{"id":62,"date":"2021-02-04T14:25:45","date_gmt":"2021-02-04T19:25:45","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/?post_type=chapter&#038;p=62"},"modified":"2024-01-25T23:25:17","modified_gmt":"2024-01-26T04:25:17","slug":"building-bond-graph-models-general-procedure-and-application","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/chapter\/building-bond-graph-models-general-procedure-and-application\/","title":{"raw":"Building Bond Graph Models: General Procedure and Application","rendered":"Building Bond Graph Models: General Procedure and Application"},"content":{"raw":"<div>\r\n<h1><a id=\"C4\"><\/a>4.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Overview<\/h1>\r\nTo demonstrate applications of BG method, we discuss the procedure for building BG models for physical systems, using the material presented in <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#C3\">chapter 3<\/a>. We use examples related to mechanical systems to establish the guidelines and steps required to build a BG model. In further chapters, we present more worked-out examples for several engineering systems and disciplines, including electrical, hydraulic systems.\r\n<h1>4.2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Steps for Building Bond Graph Models: General Guidelines<\/h1>\r\nAs mentioned, BG method can be used to build models for single- and multi-domain physical systems. The building blocks are the nine basic BG elements, including their modulated versions, and causality assignment rules (see <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#S3-4\">section 3.4<\/a>). A model for any specific system also requires definitions of relevant sign conventions for general displacement and forces. This chapter will discuss the latter and will present some worked-out examples. The following are the steps for building a BG model, in general, including for mechanical, electrical, and hydraulic systems:\r\n<ol>\r\n \t<li>Identify the physical system components in terms of their type (energy storage, source, dissipater, etc.).<\/li>\r\n \t<li>Identify the DOF (degrees of freedom) of the system. This step is optional but recommended.<\/li>\r\n \t<li>Identify and list the required BG elements.<\/li>\r\n \t<li>Identify distinct physical points\/nodes of the physical systems:\r\n<ul>\r\n \t<li>velocity or force (mechanical systems): translational<\/li>\r\n \t<li>angular velocity or torque (mechanical systems): rotational<\/li>\r\n \t<li>voltage or current (electrical systems): electrical circuits<\/li>\r\n \t<li>pressure or flow rate (hydraulic systems): fluid network<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Assign proper BG multi-port junction elements[footnote]Recall that 1- junction is a <em>flow<\/em> equalizer (or effort summator) and 0-junction is an <em>effort<\/em> equalizer (or flow summator).[\/footnote] to items from step 4:\r\n<ul>\r\n \t<li>\u201c1\u201d for velocity, angular velocity, and electrical and flow currents<\/li>\r\n \t<li>\u201c0\u201d for force, voltage, and pressure<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Connect associated elements, using BG elements and power bonds, to the items from step 5.<\/li>\r\n \t<li>Assign proper BG multi-port junction elements in between those items from step 5:\r\n<ul>\r\n \t<li>\u201c0\u201d for relative velocity and angular velocity<\/li>\r\n \t<li>\u201c1\u201d for voltage drop and pressure drop<\/li>\r\n \t<li>$TF$ and $GY$ for energy conversion<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Connect associated elements to items from step 7, using BG elements and power bonds.<\/li>\r\n \t<li>Define sign convention and connect all remaining power bonds.<\/li>\r\n \t<li>Apply all causality assignments (integral causalities must be given priority).<\/li>\r\n \t<li>Draw and build the BG model in 20-sim (when available).<\/li>\r\n \t<li>Perform simulation and design, using the obtained BG model (when required).<\/li>\r\n<\/ol>\r\nIn further sections, we will demonstrate implementation of the procedure\/algorithm mentioned above with some worked-out examples, including power bond direction, causality assignment, and sign convention.\r\n<h2>4.2.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Guidelines for Power Bond Direction<\/h2>\r\nConnecting elements in a BG model with power bonds requires compliance with the direction of energy flow in the physical system. Therefore, the directions of half-arrows are critical. The following guidelines may be helpful:\r\n<ol>\r\n \t<li>Draw power bonds from BG source elements ($S_e$ and $S_f$) toward the system, connecting to the adjacent elements.