Anders Wiik
email: anders.wiik@uia.no

Pauline Vos
email: pauline.vos@uia.no

"I want a high-educated job that pays well and is fun": Secondary students' relevance beliefs for taking advanced mathematics

Keywords: Beliefs, exchange-value (of mathematics), cultural-historical activity theory, emergent activity systems, relevance (of mathematics)

Many students experience the content of the mathematics classroom irrelevant to their lives. At the same time, achievement in mathematics is widely regarded as an important part of general education. In this paper, we report on a study regarding students' relevance beliefs. For this, we network Engeström's cultural-historical activity theory with Skovsmose's theory on foregrounds, and introduce the terms 'realized' and 'emergent' activity systems. Students provided a wide variety of beliefs, in which the belief of mathematics as door opener to higher education and key to options in life was leading. The belief in the exchange-value of mathematics overshadowed beliefs on the use-value of school mathematics (for other school subjects, for undefined jobs or for mental training). We recommend teachers and textbook authors to include more information on the relevance of mathematics in future studies and jobs.

Introduction

Many teachers of mathematics encounter the question "why do we need to learn this?". While sitting in the mathematics classroom, listening to the teacher, doing tasks and tests, many students find little relevance in the mathematical activities they are made to engage in [START_REF] Bishop | Mathematical enculturation: A cultural perspective on mathematics education[END_REF][START_REF] Onion | What use is maths to me? A report on the outcomes from student focus groups[END_REF]. Rather than learning about the usefulness of mathematics or using mathematics to solve authentic problems from out-of-school contexts, the students learn that mathematics is about mathematical objects (numbers, triangles, x, etc.), and that they must find an appropriate procedure to reach a correct answer (and only one) within a short time frame [START_REF] Dowling | The sociology of mathematics education: Mathematical myths/pedagogic texts[END_REF].

The finding that students ask this 'why?' question has been consistent for several decades. It opens a range of questions: why is it that many students find school mathematics irrelevant, how can mathematics teachers convey relevance, can mathematical modelling improve students' experiences of relevance, do experiences of relevance influence performances, and so forth. We cannot cover all these issues. In the present study we focus on students and their reasons for engaging in advanced mathematics; if they chose that path, then they must somehow see it as relevant.

Students' reasons for engaging in advanced mathematics are part of their belief system. Students have beliefs about themselves, about the school, about their peers, about life, and so forth. According to [START_REF] Skott | The promises, problems, and prospects of research on teachers' beliefs[END_REF], beliefs are mental constructs that are subjectively true for the person in question, beliefs are considered relatively stable, and beliefs affect the person's interpretations of experiences.

Following [START_REF] Hernandez-Martinez | A study on students' experiences of the relevance of mathematical modelling activities[END_REF], relevance is a connection between (1) a person's subjective perception, (2) an activity by the same or another person, (3) an objective (an aim), and (4) a value. In the present study, we consider relevance beliefs as a category of students' beliefs connected to a deliberate activity, in this case, choosing a certain course. Students' relevance beliefs can be mediated by the mathematical content, but also by interactions with others. A different category of relevance beliefs is part of teachers' beliefs and pertains to a teacher's perception of the relevance of mathematics to their students.

In the Norwegian school system, students in upper secondary can choose among several mathematics courses with differing aims, scope and difficulty. Students make a selection while negotiating contradictions, such as those between the apparent irrelevance of school mathematics, and the image of mathematics that is drawn out in the media as a subject of paramount importance for the future economic development of society [START_REF] Lange | Policy production through the media: The case of more mathematics in early childhood education[END_REF][START_REF] Valero | Mathematics for all, economic growth, and the making of the citizen-worker[END_REF]. This duality puts the students who engage in the hardest mathematics in an interesting position: how do they reason about selecting and pursuing the most advanced mathematics course available to their age group? Once they are in this course, they encounter mathematical topics such as derivatives, integrals, and trigonometry. In the school system, they encounter little information on how these mathematical topics are useful out-of-school. The few situated tasks are quite inauthentic. Also, teachers of such courses, generally, have a background in university mathematics with little emphasis on the mathematics' use-value outside of the school context. Our research on students' relevance beliefs regarding school mathematics was operationalized by the research question 'what reasons do students provide for engaging in a challenging mathematics course?'

Literature review

The discussion on the reasons for teaching mathematics has developed over the past two decades. According to [START_REF] Heymann | Why teach mathematics? : A focus on general education[END_REF], mathematics is taught as preparation for later life, for promoting cultural competence, developing an understanding of the world, developing critical thinking, developing a willingness to assume responsibility, as practice in communication and cooperation, and to enhance student's self-esteem. [START_REF] Ernest | Platform: Why teach mathematics?[END_REF] provides a similar list, although he differentiates between reasons to teach practical versus advanced mathematics. In either case, it remains questionable whether students get to 'see' these reasons. Also, it has been argued that the above reasons remain at the pupil's level without considering the social role of school mathematics. For example, [START_REF] Pais | An ideology critique of the use-value of mathematics[END_REF] suggests that school mathematics is important not only for its intrinsic purposes (use-value, aesthetics, etc.) but also because it has exchange value, for example as a testing and grading device that gives some students access to prestigious positions in society, and other students not. In the public discourse, school mathematics is typically associated with socioeconomic development, in particular with the development of advanced technology [START_REF] Lange | Policy production through the media: The case of more mathematics in early childhood education[END_REF] and economic parameters on the individual and national level [START_REF] Valero | Mathematics for all, economic growth, and the making of the citizen-worker[END_REF]. In this cultural-political tug-of-war, it seems that school mathematics is struggling to communicate the usevalue of mathematics.

A crucial challenge is the dialectics of two processes, the mathematization of knowledge and the demathematization of society [START_REF] Jablonka | Mathematisation-demathematisation[END_REF]. The first has to do with mathematics being increasingly important in social, economic and political issues, for example in dealing with complex data through modelling and visualizations. This mathematization process is accompanied by a demathematization process where mathematics is hidden and trivialized, for example when software encapsulates mathematical practices and datavisualizations can be generated in standardized formats by a few clicks.

Various teaching approaches to mediate relevance have been proposed. For example, in a study with engineering students, Hernandez-Martinez and Vos (2018) used tailored mathematical modelling activities and an invited talk by a professional to demonstrate how mathematics is used and valued amongst engineers. They found that several students experienced these activities as relevant to their future career, but some s found the extra activities irrelevant, because their primary aim was to obtain a passing grade for the course.

Theoretical background: Cultural-historical activity theory

To study relevance beliefs, we consider students as having different backgrounds (where they come from) and foregrounds (aspirations, opportunities). The term foreground was developed by [START_REF] Skovsmose | Foregrounds and politics of learning obstacles[END_REF] as analytic construct for interpreting learning processes, and is framed within the students' current political, cultural and social context. Additionally, we wanted to capture students' engagement in activities (doing mathematical tasks), which take place within a community (a classroom, peers, etc.), in which there are norms and conventions (e.g. the answers must be correct), tools (incl. language and symbol systems). To engage into an activity, a person is driven by a motive. In order to capture how the student (subject) is connected to the community and the object (aim of the activity), we adopted cultural-historical activity theory (CHAT) [START_REF] Engeström | Learning by expanding: An activity-theoretical approach to developmental research[END_REF], in which an activity system is a basic unit of analysis, see Figure 1. To analyse students' relevance beliefs, in particular their reasons for engaging with advanced mathematics, we assume that these can be affected by (1) the students' backgrounds (e.g. having a parent who uses mathematics in her workplace), (2) the environment in which the learning of mathematics takes place (the activity system of the classroom), and (3) the students' foreground [START_REF] Skovsmose | Foregrounds and politics of learning obstacles[END_REF]. This foreground can entail complex imagined activities, of which the student may not know much about; a student imagining herself as a future engineer may not know anything about the daily activities of an engineer, but nevertheless can aspire to become one. Thus, the activities envisioned in a student's foreground are not realized activity systems but emergent activity systems.

Method

To find participants, we contacted a teacher from an urban public upper secondary school. The teacher and the school were well-known to the first author, as he had previously worked there. The teacher agreed and informed the students in his class about the project in advance. Thus, students had time to consider whether or not they wanted to participate. When the interviewer visited the class, eight students (3 f, 5 m) agreed to participate. The interviews were conducted three days later.

The students were in grade 12, which in Norway is the second year in upper secondary school, and the year before the final exams. All had chosen the most challenging mathematics course available, referred to as R1, the name used in the local context. The students were interviewed in pairs for mutual support and comfort. All student names were anonymized, and the pairs follow the alphabet: A(ndré) and B(jørn); C(hristian) and D(aniel) and so on. The student pairs were interviewed for approximately 20 minutes on their back-and foregrounds. The interviews were semi-structured [START_REF] Wellington | Educational research: Contemporary issues and practical approaches[END_REF] and the interview guide followed [START_REF] Kvale | Det kvalitative forskningsintervju[END_REF].

All interviews were audio-taped, transcribed, and analyzed using [START_REF] Skovsmose | Foregrounds and politics of learning obstacles[END_REF] constructs of back-and foreground, and the CHAT frame by identifying activity systems, particularly communities and associated objects and whether the activity systems were emergent or not. The statements were then interpreted in terms of relevance beliefs based on categories from the literature (exchange-value, use-value) and themes emerging from the data [START_REF] Wellington | Educational research: Contemporary issues and practical approaches[END_REF].

Results

In the following, we present excerpts from the interviews. Due to space limitations, some answers very similar to others were left out. First, we asked the students why they chose the R1 course: Bjørn:

I like mathematics and get good grades.

