by synchronous demodulation (see [22] for details). This amplitude was
controlled by adjusting the voltage amplitude of the harmonic excitation
signal which was passed to the shaker’s amplifier using a PI-controller
(gains: $K_{\mathrm{p}}=0.03\mathrm{Vs}^{2}/\mathrm{m},\
K_{\mathrm{i}}=0.13\mathrm{Vs}/\mathrm{m}$). The signal measured by the
impedance head was used for this control task rather than the one of the
accelerometer due to the higher sensitivity of the impedance head.

##  3 Refining and updating the model of the beam without slider

In Subsection 3.1, a refined mathematical model of the geometrically nonlinear
beam is presented, which also accounts for the finite stiffness of the
clamping. The stiffness parameters and also the modal damping are updated as
described in Subsection 3.3. The updating procedure relies on the results of
an Experimental Nonlinear Modal Analysis (ENMA). The ENMA is described in
Subsection 3.2.

###  3.1 Geometrically nonlinear beam with finite-stiffness clamping: Equation
of motion

In [17, 16], the beam’s boundaries were idealized by assuming infinite
stiffness of the clamping. As will be shown in Section 3.3, this approximation
yields large errors with regard to the linearized natural frequency and the
extent of the hardening effect. To account for the finite stiffness of the
clamping in the model, the boundary conditions are relaxed, as illustrated in
Fig. 3. They are modeled by hinges with rotational springs (stiffness
$k_{\mathrm{r}}$), where the right hinge is horizontally movable and
constrained by a transversal spring (stiffness $k_{\mathrm{t}}$).

Figure 3: Model of beam with finite-stiffness clamping

In the following, the ordinary differential equation describing transverse
elastic deformations of the beam is briefly derived. Using the conventional
assumptions (small rotations, negligible longitudinal and rotational inertia,
Hooke’s law, Bernoulli assumptions, second-order kinematics), one can derive
the partial differential equation,

$\displaystyle\rho
A\ddot{w}+EIw^{\prime\prime\prime\prime}-Hw^{\prime\prime}=-\rho
A\ddot{w}_{0}\,,$ (2)

for the case of an imposed base motion $w_{0}(t)$. Overdot and prime denote
derivation with respect to time $t$ and spatial coordinate $x$, respectively.
$EI$ and $\rho A$ denote the bending stiffness and mass per unit length of the
beam. $H$ is the axial force which is determined as follows. First, the
derivative of the horizontal deflection $u(x,t)$ is considered:

$\displaystyle
u^{\prime}=\frac{H}{EA}-\frac{1}{2}\left(w^{\prime}\right)^{2}\,.$ (3)

Assuming that the horizontal motion of the right hinge is negligible compared
to the beam’s length, integrating over the beam’s length and evaluating the
resulting expression at the right end, where $H=-k_{\mathrm{t}}u(L,t)$, yields

$\displaystyle
H=\left(1+\frac{EA}{Lk_{\mathrm{t}}}\right)^{-1}\frac{EA}{2L}\int_{0}^{L}\left(w^{\prime}\right)^{2}\mathrm{d}x\,.$
(4)

Now, truncation to the $N$ lowest-frequency modes is applied, using the ansatz

$\displaystyle w(x,t)\approx\sum_{n=1}^{N}\varphi_{n}(x)q_{n}(t)\,.$ (5)

Here, $\varphi_{n}(x)$ are the mass-normalized mode shapes of the underlying
linearized system ($Hw^{\prime\prime}=0$, Euler-Bernoulli theory) and the
boundary conditions

$\displaystyle w(0)=0\,,\quad
EIw^{\prime\prime}(0)=k_{\mathrm{r}}w^{\prime}(0)\,,\quad w(L)=0\,,\quad
EIw^{\prime\prime}(L)=-k_{\mathrm{r}}w^{\prime}(L)\,.$ (6)

Substituting Eqs. (4) and (5) into (2), and requiring that the residual is
orthogonal with respect to the mode shapes, $\varphi_{n}$, yields a system of
ordinary differential equations in terms of the sought modal coordinates,
$q_{n}$,

$\displaystyle\ddot{q}_{n}+2D_{n}\omega_{n}\dot{q}_{n}+\omega_{n}^{2}q_{n}+\sum_{j=1}^{N}\sum_{k=1}^{N}\sum_{l=1}^{N}b_{j,k,l}^{(n)}q_{j}q_{k}q_{l}=\gamma_{n}\hat{\ddot{w}}_{0}\cos(\Omega
t)\quad\quad\mathrm{with}\quad n=1,2,\ldots,N\,.$ (7)

