{"id":5674,"date":"2021-05-07T20:08:42","date_gmt":"2021-05-07T18:08:42","guid":{"rendered":"https:\/\/blogs.mathworks.com\/student-lounge\/?p=5674"},"modified":"2021-05-07T20:08:42","modified_gmt":"2021-05-07T18:08:42","slug":"predicting-dynamic-behavior-of-a-continuously-variable-transmission-using-matlab","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/student-lounge\/2021\/05\/07\/predicting-dynamic-behavior-of-a-continuously-variable-transmission-using-matlab\/","title":{"rendered":"Predicting Dynamic Behavior of a Continuously Variable Transmission using MATLAB"},"content":{"rendered":"

Todays\u00a0guest\u00a0bloggers are\u00a0<\/span>Alex Silva da Purifica\u00e7\u00e3o<\/span><\/a>\u00a0and\u00a0<\/span>Lucas Martins Ricardi<\/span><\/a>\u00a0from\u00a0the\u00a0<\/span>Un<\/span>i<\/span>versity o f Bras\u00edlia<\/span><\/a>\u00a0\u2013\u00a0<\/span>Piratas do Cerrado Baja SAE Team<\/span><\/a>.<\/span>\u00a0They\u00a0will\u00a0ex<\/span>plain\u00a0how\u00a0they\u00a0have\u00a0utilized\u00a0MATLAB for\u00a0predicting\u00a0the\u00a0dynamic\u00a0behavior\u00a0of\u00a0a\u00a0contin<\/span>u<\/span>ously\u00a0variable\u00a0transmission\u00a0(CVT).\u00a0<\/span>\u00a0<\/span><\/p>\n

Introduction<\/span>\u00a0and Motivation<\/span>\u00a0<\/span><\/h1>\n

The continuously variable transmission (CVT) is an automotive automatic transmission concept capable of varying the transmission ratio continuously, with the possibility of keeping the engine rotation constant, and thus its torque and power delivery. The use of CVT is\u00a0<\/span>quite<\/span>\u00a0common<\/span>\u00a0in\u00a0<\/span>Baja SAE<\/span><\/a>\u00a0teams<\/span>, because of its ability to replace the use of\u00a0<\/span>the\u00a0<\/span>conventional clutch and the possibility of\u00a0<\/span>being adjusted<\/span>\u00a0for different applications.<\/span> \u00a0<\/span><\/p>\n

In this context, we from the\u00a0<\/span>Piratas<\/span>\u00a0do\u00a0<\/span>Cerrado<\/span>\u00a0Baja SAE<\/span><\/a>\u00a0<\/span>team at the\u00a0<\/span>University of Bras\u00edlia<\/span><\/a>, have dedicated ourselves for several seasons in adjusting sets of flyweights and springs to define the best set<\/span>\u00a0<\/span>up to provide better longitudinal performance for our car. However, over the years, this setup\u00a0<\/span>was carried out<\/span>\u00a0mostly on an experimental basis, which required a lot of testing time and financial resources<\/span>.<\/span>\u00a0<\/span><\/p>\n

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Therefore, in the last\u00a0<\/span>year<\/span>,<\/span>\u00a0we decided to carry out an ambitious project as our\u00a0<\/span>graduation thesis<\/span>, to develop an algorithm in MATLAB to predict the dynamic behavior of CVT, from the modeling of control mechanisms and belt.\u00a0<\/span>C<\/span>onsequently,<\/span>\u00a0it would be possible to change the inputs of<\/span>\u00a0mass and inertia of the flyweights, spring stiffness\u00a0<\/span>and analyze the outputs of secondary pulley rotation and torque.\u00a0<\/span>Thereby<\/span>\u00a0the CVT design and setup process become\u00a0<\/span>cheaper<\/span>\u00a0and\u00a0<\/span>less\u00a0<\/span>exhaustive.\u00a0\u00a0<\/span>\u00a0<\/span><\/p>\n

The original goal was\u00a0<\/span>to\u00a0<\/span>design our own CVT<\/span>\u00a0<\/span>with\u00a0<\/span>this<\/span>\u00a0model<\/span>, but after seeing the complexity and the various possibilities of the algorithm, we realized that to succeed we needed a full understanding of the model variables. Then, we scaled back to focus\u00a0<\/span>just\u00a0<\/span>on kinematic<\/span>\u00a0and dynamic analysis<\/span>.<\/span>\u00a0<\/span><\/p>\n

We will try in the following topics\u00a0<\/span>to\u00a0<\/span>briefly present the methodology and results we obtained in the current phase of the development of our project, as well as future implementations and projects.<\/span>\u00a0<\/span><\/p>\n