Daniel: It's pressure from parents of course, and also some from my friends. Many of them are performing very well. I want to be on the same level, you know. Also, I suppose, career aspirations and further studies are… about having a relatively high-educated job which pays well... and is fun. I think that's very important.

Fredrik:

The reason I chose R1, it is kind of a continuation. I always performed well in maths … and it went well in grade 11, so R1 was the natural choice.

Gjertrud: I think math is fun. It's great fun when you get the same answer as the answer list. There aren't tons of pages about "it may be this", or "it may be that". It's great! We see that students reacted very differently to this question. Two students, Gjertrud and Bjørn, answered to have chosen R1 because they liked mathematics, whereby Gjertrud emphasizes the clarity, and Bjørn likes getting good grades. Their answers are within the activity system of the present classroom, unlike the other students, who mention their background (e.g. parents' pressure, having been good in maths in the past), or their foregrounds (e.g. keeping options open, getting a well-paid job). The second question asked the students explicitly about wishes and ambitions for future studies and career (foregrounds). They answered as follows:

André: Likely that it'll be something related to science or technology. Or I'd study sports! Bjørn: I'm very unsure about further education, but probably I will do some university studies, and that will most likely be within science or technology. But I don't know much more than that, really, I'm a little unsure. It could be sports but it could also be social anthropology or psychology, those are things that I'm also interested in.

Christian: I always had a dream of going into . Most of the students considered their foreground. They mentioned that mathematics can be useful in future activities, for example in future studies. One student, Daniel, expressed concern that some mathematical topics may be useless. Only one student, Bjørn, considered the present activity system of the classroom when mentioning that the subject of mathematics has the purpose to train and test students' mental abilities. Emilie considered her present and emerging family activity system when she mentioned helping her kids and little brother. However, we must acknowledge that by first asking the students about their foregrounds, we primed the students to focus on their foregrounds thus biasing them towards focusing on the exchange value of mathematics.

Conclusion, Discussion, Recommendations

The present study on relevance beliefs was guided by the research question: what reasons do students provide for engaging in a challenging mathematics course, R1?. The interviews revealed that for some students, their backgrounds (parents, earlier success) and their present achievement and enjoyment in mathematics play a role. However, most students mentioned the belief that the R1 course is relevant for their future. Here, we saw different patterns emerging. For example, Daniel ("I want a relatively high-educated job which pays well... and is fun") did not have a concrete plan for his future and strived to "keep his doors open", he aspired a high-status job, possibly within engineering. Similarly, Christian ("I always had a dream of going into engineering and be a marine engineer") looked beyond the course. The object of their engagement in the mathematics classroom had an exchange-value for getting into some form of higher education that led to a desired job. Their image of activities in higher education and workplaces were emerging activity systems, rather than realized activity systems. So, two subsequent emerging activity systems are in their foreground, see Figure 2 (left). For other students, such as Emilie ("I don't know what I want to become, but I'm taking IB and then perhaps NTNU") and Gjertrud ("It will be something within natural sciences, everything is exciting, I can't really decide"), the R1 course was relevant as having exchange-value to enter some form of higher education and keep options open. They looked at a nearer future in higher education without concrete ambitions regarding professions. Thus, their foreground was one or more emergent activity systems at early stages, see Figure 2 (right).

Two students believed mathematics to be useful, for other subjects (Christian mentions physics) or in an undefined use (Gjertrud: "we're learning something we can use"). Not one mentioned usefulness for their daily life, nor did they say anything about how mathematics can be useful, for example in modelling, data analysis or data visualization. Others believed that some mathematics could be useless (Daniel: "logarithms, I'm not sure of whether it is useful"). They also mentioned the mental strengthening function of mathematics (Bjørn: "exercising the mind"). The most frequently given reason for choosing the R1 course was that they believed it keeps their options open, apparently knowing the hurdle function of mathematics. This exchange-value of mathematics overshadowed its use-value, as already indicated by earlier research on Scandinavian secondary students [START_REF] Nosrati | Ten years of mathematics education: Preparing for the supermarket[END_REF][START_REF] Pedersen | I need advanced mathematics to pursue the career of my choice": Norwegian students' motivations for enrolling in mathematics and plans for post-secondary studies[END_REF]. This suggests that a demathematization of school mathematics has happened, where the usefulness of mathematics has become obscured, and beliefs about other social functions of mathematics, for example its exchange-value, have taken over. The networking of Engeström's ( 2015) and [START_REF] Skovsmose | Foregrounds and politics of learning obstacles[END_REF] theories proved fruitful in our analysis for connecting relevance beliefs to an activity (choosing a particular mathematics course).

As a result of this, we found that for research on relevance beliefs regarding school mathematics, it is necessary to take into account students' (1) background, (2) current situation and (3) foreground.

Among the items on [START_REF] Heymann | Why teach mathematics? : A focus on general education[END_REF] list of reasons for teaching mathematics, the students on our study are clearly aware of its role in preparing for later life, although exclusively for further education and future workplace activities. The frequent reference to 'keeping all doors open' shows clearly that the students were aware of school mathematics' exchange value: a passing grade in this course is critical to many study programs that are otherwise not accessible. Furthermore, Daniel's assertion that the advanced mathematics course might lead to a "high-educated, well-paid and fun job" indicated an awareness of Pais' (2013) suggestion that school mathematics has a role in distributing prestigious positions in society. Based upon our research, we recommend, first, that beliefs may be divided into different categories, and 'relevance beliefs' regarding the usefulness of school mathematics is a separate category within students' belief system. Second, teachers, teacher educators and text book authors ought to create resources for students to inform them about the usevalue of mathematics, not only within professions typically associated with mathematics (e.g. engineering and economy) but also its use-value in medical and social disciplines for purposes of modelling, analyzing, communicating and visualizing data. Clearly, many of the reasons for teaching mathematics mentioned in the literature have not reached the students.

Figure 1 :

 1 Figure 1: Engeström's (2015) model for an activity system

  to keep the possibilities open because I don't know what to do next.

Figure 2 :

 2 Figure 2: Students realized activity systems in the mathematics classroom, with emergent activity systems as objects; the case of Daniel (left) and the case of Gjertrud (right)

  .. engineering direction for example ... be a marine engineer. That's what I have always wanted. But ... I don't know yet. I like to draw things, be creative, look at buildings, how they are built. And boats, how they break waves, how they manage to stay on water, not go down, like that.The question on students' foreground showed that all students mentioned a future, but the range and focus differed. With the exception of Christian who had a clear goal, all were insecure and wished to keep options open. Many of them indicated future studies, most of which were within science and technology, and no one mentioned economy or business administration. Above, we see only Christian and Emilie mentioning a future profession beyond their studies, although for Emilie it was vaguer than for Christian. Towards the end of the interview, we asked why they thought they had to learn mathematics: lot of mathematics we're learning is a little redundant. Of course, I see that many topics can be useful. Like what we're learning about now with differentiation, I think finding rate of change everywhere, can be relevant. But there are also the unnecessary things. Logarithms are actually one of the things that I'm not so sure of whether it is useful or not. that what we're learning is, like, further, that we're learning something we can use for … Just like when you learned the multiplication table. Why do we need to learn multiplication? Now I use it all the time.

	Emilie: Fredrik: Gjertrud: Hilde: André: Bjørn: Christian: Daniel: I think a Emilie: I don't know what I want to become, but I'm taking IB [International Baccalaureate, a prestigious examination program] next year and then perhaps NTNU [Norwegian University of Science and Technology]. I just try to keep all the doors open. I think it will be something within science, like biology. But I also want to study psychology, I think everything is very exciting and I can't decide! Nothing (…) I don't know. I think [maths] is mostly about … just having the foundation for further study. I think maths is really only about testing understanding and testing how the mind can assemble patterns and stuff. (…) it's about exercising the mind, IQ-testing … … ehh I think especially the maths we are learning now, for example, we will need it in physics. It is ideal for the ... calculations there ... to figure things out there. I can help my kids, my little brother [with homework]. Gjertrud: I really feel


Behiye Ubuz
email: ubuz@metu.edu.tr

Koeno Gravemeijer

Michelle Stephan
email: michelle.stephan@uncc.edu

Patrick Capraro
email: capraro@mathematik.uni-kl.de

Introduction to TWG26: Mathematics in the context of STEM education

Keywords: Numbers and Operations, Geometry (i.e., angles, parallel lines), and Data Organisation and Processing, secondorder differential equations, matrices, linear equations)

.

What level of integration should be strived for, however, is not undisputed. Especially, if and how mathematics fits such an integrated approach is debatable. A specific point of attention concerns the fact that mathematics education asks for careful vertical planning, which might be compromised in an integrated approach. Mathematics encompasses more fields of application than science, technology, and engineering alone. So, real-life applications involving modeling, engineering design and the integration of various disciplines has to be taken into account in mathematics as it prepares students for the future. This working group particularly aimed at addressing mathematics in the context of STEM education. Mathematics encompasses more fields of application than science, technology, and engineering alone. So, the integration of a broader variety of disciplines has to be taken into account in mathematics to prepare students for the future. Papers and posters submitted to the group focused on tasks, courses, curriculum, students, and in-service teachers mainly using qualitative methodology and interpretative approach. The issues and ideas emerged are discussed in reference to seven central themes. The mathematical contents concerned various topics (e.g.,

Thematic areas

Here we present the issues and ideas that emerged in reference to the seven central themes.

Mathematical preparedness and value

Within this theme two papers were presented. First paper by Lave and Walshe concerned the preparedness of high-school students for tackling the mathematical aspects of STEM courses in tertiary education. The second paper by Nelleke den Braber and et al centered on the value of mathematics for an inter-disciplinary STEM course.