Herein, linear modal damping with damping ratios $D_{n}$ was also added, and
the imposed base motion is assumed as harmonic,
$w_{0}(t)=\hat{\ddot{w}}_{0}\Omega^{-2}\cos(\Omega t)$. $\omega_{n}$ is the
natural (angular) frequency and $\gamma_{n}=\rho
A\int_{0}^{L}\varphi_{n}\mathrm{d}x$ the excitation coefficient of the $n$-th
linearized mode. Note that equations (7) are coupled by the nonlinear term
describing bending-stretching coupling with the coefficients

$\displaystyle
b_{j,k,l}^{(n)}=-\left(1+\frac{EA}{Lk_{\mathrm{t}}}\right)^{-1}\frac{EA}{2L}\int_{0}^{L}\varphi^{\prime}_{j}\varphi^{\prime}_{k}\mathrm{d}x\int_{0}^{L}\varphi^{\prime\prime}_{l}\varphi_{n}\mathrm{d}x\,.$
(8)

Before deriving the actual model updating procedure in Subsection 3.3, the
Experimental Nonlinear Modal Analysis is explained in Subsection 3.2, as this
is used to gather the data basis for the model updating. The focus will be
placed on the fundamental bending mode. For brevity, the associated quantities
are simply expressed as $\omega:=\omega_{1}$, $\varphi(x):=\varphi_{1}(x)$,
$D:=D_{1}$ and $b:=b_{1,1,1}^{(1)}$ in the following.

###  3.2 Experimental Nonlinear Modal Analysis (ENMA) of the beam without
slider

In this work, the amplitude-dependent modal properties of the lowest frequency
mode of the beam without slider are identified from steady-state vibration
data obtained under feedback-controlled phase-resonance testing. The procedure
is briefly described in the following; details can be found in [21].
The beam was clamped to the frame and mounted to a shaker (Brül & Kjær type
4809, driven by amplifier 2718), similar to the setup described in Section 2.1
but without attaching the slider. The target is to achieve phase resonance
between the harmonic base excitation and the beam’s transverse displacement
relative to the base (response). The excitation was measured by aligning a LDV
(Polytec, OFV-505) to the left clamping block. To measure the response, a
second LDV (Polytec, OFV-552) with reference head was used. The measurement
head of this second LDV was aligned to a point on the beam close to the
clamping and the reference head to the left clamping block where also the
excitation was measured. To achieve phase resonance, a phase-locked loop was
used with a PID controller (gains set as $K_{\mathrm{p}}=200\ 1/\mathrm{s}$,
$K_{\mathrm{i}}=100\ 1/\mathrm{s}^{2}$, $K_{\mathrm{d}}=10$).
Under the condition of phase resonance, the excitation frequency, $\Omega$,
corresponds to the natural frequency, $\tilde{\omega}(a)=\Omega$, where $a$ is
the modal amplitude and $\tilde{\omega}$ is the amplitude-dependent modal
frequency. By varying the excitation level, one can thus directly obtain the
amplitude-dependent natural frequency. In contrast, the amplitude-dependent
modal damping ratio is more difficult to obtain, because the applied
excitation force cannot be directly measured (to determine the power input to
the structure that is in balance with the dissipated power). The method
developed in [21] was used to quantify the amplitude-dependent modal damping
ratio using only response measurements. More specifically, the following
estimate was used:

$\displaystyle\tilde{D}(a)\approx\frac{\left|\hat{\boldsymbol{w}}^{\mathrm{H}}\boldsymbol{b}\hat{\ddot{w}}_{0}\right|}{2\hat{\boldsymbol{w}}^{\mathrm{H}}\hat{\boldsymbol{w}}\Omega^{2}}\,.$
(9)

Herein, $\hat{\boldsymbol{w}}$ is the complex-valued amplitude vector of the
beam’s transversal displacement relative to the base,
$\boldsymbol{b}=[1,1,\ldots,1]^{\top}$ and $()^{\mathrm{H}}$ denotes the
Hermitian (complex-conjugate transpose). The estimation converges with the
number of response sensors. A multi-point laser-Doppler vibrometer was used to
measure the response at five equidistant sensor points,
$\hat{\boldsymbol{w}}=[\hat{w}(L/6),\hat{w}(2L/6),\ldots,\hat{w}(5L/6)]^{\top}$.
With this spatial resolution, the error made by the estimation in Eq. (9) was
shown to be negligible for beams under different boundary conditions (pinned-
pinned, clamped-clamped) [21].
The results of the ENMA are illustrated in Fig. 4. Here, the data is depicted
for consecutive up-and-down stepping of the excitation level. The experiment
was started at low level and stepped upwards first (yielding result #1),
directly followed by stepping downwards (#2). The experiment was repeated
after several minutes of rest (#3 and #4). Each excitation level was held 20
seconds to reach steady state and record the last 300 periods. The total test
time of a consecutive up-and-down stepping was 920 seconds.

