Zeerak Talat

Aurélie Névéol

Stella Biderman

Miruna Clinciu

Manan Dey

Shayne Longpre

Sasha Luccioni

Maraim Masoud

Margaret Mitchell

Dragomir Radev

Alexandra Sasha Luccioni

Shanya Sharma

Arjun Subramonian

Jaesung Tae

Samson Tan

Deepak Tunuguntla

Oskar Van Der Wal

Allen Hamilton

Eleutherai

You Reap What You Sow: On the Challenges of Bias Evaluation Under Multilingual Settings

Keywords:

come

Introduction

Machine learning (ML) systems, especially large language models (LLMs), are prone to (re)produce harmful outcomes and social biases [START_REF] Bender | On the dangers of stochastic parrots: Can language models be too big?[END_REF]Raji et al., 2021;[START_REF] Su | Language (Technology) is Power: A Critical Survey of "Bias" in NLP[END_REF][START_REF] Aguera Y Arcas | Do algorithms reveal sexual orientation or just expose our stereotypes?[END_REF]. Despite recent advances in LLMs [START_REF] Bender | Climbing towards NLU: On Meaning, Form, and Understanding in the Age of Data[END_REF], they have shown to disproportionately produce harmful content when addressing certain topics [START_REF] Samuel Gehman | RealToxi-cityPrompts: Evaluating Neural Toxic Degeneration in Language Models[END_REF][START_REF] Lin | TruthfulQA: Measuring How Models Mimic Human Falsehoods[END_REF] and demographics [START_REF] Sheng | The Woman Worked as a Babysitter: On Biases in Language Generation[END_REF][START_REF] Liang | Towards Understanding and Mitigating Social Biases in Language Models[END_REF]Dev et al., 2021a)-in part due to the training data used [START_REF] Dunn | Mapping languages: the Corpus of Global Language Use[END_REF][START_REF] Gao | The Pile: An 800GB Dataset of Diverse Text for Language Modeling[END_REF][START_REF] Bender | On the dangers of stochastic parrots: Can language models be too big?[END_REF], and the design of modeling processes [START_REF] Talat | Disembodied Machine Learning: On the Illusion of Objectivity in NLP[END_REF][START_REF] Hovy | Five sources of bias in natural language processing[END_REF]. In response, previous work has explored ways in which such social biases can be measured and counteracted [START_REF] Nangia | CrowS-pairs: A challenge dataset for measuring social biases in masked language models[END_REF][START_REF] Samuel Gehman | RealToxi-cityPrompts: Evaluating Neural Toxic Degeneration in Language Models[END_REF][START_REF] Czarnowska | Quantifying Social Biases in NLP: A Generalization and Empirical Comparison of Extrinsic Fairness Metrics[END_REF]. Typically, these issues have been addressed either by conceptualizing the underlying systemic discrimination as "bias" or by developing evaluation datasets that shed light on how LLMs produce harmful social outcomes. However, in the former case, as [START_REF] Su | Language (Technology) is Power: A Critical Survey of "Bias" in NLP[END_REF] points out, these conceptualizations often lack clear descriptions, e.g., type of systemic discrimination and affected demographics. This results in a highly under-specified "bias", which could lead to a downstream issue in the validity of the technical approaches that are developed (Blodgett et al., 2021). Similarly, the ill-defined "bias" is further compounded by the specifics of many benchmarks. Often, benchmarks exhibit discrepancies between understandings of the unobservable theoretical constructs against which "bias" is being measured and their operationalization [START_REF] Jacobs | Measurement and Fairness[END_REF][START_REF] Sorelle | The (Im)possibility of fairness: different value systems require different mechanisms for fair decision making[END_REF]. Furthermore, many prior benchmark datasets were developed with specific modeling architectures in mind [START_REF] Nangia | CrowS-pairs: A challenge dataset for measuring social biases in masked language models[END_REF]. They are limited to English and are culturally Anglo-centric. 1In this position paper, we present an overview of the current state-of-the-art concerning challenges and measures taken to address bias in language models. Specifically, we document the challenges of evaluating language models, with a focus on the generation of harmful text. By engaging our challenges with the relevant social scientific literature, we propose (1) a more transparent evaluation of bias via scoping and documentation, (2) focusing on the diversity of stereotypes for increased inclusivity, (3) careful curation of culturally aware datasets, and (4) creation of general bias measures that are independent of model architecture but capture the context of the task.

We recognize that many of the challenges that we have encountered and described here are large open problems that will require joint work to address. Our goal is to analyze these challenges and provide scaffolding for future work.

Grounding Bias, Fairness and Social Impact across Disciplines

Considering biases in socio-technical systems as a purely technical construct is an insufficient consideration of the problem [START_REF] Su | Language (Technology) is Power: A Critical Survey of "Bias" in NLP[END_REF]. In this section, we situate LLMs, and their applications, within the wider interdisciplinary literature on social harms and discrimination.

Social Discrimination

Issues of socially discriminatory (human and technological) systems have long been the subject of study for scholars across disciplines, e.g. in Science and Technology Studies [START_REF] Haraway | Situated Knowledges: The Science Question in Feminism and the Privilege of Partial Perspective[END_REF], discard studies [START_REF] Lepawsky | No insides on the outsides[END_REF], social anthropology [START_REF] Douglas | Purity and danger: an analysis of the concepts of pollution and taboo[END_REF], philosophy of democracy [START_REF] Fraser | Rethinking the Public Sphere: A Contribution to the Critique of Actually Existing Democracy[END_REF], gender and

LGBTQIA+ studies [START_REF] Spade | Normal Life: Administrative Violence, Critical Trans Politics, and the Limits of Law[END_REF][START_REF] Rajunov | Nonbinary: Memoirs of Gender and Identity[END_REF][START_REF] Keyes | You keep using that word: Ways of thinking about gender in computing research[END_REF][START_REF] Ignazio | Data feminism. Strong ideas series[END_REF], media studies [START_REF] Gitelman | Raw data is an oxymoron[END_REF], archival studies [START_REF] Agostinho | Uncertain Archives: Approaching the Unknowns, Errors, and Vulnerabilities of Big Data through Cultural Theories of the Archive[END_REF], sociolinguistics [START_REF] Labov | The social stratification of (r) in new york city department stores[END_REF][START_REF] Cheshire | An untitled review of "style and sociolinguistic variation[END_REF], and critical race theory [START_REF] Umoja | Algorithms of oppression: how search engines reinforce racism[END_REF][START_REF] Benjamin | Race after technology: abolitionist tools for the new Jim code[END_REF]. 2 Scholars argue that technical systems are embedded in social contexts [START_REF] Lepawsky | No insides on the outsides[END_REF][START_REF] Haraway | Situated Knowledges: The Science Question in Feminism and the Privilege of Partial Perspective[END_REF] and are therefore necessarily evaluated as socio-technical systems interacting with complex social hierarchies [START_REF] Winner | Do artifacts have politics? Daedalus[END_REF][START_REF] Benjamin | Race after technology: abolitionist tools for the new Jim code[END_REF][START_REF] Costanza-Chock | Design Justice, A.I., and Escape from the Matrix of Domination[END_REF][START_REF] Sorelle | The (Im)possibility of fairness: different value systems require different mechanisms for fair decision making[END_REF]. When technological systems prioritize majorities, there is a risk they oppress minorities at the personal, communal, and institutional levels [START_REF] Costanza-Chock | Design Justice, A.I., and Escape from the Matrix of Domination[END_REF]. [START_REF] Haraway | Situated Knowledges: The Science Question in Feminism and the Privilege of Partial Perspective[END_REF] argues that researchers default to a "view from nowhere", without reflecting on the context or use of their research. This default view often represents the interests of dominant majorities, disregarding knowledges from marginalized communities. Considering machine learning systems, Chun (2021) argues that the development of such technological systems relies on faulty assumptions (e.g., that past data collections can adequately and fairly predict future human behavior) which can lead to embedded social biases. Situating ourselves in the wider academic literature of social discrimination and marginalization, compels us to recognize that our technical systems must be considered in the social context in which they exist.

Machine-learned Systems in Social Context

On the topic of socially discriminatory systems within machine learning, [START_REF] Curran Associates | Gender Shades: Intersectional Accuracy Disparities in Commercial Gender Classification[END_REF] and [START_REF] Inioluwa | Actionable Auditing: Investigating the Impact of Publicly Naming Biased Performance Results of Commercial AI Products[END_REF] show that there are significant disparities along gendered and racialized lines in commercially available facial recognition and analysis systems. Similar issues of discriminatory social biases in natural language processing (NLP) systems have resulted in emerging research dedicated to the identification, quantification (e.g. [START_REF] Rudinger | Gender bias in coreference resolution[END_REF][START_REF] De-Arteaga | Bias in Bios: A Case Study of Semantic Representation Bias in a High-Stakes Setting[END_REF][START_REF] Czarnowska | Quantifying Social Biases in NLP: A Generalization and Empirical Comparison of Extrinsic Fairness Metrics[END_REF], and mitigation of bias [START_REF] Bolukbasi | Man is to computer programmer as woman is to homemaker? debiasing word embeddings[END_REF][START_REF] Sun | Mitigating Gender Bias in Natural Language Processing: Literature Review[END_REF][START_REF] Garimella | He is very intelligent, she is very beautiful? On Mitigating Social Biases in Language Modelling and Generation[END_REF] in NLP systems.

However, these methods tend to obscure rather than remove social biases [START_REF] Gonen | Lipstick on a Pig: Debiasing Methods Cover up Systematic Gender Biases in Word Embeddings But do not Remove Them[END_REF], and are particularly brittle when applied to complex, contextual language representations [START_REF] Dev | On measuring and mitigating biased inferences of word embeddings[END_REF].