The first paper within this theme concerned the high non-completion/dropout rates in STEM courses at tertiary level in Ireland. It was noticed that even students with good marks might struggle with the mathematical aspects of third-level science and engineering courses. Lave and Walshe therefore searched for other factors that might affect the preparedness of the students. They investigated the perceptions of three stakeholders, teachers, students and lecturers with questionnaires. They further aimed to clarify what teachers and students themselves currently understand by interdisciplinary STEM education.

Factors that were taken into account are, -students have to know how to use math knowledge in other subjects;

-mathematics has to be taught in an integrated way;

-high-school teachers should teach with students' future third-level degrees in mind; -teachers are unaware of the benefits of integration.

In the reaction, it was argued that an alternative explanation might be in how mathematics is taught. In this respect a reference was made to Freudenthal, who argued that mathematics should be taught as to be useful. He argued that, "In an objective sense the most abstract mathematics is without doubt also the most flexible. But not subjectively, since it is wasted on individuals who are not able to avail themselves of this flexibility" (Freudenthal, 1968, p.5). Another remark was that next to benefits of integration of STEM subjects, there are also costs. These concern sustained facilities at school level (time, funding, schedule, room), teacher commitment, professional development and teacher support [START_REF] Gresnigt | Promoting science and technology in primary education: A review of integrated curricula[END_REF].

The second paper elaborated on the value of mathematics in an interdisciplinary STEM course. The key concept in this paper was that of disciplinary perspectives. It was argued the problems we face in today's world in the context of STEM call for perspectives and knowledge from many different areas. Against this background the researchers tried to get a handle on the value of mathematics in the context of a successful interdisciplinary STEM course in the Netherlands. With open questions Nelleke den Braber and et al investigated how students and teachers think about the value of mathematics. They introduced a model for the value of mathematics for interdisciplinary STEM courses. This encompassed, an overall sense of what mathematics is, ways of working and thinking, modeling, reasoning and problem solving. This model was used to analyze answers of students and teachers on an open questionnaire. It showed that both teachers and students had little awareness of the value of mathematics as a discipline. If the students mentioned mathematics, it was mainly that it was used. It seemed that the role of mathematics was not really noticed by them.

In the reaction in was noticed that the value of mathematics appeared to be equated with its usefulness. Which appeared not to be the intent. It was further argued that in practice the use of mathematics by workers was limited. As the actual mathematical work would be delegated to experts or machines. Therefore, being able to communicate about the mathematics might be more important. Following [START_REF] Kaput | Rethinking calculus: Learning and thinking[END_REF] it was further argued that one should understand the key underlying ideas and mechanisms. Freudenthal was quoted again to make a distinction between mathematical activity as "organizing subject matter in reality and organizing mathematical subject matter". The latter seemed to be missing in the model of the value of mathematics. It was further noticed that Freudenthal would not be surprised about the lack of awareness of the role of mathematics. In 1977 he predicted that, "In 2000 mathematics will have been gone as a separate subject in education. Mathematics is there to be experienced, and lived, just like reading, writing, tinkering, drawing, singing, breathing, in integrated education."

Product and STEM

In this theme two papers were presented. The first one by Wohak and Frank revolved around the value of inverse problems in making students aware of the role of mathematics in applications. The second one by Bock, Bracke and Capraro aimed at such awareness but tried to realize this with product-oriented tasks.

The first paper made a case for the development of interactive, problem-oriented material for high school students, in order to make students see the potential of mathematics for solving real life problems. The example in the paper concerned so-called inverse problems. The concrete case being the problem of computer tomography. Computer tomography is used to establish the internal structure of an object-such as a human body-on the basis of the entry and exit intensity of the Xray radiation. The objects are irradiated with parallel beams at different fixed angle settings, and the challenge is to infer the structure of the object from the patterns of the imprint the X-ray make after passing through the object. This is an inverse problem as it is the inverse of predicting what the imprint would be when the structure of the object is known. Where the latter is rather straightforward, the former is complex and not always possible as small variations may cause big errors. Wohak and Frank show that the mathematics for solving a simplified version of this type of problem is doable for high school students. They argue that by working on this problem, students will come to see the potential of mathematics to solve real-life problems.

One of the points of critique was on the strong focus on the mathematical solution of the problem. It was argued that today mathematics is done by computers and that the remaining mathematical work resides in the translation of the practical problem into a mathematical problem. It was further noted that there was little attention for the capabilities that are needed for mathematical modeling. In this respect the detailed list of capabilities that the PISA Mathematics framework offers (OECD, 2017) was mentioned.

The second paper focused on product-oriented modeling. The starting point being the task of making a product brings with it unique characteristics-which are lacking in school problems. For, the success criteria are dictated by whether the solution is actually working and feasible, often decisions have to be made, and restrictions apply. Two product-oriented modeling tasks were discussed. Both were aiming at showing the usefulness of Fourier analysis. The first concerned the task of making a musical fountain. Two sub-groups decided not to solve the original task but focused on loudness & rhythm instead. So, there was no need for Fourier analysis. One sub-group did find mathematical techniques to smooth the highly irregular amplitude-time-function. The other sub-group worked on an algorithm that could find an underlying rhythm in a song. The second task involved building a light organ. Here also two sub-groups emerged. One worked with electronic frequency filters, and thus did not need Fourier analysis. The other group wanted to use mathematics and was supported in using Fourier analysis (in a piece-by-piece manner).

One of the questions that was brought up by the discussing was whether the students of the last group really understood the Fourier analysis in terms of the underlying concepts and mechanisms. Further similar remarks could be made about the intended use of Fourier analysis as were made on the reverse-problems paper. In this case, however, part of the students was involved in translating practical problems into mathematical problems. As was the case with the loudness and rhythmproblems. Doubts were expressed about the feasibility of upscaling such projects.

Engineering design and mathematical modelling processes

Three papers were presented in this thematic area. The first paper by Abou-Hayt, Dahl and Rump concerned with the exemplification of the integration of the engineering design process (EDP) and the mathematical modelling process in two university students' projects. Abou-Hayt, Dahl and Rump used the characteristics of engineering design process [START_REF] Tayal | Engineering design process[END_REF] and mathematical modelling process [START_REF] Blomhøj | Developing mathematical modelling competence: Conceptual clarification and educational planning[END_REF] to analyze the projects. Their conclusion was that the both processes were visible in the projects. This paper led to raising some further questions: What is the sign for showing student learn best when they are actively participate and apply theory?; What is students learning in this context?; How is the interaction between the lecturer and students?; How can we compare traditional and this form of teaching outcome?; and What is the meaning of mathematics in abstract and real world?

The second paper by Costa and Domingos elaborated on the development and the implementation of mathematical interdisciplinary task related to STEM integration, in the context of a collaborative Continuing Professional Development (PD) Program targeted at primary school teachers. Costa and Domingos concluded that a PD program can only be successful if teachers can implement the task developed with their students. The following questions led the discussion regarding to this paper: What is the meaning of STEM integration in the context of a collaborative Continuing Professional Development Program targeted to primary school teachers?; How can teacher educators help teachers to develop mathematical interdisciplinary tasks related to STEM integration? Are they different for prospective teachers´ education and in-service teachers?; What are the expectations from the teachers while developing mathematical interdisciplinary tasks related to STEM integration?; What are the expectations from the teachers while implementing mathematical interdisciplinary tasks related to STEM integration?; How could different types of knowledge and skills regarding the integration of STEM be promoted in pre-and in-service teacher education?

In the third paper, Ubuz used the characteristics of EDP outlined by [START_REF] Berland | High school student perceptions of the utility of the engineering design process: Creating opportunities to engage in engineering practices and apply math and science[END_REF] to analyze the curriculum learning outcome of a Technology and Design course offered to 7th grade students in Turkey. By identifying the most common verbs and objects contained within the learning outcomes of the curricular documents guiding the 7th grade course, Ubuz found that 23 of the learning outcomes contained EDP knowledge and 29 of them concerned EDP skills. The conclusion is that the EDP knowledge and skills recommended by [START_REF] Berland | High school student perceptions of the utility of the engineering design process: Creating opportunities to engage in engineering practices and apply math and science[END_REF] are largely present in the 7th grade curriculum. Ubuz suggests that these findings can help teachers integrate similar EDP knowledge and skills in science and mathematics curriculum in middle school.

Design research

A paper by Pugalenthi, Stephan, and Pugalee was presented in this thematic area. This paper sought to capture the students' conception of angles and parallel lines in order to design engineering context based instructional sequences for middle grades (7th grade) mathematics classroom. This research was comprised of a pre-interview to assess the existing conceptions of angles and parallel lines, design and implementation of engineering-based instructional sequences and a post-interview to assess the changes in conceptions of angles and parallel lines. This research paper, however, focused mainly on the analysis of the pre-interview questions dealing with angles and parallel lines along with preliminary discussion of the design of the instructional sequence regarding designing a residential community. The students' conceptions of angles and parallel lines were traced under three themes: Prototypes or reference images, tracing of the lines, and decoupling versus decomposing. This paper leads to raising some further questions: What is mathematical instructional task with an engineering problem?; How the difficulties identified could be prevented through mathematical instructional tasks with an engineering problem?; How mathematical instructional tasks with an engineering problem let to dive deeply into a topic and use mathematics to create a solution?; How the difficulties identified do lead us to develop mathematical instructional tasks with an engineering problem such as designing a residential community?; How could the mathematical difficulties identified be prevented through designing a residential community?; What is effective STEM integration?; How could we provide training or prior skills regarding engineering content and principles to the teachers?