Figure 4: Results of ENMA and updated model: (a) modal frequency vs.
amplitude, (b) modal damping ratio vs. amplitude, (c) modal deflection shape,
color of markers corresponds with amplitude level of experiment, blue: low
level, yellow: high level

The curves #1 and #2 are slightly off at low level, which may be explained by
an increase of temperature during the test. The second run (#3 and #4) is in
very good agreement with the first one. The expected hardening behavior, which
is dominant over the repetition variance, yields a frequency shift of more
than 8% in the considered amplitude range. Damping is very light. The slight
increase with amplitude (less than 0.055% of critical damping) is attributed
to dry micro-slip friction in the bolted joints at the beam’s clamping.
The beam’s modal deflection shape (normalized to unity) is shown in Fig. 4c.
Results of #1–#4 are plotted as markers, where the amplitude level is
indicated by color. Only a minor change of the deflection shape can be
observed throughout the tests. Consequently, the modal deflection shape is
very similar to that of the linearized case at low amplitudes. Moreover, the
periodic vibration response is in good approximation harmonic: The strongest
higher harmonic of the measured velocity is ${<}5\%$ of the fundamental
harmonic throughout all sensors and amplitude levels.

###  3.3 Updating of the refined beam model

The dominance of the (single) linearized modal deflection shape and the
fundamental temporal harmonic, observed in the ENMA, allows one to pursue a
rather simple model updating procedure. In particular, truncating Eq. (7) to a
single linearized mode, $N=1$, and applying single-term Harmonic Balance
yields a closed-form expression for the amplitude-dependent natural frequency
(see e. g. [24]):

$\displaystyle\tilde{\omega}^{2}(\tilde{a})=\omega^{2}+\frac{3}{4}b^{*}\tilde{a}^{2}\,,$
(10)

with

$\displaystyle\quad
b^{*}:=\frac{bh^{2}}{\left(\varphi(L/2)\right)^{2}}\,,\quad\tilde{a}:=\frac{\hat{w}(L/2)}{h}\,.$
(11)

From the ENMA, $\tilde{\omega}^{2}(\tilde{a})$ is known, and this allows one
to determine the linear natural frequency $\omega$, and nonlinear coefficient
$b^{*}$ via Eq. (10) using a least-squares fit:

$\displaystyle\begin{bmatrix}\omega^{2}\\\
b^{*}\end{bmatrix}\approx\begin{bmatrix}1&\frac{3}{4}\tilde{a}_{1}^{2}\\\
1&\frac{3}{4}\tilde{a}_{2}^{2}\\\ \vdots&\vdots\\\
1&\frac{3}{4}\tilde{a}_{K}^{2}\end{bmatrix}^{+}\begin{bmatrix}\tilde{\omega}_{1}^{2}\\\
\tilde{\omega}_{2}^{2}\\\ \vdots\\\ \tilde{\omega}_{K}^{2}\end{bmatrix}\,.$
(12)

Here $K$ is the total number of phase-resonant points measured during the ENMA
and $()^{+}$ denotes the Moore-Penrose pseudo inverse. Evaluating Eq. (10)
with fitted $\omega$ and $b^{*}$ yields excellent agreement with the measured
data (cf. Fig. 4a), which further supports the validity of the single-mode-
single-harmonic truncation.
Next, it is assumed that the model uncertainty resides in the clamping
stiffness parameters, $k_{\mathrm{r}}$, $k_{\mathrm{t}}$, whereas the beam’s
geometric and material properties correspond to the nominal ones as specified
in Tab. 1. The underlying linear model is not affected by the transversal
elasticity $k_{\mathrm{t}}$. Hence, the rotational spring $k_{\mathrm{r}}$ can
be simply determined by solving

$\displaystyle
k_{\mathrm{r}}=\mathrm{argmin}\left(\left|\omega-\omega_{\mathrm{mod}}(k_{\mathrm{r}})\right|\right)=125.0\frac{\mathrm{Nm}}{\mathrm{rad}}\,,$
(13)

where $\omega$ is the fitted value and $\omega_{\mathrm{mod}}(k_{\mathrm{r}})$
is the lowest natural frequency of the linear Euler-Bernoulli beam in
accordance with the boundary conditions specified in Eq. (6). Subsequently,
the stiffness of the transversal spring can be determined from Eq. (8)
yielding

$\displaystyle
k_{\mathrm{t}}=\frac{EA}{L\left({b^{*}_{\infty}}/{b^{*}}-1\right)}=1.93\cdot
10^{7}\frac{\mathrm{N}}{\mathrm{m}}\,.$ (14)