Further, operationalization of under-specified "bias"

2 Many recent works on socially biased technological systems are interdisciplinary, e.g., 'Race After Technology: The New Jim Code' [START_REF] Benjamin | Race after technology: abolitionist tools for the new Jim code[END_REF] spans critical race theory, science and technology, Black feminism, and media studies.

has varied widely across studies, and in some cases has been internally inconsistent with their stated goals [START_REF] Su | Language (Technology) is Power: A Critical Survey of "Bias" in NLP[END_REF][START_REF] Jacobs | Measurement and Fairness[END_REF]. The recent surge of LLMs is no exception to such concerns. [START_REF] Hovy | Five sources of bias in natural language processing[END_REF]; [START_REF] Talat | Disembodied Machine Learning: On the Illusion of Objectivity in NLP[END_REF], and Cao and Daumé III (2020) argue that socially discriminatory biases can be encoded in several stages of the LLM development process [START_REF] Biderman | Pitfalls in machine learning research: Reexamining the development cycle[END_REF], including data sampling, annotation, selection of input representations or model, research design, and how the models are situated with regards to the language communities that they are applied to. Language generation models, despite their inference-time flexibility, are particularly susceptible to reproducing hegemonic social biases and generating offensive language, even when not explicitly prompted to do so [START_REF] Sheng | Societal Biases in Language Generation: Progress and Challenges[END_REF][START_REF] Wallace | Universal adversarial triggers for attacking and analyzing NLP[END_REF][START_REF] Bender | On the dangers of stochastic parrots: Can language models be too big?[END_REF].

In efforts to address the expression of such social biases, a number of bias evaluation benchmarks have been proposed (Dev et al., 2021b;[START_REF] Zhao | Gender bias in coreference resolution: Evaluation and debiasing methods[END_REF][START_REF] Cao | Toward Gender-Inclusive Coreference Resolution[END_REF]. However, common evaluation benchmarks are fraught with pitfalls in their conceptualization of bias, stereotypes, and harms, including meaningless or poorly formed stereotype constructions, non-intersectional examples, contexts that don't reflect downstream use, and reliance on specific model architectures (Blodgett et al., 2021;[START_REF] Jin | On transferability of bias mitigation effects in language model fine-tuning[END_REF]. Furthermore, bias evaluation benchmarks often make strong assumptions about the validity, reliability, and existence of observable properties, e.g. pronouns, as signals for unobservable theoretical constructs such as gender [START_REF] Jacobs | Measurement and Fairness[END_REF]. This is particularly problematic when building benchmarks for biases against communities that resist categorization based on observable characteristics (e.g. LGBTQIA+ and racialized people) and leads to reliance on existing stereotypes [START_REF] Tomasev | Fairness for Unobserved Characteristics: Insights from Technological Impacts on Queer Communities[END_REF]Dev et al., 2021a).

This rapid development of NLP resources and tools have further yielded a non-inclusive environment, skewed heavily towards English and Anglo-centric biases [START_REF] Joshi | The state and fate of linguistic diversity and inclusion in the NLP world[END_REF]. [START_REF] Sambasivan | Re-imagining Algorithmic Fairness in India and Beyond[END_REF] and [START_REF] Chan | The Limits of Global Inclusion in AI Development[END_REF] contend there remains a significant gap between the communities governing and governed by AI, and advocate for a redistribution of powers and responsibilities in developing responsible AI.

Considering gender bias, [START_REF] Stanczak | A Survey on Gender Bias in Natural Language Processing[END_REF] show that existing methods (1) largely avoid ethical considerations or evaluations of gender bias, (2) focus primarily on binary gender treatment, in mostly Anglo-centric settings, and (3) employ limited or flawed evaluation methodologies. Such issues are in part exacerbated by the general poverty of documentation of datasets [START_REF] Gebru | Datasheets for Datasets[END_REF][START_REF] Bender | Data Statements for Natural Language Processing: Toward Mitigating System Bias and Enabling Better Science[END_REF] and machine learning models [START_REF] Mitchell | Model Cards for Model Reporting[END_REF]. One way to mitigate these biases includes creating diverse teams with varied backgrounds and life experiences to assure the expression of diverse perspectives [START_REF] Monteiro | Ruined by design: how designers destroyed the world, and what we can do to fix it[END_REF][START_REF] Nekoto | Participatory research for low-resourced machine translation: A case study in African languages[END_REF]. However, as critiqued by [START_REF] Talat | Disembodied Machine Learning: On the Illusion of Objectivity in NLP[END_REF]; [START_REF] West | Discriminating Systems: Gender, Race, and Power in AI[END_REF], incorporating the diversity factor may be inadequate. Biases in language representations and task models can not only reflect, but also amplify bias present in the datasets [START_REF] Barocas | Big Data's Disparate Impact[END_REF][START_REF] Wang | Balanced Datasets Are Not Enough: Estimating and Mitigating Gender Bias in Deep Image Representations[END_REF]. These biases have been investigated and attempts made at creating interpretable representations and providing post-hoc explanations of model predictions.

Bias, Fairness, and Explainability

Given the grave consequences that inherent or conceptualized biases in ML systems can inflict, responsible AI has received a growing amount of research attention [START_REF] Amershi | Research Supporting Responsible AI[END_REF]. Responsible AI refers to the creation of ethical principles for AI and the development of AI systems based on these principles [START_REF] Dignum | Responsible artificial intelligence: Designing ai for human values[END_REF][START_REF] Schiff | Principles to Practices for Responsible AI: Closing the Gap[END_REF]. Colloquially, responsible AI encompasses distinct machine learning fields such as fairness, explainability, privacy, and interpretability. Concretely, how can responsible AI principles best contribute to the development of equitable systems?

Examining this question, Friedler et al. ( 2021) propose that building just ML systems requires an a priori definition of fairness. However, contemporary decisionmaking systems build on a so-called what-you-see-iswhat-you-get (WYSIWYG) approach that implicitly imbibes multiple fairness definitions or world views, leading to a system based on the conflict between the underlying value systems. To tackle this issue, ML engineers should explicitly state the underlying systemic values, as systems will inevitably comprise certain assumptions [START_REF] Birhane | The Values Encoded in Machine Learning Research[END_REF]. Thus, implying that biases as inherent to these decision-making systems and should be clearly articulated [START_REF] Bender | On the dangers of stochastic parrots: Can language models be too big?[END_REF] by explaining the whys and whats (explainability).

However, a more promising course of action for researchers would be to prioritize fairness in the entire life cycle of a language model. The tendency to consider and mitigate undesirable biases in models after training has completed leaves harmful residues that affect the communities we seek to protect (Dev et al., 2021a). Hence, a fruitful approach could be to reduce systemic unfairness by grounding the discussion on clear definitions of fairness based on input from the communities that could be harmed by the system (Liao and [START_REF] Liao | Enabling value sensitive ai systems through participatory design fictions[END_REF], explaining the inherent biases, and, if possible, minimizing bias issues by employing the measures discussed in, both, the previous and the following sections.

Challenges of Bias

Evaluating the social impacts and harmful biases LLMs exhibit is an important development step. However, despite the increased interest in developing bias benchmarks, the field still faces various challenges in evaluating LLMs with off-the-shelf benchmarks. In this section, we provide examples of existing bias measures currently used in NLP. We then discuss the challenges that originate from these: (1) they rely on vague definitions of bias, (2) are restricted to particular model architectures, (3) have limited relevance for different cultural contexts, and (4) are difficult to validate and interpret.

Examples of Bias Measure Studies

Recently, researchers and practitioners have begun to pay more attention to bias measures in NLP systems [START_REF] Su | Language (Technology) is Power: A Critical Survey of "Bias" in NLP[END_REF]Dev et al., 2021b). One line of work has focused on identifying bias in word embeddings: The Word Embedding Association Test (WEAT, [START_REF] Caliskan | Semantics derived automatically from language corpora contain human-like biases[END_REF] measures bias by comparing the relative distances of two sets of target words (e.g. occupation words: nurse, doctor) with respect to two sets of attribute words (e.g., gender attributes: male, female)and has inspired other similar approaches [START_REF] Kurita | Measuring Bias in Contextualized Word Representations[END_REF][START_REF] May | On measuring social biases in sentence encoders[END_REF][START_REF] Dev | On measuring and mitigating biased inferences of word embeddings[END_REF].

Although word embeddings may help identify biases in the context of LLMs, it is often difficult to access the learned contextual language representations of the model [START_REF] Abid | Persistent Anti-Muslim Bias in Large Language Models[END_REF][START_REF] Dev | On measuring and mitigating biased inferences of word embeddings[END_REF]. Furthermore, such methods are developed to address static word embeddings rather than the dynamic contextual word embeddings LLMs rely on [START_REF] Subramonian | Allennlp: Fairness and bias mitigation[END_REF].

Another research direction is the use of causal inference for measuring biases in LLMs, for example to analyze if the generated text by an LLM is affected considerably by only changing the protected attributes or categories in the input [START_REF] Po | Reducing sentiment bias in language models via counterfactual evaluation[END_REF][START_REF] Madaan | Generate your counterfactuals: Towards controlled counterfactual generation for text[END_REF][START_REF] Cheng | Causal Learning for Socially Responsible AI[END_REF]. In line with this idea, Huang et al. ( 2020) used a sentiment classifier to quantify and reduce the sentiment bias existent in LLMs. Similarly, the CrowS-Pairs benchmark [START_REF] Nangia | CrowS-pairs: A challenge dataset for measuring social biases in masked language models[END_REF] leverages the paradigm of minimal pairs to contrast sentences expressing stereotypes against social categories with the same sentences addressing different social categories. Crows-Pairs is designed such for language models to be probed for disparate behavior between the sentences pairs, with the hypothesis that systematic difference in the treatment reflecting the preference for stereotype indicates the presence of bias in the language models. Other examples of bias measures benchmarks include StereoSet [START_REF] Nadeem | Stereoset: Measuring stereotypical bias in pretrained language models[END_REF], WinoMT [START_REF] Stanovsky | Evaluating gender bias in machine translation[END_REF], BBQ [START_REF] Parrish | Bbq: A hand-built bias benchmark for question answering[END_REF], BOLD (Dhamala et al., 2021), and Toxicity Comment Classification competition (Jigsaw, 2017).

Defining Bias

The term "bias" is overloaded in the ML and NLP communities, as it is used in the lay (a prejudice towards or against some entity) and the statistical sense (a systematic deviation from a distribution's mean) [START_REF] Campolo | AI now 2017 report[END_REF]. Moreover, researchers often refer to vague definitions of bias and gloss over the details, which results in methods that lack specificity [START_REF] Su | Language (Technology) is Power: A Critical Survey of "Bias" in NLP[END_REF]. When discussing methods to address bias, it is critical to be precise about the bias being addressed.