Statistics in STEM

Under this theme one paper was presented. The paper written by Oliveira, Henriques, and Batista dealt with the preservice teachers (PT) perspectives about the role of statistics in a learning scenario with one 8 th grade class, fostering the integration of physics and statistics. It was claimed that successful STEM integration depends on teachers' perspectives and their competence to choose learning materials which are suitable in this context. Previous studies showed that teachers had a negative perception of STEM education based on the feeling that they were unprepared to teach within an interdisciplinary curriculum.

The data (group lesson plans and PT's reflections regarding planning and enacting learning scenario) were analysed with a focus on the dimensions of the model of authentic integration [START_REF] Treacy | Authentic integration: A model for integrating mathematics and science in the classroom[END_REF]: Application to real-world scenarios, high order thinking processes, and knowledge development, synthesis and application. According to the paper the PTs concluded that the mathematical content (in this case statistics) was not only incidental to the STEM context, but had a great centrality in the learning scenario. Mathematics did not only appear in the sense of calculation methods, since high order thinking processes took place that were supported by statistics. Furthermore, the role of technology, in this case with statistical software, was pointed out.

The collaboration with colleagues of other subjects was perceived as beneficial for the PTs.

Questions raised during the discussion were: If teachers gain experience with interdisciplinary classroom activities, do they perform better in a future approach, even if the topic is different? Is it a lack of knowledge from other disciplines or rather a lack of experience that makes teachers feel underprepared?

Approaches to introduce STEM

In this theme two papers were presented. Both had in common that they used game-like structures to enhance the students' motivation in their particular learning environment. One paper written by Viamonte and Figueiredo was about gamification in an e-learning tool for university students. In that paper dropout rates as well as the personal opinion of the students who used the e-learning tool were examined. A significantly lower dropout rate compared to earlier years could be measured, which supports the claim that gamification can be used to stimulate motivation. Also, the students' responses concerning the competitive elements seemed to confirm the expected influence, although not all reactions of the students had been positive. The issues raised to be discussed were:

 Although the statistical data concerning dropout rates show a significant impact, it could be possible that there are effects involved that can't be measured easily (e.g. a new teaching concept increases motivation, but the effect decreases over a longer period of time). One could think about examining further iterations, maybe with slight variations.

 Similarly, could the effect be decreasing if gamification was used in a broader extent (i.e. not only one lecture, but all lectures the students attend during the same semester)?

 The data are probably not representative for the whole university population. The participants were engineering students only, and furthermore most of them (93%) were male. Is it possible to transfer the conclusions to a broader range of students?

In the other paper written by Abboud, Hoppenot, and Rollinde a classroom situation was analysed, where students experienced the orbits of astronomical objects from our solar system in a roleplaying activity. The aim was to enhance activity theory [START_REF] Abboud | Use of activity theory to make sense of mathematics teaching: A dialogue between perspectives[END_REF][START_REF] Vandebrouck | Mathematics classrooms students' activities and teachers' practices[END_REF]) by the notion of bodily perception. Several implementations of the Human Orrery project were analysed qualitatively with questionnaires and interviews. It was suggested that participating students experience STEM-related concepts from an emotional perspective and describe them as "more real". With regard to this paper the issues raised to be discussed were:

 How can we measure the impact of bodily perception on the motivation at a quantitative level?

 How is it possible, after a certain STEM-related concept was experienced in this framework, to have a smooth transition to a more theoretical approach to the topic? This might be interesting when working with older students.

STEM teacher preparation

In this theme, one collaborative team consisting of mathematics and physics educators from Vietnam and Germany aimed to create a teacher preparation program that invited mathematics and physics teachers to participate in classes that utilize both theoretical and practical aspects of teaching these two disciplines. The theoretical part would provide opportunities for physics and mathematics teachers to compare and contrast epistemological foundations of each discipline, thereby illuminating the differences and similarities in the mathematics and physics. In particular, teachers would explore the meaning and practices of modelling from mathematics and physics perspectives. These theoretical foundations would then support teachers to create lessons that potentially integrate physics and mathematics that they would then use in their classrooms. Although the project is still in development, Krause and et al intend to videotape the implementation of these lessons by project teachers and determine to what extent physics teachers focused on mathematics in their class and how much mathematics teachers integrated physics into their class. The critical issues raised by this paper harken back to earlier papers in which questions about the extent to which modelling from different disciplines can be mutually supportive. It also calls to the forefront what it means to genuinely integrate science and mathematics. Finally, the crucial problem that Krause and et al attempt to solve is how to design appropriate instruction for teachers who are knowledgeable about physics and mathematics separately so that they are knowledgeable about the means of integrating the two.

Critical issues

We would like to close our introduction by including some critical issues that the participants expressed during the sessions.

 Experts within and across the STEM disciplines model for different purposes and, thus, in different ways; given this diversity of modeling, how do we characterize modeling in STEM?

 Teachers are not experts in all STEM disciplines. In the real world, there is also no such thing as a STEM expert. Rather, discipline-specific experts are invited into a project when there is a need for their expertise. This implies that teachers should work collaboratively to design and implement STEM units. Does this change the goal of STEM education from STEM integration to STEM collaboration?

 How do we design curriculum so that mathematics is not just a tool for doing the other three content areas? And what is really meant by technology use in curriculum?

 What kinds of research experiments are needed to understand STEM education? While creating nice STEM lessons has potential, there is a need to go beyond studies that make claims about the success of these lessons that are based upon one student. Rather, future research on effective STEM curricula should involve collective student learning.


Simon Bäuerle
email: baeuerle@uni-kassel.de

Hartmut Hetzler

S Y Mpo

O Tatin

Influence of seal compliance on the dynamics of an unbalanced rotor

Keywords:

Compliant hybrid seals offer a comparatively new approach to reduce leakage in turbo machinery, leading to higher efficiency even under transient operation. ese properties are achieved by a reduced sealing gap along with added seal flexibility to minimize the risk of surface rubbing. is contribution investigates the rotordynamic properties by means of a minimal model consisting of a Jeffco rotor and a visco-elastically supported seal. e steady state stability, the bifurcation behavior and the resulting limit cycles of an unbalanced rotor are being discussed and compared to the results for a balanced rotor and the case of a stiff seal support. e compliant support can either enlarge or reduce the stable operation range depending on design parameters. e permi ed seal movability minimizes the risk of rotor-seal contact during passage through the resonance. Rotor and seal orbits induced by unbalance can loose their stability via Neimark-Sacker bifurcations but then also resynchronize.

. INTRODUCTION

Seals are important components to reduce leakage in all kinds of rotating machinery and have been used for a long time. Still, these seemingly simple elements face new design challenges ranging from the demand for higher efficiency to the demand for long inspection intervals. In present industrial practice, the mainly used seals for rotordynamic application are labyrinth and brush seals. Labyrinth seals are comparatively inexpensive and in theory wear-free. In practice, though, rotor-seal contact is not uncommon [ ] resulting in decreased leakage performance. e fluid flow through the seal leads to cross-coupled stiffnesses and, therefore, rotordynamic instabilities above a certain threshold speed. Instabilities can be shi ed to higher rotational speeds by surface texturing or swirl brakes [ ] but not entirely be prevented. Brush seals on the contrary can further reduce leakage and adapt to sha movement [ ]. Rotordynamic impact due to fluid forces is negligible and proper design can avoid thermally induced vibrations (Newkirk-effect [ ]) by localized frictional heating. Yet, low axial stiffness and bristle wear with subsequent increased leakage [ ] are challenges. A comparatively new approach trying to combine advantages of labyrinth and brush seals are compliant hybrid seals (e.g. H

seal [ ], G seal [ ]): a contact-free design leads to low wear while compliance in the seal structure permits rotor deflection without surface rubbing. e adaptability allows for smaller sealing gaps and, thus, lowered leakage with subsequent increased efficiency. In addition, H and G seal narrow the gap width with increas-ing pressure giving a rotor the needed space for movement during start-up or shut-down runs. H seals are already being used and can reduce leakage up to % compared with labyrinth seals [ ].