Here $b^{*}_{\infty}$ is the value with neglected horizontal elasticity of the
frame (obtained from Eq. (8) with $k_{\mathrm{t}}\to\infty$ and normalization
as in Eq. (11)) and $b^{*}$ is the fitted value. Compared to the beam model
with ideal clamping ($k_{\mathrm{r}},\ k_{\mathrm{t}}\to\infty$), the obtained
linear frequency $\omega$ and nonlinear stiffness coefficient $b^{*}$ are
5.06% and 52.17% lower respectively. This shows that a model with ideal
clamping is not suitable to accurately describe the dynamic behavior of the
beam without slider.
After having found $k_{\mathrm{r}}$ and $k_{\mathrm{t}}$, the model is fully
determined, except for the damping. The measured damping is generally weak and
has only moderate amplitude dependency. It is expected that this beam-only
damping plays a minor role once the slider is attached, because then
unilateral and frictional contact interactions are expected to provide
substantial dissipation. For this reason, the beam-only damping is simply
modeled as linear modal damping, using the mean value of the damping ratio
from the ENMA, $D_{1}=0.102\%$. With the same argument, this damping ratio is
used also for the other modes; i. e., $D_{n}=D_{1}$ for $n=2,\ldots,N$.

##  4 Modeling the contact interactions with the slider and simulation of the
coupled beam-slider system

Except for the refinement of the beam model with regard to the finite-
stiffness clamping, modeling and simulation approach are largely adopted from
the authors’ previous work [17, 16]. For self-consistency of the present
paper, a brief recap is given in this section along with the specifics of the
model and the simulations.

###  4.1 Slider model, contact model, kinematics and numerical simulation

The system is idealized as a planar one and the slider is modelled as rigid
body with horizontal, vertical and rotational degrees of freedom (Fig. 1).
Gravity is taken into account as indicated in Fig. 1 in the slider model111It
was found that the effect of gravity on the beam (due to its own weight)
negligible.. The slider’s inner geometry is such that contact may occur only
at the four points indicated in Fig. 1b. Considering the contact kinematics,
the approximation $x_{i}^{*}\approx x_{P_{i}}$ is newly introduced. Here
$x_{P_{i}}$ is the horizontal coordinate of the considered contact point
$P_{i}$ of the slider and $x_{i}^{*}$ the coordinate of the point one obtains
when projecting $P_{i}$ onto the beam’s center line. As a consequence of small
rotations, this approximation has negligible effect on the simulation results,
which was ensured by a prestudy. The approximation makes the numerical
solution for $x_{i}^{*}$ obsolete, which reduces the overall computation cost
by about 50%. Unilateral and dry frictional contact interactions between beam
and slider are modeled by the Signorini and Coulomb laws combined with
Newton’s impact law. These are set-valued contact laws, which are associated
with velocity jumps, and thus standard time integration algorithms for
continuous ordinary differential equation systems cannot be applied. Instead,
the equation of motion is transformed to a measure differential inclusion,
which is solved numerically by Moreau’s time stepping scheme (see e. g. [25]).
The dynamic contact problem is resolved using an augmented Lagrangian
approach. In accordance with preliminary convergence studies, the beam model
is truncated to $N=5$ modes222Besides the modal convergence of the depicted
results it was also verified that the retained set of beam-only modes can
properly represent the mode shape of the beam with rigidly attached mass. and
a fixed time step of $\Delta t=2{\cdot}10^{-5}\mathrm{s}$ is used in all
simulations. This time step size is sufficiently small to resolve a vibration
period corresponding to the highest linear frequency ($\omega_{5}$) with ten
time levels.

###  4.2 Contact parameters

The coefficient of friction was determined experimentally as follows. The
frame was inclined slowly and the angle $\psi$ at which the slider starts to
slip was measured. From this the static friction coefficient can be determined
via the relation $\mu_{\mathrm{s}}=\mathrm{tan}(\psi)$. The lower sliding part
(and the additional weight) was removed for these tests in order to exclude
the effect of wedging. This analysis was repeated for ten equidistant starting
positions along the beam’s half in which the slider is supposed to move during
the self-adaptive process. The entire procedure was repeated three times (30
tests in total). The resulting coefficient is in the range
$\mu_{\mathrm{s}}\in[0.22,\ 0.31]$ with a mean value of 0.26. Unavoidable
imperfections of the beam surface stemming from the manufacturing process may
be one reason for the variance. However, variance is observed even for
nominally identical test with the same starting position. The coefficient of
sliding friction is expected to be about one third lower than the static one
for steel-steel combinations (see e. g. [26]). Coulomb’s law assumes the same
coefficient for sticking and sliding friction, which is approximated by
$\mu=0.2\approx(\mu_{\mathrm{s}}+2\mu_{\mathrm{s}}/3)/2$ here. The effect of
$\mu$ on the results is discussed in Section 5.5.
As in [17, 16], $\varepsilon_{\mathrm{n}}=0.5$ is used as normal coefficient
of restitution. A parameter study showed that varying
$\varepsilon_{\mathrm{n}}$ in the range $0.4\leq\varepsilon_{\mathrm{n}}\leq
0.6$ has no significant effect on the simulation results.