Methodology<\/span>\u00a0<\/span><\/h1>\n

I<\/span>n order t<\/span>o<\/span>\u00a0model<\/span>\u00a0the global CVT\u00a0<\/span>behavior<\/span>,<\/span>\u00a0we divided the work in<\/span>to<\/span>\u00a0two different<\/span>\u00a0models, one for the belt and the\u00a0<\/span>other for<\/span>\u00a0the drive mechanism<\/span>.<\/span>\u00a0For<\/span>\u00a0the<\/span>\u00a0first model, we discretized the belt in node<\/span>s joined by<\/span>\u00a0springs and dampers, then<\/span>,<\/span>\u00a0the node posi<\/span>tion on the time is determined solving<\/span>\u00a0the force balance in each node.<\/span>\u00a0Figure 1 shows belt discretization and the forces considered acting on the\u00a0<\/span>nodes<\/span>\u00a0[1]<\/span>,\u00a0<\/span>and<\/span>\u00a0equation 1 the sum of forces in each node.<\/span>\u00a0<\/span><\/p>\n

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Figure 1: Forces acting on the nodes<\/span><\/span>\u00a0[1]<\/span><\/span>\u00a0<\/span><\/em><\/p>\n

\"\"<\/p>\n

With the sum of forces in each node and dividing this by the mass, we have the dynamic equation of motion<\/span><\/span>. Integrating equation 2 twice in time, we can find the position values of the nodes.<\/span><\/span>\u00a0<\/span><\/p>\n

\"\"<\/p>\n

Thus, with\u00a0<\/span>the\u00a0<\/span>input of primary pulley\u00a0<\/span>rotation data, axial force on the pulleys<\/span>,<\/span>\u00a0and resistive torque submitted to the system,\u00a0<\/span>we\u00a0<\/span>are able to<\/span>\u00a0<\/span>find<\/span>\u00a0the position and speed of each<\/span>\u00a0<\/span>discretized nodes and\u00a0<\/span>consequently<\/span>\u00a0determine the rotation and torque on th<\/span>e secondary pulley.<\/span>\u00a0<\/span><\/p>\n

W<\/span>e<\/span>\u00a0use a mechanical CVT<\/span>\u00a0<\/span>in our Baja<\/span>,\u00a0<\/span>so<\/span>\u00a0the\u00a0<\/span>balance of forces between<\/span>\u00a0the springs and<\/span>\u00a0the centrifugal force under the flyweights generated by<\/span>\u00a0the rotation of the engine\u00a0<\/span>gives the\u00a0<\/span>Drive<\/span>\u00a0<\/span>axial force<\/span>, figure 2<\/span>,<\/span>\u00a0<\/span>which<\/span>\u00a0<\/span>is input to the belt model<\/span>.\u00a0<\/span>Thus,<\/span>\u00a0<\/span>we developed a model for the control mechanism and coupled this system to the belt code.\u00a0<\/span>\u00a0<\/span><\/p>\n

\"\"<\/p>\n

Figure 2<\/span><\/span>:\u00a0<\/span><\/span>Free-body diagram on the primary pulley<\/span><\/span>\u00a0<\/span><\/em><\/p>\n

With that, we have all the models for the construction of our script. The flowchart in figure 3 shows the logic sequence of the algorithm. With the input of the initial\u00a0<\/span><\/span>conditions,<\/span><\/span>\u00a0we calculated the balance of forces<\/span><\/span>,\u00a0<\/span><\/span>and with the Runge-<\/span><\/span>Kutta<\/span><\/span>\u00a0method of solving differential equations, we have as output analysis the shift curves, torque on the secondary pulley<\/span><\/span>,<\/span><\/span>\u00a0and belt position.<\/span><\/span>\u00a0<\/span><\/p>\n

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Figure 3:\u00a0Logical sequence of the algorithm\u00a0<\/em><\/p>\n

In this way, we have a model capable of sensing the variation of the CVT control parameters, a<\/span>nd thus we can analyze\u00a0<\/span>their<\/span>\u00a0interference<\/span>\u00a0in the results of coupling speed, output torque<\/span>,<\/span>\u00a0and shift ratio.\u00a0<\/span>Consequently,<\/span>\u00a0the process of setting up<\/span>\u00a0<\/span>our<\/span>\u00a0car is faster and less expensive, since we can tes<\/span>t several combinations in MATLAB<\/span>\u00a0that would not be possible experimentally.<\/span>\u00a0<\/span><\/p>\n

Results and conclusion<\/span>\u00a0<\/span><\/h1>\n

R<\/span>unning the complete model\u00a0<\/span>several analyzes can be performed.\u00a0<\/span>In video 1<\/span>, it is possible to analyze the transmission behavior during the continuous variation of the\u00a0<\/span>shift<\/span>\u00a0ratio. As expected, the application of the axial force generated by the drive control mechanism induces a tight-side tension and a loose-side tension, which produces the transmission of torque to the secondary pulley.<\/span>\u00a0<\/span>\u00a0<\/span><\/p>\n

\"\"<\/p>\n

Video 1:\u00a0Belt traction during continuous transmission variation<\/em>\u00a0<\/span><\/p>\n