Bias can, for instance, be made more specific by being defined along socially relevant dimensions. [START_REF] Nangia | CrowS-pairs: A challenge dataset for measuring social biases in masked language models[END_REF] consider the protected categories from the US Equal Employment Opportunities Commission and Queer in AI uses a similar list (gender identity and expression, sexual orientation, disability, neurodivergence, skill set, physical appearance, body size, race, caste, age, nationality, citizenship status, colonial experience, religion), yet other characteristics may be relevant elsewhere in the world (e.g. illness, migrant, and social status).3 However, protected classes are only one dimension along which to define bias; researchers should also be mindful of political biases and biases resulting from the focus on prestigious, highly resourced language varieties, in additions to the intersections of multiple dimensions [START_REF] Kearns | Preventing fairness gerrymandering: Auditing and learning for subgroup fairness[END_REF][START_REF] Curran Associates | Gender Shades: Intersectional Accuracy Disparities in Commercial Gender Classification[END_REF][START_REF] Crenshaw | Mapping the margins: Intersectionality, identity politics, and violence against women of color[END_REF].

With respect to any of the aforementioned dimensions, a "bias" is a preferential disposition towards or against an entity. Colloquially, it is perceived negatively and considered to be unfair treatment. As pointed out by [START_REF] Barocas | The problem with bias: from allocative to representational harms in machine learning. special interest group for computing[END_REF], biases in language models can manifest in the form of quality-of-service and representation disparities. As quality-of-service bias describes subpar performance of a language model when used by a particular group. For example, LLM-driven machine translation systems provide significantly better support for "prestigious", high-resource languages, and consequently deny quality performance to individuals who do not speak these languages [START_REF] Nekoto | Participatory research for low-resourced machine translation: A case study in African languages[END_REF]. Furthermore, in fundamental NLP tasks such as coreference resolution, LLMs can fail for people who use neopronouns, and often capture meaningless representations for language associated with trans and non-binary individuals. (Cao and Daumé III, 2020;Dev et al., 2021a). Additionally, [START_REF] Su | Twitter Universal Dependency parsing for African-American and mainstream American English[END_REF] show that parsing systems trained primarily on White Mainstream American English exhibit disparate performance on African American English and [START_REF] Tan | It's morphin' time! Combating linguistic discrimination with inflectional perturbations[END_REF] show that English question answering and machine translation systems often fail on the morphological variation that is often present in non-prestige and Learner Englishes.

Representation biases consist of stereotypes and under-representation (or over-representation) of data or model outputs. Stereotyping is a cognitive process that manifests from often negative cultural norms about a characteristic; stereotyping permeates what people do, say, or write. A long line of work has shown that language models capture social stereotypes, for example, with respect to binary gender and occupations [START_REF] Zhao | Gender bias in coreference resolution: Evaluation and debiasing methods[END_REF][START_REF] Bordia | Identifying and Reducing Gender Bias in Word-Level Language Models[END_REF][START_REF] De | Stereotype and Skew: Quantifying Gender Bias in Pre-trained and Fine-tuned Language Models[END_REF]. With regard to (under)representation, in MIMIC-III, a clinical notes dataset, only 1.9% of patients identify as Asian, in comparison to 71.5% who identify as white [START_REF] Chen | Exploring text specific and blackbox fairness algorithms in multimodal clinical NLP[END_REF]. Furthermore, blocklists in the Colossal Clean Crawled Corpus (C4) dataset disproportionately filter words related to queerness and language that is not White-aligned English [START_REF] Dodge | Documenting large webtext corpora: A case study on the colossal clean crawled corpus[END_REF]. Notably, quality-of-service and representation biases are not mutually exclusive; for instance, the brittle representations learned by a LLM for language associated with trans and non-binary individuals largely stems from the severe under-representation of this in training data (Dev et al., 2021a;[START_REF] Barocas | Big Data's Disparate Impact[END_REF].

The breakdown of biases into quality-of-service and representation disparities is only one of many possible lenses. It is also critical to explicitly consider biases stemming from disparities in resources, broadly defined in terms of data availability, time to invest into dataset curation, access to compute resources, financial resources, and more [START_REF] Bender | On the dangers of stochastic parrots: Can language models be too big?[END_REF].

Overreliance on Model Architectures

Current benchmarks often measure bias in specific downstream tasks (e.g. Machine Translation [START_REF] Stanovsky | Evaluating gender bias in machine translation[END_REF], Question Answering [START_REF] Parrish | Bbq: A hand-built bias benchmark for question answering[END_REF], or Text Generation (Dhamala et al., 2021)), while others focus on bias in LLMs more generally (e.g [START_REF] Kurita | Measuring Bias in Contextualized Word Representations[END_REF][START_REF] Nadeem | Stereoset: Measuring stereotypical bias in pretrained language models[END_REF][START_REF] Nangia | CrowS-pairs: A challenge dataset for measuring social biases in masked language models[END_REF]. This has the advantage of being more widely applicable, as many NLP systems are based on LLMs, and it avoids the need for creating and validating a new benchmark for each possible downstream task. Yet, when the benchmarks heavily rely on the model architecture rather than the task specification, quantitative comparison between different models based on these benchmarks is no longer possible. In such cases, it also becomes more difficult to assess the validity of the bias measure in how it relates to other benchmarks (criterion validity) and the more abstract notion of fairness (construct validity).

Some researchers circumvent this problem by adapting the original bias metric, but care should be taken when doing so. For instance, bias metrics originally developed for masked language models have been adapted by using perplexity (e.g. [START_REF] Nadeem | Stereoset: Measuring stereotypical bias in pretrained language models[END_REF] or prompting (e.g. [START_REF] Gao | Making pre-trained language models better few-shot learners[END_REF][START_REF] Sanh | Multitask Prompted Training Enables Zero-Shot Task Generalization[END_REF] instead. While these could still result in important insights, they also open new questions. Are the underlying assumptions of the bias measure still valid? Can you compare the bias metrics across different (future) types of models? Do the results of the initial validation of the benchmark still hold? And how does the kind of training data impact the evaluation that assumes a different training domain (e.g., legal texts vs. social media)?

While bias is ideally defined independently of the particular model architecture-not least because implementations change over time-we should not fall into a generalization trap either. As argued before, bias is inherent to systems and context-sensitive, and we should not strive for a panacea bias measure. Instead, the goal should be to develop methods that are task-specific yet independent of a given architecture, to the degree that this is possible. Researchers should keep this tension between task-and architecture-specific measures in mind when designing methods for measuring biases in LLMs.

Bias Measures are Anglo-centric

Despite the need for evaluating LLMs for a wide range of languages, bias benchmarks that cover non-English languages are rare [START_REF] Zhou | Examining Gender Bias in Languages with Grammatical Gender[END_REF][START_REF] Joshi | The state and fate of linguistic diversity and inclusion in the NLP world[END_REF]. As a solution, simply translating existing English benchmarks is not ideal: manual translation is a labor-intensive and highly skilled task, while automated translations are prone to errors and could potentially introduce new algorithmic sources of bias. Moreover, translated benchmarks may only test for Anglo-centric biases, which do not necessarily hold in many non-Western cultural contexts. For instance, many gender bias evaluations focus on Western professions, which are grammatically gendered in some languages [START_REF] Chen | Gender Bias and Under-Representation in Natural Language Processing Across Human Languages[END_REF][START_REF] Zhou | Examining Gender Bias in Languages with Grammatical Gender[END_REF] or may not cover other prevalent occupations outside the U.S. (Escudé Font and Costa-jussà, 2019). WinoMT [START_REF] Stanovsky | Evaluating gender bias in machine translation[END_REF] is one of the few benchmarks that covers multiple languages, but it comes with its own downsides. The sentences are generated from templates that capture a limited range of actual language use; the samples are translated from English examples, which may not reflect how stereotypes would occur in other languages; and the scope is limited to machine translation systems, and therefore WinoMT may not be suitable for multilingual models that are not trained on this specific task. The tightly coupled nature of bias and cultural context should be emphasized when designing a multilingual bias benchmark.

Validity of Bias Measures

Towards making NLP systems more just, we must understand the flaws of common bias measures and develop better guidelines to address biases. According to [START_REF] Jacobs | Measurement and Fairness[END_REF] and Blodgett et al. (2021), bias measures are measurement models which link observable properties, e.g., quality-of-service and representational biases, with unobservable theoretical constructs such as social discrimination, power dynamics, and systemic oppression. Consequently, bias measures are deeply political. Notably, a vast majority of bias measures themselves rely on other measurement models, such as the presence of gendered pronouns, to infer theoretical protected categories, e.g., gender. Moreover, bias measures may cause further epistemic violence onto the marginalized by creating a veneer of fairness, in spite of ongoing marginalization [START_REF] Gonen | Lipstick on a Pig: Debiasing Methods Cover up Systematic Gender Biases in Word Embeddings But do not Remove Them[END_REF][START_REF] Talat | Disembodied Machine Learning: On the Illusion of Objectivity in NLP[END_REF][START_REF] Jacobs | Measurement and Fairness[END_REF]. In ensuring the reliability, validity, and correct interpretation of bias measures, it is critical to examine all components in a bias measurement method.

Upstream measurement models that infer protected categories can be unreliable or even non-existent. For instance, pronouns and gendered names are usually em-ployed as proxies for binary gender, which is problematic (Dev et al., 2021a). Furthermore, characteristics like sexuality and disability are usually unobservable, which can lead to a reliance on hegemonic stereotypes and unnatural language in bias evaluation benchmarks [START_REF] Tomasev | Fairness for Unobserved Characteristics: Insights from Technological Impacts on Queer Communities[END_REF][START_REF] Hutchinson | Social biases in NLP models as barriers for persons with disabilities[END_REF].

With regard to validity, Blodgett et al. ( 2021) reviews how bias measures often rely on operationalization of stereotypes that are invalid for reasons such as misalignment and conflation. Additionally, the mathematical formalization of most bias measures is based on notions of parity-based fairness and do not reflect other conceptualizations of fairness such as distributive justice [START_REF] Jacobs | Measurement and Fairness[END_REF]. Another source of invalidity of bias measures lies in the purported generality of associated benchmarks. Raji et al. ( 2021) argue that the "instantiation [of benchmarks] in particular data, metrics and practice" undermines the validity of their construction to have "general applicability." Moreover, measurement models for protected categories fallaciously assume that the identities being indirectly observed can be discretized. Hence, Dev et al. (2021b) advocate for documenting the limitations of bias measures and related data in terms of their validity. In this process, it is critical to describe the relationship between the context of the data, model usage, and bias measure at stake.