<\/li>\r\n \t<li>Draw power bonds toward BG passive elements (i.e., $I$, $C$, and $R$)<\/li>\r\n \t<li>Draw power bonds to and from BG junction elements (\u201c1\u201d and \u201c0\u201d) according to a previously defined sign convention (see <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#S3-4.5\">section 3.4.5<\/a>).<\/li>\r\n \t<li>Draw remaining power bonds to have all BG elements connected.<\/li>\r\n \t<li>Some simplifications of the BG model may be justified, but not required.<\/li>\r\n<\/ol>\r\nAfter drawing all power bonds for the model, assign the causality strokes. The next section provides a list of guidelines for causality assignments.\r\n<h2>4.2.2 \u00a0 \u00a0 \u00a0\u00a0 Guidelines for Assigning Causality Strokes<a id=\"S4-2.2\"><\/a><\/h2>\r\nThe assignment of causality strokes is a required step in building any BG model. The following steps help with achieving this requirement.\r\n<ol>\r\n \t<li>Assign causality to BG source elements.<\/li>\r\n \t<li>Assign causality assignments with preferred integral causality strokes to $I$- and $C$- elements.<\/li>\r\n \t<li>As far as possible, extend the causality assignments to other power bonds, using the causality requirements for connecting elements (e.g., 1, 0, $TF$ and $GY$)<\/li>\r\n \t<li>Assign causality assignments to $R$-elements that accept neutral causality stroke assignment.<\/li>\r\n \t<li>As far as possible, using the causality requirements for connecting elements, extend the causality assignments to all remaining power bonds in the model.<\/li>\r\n<\/ol>\r\nIf execution of step 5 from the above list cannot be completed, then the BG model contains some specific mathematical properties\u2014<em>algebraic loop<\/em> or <em>differential\/derivative causality<\/em> (see <a href=\"\/engineeringsystems\/chapter\/miscellaneous-topics\/\">chapter 11<\/a>). The application 20-sim automatically assigns the causality strokes with prioritizing integral causalities, and if present, identifies the derivative or algebraic loops causalities in the model with red-colour strokes. In further sections, we will explore these features, with some examples.\r\n<h1>4.3 \u00a0 \u00a0 \u00a0\u00a0 Example: BG Model for a One-DOF Mass-Spring-Damper Mechanical System<a id=\"S4-3\"><\/a><\/h1>\r\nA mechanical system consists of mass $m$ [kg], spring $k$ [N\/m], and damper $b$ [N.s\/m]. The applied force on mass is $F(t)$. Build a BG model for this system as shown in <a href=\"#F4-1\">Figure 4\u20111<\/a>, neglecting friction of the rollers.<a id=\"F4-1\"><\/a>\r\n\r\n[caption id=\"attachment_1156\" align=\"aligncenter\" width=\"302\"]<img class=\"wp-image-1156 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-1.jpg\" alt=\"\" width=\"302\" height=\"167\" \/> Figure 4-1 A mass-spring-damper mechanical system[\/caption]\r\n\r\nSolution:\r\n\r\nDOF = 1 (1D translational motion of one mass) and the required BG elements are: $I$ (representing the mass), $C$ (representing the spring), $R$ (representing the damper), $S_e$ (representing force $F$) and $S_f$ (representing the wall velocity). Also, we are required to have junctions \u201c1\u201d and \u201c0.\u201d\r\n<ol>\r\n \t<li>Distinct velocity nodes are the mass and the wall (although the wall usually is stationary). Hence, we need two \u201c1\u201d junctions to represent common velocity for all elements attached to the mass and the wall.<\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 40px\">We draw them as<\/p>\r\n<img class=\"aligncenter wp-image-1370 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-1.jpg\" alt=\"\" width=\"1194\" height=\"234\" \/>\r\n<p style=\"padding-left: 40px\">As well, for each junction, it is useful to assign a name related to its representation.<\/p>\r\n\r\n<ol start=\"2\">\r\n \t<li>We draw all elements connecting to the junctions which have the same distinct velocity values and connect them with power bonds. Therefore, for the wall-velocity junction, we use flow source $S_f$ and for mass-velocity junction and inertial element $I$, representing the mass $m$ and a $S_e$ representing the applied force $F(t).$ Note that $I$-element should receive power (passive element), and sources send power to the system (active elements).