###  4.3 Recap of time scale separation, super-slow manifold, system with
pseudo-constrained slider, explanation of amplitude jumps

Both experimental and numerical results show that the self-adaption process of
the beam-slider system happens on three well-separated time scales: The beam’s
vibration takes place on the fast time scale. Modulations of the beam’s
vibration take place on an intermediate slow time scale. The adaption of the
slider position as well as the change of the overall vibration amplitude take
place on a super-slow time scale. By constraining the horizontal slider
position and determining the steady-state vibration, a subspace is obtained,
that will be referred to as _super-slow manifold_. The transient dynamics of
the self-adaptive system with free slider closely follows the super-slow
manifold. The time scale separation and the notion of the super-slow manifold
are crucial for the interpretation of the intricate nonlinear dynamics
discussed in Section 5. Before proceeding to the confrontation of experimental
and simulation results, it is useful to describe how the super-slow manifold
is computed and explain its characteristic shape.
To directly compute the super-slow manifold, the horizontal slider position
needs to be treated as a control parameter rather than as a degree of freedom.
In the past, this was achieved simply by prescribing the horizontal
displacement of the slider’s center of mass (i. e., imposing a vertical
guidance). As it turned out, this is too intrusive in the sense that the mean
horizontal reaction force of the guidance was inconsistent with the actual
movement of the slider in some parameter regimes. Instead, the slider is let
horizontally free, but the prescribed (relative) slider position is always fed
to the algorithm that evaluates the contact kinematics. This can be
interpreted as moving the beam horizontally according to the slider’s motion.
This is referred to as model with pseudo-constrained slider (PCS model) in the
following, which is to be distinguished from the model with free slider (FS
model). In Section 5, the super-slow manifold is computed using the PCS model
and plotted to interpret the dynamics.

Figure 5: Bifurcation diagram illustrating the beam’s response level under
variation of slider position (super-slow manifold). PCS model evaluated with
different modal truncation orders, $f_{\mathrm{ex}}=124\mathrm{Hz}$ (a)
excitation level $\hat{\ddot{w}}_{0}=14\mathrm{m/s}^{2}$ (b)
$\hat{\ddot{w}}_{0}=8\mathrm{m/s}^{2}$

In Fig. 5, a typical super-slow manifold obtained using the PCS model is
depicted in the plane spanned by the beam’s vibration level and the slider
position. More specifically, the slider position, $s$, is normalized by the
beam’s length, $L$, and the beam’s transversal displacement amplitude is
determined at the location $4/7L$ and normalized by the beam’s thickness.
Apparently, there are co-existing steady vibration states in a wide range of
slider positions around the beam’s center. To resolve both stable branches of
the super-slow manifold, a sequential continuation procedure was used: The
slider was initially placed near the beam’s center (near the clamping), the
simulation was run until steady state, and then the slider position is
stepwise decreased (increased). Increments of $\Delta s/L={0.005}$ were found
as reasonable compromise between resolution and effort, and a number of 300
excitation periods was found to ensure a reasonably steady state. The upper
(high-amplitude) branch extends from the clamping towards the beam’s center.
The lower (low-amplitude) branch extends from the beam’s center until it
reaches a turning point, and thus a jump to the high-amplitude branch occurs
during the sequential continuation towards the clamping.
The super-slow manifold is depicted in Fig. 5 with different modal truncation
orders. Fig. 5a suggests that $N=1$ is sufficient in the regions close to the
clamping and the beam’s center, whereas at least $N=2$ is needed to properly
describe the intermediate range. However, the results in Fig. 5b (lower
excitation level) indicate that $N=2$ is not sufficient either: The high
amplitude branch corresponding to $N=5$ terminates at $s/L\approx 0.32$ and
the amplitude jumps down. This behavior is crucial to understand one type of
behavior of the system with free slider (relaxation oscillation) addressed in
Section 5. One possible explanation for the jump down may be a modal
interaction which is not captured in models with $N\leq 2$. A more thorough
analysis of this phenomenon is considered as beyond the scope of the present
work. Further increasing the number of retained modes ($N>5$) does not
significantly affect the results, as found in a preliminary study.

##  5 Confronting experimental and simulation results

In this section, a validation of the model with horizontally free slider (FS
model) by means of the experiment is presented. Of course, the initial slider
position is an important parameter, which was varied in a wide range. In the
present work, however, results are shown for initial slider position $s/L=0.3$
(unless otherwise stated). As will be shown, all types of behavior can then
still be reached by varying the excitation frequency and excitation level
(_operating point_). Starting at static equilibrium, each operating point is
driven for 120 seconds which is sufficient to reach a reasonably steady state.
Representative results measured from the experiments are available in a
repository [27]. Four qualitatively different types of behavior are
distinguished, as mentioned in the introduction: A successful adaption to high
vibrations in the form of the signature move, the trivial adaption to high
vibrations, a relaxation oscillation and self-damping behavior. Representative
results for each type of behavior are shown in the following Subsections
5.1-5.4. A very good agreement of experiment and simulation is observed for
all cases. Minor deviations are explained by high sensitivity with respect to
uncertain system parameters in Subsection 5.5. A more global picture is
presented in Subsection 5.6, showing which type of behavior is reached for
what operating conditions.