The Elephant in the Room: Power, Privilege, and Point of View

Throughout the paper, we have primarily discussed bias in language models as a mechanical phenomenon. However, it is important to situate these discussions within the context and power dynamics of the way that NLP is practiced -both in research and in application [START_REF] Miceli | Studying Up Machine Learning Data: Why Talk About Bias When We Mean Power? Proceedings of the[END_REF]. In this section, we discuss sociopolitical influences on AI ethics and bias research in NLP. We argue that contemporary developments of LLMs have been an exercise in financial, institutional, ecological, linguistic, and cultural privilege. They are the consequence of the political will to create totalizing technologies and evaluation of bias, fairness and social impact should be viewed as a countervailing power mechanism, although in some cases serve to obscure these.

Large Language Models are Expensive

The current dominant paradigm in natural language processing is driven by the creation of ever-larger pretrained transformer models [START_REF] Brown | Language Models are Few-Shot Learners[END_REF]. As the size of LLMs increases, so do the requirements for hardware, energy, and time. For example, GPT-NeoX 20B [START_REF] Black | GPT-NeoX-20B: An Open-Source Autoregressive Language Model[END_REF] was trained for 1830 hours on 96 A100 GPUs, consuming 43.92 MWh of electricity and emitting 23 metric tons of CO 2 . Based on the current price listing of the cloud provider the model was trained on, training such a model would cost between 250,000 and 500,000 USD. 4 While this is not on the scale of the largest research programs, it is a significant amount of money and beyond the funding of many institutions, or beyond their political will to spend. While the development of such models can contribute towards improving the ability of people with less resources to pursue cutting edge downstream research, such pursuits have significant costs and barriers to entry for upstream research. This creates a stratification of research, wherein money is a barrier of entry for some forms of research but not for others.

Language is Multicultural, Language Models are Not

Although there are thousands of spoken languages in the world, the overwhelming majority of LLMs are monolingual and encode white respectability politics [START_REF] Thylstrup | Detecting 'Dirt' and 'Toxicity': Rethinking Content Moderation as Pollution Behaviour[END_REF][START_REF] Kerrison | Your Pants Won't Save You": Why Black Youth Challenge Race-Based Police Surveillance and the Demands of Black Respectability Politics[END_REF] onto minoritized variants of English [START_REF] Samuel Gehman | RealToxi-cityPrompts: Evaluating Neural Toxic Degeneration in Language Models[END_REF].

In this way, the cost of the developing LLMs extends from externalizing computational and infrastructural costs, to externalizing languages and language variants [START_REF] Lau | Artificial intelligence language models and the false fantasy of participatory language policies[END_REF]. Specifically, the vast majority of LLMs are trained to operate on an unspecified variant of "English" [START_REF] Bender | The #BenderRule: On Naming the Languages We Study and Why It Matters[END_REF], and in some cases Chinese (see Table 1 for a detailed overview of the top 25 LLMs). The dominance of English, and to a lesser degree Chinese, reifies cultural hegemonies and precipitates technological imperialism. Even when researchers seek to include other languages, these purportedly multilingual models often underserve certain languages and communities [START_REF] Kerrison | Your Pants Won't Save You": Why Black Youth Challenge Race-Based Police Surveillance and the Demands of Black Respectability Politics[END_REF][START_REF] Virtanen | Multilingual is not enough: BERT for Finnish[END_REF][START_REF] Kreutzer | Quality at a Glance: An Audit of Web-Crawled Multilingual Datasets[END_REF][START_REF] Gururangan | Whose Language Counts as High Quality? Measuring Language Ideologies in Text Data Selection[END_REF]. We also note that few of these models have been assessed for bias or fairness (see table 1). This act relies on two foundations. First, LLMs should only be used for languages that they have been developed for, with the cultural stereotypes that they have been trained on, thus limiting LLMs to be used within a small set of cultural contexts, or casting cultural contexts for which they are trained onto ones that they are not developed for. Second, should a multilingual LLM be trained, its primary data sources will still be in English, whereas the remaining languages will only be incidental to it. Such cultural imperialism is evident from the fact that only 2 of the 14 organizations involved in developing LLMs have teams in multiple countries (see table 1). Further, all multinational LLM efforts, except for one, draw their membership from the USA, UK, Germany, & Australia. GPT-NeoX 20B [START_REF] Black | GPT-NeoX-20B: An Open-Source Autoregressive Language Model[END_REF]) is an exception, as it also includes authors from India. A commonly-used resource for developing LLMs, CommonCrawl, relies on data that primarily stems from the US [START_REF] Dodge | Documenting large webtext corpora: A case study on the colossal clean crawled corpus[END_REF] and is written in privileged dialects of English [START_REF] Dunn | Mapping languages: the Corpus of Global Language Use[END_REF]. This prioritization is reflected by 16 teams being physically located in the U.S. Consequently, the current state of LLM development is a totalizing endeavor (Talat et al., the upper end reflects the sticker price of the systems. 2021), which engages in externalization across a number of axes, as is apparent from the infrastructural and development practices and the efforts to evaluate and mitigate social harms that arise from such technologies.

Large Language Models Allow Powerful

Actors to Control NLP Research

Due to the costs involved with training large language models and the small number of actors who have decided to train them, the overwhelming majority of research studying their properties is not carried out by people who train LLMs. When the actors that do possess the models choose to not publicly release them, model trainers are afforded control over the research that can be conducted with and by these models. Famously, OpenAI's initial announcement of GPT-3 asserted that access to the model would be heavily restricted while the company continued to research ethical interventions in their model. OpenAI is not alone in this; the idea that it is inherently dangerous to release models to the public has been put forth by several other actors in this space (Weidinger et al., 2021a;[START_REF] Askell | A general language assistant as a laboratory for alignment[END_REF].

It is essential to recognize that the decisions regarding access and the kind of research that can be conducted on large language models (or any ML models, for that matter) is an inherently political one [START_REF] Leahy | The hard problem of aligning AI to human values[END_REF]. Regardless of the truth of the aforementioned claims, they are highly contentious political claims and should be treated as such rather than passively accepted.

Direct access to LLMs is important to perform independent research on their datasets, functions, and societal impact [START_REF] Kandpal | Deduplicating Training Data Mitigates Privacy Risks in Language Models[END_REF][START_REF] Carlini | Quantifying memorization across neural language models[END_REF]. While language models produced by the academic research community are widely available for critical examination, commercial systems are often only available through APIs provided by the developers (see table 1 for an overview on access for the 25 largest pretrained language models. Such restrictions to access to the models and resources that they are developed for provide a significant barrier to a) principles of open science and b) research on how the datasets and language models themselves embed and amplify social biases.

Addressing Bias

Researchers have developed various strategies to address bias in large language models. As discussed in earlier sections, however, these strategies are insufficient to tackle multiple dimensions of bias. Below, we enumerate a few ways in which bias can be addressed by the research community to effectively engage with our aforementioned concerns: (1) moving towards a more transparent way of evaluating bias, (2) focusing on the diversity of stereotypes and increasing inclusivity, and (3) considering the impact of linguistic and cultural differences on the identification and mitigation of bias in designing culturally comparable datasets. We would like to highlight that these suggestions are not exhaustive. They will, however, guide the work in this area.

Transparency Through Documentation

Stereotypes and biases cover a broad definition and vary in conceptualization across geographical and cultural contexts. To ensure that the nuances are well communicated and that practitioners understand the applicability of the evaluation approach, we suggest documenting a thorough analysis of the scope. Below, we provide a starting point based on [START_REF] Mitchell | Model Cards for Model Reporting[END_REF][START_REF] Gebru | Datasheets for Datasets[END_REF]; Dev et al. (2021b);[START_REF] Su | Language (Technology) is Power: A Critical Survey of "Bias" in NLP[END_REF].

Defining the scope of the approach Blodgett et al. (2020) found that works around bias "often fail to explain what kinds of system behaviors are harmful, in what ways, to whom, and why." It thus becomes imperative to question what underrepresented groups would benefit more from a given evaluation benchmark. We therefore urge researchers and practitioners to clearly specify the demographic a particular method is relevant for. Moreover, given how social hierarchies intertwine tightly with language and may present themselves through its peculiarities, we also encourage researchers to specify the limitations and scope of their approaches.

As an example, we consider the gender bias evaluation in English [START_REF] Zhao | Gender bias in coreference resolution: Evaluation and debiasing methods[END_REF][START_REF] Stanovsky | Evaluating gender bias in machine translation[END_REF][START_REF] Levy | Collecting a large-scale gender bias dataset for coreference resolution and machine translation[END_REF][START_REF] Sharma | Evaluating gender bias in natural language inference[END_REF], where the bias might present itself through strong associations between grammatical constructs like pronouns. The same does not hold true for genderless languages, despite the existence of the bias [START_REF] Zmigrod | Counterfactual Data Augmentation for Mitigating Gender Stereotypes in Languages with Rich Morphology[END_REF]. Thus, evaluation benchmarks and approaches do not always transfer well to other languages. Additionally, while such benchmarks use gender associations to professions for their evaluation, this method covers only one aspect of the social hierarchy, and does not address gender bias in language in its entirety. By being binary in nature and tightly coupled to Anglo-centric contexts (see §3) benchmarks are limited in their scope and relevance. While most recent works do include ethical considerations, the limitations and scope are only vaguely specified. We advocate for such limitations to be highlighted and pointed out for the community to have a clearer picture about the steps that need to be taken towards greater inclusivity.

Documenting the demographics Previous work has highlighted the importance of engaging with individuals on the receiving end of the bias [START_REF] Bender | On the dangers of stochastic parrots: Can language models be too big?[END_REF]. It thus becomes important to understand the demographics of those involved in the creation of the benchmarks. As previously shown [START_REF] Hala | Identifying and measuring annotator bias based on annotators' demographic characteristics[END_REF] there exists a relation between annotators' identities and toxicity/bias in dataset. On this basis, we urge the researchers to collect and document the demographic information and annotator attitude scores [START_REF] Sap | Annotators with Attitudes: How Annotator Beliefs And Identities Bias Toxic Language Detection[END_REF]. Building upon the same, we encourage the collection and reporting of this information about the researchers involved.

Diversity Beyond Gender Bias

The majority of previous work on bias has focused particularly on gender bias [START_REF] Zhao | Gender bias in coreference resolution: Evaluation and debiasing methods[END_REF][START_REF] Stanovsky | Evaluating gender bias in machine translation[END_REF][START_REF] Levy | Collecting a large-scale gender bias dataset for coreference resolution and machine translation[END_REF][START_REF] Sharma | Evaluating gender bias in natural language inference[END_REF] and the very few works [START_REF] Nadeem | Stereoset: Measuring stereotypical bias in pretrained language models[END_REF][START_REF] Nangia | CrowS-pairs: A challenge dataset for measuring social biases in masked language models[END_REF] that take other dimensions of biases into account, have their own shortcomings, as discussed in Section 3. It thus becomes important to diversify the range of bias and stereotypes that are being investigated by research, and covered by a certain evaluation technique. In extending the coverage to more dimensions, context stands as an important aspect of bias. The contextual aspects of bias as represented in language, culture, and history hold a significant role in forming and assessing the bias itself. Hence, as a practice, we encourage researchers to consider these three aspects when constructing bias measures and datasets.