is contribution focuses on the rotordynamic behavior under the influence of compliant seal structures employing a generic fluid forces model representing in this case an incompressible fluid. Rotor instabilities due to laminar flow in journal bearings are a long known phenomenon: Newkirk was one of the first to report on this in [ ]. e destabilizing influence of seals wasn't investigated until the 's when Lomakin amongst others published on this issue in [ ]. In contrast to bearings the flow in seals is predominantly turbulent due to the comparatively large gap and due to the pressure gradient over the seal with consequential high axial velocities. us, the usage of Reynolds equations is not allowed anymore. And since coupled fluid-structureinteraction simulations with respect to the full Navier-Stokes equations are most of the time too costly, reduced models are required. Amongst others Ng and Pan [ ], Constantinescu [ ] and Hirs [ ] developed such simplified models for low and high Reynolds numbers making use of the small aspect ratio between sealing gap and rotor radius. If coupled simulations with reduced models are still too cumbersome, seal forces can alternatively be represented by force coefficient models stemming either from experimental measurements, full CFD simulations or an analytical derivations of the mentioned reduced models as i.e. done by Childs [ ]. Another way is to postulate some of the dependencies of the fluid subscripted R/ S: rotor/seal-related I : inertially fixed cartesian reference frame R : rotor fixed/ co-rotating cartesian reference frame q R,S position vector wrt R m 0 Childs-Hirs coefficient [ ] ∆q = q Rq S relative rotor deflection wrt R n empirical parameter q 0 equilibrium position wrt R n 0 Childs- Being able to describe the influence of the fluid a basic understanding of the dynamic phenomena in rotor/seal systems needs to be developed: simple Jeffco rotor models are used frequently as a first step. Ding [ ] used a rotor/seal model with stiff bearings and described the seal forces with the Muszynska model. e solution of the balanced system looses its stability in a Hopf bifurcation, whereas the unbalanced system shows a Neimark-Sacker or period doubling bifurcation depending on the level of mass eccentricity. S. Li [ ] used the same model and found similar results. In addition, he could show the existence of reoccurring periodic windows within the quasi-periodic operation range. W. Li's [ ] sealed rotor is mounted in fluid film bearings and experiences an even richer dynamical behaviour, where periodic motions become double-periodic, quasi-periodic and resynchronize to periodic motions again before becoming multi-/quasi-periodic at maximum speed. All three authors used Childs' equations for incompressible fluid seal forces [ ] to compute the parameters in the Muszynska model, which is also done here. ematically similar to a flexible seal structure or support is a flexible fluid film bearing support: More recent works like Vázquez' [ ] could theoretically and experimentally show an increase in onset speed of instability and an increase in the first resonance amplitude with decreasing support stiffness as well as a continuance in the first and a decrease in the second critical speed with decreasing support stiffness. Guo and Kirk [ ] highlighted the influence of the support damping on the onset speed of instability, stating that there is no monotonous dependence but rather a range of optimal values. Bai [ ] investigated a FEM model of a turbopump comprising a seal and two rolling bearings, one being flexibly supported. Neglecting the compressibility of the gaseous working operating medium, they could show a significant dependence of the onset speed of instability on the support flexibility, where an intermediate stiffness lead to the highest stable operation speed.

= . m 0 = -. C = 0.005 • R d R = ξ = . c R = 7.92 • 10 4 N m n = L = 0.1 • R d S = 0.05 2 √ c S m S ρ = kg m 3 m R = 50 kg n 0 = . R = 15 • 10 -2 m µ = 1.295 • 10 -3 Ns m 2 τ 0 = . m S = . •m R ∆p = 5 • 10 5 N m 2
e investigation of the influence of compliant seal structures on rotordynamics is the topic of this contribution. erefore, a minimal model comprising aJeffco rotor with mass eccentricity and a visco-elastically supported stiff seal ring is considered. e non-linear Muszynska model is used to account for the incompressible fluid forces. In the modelling section the equations of motion with respect to an inertially fixed reference frame and with respect to a co-rotating reference frame are derived. e Muszynska model is discussed. e results section is subdivided in three parts: At first, the solution stability is reviewed followed by a discussion on two exemplary bifurcation paths. e contribution closes with a conclusion and an outlook.

. MODELLING With respect to an inertial cartesian system I the dynamical behavior of these rigid bodies is described by

m R 0 0 m R ≔M R r R + d R 0 0 d R ≔B R r R + c R 0 0 c R ≔C R r R = m R eΩ 2 cos(Ωt) sin(Ωt) + F f (∆r) ( )
and

m S 0 0 m S ≔M S r S + d S 0 0 d S ≔B S r S + c S 0 0 c S ≔C S r S = -F f (∆r) ( ) where ∆r = r R -r S ( )
and (.) = d dt (.) is the derivative with respect to time t. e influence of gravitational forces is not considered. F f (∆r) is the fluid force vector which originates from the third part: an incompressible, newtonian lubrication film. e flow is assumed to be fully turbulent at all times due to the axial pressure drop over the seal and the rotor rotation. e fluid force is described by the non-linear Muszynska model

F(∆r) = - m f 0 0 m f ∆ r - D f 2τ f Ωm f -2τ f Ωm f D f ∆ r - K f -m f τ 2 f Ω 2 τ f ΩD f -τ f ΩD f K f -m f τ 2 f Ω 2 ∆r. ( )
e nonlinearity originates from the terms

K f = K f 0 (1 - 1 C r R -r S 2 ) -n , ( )
D f = D f 0 (1 - 1 C r R -r S 2 ) -n ( )
and

τ f = τ f 0 (1 - 1 C r R -r S 2 ) b . ( ) d R , c R m R m S d S 2 c S 2 S e Figure .
Minimal model: Jeffco -rotor and stiff seal ring (visco-elastically connected to its surrounding). An incompressible turbulent fluid flows axially through the sealing gap.

e dependencies of F f on the relative rotor deflection ∆r and the angular speed Ω are postulated by the Muszynska model. e remaining parameters m f , D f 0 and K f 0 are determined using Childs analytical formulas for incompressible fluid forces in short plain seals [ ]. ese formulas are derived from Hirs' bulk flow theory which is based on a semiempirical approach [ ]. Hirs did experimental validations which showed good agreement of his model for an axial Reynolds-number R a = ρVC µ < 10 5 and satisfactory agreement for a circumferential Reynolds-number R c = ρRΩ µ < 10 5 . ese conditions are met at a dimensionless angular speed of η = Ω ω R ≈ 30 which will be the maximum speed for all subsequent investigations. Muszynskas coefficient form model was motivated by experimental and numerical investigations as well as heuristical considerations. e model is based on the idea that the fluid rotates with a mean circumferential speed of 0 ≪ τ f Ω < 1 2 Ω, where τ f = 1 2 is the radial mean value of a perfect laminar Coue e-flow. e quantity τ is deflectiondependent. In addition, the rotor vibrations are required to be small enough for the circumferential flow to remain unidirectional (no backward flow) and the whole flow must be temporally fully developed [ ]. Considering equations ( ) and ( ) it becomes apparent, that the fluid stiffness K f and the fluid damping D f tend to infinity when the normed relative rotor deflection ε = 1 C r Rr S approaches , since n = 2 was assumed. is modeling does not permit for rotor seal contact, which is of course physically possible. us, it must always be assured that the normed relative deflection does not take on values close to . Since the Muszynska model is generic, the parameters could also be fi ed to represent compressible fluid behavior, which was done in [ ].

Combining equations ( ) and ( ) into one formula gives

≔M M R + M f -M f -M f M S + M f r + ≔B(∆r) B R + B f -B f -B f B S + B f r + C R + C f -C f -C f C S + C f ≔C(∆r) r = m R eΩ 2         cos(Ωt) sin(Ωt) 0 0         , ( ) with r = (r ⊤ R , r ⊤ S ) ⊤ ( )
and B(∆r) or C(∆r) respectively being non-linearly dependent on the relative deflection ∆r due to the fluidic part.

is contribution is considering an unbalanced rotor, which makes it favorable to rewrite equations ( ) with respect to a rotor co-rotating cartesian frame of reference R and corotating coordinates

r R = R q R , ( )
and

r S = R q S ( )
where

R = cos(Ωt) -sin(Ωt) sin(Ωt) cos(Ωt) . ( )
Inserting equations ( ) to ( ) into equations ( ) gives

M q + [B(∆q) -2ΩQM] q + K(∆q) -ΩQB(∆q) -Ω 2 M q = m R eΩ 2 1 0 0 0 ( ) with Q = 0 1 -1 0 0 0 0 1 -1 0 ( ) and q = (q T R , q T S ), ( ) which is valid since r R -r S 2 = q R -q S 2
holds. e results are autonomous differential equations.

. RESULTS

In the following three subsections the results of the numerical investigation of equations ( ) and ( ) are presented. All results were obtained by using M ® and the continuation toolbox M

. e main focus of this contribution is the stability and bifurcation behavior of the system under the influence of unbalance. e results for a rotor with an exemplary unbalance of e = 0.15 C will be contrasted with results of an equally unbalanced rotor with stiff seal foundation and a balanced rotor with visco-elastic seal foundation, thereby illustrating the effect of unbalance and support compliance. All results of the bifurcation analysis are visualized with respect to the co-rotating reference frame R. Synchronized periodic limit cycles with respect to the inertial frame I become equilibria positions with respect to R and quasi-periodic motions become periodic ones. Consequently, the terminology in this section refers to reference frame R followed by the corresponding terms with respect to I in brackets. e depiction of the results solely by means of radii is complete since only circular orbits occur due to the geometrical symmetry of the structure of the system.

. Stability analysis

In practice, ever-present unbalance leads to periodic (synchronized) limit cycles. ese cycles can become unstable with increasing rotor speed resulting in potentially harmful quasi-periodic vibrations.

is is also known as oil whirl phenomena.

e stability of periodic limit cycles can be investigated by applying Floquet theory. By switching to co-rotating coordinates q R,S and reference frame R the stability analysis simplifies to the calculation of eigenvalues of equations ( ) linearized around the current equilibrium position q 0 (periodic limit cycle (PLC)) given by

K(∆q 0 ) -ΩQB(∆q 0 ) -Ω 2 M q 0 = m R eΩ 2         1 0 0 0         . ( ) Fig.
displays the stability behavior of the unbalanced and balanced compliant rotor-seal system (red line/ black line) as well as the unbalanced rotor-stiff seal system (blue line). In this chart the stiffness ratio κ 2 = c S c R is plo ed against the dimensionless angular rotor speed η = Ω ω R , where ω R as the rotor eigenfrequency does not include liquid effects (ω R is also known as dry critical speed). Encircled digits indicate numbers of unstable (complex conjugated) pairs of eigenvalues (Floquet Multipliers (FM)) and the le ers A und B correspond to individual unstable pairs. e solid red and black lines mark the stability border, whereas the dashed red lines separate areas of different unstable eigenvalue pairs (FM) . Looking at fig. four major conclusions can be drawn: First, comparing the vertical blue and the red/ black lines shows that added compliance may substantially increase or decrease the stability of the system. For lower stiffness ratios an area of extended stability forms to the right, where higher operational speeds can be reached. At higher κ values the stability is diminished. Secondly, the compliance of the seal foundation can be identified as the stabilizing mechanism since the stability boundaries for compliant and stiff support e same separating lines also exist for the balanced rotor (black line). ey are not shown to enhance graphic representation. and reveals in contrast that there is also a parameter range 6 < κ 2 < 10 where the unbalance has nearly no influence on the stability at all. On the other hand side, tongues of extended stability form in an upward direction. is special behavior will be discussed in subsection . .