###  5.1 Successful adaption (signature move)

Results for a representative signature move are depicted in Fig. 6. The
process can be subdivided in three phases. The transitions between these
phases are indicated by vertical dashed lines in Fig. 6a,b,e. In phase 1, the
vibration level is small and the slider moves away from the beam’s center. A
closer look to the beam’s transversal displacement during phase 1 shows a
strongly modulated response (Fig. 6c,d). At the transition from phase 1 to
phase 2, the amplitude jumps to a higher level, after which the slider moves
back towards the beam’s center and the amplitude increases further. In phase
3, the horizontal slider position and large vibration level are maintained
(steady state).
The three separate time scales described in Subsection 4.3 can be well
recognized in Fig. 6. The time history within phase 1 depicted in Fig. 6c,d
clearly shows the intermediate time scale, on which the amplitude modulation
takes place, and the fast time scale of the oscillation. The time variation of
the horizontal slider location $s$, depicted in Fig. 6e, happens on a much
slower (super-slow) time scale. Consequently, the transient dynamics of the
model with free slider (FS model) closely follows the super-slow manifold
corresponding to the steady-state dynamics of the model with pseudo-
constrained slider (PCS model), see Fig. 6f. More specifically, the FS model
initially follows the low-amplitude branch of the PCS model (phase 1). The
amplitude jumps precisely at the position predicted by the PCS model, and then
the FS model follows the high-level branch (phase 2) until steady state is
reached (phase 3). To obtain the time-dependent super-slow amplitude of the
beam’s vibration (both from the experimental data and the simulation data of
the FS model), the Hilbert transform was used and the outcome was subjected to
a moving average with a window time of 1 second.
Model and experiment are in excellent qualitative agreement with regard to all
features of the signature move, including the strongly modulated character of
the response in phase 1. Quantitatively, the amplitude level and the speed of
adaption (i. e. the speed of the slider’s horizontal movement) are in
excellent agreement for phase 1. Additionally, the speed of adaption is very
similar in phase 2. The slider’s turning point that separates phase 1 and 2 as
well as the final slider position are slightly closer to the beam’s center in
the experiment. A possible explanation for the deviation of the final slider
position is the uncertainty of the friction coefficient (systematic deviation
between static vs. dynamic friction coefficient, deterministic variation along
the beam’s length, plus aleatoric uncertainty). This inevitably leads to some
deviation, for instance in case one of the states (sticking or sliding)
dominates. The high sensitivity of the final slider position with respect to
the friction coefficient is analyzed in Section 5.5. The (averaged) steady-
state amplitude is almost identical for experiment and simulation, which one
can see in Fig. 6f. In that sub-figure, it can also be seen that the measured
super-slow trajectory is very close to the simulated one, merely shifted
slightly towards the right. A second example is depicted in Fig. 12b,
illustrating that the signature move is also possible in case the slider is
placed closer to center initially.

Figure 6: Successful adaption (signature move): Results of experiment and
simulation model at $f_{\mathrm{ex}}=124\mathrm{Hz}$,
$\hat{\ddot{w}}_{0}=14\mathrm{m/s}^{2}$ (a)-(d) beam’s deformation vs. time
(e) slider position vs. time (f) super slow dynamics

###  5.2 Successful adaption (trivial)

A representative example of trivial successful adaption is depicted in Fig. 7.
The excitation frequency is significantly lower compared to the signature move
example. In the experiment, the slider moves monotonously towards the beam’s
center first (phase 1) and then assumes a certain steady-state position (phase
2), see Fig. 7c,d. The beam’s vibration level increases during phase 1 and is
maintained at high level in phase 2 (Fig. 7a,b). This type of behavior is also
captured well by the simulation. Minor deviations exist in the the final
vibration level and the time history of the slider position. As for the
signature move, there is only a minor offset between the super-slow
trajectories of experiment and FS model.