In discussing bias, it is important to note that discrimination does not occur in a vacuum. An act of discrimination against a person may be directed towards several intersecting identities. Considering bias using a singleaxis framework makes it impossible to engage with and evaluate the harms extended to the social groups that lie at the intersection of multiple identities [START_REF] Crenshaw | Mapping the margins: Intersectionality, identity politics, and violence against women of color[END_REF]. In an Indian context, for example, even those who identify as belonging to the "same" caste [START_REF] Malik | Socially aware bias measurements for hindi language representations[END_REF], can have varied lived experiences based on class, gender, and other identities. More precisely, it is impossible to disentangle which specific identity a discriminatory act is directed against. Previous works have highlighted the importance of studying intersectional bias [START_REF] Bender | On the dangers of stochastic parrots: Can language models be too big?[END_REF][START_REF] Curran Associates | Gender Shades: Intersectional Accuracy Disparities in Commercial Gender Classification[END_REF][START_REF] Field | A Survey of Race, Racism, and Anti-Racism in NLP[END_REF][START_REF] Guo | Toward fairness in ai for people with disabilities: A research roadmap[END_REF][START_REF] Crenshaw | Mapping the margins: Intersectionality, identity politics, and violence against women of color[END_REF] but little research has been conducted around addressing such biases [START_REF] Magee | Intersectional Bias in Causal Language Models[END_REF][START_REF] Guo | Detecting Emergent Intersectional Biases: Contextualized Word Embeddings Contain a Distribution of Human-like Biases[END_REF]. We thus encourage researchers to develop measures and benchmarks which are grounded in intersectional understanding of bias and adequately address the lived experiences of various social groups, towards increased inclusivity and fairness.

Not only can the dimensions and context influence our definitions and approaches to bias, but the categories (values) assigned to each dimension (e.g., age) can also limit our understanding and solution of bias. For instance, the majority of gender-bias evaluation datasets solely deal with binary gender, i.e., male and female, with just a handful covering non-binary genders with only minimal representation (Dev et al., 2021a;[START_REF] Cao | Toward Gender-Inclusive Coreference Resolution[END_REF]. As a result, category inclusiveness is critical in the development of a high-quality bias evaluation dataset. A set of categories that can act as a starting point are provided by Queer in AI in Section 3.2.

Acknowledging Differences

Stereotype and bias formation is influenced by culture. As a result, what might be a stereotype in a given culture might not stand relevant in another. For instance, the characterization that parental leave is for mothers is considered stereotypical in the United States, but not in Sweden, where parental leave is split between both parents.

Table 1: The 25 largest pretrained dense language models, ranging from 6 billion parameters to 530 billion. Models are overwhelmingly trained by teams located in the US and on English text. Less than half of the language models were evaluated for bias by their creators.

Previous sections have criticized the Anglo-centricity in the research of NLP bias and the influence on languages other than English. In particular, the lack of culturally-aware datasets limits the degree to which future NLP algorithms can be evaluated for biases. More crucially, these unspecified languages and cultures are on the receiving end of unmanaged effects. As a result, researchers are encouraged to develop bias datasets and benchmarks for non Anglo-centric cultures and languages [START_REF] Bender | On the dangers of stochastic parrots: Can language models be too big?[END_REF]. Involving experts in related areas, especially participants with lived experiences of language-related harms, might aid decisions at all parts of this process, e.g. deciding what groups and content to include in research or dataset design [START_REF] Liao | Enabling value sensitive ai systems through participatory design fictions[END_REF]Dev et al., 2021a;[START_REF] Mcmillan-Major | Documenting Geographically and Contextually Diverse Data Sources: The BigScience Catalogue of Language Data and Resources[END_REF]. Overall, having culturally diverse and comparable datasets for a diverse set of languages (ideally covering all languages) is critical for evaluating multilingual models. Moreover, the applicability of bias measures across various languages suggests the necessity for cross-linguistic metrics or measurements that can be extended to different languages or cultures [START_REF] Zhou | Examining Gender Bias in Languages with Grammatical Gender[END_REF]Escudé Font and Costa-jussà, 2019;[START_REF] Malik | Socially aware bias measurements for hindi language representations[END_REF].

Conclusion

Recent improvements in LLMs to mimic human text have led to a surge in research that seeks to identify and address the harms arising from their training and deployment. However, the considerations on social harms that arise has been limited to narrow, Anglo-centric, contradictory, and often underspecified definitions of fairness and bias. Furthermore, the development of contemporary methods has conflated task-specific and architecture-specific designations. Compounded with the structural inequalities around resources, language, and identity, this has yielded an overreliance on prestige forms of English for developing LLMs and interrogating and addressing the social biases that they harbor. Situating these methods within such Englishes has had the consequence of over-emphasizing Western-centric social categories. Moreover, datasets for evaluating social biases in LLMs have traditionally failed to denote and specify the context within which biases are situated. Such concerns have been the cause for questions around the validity of the developed measures, and in particular for multilingual LLMs.

To address such challenges, we propose that developing methods for multilingual LLMs requires researchers to provide thorough documentation of their approaches, including documenting the scope, demographics of speakers, and potential annotators. Additionally, we also recommend that researchers situate their bias evaluation methods within the specific context of the languages that the model operates on. In doing so, bias evaluation methods can be made to specifically address biases under the conditions and contexts that they occur in each of the model's languages. Furthermore, we recommend that researchers examine diversity issues beyond gender bias, with a particular focus on intersectional issues [START_REF] Guo | Detecting Emergent Intersectional Biases: Contextualized Word Embeddings Contain a Distribution of Human-like Biases[END_REF].

Finally, we recommend that researchers are cognizant of the social and environmental harms that developing LLMs have. For instance, developing ever-larger language models that achieve marginal improvements for English may bring a smaller benefit than developing a LLM for other languages. Thus, in a consideration of developing a new language model, we implore researchers to consider ways in which harms can be limited, or the benefits can come to compensate for their costs.

For example, the BigScience biomedical working group has estimated that 82% of evaluation datasets in the biomedical and clinical field are for corpora in English(Datta et al.,

2021).

Queer in AI (http://queerinai.org/) is a grassroots D&I organization that seeks to empower queer and trans researchers in AI and advance research at the intersections of AI and queerness. Their list of categories can be found here: http://queerinai.org/code-of-conduct.

The lower end of this range reflects the common practice of giving discounts of up to

50% for large purchases, while

A Acknowledgments

The work presented in this paper is the outcome of the discussions and work within the BigScience initiative. Notably, we would like to acknowledge Ioana Baldini and Xudong Sheng, who contributed significantly to an earlier iteration and whose work served as a foundation for the specific contributions and arguments of this paper.

B Credit Author Statement

We follow the recommendations and taxonomy provided by [START_REF] Allen | How can we ensure visibility and diversity in research contributions? How the Contributor Role Taxonomy (CRediT) is helping the shift from authorship to contributorship[END_REF] to determine and outline author contributions.

Stella Biderman: Writing -Original Draft (Section 4), Writing -Review & Editing.

Miruna Clinciu: Conceptualization, Writing -Original Draft (Section 3), Writing -Review & Editing (Section 3.5).

Manan Dey: Writing -Original draft preparation (Section 5), Writing -Review and Editing.

Shayne Longpre: Writing -Original draft preparation (Section 1-2), Writing -Review & Editing (Section 3).

Alexandra Sasha Luccioni: Writing -Original Draft (Section 4), Writing -Review & Editing.

Maraim Masoud: Conceptualization, Writing -Original draft preparation (Section 4), Writing -Review & Editing (Section 4). Oskar van der Wal: Conceptualization, Writing -Original Draft (Section 3), Writing -Review & Editing (Section 3).

C Determination of Author Order

Contributors are listed alphabetically, except for Zeerak Talat and Aurelié Nevéol, who managed paper writing and chaired the working group, respectively. All authors contributed to the conceptualizing and writing of the document.


Adrien Mélot
email: adrien.melot@ec-lyon.fr

Emmanuel Rigaud

Joël Perret-Liaudet

Robust design of vibro-impacting geared systems with uncertain tooth profile modifications via bifurcation tracking

Keywords:

The present work investigates the influence of uncertain tooth profile modifications on the nonlinear dynamic response of a spur gear pair induced by a backlash nonlinearity. To this end, an original approach based on bifurcation tracking is developed. The equations of motion are solved in the frequency domain with the harmonic balance method (HBM) coupled to an arc-length continuation algorithm and a bordering technique. The evolution of the bifurcation points with respect to the uncertain parameter is computed in a deterministic way. The study focuses on minimizing the amplitude-jump instabilities induced by the backlash nonlinearity around the primary resonance peak. The proposed methodology allows for a fast and reliable estimation of the tooth profile modification that minimizes the amplitude-jump instability by defining two criteria using the results of the bifurcation tracking algorithm. Probability density functions (PDF) of various indicators of the severity of vibro-impacts can be computed with Monte-Carlo (MC) simulation with minimal computational burden. Results show that the tooth profile modification that minimizes the amplitude-jump instabilities differs from the optimum obtained with static computations.

Introduction

Vibration and noise mitigation in geared system remains to this day a major challenge and research interest. The reasons are twofold. Firstly, large vibration amplitudes induce wear, fatigue and can compromise the structural integrity of the system. Secondly, vibrations generated at the gear are transferred to the housing via the shafts and bearings, which creates an unwanted noise: whining noise in case of permanent contact [START_REF] Carbonelli | Vibro-Acoustic Analysis of Geared Systems-Predicting and Controlling the Whining Noise[END_REF], or hammering or rattle noise when the response exhibits vibro-impacts [START_REF] Pfeiffer | Hammering in diesel-engine driveline systems[END_REF][START_REF] Rigaud | Investigation of gear rattle noise including visualization of vibro-impact regimes[END_REF].