. Bifurcation behavior for κ 2 = 1:

In this section the bifurcation analysis for κ 2 = c S c R = 1 with respect to the bifurcation parameter η is presented. Fig.

shows the expected bifurcation diagram of a rotor-seal system with stiff support in dependence of the dimensionless angular rotor speed η as the bifurcation parameter.

Here, the solid (dashed) gray line marks stable (unstable) rotor equilibria positions (PLC) 1 C q R . e blue line indicates the radius 1 C q R of stable unsynchronized periodic limit cycles (quasi-periodic a ractor (QPA)). A er passing through the slightly tilted resonance peak a Hopf bifurcation (Neimark-Sacker bifurcation (NSB)) occurs which is also known as oil whirl. e fast rising amplitudes soon reach critical levels, where contact between rotor and seal becomes more likely and a safe machine operation is not guaranteed any more. e validity of the quantitative statement of the Muszynska model for such high amplitudes must be evaluated with caution. Nevertheless, qualitative conclusions may be drawn. Near η = 25 a period doubling bifurcation : First of all, the solution shows two resonance peaks within the stable operation range. e in comparison to the stiffly supported seal lower (first) critical speed is not unexpected when adding flexibility to a system and also found in [ ]. e passage through this first critical speed results in very high amplitudes of rotor and seal with a maximal magnitude of around 1 C q R,S ≈ 4. Yet, the relative deflection

1 C ∆q = 1 C q R -
q S stays below 0.25. e second resonance only results in higher amplitudes for the seal which is in correspondence with the dominant eigenmode. e course of the relative deflection in the vicinity of and a er the second resonance peak can be explained by a change in the phase angle (not shown here) between rotor and seal: first, rotor and seal conduct an in phase-motion which lasts even during the first resonance peak and changes then to a phase difference of around 2 3 π. e stability of the equilibrium (PLC) is lost in a Hopf bifurcation (NSB) leading to oil whirl motions of rotor and seal, whereas seal amplitudes rise fast and rotor amplitudes rise slow. ree interesting properties become apparent comparing the behavior of stiff and compliant support. First, as already indicated in fig. , the onset speed of instability of the compliant system is much higher. It can be operated at higher speeds with stable synchronized and, thus, lower rotor amplitudes. Secondly, the absolute rotor amplitudes in the resonance area of the compliant system are extremely high, but the relative deflection only takes on values which are % smaller than the amplitudes of the stiff seal support. A passage without rotor-seal contact through the resonance area is much more likely. irdly, the amplitudes of the periodic limit cycles (QPA) a er the loss of stability are much smaller for the compliant system. Especially the rotor amplitudes stay considerably small which could make an ongoing operation possible. Oil whip motion does not occur in either of the two systems: with the used parameters and within the considered rotational speed the whirl frequency does not coincide with one of the resonance frequencies. e gray line marks the centric rotor/ seal equilibrium position (wrt I) which looses its stability in a Hopf bifurcation.

e emerging periodic limit cycles display medium sized seal and small rotor amplitudes. e comparison of the balanced and unbalanced case emphasizes again the statement of fig.

that the unbalance stabilizes the solution and leads to higher rotational speeds with stable solutions. Despite starting from different equilibria positions (wrt R in the unbalanced and wrt I in the balanced case) both amplitudes show a qualitative similar behavior in dependence of η.

. Bifurcation behavior for κ 2 = 7:

In this section the bifurcation analysis for κ 2 = c S c R = 7 with respect to the bifurcation parameter η is presented.

e bifurcation behavior of the compliant system with unbalance is shown in fig. ). e system passes through the resonance with maximal amplitudes of around 1 C q R,S ≈ 2. e relative deflection reaches values of 1 C ∆q = 1 C q Rq S ≈ 0.5 and the tip is slightly bend to the right as expected from systems with noticeable non-linearities. Compared to the bifurcation path for κ 2 = 1 the maximal amplitudes are lower while the relative deflection ∆q is higher. A possible explanation might be the following: the damping in the seal support is low, whereas the fluid damping is approximately times higher. Keeping in mind that the seal is easier to move for low stiffness ratios κ 2 it is clear that the process at κ 2 = 1 is dominated by the low seal damping and low squeezing or energy dissipation respectively by the fluid. ere is greater fluid squeezing at high stiffness ratios since the seal is less movable and, thus, large energy dissipation limits the maximum amplitudes .

A er the decline of the amplitudes a first Hopf bifurcation (NSB) occurs which is the second interesting aspect. A er the bifurcation the amplitudes of the periodic limit cycles (QPA) rise fast just to loose their stability in a fold bifurcation (secondary fold bifurcation (SFB)). In this small η interval the amplitudes rise even higher than in the whole considered η interval for κ 2 = 1. e instability in this instance is caused by the eigenvalue pair B (FM) leading to the qualitatively completely different behavior with rotor and seal amplitudes Changing the parameter se ings for the simulation will of course lead to a changed dissipation mechanism. of similar magnitude. e stability is regained in a second fold bifurcation (SFB) and the resulting stable periodic limit cycle (QPA) collapses in another Hopf bifurcation (NSB) .

e third part is the area of stable equilibria (PLC) between Hopf bifurcation and , where the self-excited whirl motion synchronizes with the unbalance induced vibrations in the vicinity of the second resonance. is behavior is also described by Muszynska in [ ] for a stiff support. A possible explanation might be the increase in fluid damping due to the rising relative deflection in the trail of the second resonance (cf. fig. ), which forces the self-excited vibrations to collapse. Fig . shows the fourth part: the once regained stability is lost again in a third Hopf bifurcation (NSB) leading to periodic limit cycles (QPA) whose amplitudes rise qualitatively similar to the ones discussed in the second part: e same eigenvalue pair B (FM) becomes unstable. Extreme high amplitudes are encountered. Finally, crossing the area of two unstable eigenvalues (FM) to four unstable eigenvalues (cf. fig. ) enforces a fourth Hopf bifurcation (NSB) from which an unstable periodic limit cycle (quasi-periodic repellor QPR) emerges. Compared to the bifurcation behavior for the stiff seal support (cf. fig. ) it is unfavorable to operate the rotor for κ 2 = 7 from a practical point of view: e (relative) deflection in the area of the first resonance is comparable in size, the stability limit is much lower, the amplitudes of the oil whirl are fast rising and considerably bigger and, therefore, most certainly prohibiting ongoing operation. Comparing the unbalanced to the balanced behavior in fig.

shows a lot of qualitative resemblances: A er the first Hopf bifurcation the balanced systems shows fast rising amplitudes. Crossing the line between and unstable eigenvalues an unstable periodic limit cycles emerges, which qualitatively resembles the one in fig. . e periodic limit cycles (QPA) of the unbalanced system display a similar fast rise in amplitudes a er Hopf bifurcations and (NSB) and a correspondence between the unstable periodic limit cycle (QPA) a er Hopf bifurcation (NSB) can be seen. Applications, where the sealing fluids are incompressible such as pumps or sealed fluid film bearings, involve o en classically stiff rotors , which is not the case for the results presented here. Despite being more of a fundamental study, these results do nevertheless apply to systems which were initially designed to run below critical speed but actually run overcritical or near critical if the flexibility of the surrounding seal structure is taken into account [ ]. e results also apply to cases where the compressibility of the gas can be neglected as a first approximation (e.g. liquid fuel turbopump/ liquid hydrogen [ ]).

ANSI/ API Standard

: classically stiff rotors are characterized by a first dry critical speed being % -% above the continuous machine speed.

. CONCLUSION AND OUTLOOK

Alternatively to the established designs of brush and labyrinth seals, hybrid seal might feature low leakage, low wear and improved rotordynamic properties. is issue has been discussed in this contribution by means of a simple rotor-seal model comprising an unbalanced rotor and a compliant seal support with a focus on incompressible fluids. e fluid forces have been represented by the non-linear Muszynska model. e results have been compared to the same model with stiff seal support and to a balanced compliant rotor-seal model.

e enabled movability of the stiff seal ring can increase or decrease the stable operation range depending on the stiffness ratio κ 2 = c S c R . e stable operation range enlarges and stability tongues form with rising mass eccentricity. In case of lower stiffness ratios the relative rotor deflection stays small even during resonance passage and subsequent amplitudes of an oil whirl motion are bounded.

is configuration might be suitable for practical use and displays improved properties compared to the stiff support. Higher stiffness ratios lead to a decreased stable operation range and high whirl amplitudes. e rich occurring bifurcation behavior assigns this case more theoretical than practical relevance: Orbital stability is lost and regained in Neimark-Sacker bifurcations leading to quasi-periodic vibrations and resynchronization. Future work will include three major parts: e non-linear effects of the present model before and a er a loss of stability (synchronization, Arnold tongues, chaos) will be investigated. is requires a fluid model being valid at higher rotational speeds. In a second part, the understanding of the fluid mechanical interaction with the rotor shall be deepened by building up a fully coupled FEM simulation of Hirs' fluid model and incorporating compressible fluid film models. Some yet unpublished work has been conducted in this area. Manufacturing and commissioning of an already designed test rig is the last major part. [ ] Luis San Andrés and Alain Anderson. An all-metal compliant seal versus a labyrinth seal: A comparison of gas leakage at high temperatures. Journal of Engineering for Gas Turbines and Power, ( ): , .