Figure 7: Successful adaption (trivial): Results of experiment and simulation
model at $f_{\mathrm{ex}}=104\mathrm{Hz}$,
$\hat{\ddot{w}}_{0}=14\mathrm{m/s}^{2}$ (a)-(b) beam’s deformation vs. time
(c) slider position vs. time (d) super slow dynamics

###  5.3 Relaxation oscillation

A representative case of a relaxation oscillation is shown in Fig. 8. Phase 1
and 2 are similar to the first two phases of the signature move, however,
phase 2 ends with a sudden jump down to the low-amplitude branch. After this,
the process repeats approximately periodically, yielding a strongly modulated
response in terms of the beam’s vibration on the super-slow time scale. Also
this highly complex type of behavior is well-reproduced by the model.
Quantitative features, such as the amplitude and the duration of each super-
slow oscillation cycle are in very good agreement. As already discussed in
Section 4.3, a sufficient number of modes ($N\geq 5$) needs to be retained in
the modal beam model to predict the amplitude jump down.

Figure 8: Relaxation oscillation: Results of experiment and simulation model
at $f_{\mathrm{ex}}=126\mathrm{Hz}$, $\hat{\ddot{w}}_{0}=10\mathrm{m/s}^{2}$
(a)-(b) beam’s deformation vs. time (c) slider position vs. time (d) super
slow dynamics

###  5.4 Self-damping

The last presented type of behavior is self-damping, see Fig. 9. Here, the
excitation frequency is the same as in the case of the signature move, but the
excitation level is significantly decreased. Phase 1 again is similar to the
signature move; i. e., the slider moves away from the beam’s center and the
vibration amplitude increases slightly. After the jump to the high-amplitude
branch, however, the slider does not turn. Instead, it keeps moving towards
the clamping and, in accordance with the PCS model, the amplitude decreases
again (phase 2). At the end of phase 2 and during phase 3, the amplitude is so
small that the upper contact points do not lift off anymore. This can be
easily verified in the simulation. Consistently, in the experiment, the
rattling noise abruptly stops at the beginning of phase 3. Subsequently, the
beam vibrates harmonically at very low level and the slider’s position is
maintained at $s/L\approx 0.1$. Again, the simulation results agree
excellently with the measurements. The amplitude jumping is less clear in the
experimental results than in the simulated ones. In fact, several jumps up and
down are observed in the experiment. This is attributed to the poor
performance of the excitation level control in that time span. The controller
is relatively slow so that it was unable to maintain a constant excitation
amplitude. More specifically, the excitation amplitude varied by $\pm
1.6\mathrm{m/s}^{2}$ (nominal value $6\mathrm{m/s}^{2}$) in that time span. In
the simulation, the amplitude jump is smaller compared to the experiment. In
the following subsection it is shown that this deviation can be explained by
the high sensitivity of the process with respect to uncertain model
parameters.

Figure 9: Self-damping: Results of experiment and simulation model at
$f_{\mathrm{ex}}=124\mathrm{Hz}$, $\hat{\ddot{w}}_{0}=6\mathrm{m/s}^{2}$
(a)-(d) beam’s deformation vs. time (e) slider position vs. time (f) super
slow dynamics

###  5.5 Sensitivity with respect to experimental imperfections

In conclusion, all different types of behavior that are observed in the
experiments can be reproduced by the model in an excellent manner. Only minor
quantitative deviations of the super-slow dynamics, more particular the steady
slider position and the amplitude–slider position trajectory are encountered.
The small shift of this trajectory may be explained by the fact that the
experiments with attached slider and the ENMA defining the beam model could
not be done on the same day due to the large time required for setting up the
instrumentation for the ENMA and varying the excitation parameters in wide
ranges. Moreover, beam and frame needed to be disassembled and reassembled in
between the ENMA of the beam and the study of the beam-slider system. Thus, a
certain deviation of the linear natural frequency is to be expected. To show
how sensitive the super-slow manifold is with respect to small variations of
the updated beam model, the lowest linear natural frequency is deliberately
de-tuned by $\omega^{*}=\omega\pm 5\%$. This results in a shift of the turning
point by $\Delta s/L\approx{\pm}0.013$ and maximum amplitude
$\Delta\hat{w}(4L/7)/h\approx{\mp}0.088$, as illustrated in Fig. 10a.

Figure 10: (a) Sensitivity of super-slow manifold with respect to the linear
natural frequency: Experiment and PCS model at
$f_{\mathrm{ex}}=124\mathrm{Hz}$, $\hat{\ddot{w}}_{0}=14\mathrm{m/s}^{2}$ (b)
slider position versus time under variation of friction coefficient: Signature
move, same operating point (c) super-slow manifold for the case of self-
damping, reference and slightly increased coefficient of friction at
$f_{\mathrm{ex}}=124\mathrm{Hz}$, $\hat{\ddot{w}}_{0}=6\mathrm{m/s}^{2}$

The steady slider position is highly sensitive with respect to the friction
coefficient as shown in Fig. 10b. As already indicated in Section 4.2, the
coefficient of friction is an uncertain parameter and it varies along the
beam’s length. Also the super-slow manifold in the case of self-damping
depends on the friction coefficient. In the representative operating point
(shown in Section 5.4), a slightly increased friction coefficient yields a
bigger amplitude jump comparable to the one observed in the experiment, see
Fig. 10c. In conclusion, the small quantitative deviations between experiment
and model can be attributed to the uncertainties inherent to jointed
structures [28].