<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter wp-image-1371 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-2.jpg\" alt=\"\" width=\"1503\" height=\"387\" \/>\r\n<ol start=\"3\">\r\n \t<li>The spring and damper experience the same value of relative velocity, |$V_m-V_W$|, which is represented by 0-junctions. Recall that 0-junction element is a flow summator. According to the power bonds connecting the spring (or damper) to the 0-junction, we can have ([latex]V_m&gt;V_W[\/latex]) or ($V_m&lt;V_W$), considering the $x$-coordinate as given in <a href=\"#F4-1\">Figure 4\u20111<\/a>. Therefore, to specify the associated power bond directions we should define a sign convention. The common practice is to consider the spring (or damper) from the BG model and define either <em>tension<\/em> force as being positive (+T) or the <em>compression<\/em> force being positive (+C). For this example, we use the spring displacement\/velocity to demonstrate the sign convention. A similar argument applies for the damper\u2019s displacement.<\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 40px\">To represent the relative velocity, we add two 0-junctions and $C$- and $R$- elements to the model and use, e.g., (+T) sign convention, as shown below:<\/p>\r\n<p style=\"padding-left: 40px\"><b> <img class=\"wp-image-2453 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/02\/fig4-3-1-300x209.png\" alt=\"\" width=\"748\" height=\"521\" \/><\/b>Note that $C$- and $R$- elements are passive and should receive power from the system. After labelling the bonds connecting to the 0-junction associated with the $C$- element, we can write the power balance as $e_2f_2-e_3f_3-e_1f_1=0$. But $e_1=e_2=e_3$. Hence, $f_2-f_3=f_1$, or $V_m-V_W=V_C$ where $V_C$ is the spring displacement rate or velocity equal to the relative velocity. Now, to have the displacement of the spring in the +$x$ direction, we should have [latex]V_C \\ \\textgreater \\ 0[\/latex] or [latex]V_m&gt;V_W[\/latex]. This implies that the displacement\/velocity of the mass should be larger than that of the wall, for the spring is experiencing a positive tension force. Therefore, the spring is under tension and the assigned sign convention (+T) is satisfied, considering the +$x$ direction.<\/p>\r\n<p style=\"padding-left: 40px\">Now, if we change the power direction of the bonds connecting to the 0-junction, as shown in the sketch below, we have $V_W-V_m=V_C&gt;0$, or $V_W&gt;V_m$; hence, we have the spring under compression. Therefore, the (+C) sign convention is satisfied. Note that only the power direction of the four bonds associated with the two 0-juctions can change their directions since the rest are associated with source or passive elements and are unique in their directions, as shown below.<span style=\"font-size: 14.4px\">\u00a0<\/span><\/p>\r\n<p style=\"padding-left: 40px\"><img class=\"aligncenter wp-image-1373 size-large\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-4-1024x732.jpg\" alt=\"\" width=\"1024\" height=\"732\" \/>Both sign conventions are legitimate, but only one should be selected and used consistently for building a BG model. We continue, using the BG model with (+T) sign convention.<\/p>\r\n\r\n<ol start=\"4\">\r\n \t<li>Causality assignments are now applied, according to the rules discussed in <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#C3\">chapter 3<\/a>. Following the guidelines given in <a href=\"#S4-2.2\">section 4.2.2<\/a>, we start applying the causality to the source elements, followed by those for $I$- and $C$- elements. Recall that integral causalities are preferred for elements $I$ (i.e., $I$ receives effort) and $C$ (i.e., C sends effort). The causality strokes are shown with transvers lines, as shown below.<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter wp-image-1374 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-5.jpg\" alt=\"\" width=\"1116\" height=\"794\" \/>\r\n<ol start=\"5\">\r\n \t<li>Extend the causality assignments to the remaining bonds, using the rules for 1-junction (can receive only one flow signal through its strong bond) and 0-junction (can receive only one effort signal through its strong bond), as shown below in the model sketch (see <a href=\"#F4-2\">Figure 4\u20112<\/a>).