[ ] Dara W Childs. Turbomachinery rotordynamics: phenomena, modeling, and analysis. John Wiley & Sons, .

[ ] MJ Braun and VV Kudriavtsev. A numerical simulation of a brush seal section and some experimental results. In ASME International Gas Turbine and Aeroengine Congress and Exposition, pages V CT A -V CT A . ASME, .

[ ] AD Dimarogonas. Newkirk effect: thermally induced dynamic instability of high-speed rotors. In ASME International Gas Turbine Conference and Products Show, pages V T A -V T A . ASME, .

[ ] Andrew Messenger, Richard Williams, Grant Ingram, Simon Hogg, Stacie Tibos, and Jon Seaton. A dynamic clearance seal for steam turbine application. In ASME Turbo Expo , pages V T A -V T A .

ASME, .

[ ] BL Newkirk. Sha whipping. General Electric Review, ( ): -, .

[ ] FS Bedcher and AA Lomakin. Determination of the critical number of revolutions of a pump rotor with a view to the forces induced in the seals. In Construction of Steam and Gas Turbines, number , pages -, .

Figure

  Figure shows the used minimal model which consists of three parts: e first one is a classical Jeffco -rotor: It spins with the constant angular velocity Ω and the center of mass S has the eccentricity e. e second part is a stiff seal ring, which is visco-elastically connected to its inertially fixed surrounding. is support allows only for translative motions. e two rigid bodies move solely in-plane and their physical properties are indicated in figure and the used parameter values can be found in table .With respect to an inertial cartesian system I the dynamical behavior of these rigid bodies is described by

Figure . Figure .

 .. Figure . Detail from fig.

Figure .

 . Figure . Bifurcation path for κ 2 = 1 and mass eccentricity e = 0.15 C; normed rotor/ seal radii 1 C q R,S are plo ed against η = Ω ω R ; dashed lines indicate unstable solutions; all results wrt R. (secondary period doubling bifurcation (SPDB)) occurs. In addition to the systematic of labeling in fig. , do ed lines in fig. correspond to the dynamic behavior of the seal and solid lines to the one of the rotor. e dash-do ed gray line indicates the relative deflection 1C ∆q = 1 C q Rq S of the rotor.

Figure .

 . Figure . Bifurcation path for stiff seal support and mass eccentricity e = 0.15 C;; normed rotor radii 1 C q R are plo ed against η = Ω ω R ; dashed line indicates unstable solutions; all results wrt R. e figure displays a qualitatively and quantitatively completely different picture compared to its stiff counterpart in fig.: First of all, the solution shows two resonance peaks within the stable operation range. e in comparison to the stiffly supported seal lower (first) critical speed is not unexpected when adding flexibility to a system and also found in [ ]. e passage through this first critical speed results in very high amplitudes of rotor and seal with a maximal magnitude of around 1 C q R,S ≈ 4. Yet, the relative deflection

Figure .

 . Figure . Bifurcation path for κ 2 = 1 and balanced rotor; normed rotor/ seal radii 1 C r R,S are plo ed against η = Ω ω R ; dashed line indicates unstable solutions; all results wrt I.

  Fig. shows the bifurcation path for the balanced system.

  and a detail in fig. . e chosen path crosses the upwards reaching tongue of extended stability in fig. and shows the excepted rich dynamics. e solution shows five interesting parts separated by Hopf bifurcations (NSB) which will be discussed individually: e first part is the first resonance peak (c.f. fig.

Figure . Figure . Figure .

 ... Figure .Bifurcation path for stiff seal support and a balanced rotor; normed rotor/ seal 1 C r R,S radii are plo ed against η = Ω ω R ; dashed lines indicate unstable solutions; all results wrt R.Hopf and fold bifurcation are close together and, therefore, not indicated separately in the chart.

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[ ]


Wolfgang Bock
email: bock@mathematik.uni-kl.de

Martin Bracke
email: bracke@mathematik.uni-kl.de

Patrick Capraro
email: capraro@mathematik.uni-kl.de

Mathematical modeling of musical fountains and light organs -where is the M in interdisciplinary STEM projects?

Keywords: Interdisciplinarity, modeling, product orientation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Some of the most important real-life applications of mathematical modeling we find in engineering and related activities, where products are developed and improved. At the university of Kaiserslautern industrial mathematics plays an important role in research and teaching. It also had an influence in mathematical modeling activities with high school students (see [START_REF] Bock | Applied school mathematicsmade in Kaiserslautern[END_REF]. In these cases, the process of mathematical modeling is closely connected to the product that should be developed.

In Kotler et.al. (2006), p. 230, the term product is defined from an economical point of view as "anything that can be offered to a market that might satisfy a want or need". Hence a product can take various forms, it can be a device that is actually being assembled, it can be a computer software or just the answer to a question. Here the production process closely affects the modeling activity. Even a sophisticated mathematical model has its flaws, if it can't be applied in the desired situation or if it doesn't solve the given problem adequately. Furthermore, the more production steps are needed to advance from the mathematical model to the product, the more impact the production process has on the mathematical model (compare to Bock, Bracke & Capraro, 2017).

Product oriented mathematical modeling immediately calls for high interdisciplinarity and involves all aspects of STEM, especially if the product is not only a theoretical concept, but the implementation of the solution actually takes place [START_REF] Bock | How does a product-based and actionorientated modelling project promote interdisciplinary teaching and learning? To appear[END_REF].

In our technology driven world, complex computations can easily be done by computers and other electronic devices. Often there is no need for the programmer (or engineer) to know all the details of the involved mathematical tools. Here the question arises, if product oriented mathematical modeling and engineering in general can be done with a minimum of mathematical activity (see also [START_REF] Tosmour-Bayazit | Practicing engineer's perspective on mathematics and mathematics education in college[END_REF].

Here we present a modeling project, in which students have to build a musical fountain or a light organ. Both projects contain similar mathematics at a university level (Fourier analysis), which can be hidden in numeric tools and used by students with a basic mathematical knowledge at a secondary school level. The project has a strong interdisciplinary character, since programming is essential for data processing, and building a working model of the product or at least creating a simulation is requested.

We have a look at two implementations of the project. We find that even if the computer is used in a large extent, there still are enough mathematical challenges for students.

The modeling task and mathematical background

The students are given the task of modeling the regulation of a musical fountain or a light organ (both lead to similar mathematical problems). The students get a short introduction in a programming language to be able to read audio files and get access to the raw data. Furthermore we teach basic ideas of Fourier analysis to enable the students to compute frequency spectra and analyze them. Fundamental is the Fourier series, that states that a T-periodic function g (with certain conditions to integrability or Lipschitz-continuity) can be decomposed to a series of trigonometric functions where the coefficients A n , B n give the amplitude of the corresponding frequency nf. By the orthogonality of trigonometric functions, we can derive formulae for the coefficients From this the Fourier transform (3.1) arises, which gives the frequency spectrum G of the function g. If we process the data g on a computer, it isn't given as a function but rather as a vector with finitely many entries. In this case we have the discrete Fourier transform (3.2), where the integral becomes a sum Two more results about discrete Fourier transform are substantial when the spectra are going to be interpreted. For a data vector g with n real valued entries, we have G m =G n+1-m *, i.e. the second half of the spectrum is just a (complex conjugate) copy of the first. Furthermore, we have the sampling theorem, which states that for a data sample g with frame rate Fr, the spectrum can be computed for frequencies up to Fr/2. For a data set of n samples, the value G n/2 corresponds to the frequency Fr/2.

Breaking down Fourier analysis to high school level

Fourier analysis involves mathematical concepts that are located at the final years of high school education or even at university level, like integrals, complex numbers and convergence of functions. Nevertheless, if we see Fourier analysis as a numerical tool, the barriers to implement it in school are quite low. Fourier transform can be introduced as a black box. Certainly, for all well known programming languages there already exists an implementation of the code and that's all we need to know. On the other hand, if there is enough time to do so, the discrete Fourier transform (equation 3.2) can easily be understood and applied as soon as the students know the concept of trigonometric functions.

Understanding overtones and the Fourier series

The Fourier series is strongly connected to instruments and the concept of overtones: If we play a tone on an instrument, we do not only produce the fundamental frequency f of the tone itself, but also overtones with frequencies 2f, 3f, 4f, etc. From a mathematical point of view this is obvious, since acoustic waves are represented by periodic functions. The period of the acoustic wave is the same as the period of the fundamental frequency. Hence the overtones do not disturb the pitch of the observed tone.

The characteristic sound of an instrument is strongly determined by the relative intensities of these overtones. This real-life phenomenon can be used as a motivation to understand and use the ideas of Fourier series.

If the students do not know the concept of overtones, we could create and analyze frequency spectra (e.g. by playing tones on instruments, recording them and using computer software to obtain the spectra, compare to Fig. 1). The pattern in the graphs that are created could motivate a further investigation of the mathematical concepts behind. This could also be the starting point for an interdisciplinary project between mathematics, music and physics classes.

How to deal with complex numbers?

Fourier transform involves complex numbers and the complex exponential function. As we know, the exponential function with imaginary exponent decomposes in real and imaginary parts with trigonometric functions and real valued arguments. Hence, we have the interpretation, that the vector entries in G take 2-dimensional real values The intensity of the frequency represented by G m is then given by the length of the vector.