The meshing process is the origin of the transmission error and mesh stiffness fluctuations, which are assumed to be the main sources of excitation in geared systems. The former is defined as the difference between the actual position of the ouput gear and the position it would occupy, were the gear pair perfectly conjugate [START_REF] Welbourn | Fundamental knowledge of gear noise: A survey[END_REF]; the latter is expressed as the derivative of the transmitted load with respect to the transmission error. Modifications of the gear micro-geometry, i.e. the profile of the tooth flanks, are commonly employed to optimize contact conditions, compensate for manufacturing defects and minimize the gear dynamic response. Such modifications consist in an intentional removal of material from the tooth flanks so that the tooth profile differs from the theoretical perfect involute. Their effect on both the static and dynamic responses of geared systems are well known [START_REF] Gregory | Dynamic behaviour of spur gears[END_REF][START_REF] Kahraman | Effect of Involute Tip Relief on Dynamic Response of Spur Gear Pairs[END_REF][START_REF] Ozturk | Optimum profile modifications for the minimization of dynamic transmission error[END_REF][START_REF] Hotait | Experiments on the relationship between the dynamic transmission error and the dynamic stress factor of spur gear pairs[END_REF]. Most studies aim at minimizing the fluctuations of the transmission error to mitigate the dynamic response. This is traditionally carried out by optimizing the tooth profile modifications through static computations [START_REF] Tavakoli | Optimum Profile Modifications for the Minimization of Static Transmission Errors of Spur Gears[END_REF][START_REF] Bonori | Optimum profile modifications of spur gears by means of genetic algorithms[END_REF][START_REF] Carbonelli | Particle Swarm Optimization as an Efficient Computational Method in order to Minimize Vibrations of Multimesh Gears Transmission[END_REF][START_REF] Garambois | NVH robust optimization of gear macro and microgeometries using an efficient tooth contact model[END_REF]. Very few studies carry out dynamic analyses to define the best TPM, which often lead to different optimal solutions [START_REF] Liu | Dynamic Modeling and Analysis of Tooth Profile Modification for Multimesh Gear Vibration[END_REF][START_REF] Faggioni | Dynamic optimization of spur gears[END_REF][START_REF] Ozturk | Optimum profile modifications for the minimization of dynamic transmission error[END_REF]. Such works try to minimize the peak-to-peak (𝐷𝑇𝐸 𝑃 𝑃 ) or the the root mean square values (𝐷𝑇𝐸 𝑅𝑀𝑆 ) of the dynamic transmission error. However such studies rely on computations of the dynamic response over the full range of operating rotational speed for a few discrete values of tooth profile modifications which can result in a significant computational effort. Uncertainties in the system parameters are unavoidable due to manufacturing tolerances, wear, etc. They are seldom taken into account in dynamics analyses, all the more when the backlash nonlinearity is considered, since the associated computational costs are prohibitive. Tobe investigated the effect of random tooth-to-tooth errors on the dynamic behaviour of a gear pair both numerically [START_REF] Tobe | Statistical analysis of dynamic loads on spur gear teeth[END_REF] with Monte-Carlo simulations and experimentally [START_REF] Tobe | Statistical analysis of dynamic loads on spur gear teeth : Experimental study[END_REF]. Driot investigated the variability of the gearbox whining noise [START_REF] Driot | Allocation of gear tolerances to minimize gearbox noise variability[END_REF] and modal characteristics [START_REF] Driot | Variability of modal behavior in terms of critical speeds of a gear pair due to manufacturing errors and shaft misalignments[END_REF] induced by gear manufacturing tolerances. Uncertainties in the torque fluctuations were studied [START_REF] Beyaoui | Dynamic behaviour of a wind turbine gear system with uncertainties[END_REF] with polynomial chaos expansion. Guerine [START_REF] Guerine | A perturbation approach for the dynamic analysis of one stage gear system with uncertain nnparameters[END_REF] employed perturbation techniques to investigate uncertain structural parameters. Wei [START_REF] Wei | Dynamic response of a single-mesh gear system with periodic mesh stiffness and backlash nonlinearity under uncertainty[END_REF] implemented the incremental harmonic balance solver with the Chebyshev inclusion function to study the effects on uncertainties in the mesh stiffness, static transmission error, damping ratio and torque fluctuations.

Bifurcation tracking is a method to carry out parametric analyses of nonlinear systems. The first attempts at numerically tracking bifurcations date back to the 1970s. This type of dynamic analysis has gained a lot of attention in recents years, following the papers from Detroux [START_REF] Detroux | The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems[END_REF] and Xie [START_REF] Xie | Bifurcation tracking by harmonic balance method for performance tuning of nonlinear dynamical systems[END_REF]. Carrying out parametric analyses of nonlinear systems can be computationally expensive because of the required computation of the forced response curves (FRC) for all considered points of the design space. Bifurcation tracking is particularly attractive as it consists in only following the evo-lution of bifurcation points, reducing the number of computed points to the number of bifurcations. Another benefit of directly following bifurcation points is that a folding of the bifurcation curves can reveal isolated responses, i.e., responses that are not connected to the main solution branch [START_REF] Alcorta | Period doubling bifurcation analysis and isolated subharmonic resonances in an oscillator with asymmetric clearances[END_REF][START_REF] Grenat | A multiparametric recursive continuation method for nonlinear dynamical systems[END_REF].

The objectives of this work are twofold: to introduce bifurcation tracking as a efficient tool to design tooth profile modifications in order to mitigate the nonlinear dynamic response of the system and to investigate the influence of uncertainty on the optimal tooth profile modifications. The present paper is structured as follows: the mathematical model of the gear pair is introduced in Sect. 2. The computational strategy and uncertainty modelling is detailed in Sect. 3. Results are discussed in Sect. 4 and main conclusions are drawn in Sect. 5.

Mathematical model of the gear pair

Equation of motion of the gear pair

The geared system used in this paper is a spur gear pair whose gear ratio is equal to 1. The gears are modelled as two rigid disks with respective inertia 𝐼 1 and 𝐼 2 and rotational degrees of freedom (DoF) 𝜃 1 and 𝜃 2 . A nonlinear elastic force, noted 𝑓 𝑛𝑙 , representing the contact between gear teeth acts along the line of action, i.e. in the normal direction of the tooth profile.

The equations of motion of this two-degree-of-freedom model are:

{ 𝐼 1 θ1 -𝑇 1 + 𝑟 𝑏,1 𝑓 𝑛𝑙 = 0 𝐼 2 θ2 + 𝑇 2 -𝑟 𝑏,2 𝑓 𝑛𝑙 = 0 (1)
where 𝑇 1 = 𝑇 2 = 100 N⋅m are respectively the input and output torques and 𝑟 𝑏,𝑘 denotes the base radius of gear 𝑘.

The transmission error 𝑞 = 𝑟 𝑏,1 𝜃 1 -𝑟 𝑏,2 𝜃 2 can be introduced at this stage in order to remove the rigid body motion corresponding to the rotation of the gear pair. The resulting equation of motion reads:

𝑀 𝑒𝑞 q + 𝑓 𝑛𝑙 (𝑞, 𝑡) = 𝐹 𝑠 (2)
with the equilavent mass 𝑀 𝑒𝑞 and static mesh force 𝐹 𝑠 :

⎧ ⎪ ⎨ ⎪ ⎩ 𝑀 𝑒𝑞 = 𝐼 1 𝐼 2 𝑟 2 𝑏,1 𝐼 2 +𝑟 2 𝑏,2 𝐼 1 𝐹 𝑠 = 𝑇 1 𝑟 𝑏,1 = 𝑇 2 𝑟 𝑏,2 (3)

Gear coupling

The nonlinear mesh force 𝑓 𝑛𝑙 describes an elastic coupling when the gears are in contact and must be able to model potential contact loss and resulting impacts between gear teeth.

Backlash nonlinearity

Linerarizing the transmitted load around the static equilibrium, i.e. the static transmission error 𝑞 𝑠 , allows one to

𝑘 𝑚 (𝜃 1 ) = 𝜕𝐹 𝑠 𝜕𝑞 𝑠 (𝜃 1 ) (4)
Note that for non-zero rotational speeds 𝑘 𝑚 (𝜃 1 ) = 𝑘 𝑚 (Ω𝑡).

Due to the linearization, the contact force associated to the mesh stiffness equals zero for a positive displacement if:

𝑞(𝑡) ≤ 𝑔(𝑡) = 𝑏 + 𝑞 𝑠 (𝑡) - 𝐹 𝑠 𝑘 𝑚 (𝑡) ( 5
)
where 𝑏 is half backlash, considered constant in the following. The mesh force (cf. Fig. 2) including the backlash nonlinearity can therefore be expressed as:

𝑓 𝑛𝑙 (𝑞, 𝑡) = 𝑘 𝑚 (𝑡) (𝑞 -𝑔(𝑡))  (𝑞 -𝑔(𝑡)) + 𝑘 𝑚 (𝑡) (𝑞 + 𝑔(𝑡))  (-𝑞 -𝑔(𝑡)) ( 6
)
where  is the Heaviside step function.

Damping

In the following, we consider an equivalent viscous damping that does not depend on the contact state [START_REF] Kahraman | Non-linear dynamics of a spur gear pair[END_REF][START_REF] Farshidianfar | Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems[END_REF][START_REF] Mélot | Effect of gear topology discontinuities on the nonlinear dynamic response of a multi-degree-of-freedom gear train[END_REF][START_REF] Mélot | Bifurcation tracking of geared systems with parameter-dependent internal excitation[END_REF].

The damping term reads:

𝐶 = 2𝜉 √ 𝑘 𝑚 𝑀 𝑒𝑞 (7)
where 𝑘 𝑚 is the mean value of the periodic mesh stiffness and 𝜉 the damping ratio. Thus, the complete equation of motion of the gear pair is:

𝑀 𝑒𝑞 q + 𝐶 q + 𝑓 𝑛𝑙 (𝑞, 𝑡) = 𝐹 𝑠 (8) 𝐹 𝑘 𝑚 (𝑡) 𝐹 𝑠 𝑞 𝑏 𝑔 𝑡 𝑞 𝑠 (𝑡) -𝑏

Computational strategy and uncertainty modelling

Monte-Carlo simulations [START_REF] Metropolis | The monte carlo method[END_REF] are commonly used in dynamic analyses in order to propagate uncertainties. Despite their widespread use, such simulations often lead to hefty computational effort due to the large number of samples and computations of the forced response curves required to ensure convergence. To avoid such issues, the proposed approach is based on a deterministic computation of the bifurcation curves of the investigated system. This reduces the number of points to compute for each value of the uncertain parameter to the number of identified bifurcations. Herein, bifurcation tracking analysis is applied to the amount of tip relief, corresponding to an intentional removal of material at the tip of the gear teeth (see. Fig. 3). The study focuses on minimizing the dynamic response around the primary resonance, especially the amplitude-jump phenomenon induced by the folding of the response curve delimited by two saddle-node (SN) bifurcations. The location of bifurcation points with respect to this parameter are obtained with minimal computational effort using the methodology thoroughly detailed in [START_REF] Mélot | Bifurcation tracking of geared systems with parameter-dependent internal excitation[END_REF]. For the sake of completeness a short description is provided hereafter.