<a id=\"F4-2\"><\/a><\/li>\r\n<\/ol>\r\n[caption id=\"attachment_1157\" align=\"aligncenter\" width=\"1340\"]<img class=\"wp-image-1157 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-2.jpg\" alt=\"\" width=\"1340\" height=\"810\" \/> Figure 4\u20112 Bond graph model for a one-DOF mass-spring-damper system[\/caption]\r\n\r\n&nbsp;\r\n<ol start=\"6\">\r\n \t<li>To a reasonable extent, we can simplify the BG model such that it clearly resembles the physical system. The two 0-junctions represent the same value of relative velocity, $V_m-V_W$. Therefore, we can combine them into a single 0-junction and share the relative velocity value through a 1-junction element with the $C$- and $R$- elements. This simplification becomes very useful for building large BG models for more complex systems. <a href=\"#F4-3\">Figure 4\u20113<\/a> shows the resulting BG model. Note that the causality strokes should be adjusted after simplifications are made.<a id=\"F4-3\"><\/a><\/li>\r\n<\/ol>\r\n[caption id=\"attachment_1158\" align=\"aligncenter\" width=\"1234\"]<img class=\"wp-image-1158 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-3.jpg\" alt=\"\" width=\"1234\" height=\"531\" \/> Figure 4\u20113 Simplified bond graph model for a one-DOF mass-spring-damper system[\/caption]\r\n<h1>4.4 \u00a0 \u00a0 \u00a0\u00a0 Example: BG Model for a Two-DOF Mass-Spring-Damper Mechanical System<a id=\"S4-4\"><\/a><\/h1>\r\nBuild the BG model for the mechanical system as shown in <a href=\"#F4-4\">Figure 4-4<\/a>. Consider the (+C) to be the sign convention for internal forces.<a id=\"F4-4\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1159\" align=\"aligncenter\" width=\"467\"]<img class=\"wp-image-1159 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-4.jpg\" alt=\"\" width=\"467\" height=\"120\" \/> Figure 4-4 A two-DOF mass-spring-damper mechanical system[\/caption]\r\n\r\n&nbsp;\r\n\r\nSolution:\r\n\r\nThis system has two DOF and four distinct velocity points, corresponding to mass $m_1$ and $m_2$ and the two walls. Therefore, we lay out four 1-junctions to represent them in the model. The remaining required BG elements are $I$, $C$, $R$, $S_e$, $S_f$, and 1- and 0- junctions.\r\n\r\nWe follow the same guidelines demonstrated in the previous example (see <a href=\"#S4-3\">section 4.3<\/a>) and build the BG model as shown in <a href=\"#F4-5\">Figure 4-5<\/a>.<a id=\"F4-5\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1160\" align=\"aligncenter\" width=\"356\"]<img class=\"wp-image-1160 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-5.jpg\" alt=\"\" width=\"356\" height=\"183\" \/> Figure 4-5 BG model for a two-DOF mass-spring-damper mechanical system[\/caption]\r\n\r\n&nbsp;\r\n\r\nThe reader is encouraged to build this BG model and to compare the results with those provided in <a href=\"#F4-5\">Figure 4-5<\/a>. The 20-sim BG model and a screen recording are available as companion resources describing the process to build the equation model and typical results for this example.\r\n\r\nhttps:\/\/vimeo.com\/563486977\r\n<h1>4.5 \u00a0 \u00a0 \u00a0\u00a0 Example: BG Model for a Three-DOF Mass-Spring-Damper Mechanical System<a id=\"S4-5\"><\/a><\/h1>\r\nBuild the BG model for the mechanical system as shown in <a href=\"#F4-6\">Figure 4-6<\/a>. Consider the (+C) to be the sign convention for internal forces.<a id=\"F4-6\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1161\" align=\"aligncenter\" width=\"906\"]<img class=\"wp-image-1161 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-6.jpg\" alt=\"\" width=\"906\" height=\"344\" \/> Figure 4-6 A three-DOF mass-spring-damper mechanical system[\/caption]\r\n\r\n&nbsp;\r\n\r\nSolution:\r\n\r\nThis system has three DOF and five distinct velocity points corresponding to mass $m_1$, $m_2$, and $m_3$ and the two walls. Therefore, we lay out five 1-junctions to represent them in the model. The remaining required BG elements are $I$, $C$, $R$, $S_e$, $S_f$, and 1- and 0-junctions.\r\n\r\nWe follow the same guidelines demonstrated in the previous example (see <a href=\"#S4-3\">section 4.3<\/a>) and build the BG model, as shown in <a href=\"#F4-7\">Figure 4-7<\/a>.<a id=\"F4-7\"><\/a>\r\n\r\n[caption id=\"attachment_1162\" align=\"aligncenter\" width=\"1172\"]<img class=\"wp-image-1162 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-7.jpg\" alt=\"\" width=\"1172\" height=\"528\" \/> Figure 4-7 BG model for a three-DOF mass-spring-damper mechanical system[\/caption]\r\n\r\nThe reader is encouraged to build this BG model and compare the results with those provided in <a href=\"#F4-7\">Figure 4-7<\/a>. The 20-sim BG model and a screen recording are available as companion resources describing the process to build the equation model and typical results for this example.