###  5.6 Operating regimes

Finally, the operating map spanned by the excitation level and excitation
frequency in Fig. 11 is analyzed. The map illustrates the type of behavior
reached for any excitation parameter combination. The discrete values of the
excitation level are
$\hat{\ddot{w}}_{0}\in\\{4,6,8,\dots,16\\}\mathrm{m/s}^{2}$, and of the
excitation frequency,
$f_{\mathrm{ex}}\in\\{100,102,104,\dots,130\\}\mathrm{Hz}$. Besides ranges of
the already discussed four types of behavior (signature move, trivial
adaption, relaxation oscillation, self-damping), there is a small range of
_inactivity_ where the slider does not move at all. This range appears for
relatively low excitation frequency. Apparently, the beam-slider system is too
far from resonance in this range, so that the vibration level is not
sufficient to activate any locomotion of the slider. Comparing experiment and
simulation, this range appears at high and low excitation amplitude,
respectively. Simulations at lower excitation frequency (not captured in Fig.
11) yield the state of inactivity also for large excitation level. The rest of
the map is filled by well separated ranges corresponding to the four described
types of behavior. From top left to bottom right, these ranges are arranged in
the order adaption (trivial)–adaption (signature move)–relaxation
oscillation–self-damping. Experiment and model also agree with regard to this
arrangement. Considering the regime of relaxation oscillation in the model,
the grid spanned by the operating conditions needed to be refined in order to
show continuity. In the model, the range of the signature move is larger than
in the experiment. The shift of the border between trivial adaption and
signature move is explained by the slightly shifted (potential) jumping point.
It is worth emphasizing, that this border is also depending on the slider’s
start position. More precise, it is crucial wether one starts left or right of
the potential jumping point, as illustrated for one operating point in Fig.
12. Also the borders between the other operating ranges are somewhat shifted,
which is attributed to the uncertainties (in particular natural frequency,
friction coefficient) and other imperfections (for instance imperfect
excitation level control) inherent to the experiment, as already mentioned.

Figure 11: Operating map illustrating the ranges of different types of
behavior (a) experiment (b) simulation

Figure 12: Successful adaption under variation of slider’s starting position
at $f_{\mathrm{ex}}=106\mathrm{Hz}$, $\hat{\ddot{w}}_{0}=10\mathrm{m/s}^{2}$
(a) trivial adaption starting from $s/L=0.3$ (b) signature move starting from
$s/L=0.4$

##  6 Conclusions

Subject of the current work is the qualitative and quantitative validation of
a model of a self-adaptive beam-slider system. A new prototype of the system
with continuously adjustable clearance and concave inner slider geometry was
designed in order to facilitate robust locomotion. All technical drawings are
provided so that other research groups can easily reproduce the tests [23].
The new prototype yields successful adaption in a broad range of operating
conditions. In adjacent operating ranges, a relaxation oscillation and self-
damping behavior are observed. To accurately describe the dynamics of the
geometrically nonlinear beam, the model was refined by considering the finite
stiffness of the clamping. The properties of the clamping stiffness and the
modal damping were updated using the results of an Experimental Nonlinear
Modal Analysis of the beam without slider. Numerical simulation results of the
beam-slider system, accounting for the clearance and the frictional contact
interactions between the bodies, are in excellent agreement with the
experiments. Qualitatively, the model captures all characteristic types of
behavior observed in the experiments (trivial adaption, signature move,
relaxation oscillation, self-damping, inactivity) and all involved features.
More precisely:

  * (1)

Movement of the slider away from/towards the beam’s center at low/high
amplitude

  * (2)

Strongly modulated response at low amplitude

  * (3)

Super-slow movement of slider compared to beam’s vibration

  * (4)

Jumps to higher/lower vibration level

  * (5)

Maintenance of slider position and vibration amplitude at steady state after
adaption

  * (6)

Transition to harmonic vibration at very low amplitude

Quantitatively, experiment and model agree excellently in terms of:

  * (1)

Speed of the slider’s horizontal movement

  * (2)

Amplitude levels during corresponding phases

  * (3)

Final slider position in case of transition to harmonic vibration

Minor deviations of the steady slider position and the super-slow manifold
(vibration amplitude vs. slider position) are observed. These are explained by
the system’s sensitivity to inevitable uncertainties, in particular with
regard to the friction coefficient and the linear natural frequency. In
conclusion, the intended validation of the simulation model is entirely
successful. This provides a proper basis for further theoretical studies to
gain deep understanding of the intriguing dynamics of the considered system
and the model-based optimization for technical applications such as energy
harvesting.

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