Nonlinear solver

The periodic solutions of equation ( 8) are sought as truncated Fourier series:

𝑞 ≈ Re ( 𝐻 ∑ 𝑘=0 q𝑘 e 𝑖𝑘Ω𝑡 ) , (9)
Eq. ( 9) and its time derivatives are plugged into the equation of motion [START_REF] Driot | Variability of modal behavior in terms of critical speeds of a gear pair due to manufacturing errors and shaft misalignments[END_REF] and Galerkin projection is performed with the same harmonic base functions to obtain a system of nonlinear algebraic equations. An arc-length continuation procedure [START_REF] Seydel | Practical Bifurcation and Stability Analysis[END_REF] with a tangent predictor is then used to follow the evolution of the solution point with respect to the forcing (rotational) frequency Ω. At each iteration, the nonlinear mesh force is evaluated from Eq. ( 6) with the alternating frequency/time (AFT) algorithm [START_REF] Cameron | An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems[END_REF].

In order to track bifurcation points with respect to an additional parameter, a minimaly extended system is built by appending an additional equation to the system [START_REF] Govaerts | Numerical continuation of bifurcations of limit cycles in matlab[END_REF][START_REF] Detroux | The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems[END_REF][START_REF] Mélot | Bifurcation tracking of geared systems with parameter-dependent internal excitation[END_REF]. The bifurcation equation, noted 𝐺, is evaluated through the resolution of a linear system of equations which depends on the bifurcation being tracked:

[ A b d † 0 ] ( w 𝐺 ) = ( 0 1
) ( 10
)
where vectors b and d are bordering vector that ought to ensure that system (10) is nonsingular and superscript † denotes the Hermitian transpose. Matrix 𝐴 is the Jacobian matrix of the system with respect to the unknown Fourier coefficients when saddle-node (SN) bifurcations are tracked. For computational efficiency, the derivatives of the residual and arc-length equation are evaluated analytically. The derivatives of the bifurcation equation 𝐺 are computed semianalytically in parallel. The above described computational strategy relies on a Newton-Raphson solver which requires a initial guess sufficiently close to an actual solution point. For bifurcation tracking, a preliminary forced response curve is computed and bifurcation points are detected with a stability analysis based on Hill's method [START_REF] Moore | Floquet theory as a computational tool[END_REF][START_REF] Groll | The harmonic balance method with arc-length continuation in rotor/stator contact problems[END_REF][START_REF] Lazarus | A harmonic-based method for computing the stability of periodic solutions of dynamical systems[END_REF][START_REF] Guillot | A purely frequency based floquet-hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems[END_REF][START_REF] Peletan | A comparison of stability computational methods for periodic solution of nonlinear problems with application to rotordynamics[END_REF].

Static transmission error and mesh stiffness

During the computation of the bifurcations curves, the residual of the system and its derivatives have to be evaluated at values that are unknown beforehand. The computation of the nonlinear forces, given by Eq. ( 6), involves the STE and time-varying mesh stiffness (TVMS). Both heavily depend on the amount of tip relief 𝜇 and are periodic under stationary operating conditions. The strategy employed hereafter [START_REF] Mélot | Bifurcation tracking of geared systems with parameter-dependent internal excitation[END_REF] consists in expressing the STE, TVMS and their derivatives with respect to the amount of tip relief 𝜇 as truncated Fourier series with interpolated Fourier coefficients:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑞 𝑠 (𝜇, 𝑡) = 𝐻 𝑞 𝑠 ∑ 𝑘=0 𝑞
where 𝜕 𝜇 denotes the partial derivative with respect to the bifurcation tracking parameter, corresponding to the amount of tip relief in the present study. The in-house solver TERRA (Transmission ERRor Analysis) [START_REF] Rigaud | Modelling and Analysis of Static Transmission Error. Effect of Wheel Body Deformation and Interactions between Adjacent Loaded Teeth[END_REF][START_REF] Garambois | NVH robust optimization of gear macro and microgeometries using an efficient tooth contact model[END_REF] is used to compute the STE and TVMS for several amounts of tip relief. Fast Fourier transforms (FFT) are performed and coefficients 𝑞 𝑐,𝑘 𝑠 , 𝑞 𝑠,𝑘 𝑠 , 𝑘 𝑐,𝑘 𝑚 and 𝑘 𝑠,𝑘 𝑚 are interpolated. The derivatives of the Fourier coefficients are then calculated analytically, directly within the AFT algorithm. For more details, the interested reader can refer to [START_REF] Mélot | Bifurcation tracking of geared systems with parameter-dependent internal excitation[END_REF].

Uncertainty modelling

In present study, we consider an uncertain amount of tip relief 𝜇 on both gears. In order to evaluate its influence on the dynamic response of the system and resulting optimum TPM, we define two criteria. The nonlinearity can induce multiple co-existing solutions for a given set of parameters. As is well known, if the vibration amplitude is large enough, saddle-node bifurcations can appear and lead to potential amplitude-jumps under variations of the rotational speed or perturbations of the system. In this work, we define the optimum TPM as the median amount of tip relief 𝜇 𝑚𝑒𝑑 that best mitigates the severity of the amplitude jump phenomenon. We introduce two ad hoc criteria to be minimised, namely  𝑓 𝑟𝑒𝑞 and  𝑓 𝑟𝑒𝑞 :

 𝑓 𝑟𝑒𝑞 (𝜇) = ∫ 𝜇 𝑚𝑎𝑥 𝜇 𝑚𝑖𝑛 ΔΩ(𝑥)𝑓 𝑝 (𝑥)d𝑥 (12)
 𝑎𝑚𝑝𝑙 (𝜇) = ∫ 𝜇 𝑚𝑎𝑥 𝜇 𝑚𝑖𝑛 Δ𝐷𝑇𝐸(𝑥)𝑓 𝑝 (𝑥)d𝑥 (13)
where 𝑓 𝑝 (𝑥) corresponds to a probability density function, ΔΩ(𝑥) and Δ𝐷𝑇𝐸(𝑥) are respectively the frequency and amplitude interval between the upper and lower SN bifurcation on the primary resonance (see Fig. 4). 𝜇 𝑚𝑖𝑛 and 𝜇 𝑚𝑎𝑥 are the bounds of the interval of variability. The size of the latter is herein taken equal to 5 𝜇m with the lower bound 𝜇 𝑚𝑖𝑛 ∈ [0, 10] 𝜇m. Negative values are not considered here, as it would imply added material. Information regarding distribution laws of tooth profile modification is scarce, although a few sources report uniform, Gaussian or Weibull distributions. Due to the lack of information and in order to quantify the influence of the type of distribution on the results all three distributions as well as a beta distribution are considered in the following (see Fig. 5). The first two are symmetric while the Weibull and beta distributions are asymmetric in most cases, depending on the chosen parameters.

The uniform distribution depends on the lower and upper bounds of the uncertainty interval such that:

𝑓 𝑝 (𝑥) = 1 𝜇 𝑚𝑎𝑥 -𝜇 𝑚𝑖𝑛 , ∀𝑥 ∈ [𝜇 𝑚𝑖𝑛 , 𝜇 𝑚𝑎𝑥 ] (14)
The normal distribution is expressed with its mean value 𝜇 = (𝜇 𝑚𝑖𝑛 + 𝜇 𝑚𝑎𝑥 )∕2 and its standard deviation 𝜎 such that 3𝜎 = (𝜇 𝑚𝑖𝑛 -𝜇 𝑚𝑎𝑥 )∕2:

𝑓 𝑝 (𝑥) = 1 𝜎 √ 2𝜋 exp ( - 1 2
( 𝑥 -𝜇 𝜎 ) 2 ) (15)
The Weibull distribution has the following expression:

𝑓 𝑝 (𝑥) = 𝑘 𝜆 ( 𝑥 -𝜃 𝜆 ) 𝑘-1 exp ( - ( 𝑥 -𝜃 𝑘 ) 𝑘 ) , ∀𝑥 ≥ 𝜃 (16)
where 𝑘 and 𝜆 are the shape and scale parameters. Parameter 𝜃 is a location parameter allowing for an offset of the distribution on the x-axis to account for the varying lower bound of the interval of uncertainty. The shape and scale parameters chosen in this study are 𝑘 = 1.75 and 𝜆 = 2, respectively, which results in a "left-skewed" distribution. The beta distribution considered hereafter is expressed:

𝑓 𝑏𝑒𝑡𝑎 (𝑥) = Γ(𝛼 + 𝛽) Γ(𝛼)Γ(𝛽) 𝑥 𝛼-1 (1 -𝑥) 𝛽-1 , ∀𝑥 ∈ [0, 1] ( 17
)
where Γ is the gamma function and 𝛼 and 𝛽 are shape parameters whose value are 𝛼 = 2.5 and 𝛽 = 2 in this study, resulting in a "right-skewed" distribution. The beta distribution is defined for 𝑥 ∈ [0, 1]. In the following, the beta PDF is computed with 𝑥 ∈ [𝜇 𝑚𝑖𝑛 , 𝜇 𝑚𝑎𝑥 ] and normalized, i.e.:

𝑓 𝑝 (𝑥) = 𝑓 𝑏𝑒𝑡𝑎 ( 𝑥-𝜇 𝑚𝑖𝑛 𝜇 𝑚𝑎𝑥 -𝜇 𝑚𝑖𝑛 ) 𝜇 𝑚𝑎𝑥 -𝜇 𝑚𝑖𝑛 ( 18
)

Results and discussion

Optimization of the static transmission error with respect to the amount of tip relief

In order to highlight the benefits of the proposed methodology, we first discuss the choice of tip relief to minimize the excitation induced by the meshing process, as is standard practice in gear design. The static transmission error is computed for each value of the amount of tip relief in the discretized design interval  = [0 15] 𝜇m.