\r\n\r\nhttps:\/\/vimeo.com\/563487776\r\n<h1><a id=\"S4-6\"><\/a>4.6\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Example: Kinetics and Kinematics of a Mechanical System Using BG Model<\/h1>\r\n<span style=\"font-size: 1em\">As mentioned previously<\/span><span style=\"text-align: initial;font-size: 1em\">\u00a0(see <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#S3.2\">section 3.2<\/a>), one of the advantages of BG modelling method is that a BG model allows us to gain insights by visual inspection of the BG model. This can be achieved by drawing the streams of efforts (kinetics) and flows (kinematics) for a BG model.<\/span>\r\n\r\nIn this example, we use the results from the example given in <a href=\"#S4-3\">section 4.3<\/a> to demonstrate this property by explicitly drawing the effort and flow associated with each power bond in the model. First, we look at the kinetics of the system by drawing the efforts, as shown in <a href=\"#F4-8\">Figure 4-8<\/a>. As shown, the efforts\/forces associated with the spring $e_C$ and dumper $e_R$ are collected as force $e$ and transferred to the mass $m$ in addition to the applied force $F$ shown as $e_F$. Clearly, the wall receives the collected force $e=e_C + e_R$.<a id=\"F4-8\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1163\" align=\"aligncenter\" width=\"1491\"]<img class=\"wp-image-1163 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-8.jpg\" alt=\"\" width=\"1491\" height=\"759\" \/> Figure 4-8 Kinetics of a one-DOF mechanical system showing the stream of efforts with its BG model[\/caption]\r\n\r\n&nbsp;\r\n\r\nSimilarly, by drawing the flows, as shown in <a href=\"#F4-9\">Figure 4-9<\/a> the kinematics of the system can be visualized. As shown, the flows\/velocities associated with the mass $f_m$ and wall $f_w$ are collected as velocity $f$ and transferred to the spring and damper. Clearly, these elements receive the relative velocity $f=f_m+ f_w$ due to the motion of mass and the wall (if stationary, $f_w=0$).<a id=\"F4-9\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1164\" align=\"aligncenter\" width=\"1495\"]<img class=\"wp-image-1164 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-9.jpg\" alt=\"\" width=\"1495\" height=\"754\" \/> Figure 4-9 Kinematics of a one-DOF mechanical system showing the stream of flows with its BG model[\/caption]\r\n<h1>4.7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Modelling and Simulation Approaches in Engineering: Modern vs. Traditional<\/h1>\r\nConsidering BG\u2014our focus in this textbook\u2014as the modelling method, once we have the corresponding BG model, we can proceed to simulation, and hence, design of a system. One can take two approaches to perform this task: traditional or modern. As mentioned, the main objective of modelling and simulation is to help with more effective design of the systems in terms of their cost, function, material consumption, etc. Therefore, any modelling method, including bond graph, should result in a mathematical model consisting of the systems\u2019 equations. The solution of the equations can be used for system simulation and analysis to support effective system design. In the following sections, we briefly describe possible approaches and the reason for choosing the modern approach over the more traditional one.\r\n\r\n&nbsp;\r\n<h2>4.7.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Traditional Approach<\/h2>\r\nOnce the BG model is available for a system, we can derive\/extract the system equations from the BG model\u2014usually a laborious task\u2014and then using numerical methods, we can obtain their solutions. This task is usually achieved with the help of computer programs (usually developed from scratch) based on a selected numerical method. This approach, along with both its system equation extraction and especially the computer coding, is limited in practical application, being specific from one problem to another one, and is laboriously time consuming. <a href=\"#F4-10\">Figure 4-10<\/a> shows the major steps of the traditional approach.