Deriving formulae for the Fourier coefficients

Equation ( 2) is a consequence of the orthogonality relations of trigonometric functions Older students who know the concept of integrals can prove these identities. The basic idea is to use integration by parts two times, until right hand side and left-hand side show the same integral (with a prefactor and an additive constant).

With equation ( 2) the definition of the Fourier transform can easily be understood. Hence in this case there is no need to see the Fourier transform as a black box tool or a formula given by the teacher without any theoretical background.

Sampling theorem

If the students are using a discrete Fourier transform to obtain frequency spectra, they surely will encounter two questions: Why is there an axial symmetry in the spectrum? And how can we read the frequencies on our axes? To answer these questions, we can give the students the above mentioned results about the Fourier transformed data vector G. A quantitative discussion why these results are true is omitted due to the complexity of the involved mathematics.

Two examples for interdisciplinary modeling projects

In Kaiserslautern, mathematical modeling activities involving acoustic phenomena have been implemented in various situations with high school students and university students. Here we present two cases.

Modeling of a musical fountain with a time frame of 15 hours

This implementation was done in August 2018 with 6 high school students who participated at a Fraunhofer math talent school for girls. The participating students (upper secondary level) came from different schools in Germany and worked on 4 different projects at the Fraunhofer institute for industrial mathematics in Kaiserslautern. The students had the opportunity to vote for their preferred projects. In the musical fountain project 5 of 6 students told us that they had a strong connection to music (all 5 played an instrument) and therefore chose this project. Also, the sixth student voted for the project as her first choice. Some of the students had programming experience, some of them didn't. The group was supervised by two university members.

There were 4 working phases (3 hours each) at the first two days. At the third day, there were 3 hours for preparation and execution of the presentations. Due to the restricted time frame, the students were given selected mathematical input at the beginning of the course.

First working phase: Introduction to Fourier analysis (frontal instruction); reading audio files and plotting spectra with python (frontal instruction and exercise); after a strongly teacher driven opening, the students had the chance to create audio files and analyze them with python or other computer software, like audacity. They did this in groups of 2 students.

Second working phase: The students were asked to develop mathematical criteria that could be used for the regulation of the fountain, but had no further instructions. They chose to stay in the previously formed groups and analyze different aspects of the graphs they could create (group work). After first ideas were established, an additional programming course in python was given (frontal instruction and exercise).

Third working phase:

The ideas of the second working phase were implemented in python code. Some ideas were dropped, others could be improved. One group found mathematical techniques to smooth the highly irregular amplitude-time-function, another one worked on an algorithm that could find an underlying rhythm in a song.

Fourth working phase: It was discussed, if the students wished to build a real-life model of a small fountain with 4 solenoid valves. The students preferred to do a computer visualization of their mathematical results instead. The students used the remaining time to further develop their python code and to decide how their result should be used to regulate the fountain. They also started to work on the visualization.

Presentation day:

The students had 2 hours to prepare their presentations and finish the work on the visualization of the fountain regulation. Each of the 4 project groups had 15 min. for their presentation.

In the simulation, the analyzed song and the visualization of the processed data where replayed simultaneously. It became apparent that the data and the perceived sound matched adequately.

Modeling of a light organ with a time frame of 5 days

The light organ was one of 8 projects at a mathematical modeling week in February 2018. At a modeling week the students (upper secondary level) spend 5 days at a youth hostel and work on their own responsibility for 4 days (from 9 am to 18:30 pm with several breaks for lunch and coffee as well as an excursion on one afternoon). On the 5 th day the presentations are given.

The students can choose their preferred project. There were 4 students working on the light organ. They chose the project with different motivations, concentrating on different aspects of STEM. They were supervised by one university member and by 2 teachers. For the teachers, the participation was part of a teacher training.

With a rather generous time frame, the projects at the modeling week are open to a great extent. The students were asked to build a light organ that would work without a microphone. Instead they should get the data directly from the electronic device, where the song is played, or use digital information. They were provided with electronic components such as a Raspberry Pi, Arduino microcontrollers, LEDs, cables, resistors, etc.

One of the students immediately took the initiative and started to work on his idea of building electric circuits that would work as frequency filters. He started with some first experiments involving the Arduino microcontroller and the question how to record the audio signal. The other students helped him but didn't participate in the creative process.

During the last hours of the first day the group experienced a crisis: 2 of the students were unsatisfied with the progress that was mainly based on ideas from physics and electrical engineering. Both wanted to focus on mathematical and computational aspects of the project. They discussed their ideas with the supervisor. Together a solution was worked out that would be based on the audio information that is stored in music files. From this point on the students worked on two different solutions.

At the second day, the students with the mathematical and computational focus asked for some mathematical background about frequencies. They were given the same information as in the musical fountain project. Only this time not as frontal instructions, but rather in a piece by piece manner. The students were given some mathematical input and had the chance to do some experiments or online research. When they asked deeper questions, they received further details.

Later they were able to implement a Fourier transform in python and run the program on a Raspberry Pi. At this point the students worked on 2 tasks simultaneously, one improving the python program, the other controlling an RGB LED on the Raspberry Pi. Questions to the supervisor were mainly about the involved programming languages and the operating system of the Raspberry Pi. At the end, they could split a piece of music in time intervals, compare the intensity of different frequency intervals and assign color and brightness of the LED to the frequency values.

From a mathematical point of view, the students had to deal with problems related to esthetical questions: How should the frequencies be assigned to different colors, such that the light effects won't be too monotonous? If the brightness of the lights is going to be related to the loudness of the music, which level of loudness should be defined as 100%? Are these thoughts independent of the music To questions like these there is no right answer, but rather a wide range of possible approaches.

Here mathematical experience helps us to give a structure to our problems and quantize the vague questions asked above. Mathematical skills help us to evaluate the information that is hidden in the given data

The other group with the focus on electrical engineering worked almost without supervision. Their main problem was the lack of capacitors with a suitable capacity. This could be solved when one of the supervising teachers asked at a nearby school if she could borrow some material.

In this group mathematical activities took place on a rather basic level, e.g. when the students had to choose the threshold of their frequency filters and compute the resistance and capacity of the electric components that were needed.

Misconceptions of the students about algorithms and computers

When confronted with a subject that most people know very well from their everyday life (like music in our case), the students come up rather quickly with interesting ideas. Some of them are based on the misconception that computers can do complicated tasks on their own, while the programmer doesn't need to fully understand the situation.

In the musical fountain project, there were ideas about recognizing different instruments in the music or finding certain patterns in the rhythm automatically. Those tasks are difficult to implement in an algorithm. The students were surprised how complicated their mathematical model had to be to find a quite simple rhythm in their song (and they didn't even elaborate further steps to make the algorithm work for arbitrary music). In the light organ project, similar situations took place.

The project clearly helped the students to get a better idea of what algorithms are capable of and where the limitations could be. One of the main consequences in the project was that if we want an automated solution to a problem we have to provide a detailed mathematical description.

Conclusion

Using acoustic phenomena in terms of mathematical modeling is a rich field for interdisciplinary activities. It naturally includes physics and music. The link between both topics we find in Fourier analysis, where concepts of music theory, like overtones, become obvious in the mathematical description of waves.

Computer science (or rather programming) is not only helpful, but essential to manage huge amounts of data and extensive computations. Technical aspects can be included by posing a product oriented problem. Hence all aspects of STEM can be addressed.

If all calculations are done by computers, one might fear that the mathematical aspects of such a project will only appear in theory lessons, but won't be applied by the students. In our examples, we have seen that there still is a strong potential to include mathematical aspects that go beyond the computational part. The raw data could be in a bad state for data processing (compare to Figure 2), where elaborate mathematical ideas are needed to overcome the problems. On the other hand, like in the case of the light organ, assumptions have to be made and decisions have to be taken, where an intuitive evaluation of the given data is necessary.

Of course, with a modeling task that is openly posed, there always is the possibility that students find another way of solving the problem. One part of the light organ group used a completely technology based approach, where no mathematics beyond basic arithmetic was needed. On the other hand, from our experience it is essential to let the students choose their approach with the least amount of regulation. Otherwise, if the teacher tries to force certain solutions and mathematical tools, this could have a devastating impact on the students' motivation.

This can lead to serious difficulties for the teacher, since he or she must be able to adapt the ideas of the students quite fast and try to figure out possible outcomes or obstacles for an approach that wasn't considered beforehand. If this new approach requires expert knowledge from another discipline, either the teacher has to have that knowledge (or acquire it fast enough). If this isn't the case, the teacher must be confident that the students can work on their solution relying on their own expertise (why would they have chosen this approach if they don't have any knowledge about the topic?). Additionally, not knowing all the details of a solution for a given problem is an unfamiliar situation for most teachers. It requires a certain amount of self-confidence to let the students proceed anyway.

Listing 1 :

 1 A code example in python. The program reads a wav-file and plots the frequency spectrum.

Figure 1 :

 1 Figure 1: The frequency spectrum of a violin. The fundamental frequency is 440 Hz. The peaks around 0 Hz are possibly due to edge effects or noise.

Figure 2 :

 2 Figure 2: Loudness of a song vs. time. The students wanted to use increasing and decreasing loudness to control the fountain. Since the irregularity of the sample data caused problems, they found methods to smooth the data (left: first iteration of the smoothing process, right: second iteration).

Figure 3 :

 3 Figure 3: The light organ. The yellow LED is controlled by the loudness, the RGB LED by the frequency-spectrum.

Acknowledgements

The close collaboration with the Fraunhofer Institute for Industrial Mathematics as well as the financial support of the European Social Fund for the modeling weeks are thankfully acknowledged.