Figure 6 shows the evolution of the peak-to-peak STE with respect to the amount of tip relief for the input torque 𝑇 = 100 N⋅m. The STE 𝑃 𝑃 first decreases as the amount of tip relief is increased. The peak-to-peak value decreases from 2.1 𝜇m with no tip relief to 1.2 𝜇m when the amount of tip relief reaches 3 𝜇m, which will be henceforth referred to as static optimum. The STE 𝑃 𝑃 then increases with the amount of tip relief.

Optimization of the dynamic behaviour with respect to the uncertain amount of tip relief

The first step of the proposed methodology is to compute the bifurcation curves with respect to the uncertain parameter. The number of sampling points used in the AFT algorithm is 𝑁 𝑠 = 2 11 and the list of harmonics for the HBM is 𝐻 = (0, 50, 100, 150, 200, 250, 300, 350), corresponding to harmonics of the meshing frequency 𝐻 50 . The bifurcation tracking algorithm requires an initial guess close to the location of a bifurcation point in the design space. Two preliminary forced response analyses are carried out at each end of the interval of variability, i.e. for 𝜇 = 0 𝜇m and 𝜇 = 15 𝜇m. A stability analysis allows one to detect two SN bifurcations on each curve. The SN bifurcations located at the top of the primary resonances are chosen as initial guesses.

Figure 7 shows the SN bifurcation curves computed with the tracking algorithm with respect to the excitation (rotational) frequency and amount of tip relief. The root mean square value of the fluctuations of dynamic transmission error, 𝐷𝑇𝐸 𝑅𝑀𝑆 , is chosen as a measure of the response amplitude. Figures 7b and7c show the bifurcation curves in the frequency-tip relief and amplitude-tip relief planes, respectively. The tracking reveals two unstable zones delimited by the saddle-node bifurcations, i.e. the red lines, where multiple solution coexist and a small zone that doesn't exhibit amplitude-jump instability in between. The first unstable zone exists for values of tip relief 𝜇 ∈ [0 2.4] 𝜇m, which is quite close to the optimal tip relief computed with static analyses. The second unstable zone appears for 𝜇 ≥ 5.3 𝜇m. Interestingly, the shapes of the unstable zones in the frequency-tip relief plane (cf. Fig. 7b) are quite similar since the width ΔΩ seems to stabilize at approximately ΔΩ = 100 rad/s. Nevertheless, the first unstable zone shows a rapid decline in the amplitude-tip relief plane (cf. Fig. 7c). Indeed, the width of the unstable zone is Δ𝐷𝑇𝐸 = 4.8 𝜇m at 𝜇 = 0 𝜇m and decreases rapidly until the two SN bifurcations coalesce and disappear at 𝜇 = 2.4 𝜇m. The width of the second unstable zone grows significantly slower as the tip relief increases. For instance, the SN bifurcations emerge at 𝜇 = 5.3 𝜇m and the width only reaches Δ𝐷𝑇𝐸 = 3.1 𝜇m at 𝜇 = 10 𝜇m.

To carry out the uncertainty analysis, the interval  = [0, 10] 𝜇m, in which the lower bound of the uncertainty interval varies, is discretized with 100 points. The uncertainty interval 𝜇 𝑚𝑎𝑥 -𝜇 𝑚𝑖𝑛 itself is discretized with an odd number of points, 101 in this study, for an accurate representation of the probability density functions. The criteria defined by Eq. ( 12) and ( 13) are computed for each value of the lower bound of the uncertainty interval with almost no additional computational effort using the results of the bifurcation tracking analysis. For each value of the lower bound 𝜇 𝑚𝑖𝑛 , the median value of tip relief for each probability density function is stored.

Figure 8 shows the evolution of both criteria with respect to the median value of the amount of tip relief 𝜇 𝑚𝑒𝑑 . Solid, dashed and dotted lines are associated to the criteria calculated with the different probability density functions. The peak-to-peak values of the STE are superimposed for easier comparison. The optimum amount of tip relief, i.e. the median tip relief associated to the smallest value of the criteria, for each considered probability density function is denoted with a red circle marker.

One can see that the optimal tip relief is shifted towards larger values compared with the STE computations. Interestingly, the type of probability density function used for the calculation has a moderate influence of the optimal value of tip relief, with all optimums located approximately at 𝜇 = 4 𝜇m. Overall, both criteria have a similar evolution with the tip relief and yield somewhat similar results. One can also note that both criteria reach values close to zero at the optimal value of tip relief, indicating that amplitude-jump instabilities are well mitigated and almost non-existent. This is confirmed by carrying out Monte-Carlo simulations from the bifurcation tracking results. For each distribution of the amount of tip relief, a Monte-Carlo simulation is performed within the uncertainty interval associated to the optimum amount of tip relief computed with the proposed methodology. Note that when the optimums obtained with the two criteria differ, the one minimizing  𝑎𝑚𝑝𝑙 is used. Figure 9 shows the probability density functions of Δ𝐷𝑇𝐸 and ΔΩ computed with 𝑛 𝑀𝐶 = 100000 samples. For reference, MC simulations with the uncertainty interval associated to the static optimum, i.e. with a median equal to 𝜇 = 3 𝜇m, is also carried out (cf. Fig. 9a-9d and Fig. 9i-9l). One can see that, for all cases, the probability density functions of both Δ𝐷𝑇𝐸 and ΔΩ are utterly different from the probability density functions of the amount of tip relief. Besides, the majority of samples are characterized by an absence of amplitude jump as indicated by a median equal to zero. Furthermore, it appears that the dynamic optimum induces a shift of the mean value of both Δ𝐷𝑇𝐸 and ΔΩ towards zero as well as a smaller standard deviation. Thus, amplitude-jump instabilities are better mitigated.

Conclusion

A computational procedure based on bifurcation tracking was proposed to design tooth profile modifications in the presence of uncertainties. The proposed approach allows one to analyze the influence of uncertainty in tooth profile modifications on the nonlinear dynamic response of a gear pair and to define a optimal design. It was found that considering uncertainties leads to a different optimal value of the tooth profile modification compared to the one obtained with static computations. Four probability density functions were considered in the analysis. Results suggest that their influence on the optimal value is limited.

The methodology was applied to the amount of tip relief but can be implemented with any type of tooth profile modification. At the moment, only one uncertain parameter can be investigated at a time. Future work may look into extending the methodology to higher co-dimensions. It can also be extended to large-scale geared system models using model order reduction techniques [START_REF] Mélot | Vibro-impact dynamics of large-scale geared systems[END_REF] without the need to resort to parametric reduced-order models [START_REF] Krack | Reliability optimization of friction-damped systems using nonlinear modes[END_REF] since the gear coupling is treated in physical coordinates. Finally, the approach can be extended to other types of bifurcation (Neimark-Sacker, period-doubling) for better designs.

Figure 1 :

 1 Figure 1: Nonlinear gear model.

Figure 2 :

 2 Figure 2: Nonlinear force model for a given angular position of the gear pair.

Figure 3 :

 3 Figure 3: Schematic representation of a tip relief modification. 𝜇 is the amount of tip relief, 𝜙 𝑡𝑟 is starting radius of the profile modification (𝜙 𝑡𝑟 = 50.26 𝜇m).

Figure 4 :

 4 Figure 4: Example of forced response curve without tip relief modification (𝜇 = 0 𝜇m). Solid and dashed lines denote stable and unstable responses, respectively. Saddle-node bifurcation points are represented by red circle markers.

Figure 5 :

 5 Figure 5: Probability density functions considered for the uncertain tip relief parameter. (a) Uniform distribution (b) Normal distribution, (c) Weibull distribution with parameters 𝜆 = 1.75 and 𝑘 = 2, (d) beta distribution with parameters 𝛼 = 2.5 and 𝛽 = 2.

  -peak STE -[ m]

Figure 6 :

 6 Figure 6: Evolution of the peak-to-peak static transmission error with respect to the amount of tip relief. The red dot denotes the optimum solution, corresponding to the smallest peak-to-peak value of the static transmission error

  (a) 3D space (b) Frequency-tip relief plane (c) Amplitude-tip relief plane

Figure 7 :

 7 Figure 7: Saddle-node bifurcation curves. The blue lines represent the forced response curves; saddle-node bifurcation points are represented by red circle markers and red lines denote the bifurcation curves.

Figure 8 :

 8 Figure 8: Evolution of the frequency criterion (a) and amplitude criterion (b) with respect to the amount of tip relief. Red circle markers denote the optimal value of tip relief. The evolution of the peak-to-peak STE is represented as a black line with cross markers.

Figure 9 :

 9 Figure 9: Probability density functions of the amplitude (a, b, c, d, e, f, g, h) and frequency intervals (i, j, k, l, m, n, o, p) between upper and lower SN bifurcations. PDF computed with MC simulations in the interval whose median equals the optimum obtained from static computations(a, b, c, d, i, j, k, l) and dynamic computations (e, f, g, h, m, n, o, p). (a, e, i, m) Uniform distribution, (b, f, j, n) Normal distribution, (c, g k, o) Weibull distribution with parameters 𝜆 = 1.75 and 𝑘 = 2, (d, h, l, p) beta distribution with parameters 𝛼 = 2.5 and 𝛽 = 2. The red solid lines denote the mean values while the red dashed lines represent the medians.

Table 1

 1 Characteristics of the gear pair

	Name		Gear 1	Gear 2	Unit
	Module	𝑚	2		mm
	Number of teeth	𝑍	50	50	-
	Pressure angle	𝛼	20		deg
	Base radius	𝑟 𝑏	46.984	46.984	mm
	Profile shift coefficient	𝑥	0	0	-
	Addendum coefficient	ℎ 𝑎	1	1	-
	Dedendum coefficient	ℎ 𝑑	1.25	1.25	-
	Face width	𝑏 𝑓	20		mm
	Inertia	𝐼 1 1.52e-3 1.52e-3 kg.m 2
	define, for each angular position 𝜃 1 , a mesh stiffness 𝑘 𝑚 (𝜃 1 )
	coupling the two gears in contact:

Acknowledgements

This work was performed within the LabCom LADAGE (LAboratoire de Dynamique des engrenAGEs), created by the LTDS and the Vibratec Company and operated by the French National Research Agency (ANR-14-LAB6-0003). It was also performed within the framework of the LABEX CeLyA (ANR-10-LABX-0060) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-16-IDEX-0005) operated by the French National Research Agency (ANR).

Conflict of interest

The authors declare that they have no conflict of interest.

CRediT authorship contribution statement