<a id=\"F4-10\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1165\" align=\"aligncenter\" width=\"1445\"]<img class=\"wp-image-1165 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-10.jpg\" alt=\"\" width=\"1445\" height=\"583\" \/> Figure 4-10 Traditional approach for system simulation and design[\/caption]\r\n\r\nIn practice, it is inefficient to develop computer codes for each specific design: the amount of person power, computer power, and other resources become overwhelming for the fast-paced engineering design needs of today\u2019s industries. Therefore, a huge effort has been made to develop commercially available software tools to help meet these objectives and to make the whole process of system design more effective, economically viable, and efficient.\r\n<h2>4.7.2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Modern Approach<\/h2>\r\nAlternatively\u2014or rather, preferably\u2014the modern approach in engineering and system design employs related software tools. These tools provide opportunities to perform systems simulation immediately after obtaining the BG model. The software tool that we introduce and use in this textbook, 20-sim, helps with extracting the system equations from the BG model seamlessly and provides facilities for system simulation and parametric analysis. This modern approach is more effective in engineering practice and provides more and quicker insights into engineering systems design. In addition, the modern approach helps to respond more effectively to the fast-paced engineering demands in industry and is recommended for engineers in practice. <a href=\"#F4-11\">Figure 4-11<\/a> shows the major steps of the modern approach. Note that verification and validation should be considered in the modelling step as well.<a id=\"F4-11\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1166\" align=\"aligncenter\" width=\"1440\"]<img class=\"wp-image-1166 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-11.jpg\" alt=\"\" width=\"1440\" height=\"160\" \/> Figure 4\u201111 Modern approach for system simulation and design[\/caption]\r\n\r\nIn the next section, we introduce the 20-sim software package with a focus on bond graph modelling, simulation, and time and frequency analysis for engineering systems and design. In further sections, we use 20-sim to build BG models and their simulations and to study their dynamical behaviour.\r\n<h1>Exercise Problems for Chapter 4<\/h1>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercises<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li style=\"text-align: left\">Build the BG model, including causality assignment, for the example given in <a href=\"#S4-4\">section 4.4<\/a> considering (+T) as the sign convention for internal forces.\r\n<ol style=\"list-style-type: lower-alpha\">\r\n \t<li style=\"text-align: left\">Draw a kinetic map of the system, using the stream of efforts.<\/li>\r\n \t<li style=\"text-align: left\">Draw a kinematic map of the system, using the stream of flows.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li style=\"text-align: left\">Build the BG model, including causality assignment, for the example given in <a href=\"#S4-5\">section 4.5<\/a> considering (+T) as the sign convention for internal forces.\r\n<ol style=\"list-style-type: lower-alpha\">\r\n \t<li style=\"text-align: left\">Draw a kinetic map of the system, using the stream of efforts.<\/li>\r\n \t<li style=\"text-align: left\">Draw a kinematic map of the system, using the stream of flows.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li style=\"text-align: left\">Discuss the benefits of modern vs. traditional approaches for simulation and design of systems, from a practical point of view. Include speed of calculations, and economical aspects of the two methods in your discussion.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div>\n<h1><a id=\"C4\"><\/a>4.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Overview<\/h1>\n<p>To demonstrate applications of BG method, we discuss the procedure for building BG models for physical systems, using the material presented in <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#C3\">chapter 3<\/a>. We use examples related to mechanical systems to establish the guidelines and steps required to build a BG model. In further chapters, we present more worked-out examples for several engineering systems and disciplines, including electrical, hydraulic systems.