This difference in tension between the sides of the belt can be clearly seen in the distribution of tractive force along the nodes, figure 4.\u00a0<\/span>T<\/span>he regions\u00a0<\/span>in red are the nodes on<\/span>\u00a0the primary pulley, in blue\u00a0<\/span>the nodes on<\/span>\u00a0the secondary pulley<\/span>,<\/span>\u00a0and\u00a0<\/span>in\u00a0<\/span>green the nodes outside the pulleys.<\/span>\u00a0<\/span><\/p>\n

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Figure 4:\u00a0Distribution of tractive force along the nodes\u00a0\u2013 Nodes on the primary pulley (red), on the secondary pulley (blue), outside the pulleys (green).\u00a0<\/em><\/p>\n

The slack dynamic behavior analysis during the transmission is very useful because it is possible to characterize the efficiency of the transmission, which is greatly influenced by the slip between the belt and the pulleys. This behavior would not be possible to\u00a0<\/span>analyze<\/span>\u00a0in a static model, without the discretization of the belt.<\/span>\u00a0<\/span><\/p>\n

The dri<\/span>ve ratio curve in time, figure 5<\/span>, is very important for us because this tr<\/span>ansmission variation determines<\/span>\u00a0the a<\/span>cceleration behavior of our car<\/span>. This curve\u00a0<\/span>serves as input<\/span>\u00a0<\/span>into a longitudinal dynamics algorithm that gives us the car’s speed and acceleration over time.<\/span>\u00a0<\/span><\/p>\n

\"\"<\/p>\n

Figure 5<\/span><\/span>: Drive ratio over time<\/span><\/span>\u00a0<\/span><\/em><\/p>\n

With the implementation of this model in the dimensioning calculation of our project, it will be possible to simulate conditions that would not be possible experimentally. Furthermore, it provided\u00a0<\/span>us great evolutions in the<\/span>\u00a0theoretical knowledge about the dynamic behavior of the CVT<\/span>, and how parameters such<\/span>\u00a0stiffness, mass, inertia<\/span>,<\/span>\u00a0and pre-tension interfere in the longitudinal dynamics of the car.<\/span>\u00a0<\/span><\/p>\n

Future scope<\/span>\u00a0<\/span><\/h1>\n

For the continuation of our work, we intend to implement this algorithm to the longitudinal dynamics model of the car already developed by the team. So, in the same program in MATLAB, will be possible to analyze the interference of the CVT adjustment parameters in the global acceleration behavior and speed of the entire vehicle.<\/span>\u00a0<\/span>Besides<\/span>, we hope in the future to carry out optimization analyzes of the control parameters based on fixed variables,\u00a0<\/span>to<\/span>\u00a0have\u00a0<\/span>the\u00a0<\/span>knowledge to develop our own CVT with design optimization for application in BAJA competitions.<\/span>\u00a0<\/span><\/p>\n

To get an update\u00a0<\/span>on<\/span>\u00a0our progress or take any question<\/span>s<\/span>\u00a0about our dynamic CVT model,\u00a0<\/span>feel free to contact us on\u00a0<\/span>Link<\/span><\/a>edIn<\/span><\/a>\u00a0or the team\u2019s\u00a0<\/span>Instagram<\/span><\/a>.<\/span>\u00a0<\/span><\/p>\n

Thanks for reading and I hope you enjoyed<\/span>\u00a0it<\/span>.<\/span>\u00a0<\/span>Cheers!<\/span>\u00a0<\/span><\/p>\n

References<\/span>\u00a0<\/span><\/h2>\n

[1]\u00a0<\/span>JULIO, G.; PLANTE, J.-S. An\u00a0<\/span>experimentally-val<\/span>idated<\/span>\u00a0model of rubber-belt\u00a0<\/span>cvt<\/span>\u00a0mechanics.\u00a0<\/span>Mechanism and Machine Theory – MECH M<\/span>ACH THEOR, v. 46, p. 1037\u20131053,\u00a0<\/span>08 2011.<\/span>\u00a0<\/span><\/p>\n

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Todays\u00a0guest\u00a0bloggers are\u00a0Alex Silva da Purifica\u00e7\u00e3o\u00a0and\u00a0Lucas Martins Ricardi\u00a0from\u00a0the\u00a0University o f Bras\u00edlia\u00a0\u2013\u00a0Piratas do Cerrado Baja SAE Team.\u00a0They\u00a0will\u00a0explain\u00a0how\u00a0they\u00a0have\u00a0utilized\u00a0MATLAB… read more >><\/a><\/p>\n","protected":false},"author":174,"featured_media":5676,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[4,9,13],"tags":[70,275,498,496,443,117],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/posts\/5674"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/users\/174"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/comments?post=5674"}],"version-history":[{"count":3,"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/posts\/5674\/revisions"}],"predecessor-version":[{"id":5698,"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/posts\/5674\/revisions\/5698"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/media\/5676"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/media?parent=5674"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/categories?post=5674"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/student-lounge\/wp-json\/wp\/v2\/tags?post=5674"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}