<\/p>\n<h1>4.2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Steps for Building Bond Graph Models: General Guidelines<\/h1>\n<p>As mentioned, BG method can be used to build models for single- and multi-domain physical systems. The building blocks are the nine basic BG elements, including their modulated versions, and causality assignment rules (see <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#S3-4\">section 3.4<\/a>). A model for any specific system also requires definitions of relevant sign conventions for general displacement and forces. This chapter will discuss the latter and will present some worked-out examples. The following are the steps for building a BG model, in general, including for mechanical, electrical, and hydraulic systems:<\/p>\n<ol>\n<li>Identify the physical system components in terms of their type (energy storage, source, dissipater, etc.).<\/li>\n<li>Identify the DOF (degrees of freedom) of the system. This step is optional but recommended.<\/li>\n<li>Identify and list the required BG elements.<\/li>\n<li>Identify distinct physical points\/nodes of the physical systems:\n<ul>\n<li>velocity or force (mechanical systems): translational<\/li>\n<li>angular velocity or torque (mechanical systems): rotational<\/li>\n<li>voltage or current (electrical systems): electrical circuits<\/li>\n<li>pressure or flow rate (hydraulic systems): fluid network<\/li>\n<\/ul>\n<\/li>\n<li>Assign proper BG multi-port junction elements<a class=\"footnote\" title=\"Recall that 1- junction is a flow equalizer (or effort summator) and 0-junction is an effort equalizer (or flow summator).\" id=\"return-footnote-62-1\" href=\"#footnote-62-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> to items from step 4:\n<ul>\n<li>\u201c1\u201d for velocity, angular velocity, and electrical and flow currents<\/li>\n<li>\u201c0\u201d for force, voltage, and pressure<\/li>\n<\/ul>\n<\/li>\n<li>Connect associated elements, using BG elements and power bonds, to the items from step 5.<\/li>\n<li>Assign proper BG multi-port junction elements in between those items from step 5:\n<ul>\n<li>\u201c0\u201d for relative velocity and angular velocity<\/li>\n<li>\u201c1\u201d for voltage drop and pressure drop<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-c83edd8a73e25b889812de87029ee455_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"22\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-9b140dab83603fb3b9e9fbd26dfbdba8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"23\" style=\"vertical-align: 0px;\" \/> for energy conversion<\/li>\n<\/ul>\n<\/li>\n<li>Connect associated elements to items from step 7, using BG elements and power bonds.<\/li>\n<li>Define sign convention and connect all remaining power bonds.<\/li>\n<li>Apply all causality assignments (integral causalities must be given priority).<\/li>\n<li>Draw and build the BG model in 20-sim (when available).<\/li>\n<li>Perform simulation and design, using the obtained BG model (when required).<\/li>\n<\/ol>\n<p>In further sections, we will demonstrate implementation of the procedure\/algorithm mentioned above with some worked-out examples, including power bond direction, causality assignment, and sign convention.<\/p>\n<h2>4.2.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Guidelines for Power Bond Direction<\/h2>\n<p>Connecting elements in a BG model with power bonds requires compliance with the direction of energy flow in the physical system. Therefore, the directions of half-arrows are critical. The following guidelines may be helpful:<\/p>\n<ol>\n<li>Draw power bonds from BG source elements (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-ed63991f05623afc79c0427a3c722cec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-71c9985fb7e53bd022c3f0c6e2775281_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: -5px;\" \/>) toward the system, connecting to the adjacent elements.<\/li>\n<li>Draw power bonds toward BG passive elements (i.e., <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-25cfe7b772dea23f45d0cdd4f5c10d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/>)<\/li>\n<li>Draw power bonds to and from BG junction elements (\u201c1\u201d and \u201c0\u201d) according to a previously defined sign convention (see <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#S3-4.5\">section 3.4.5<\/a>).<\/li>\n<li>Draw remaining power bonds to have all BG elements connected.<\/li>\n<li>Some simplifications of the BG model may be justified, but not required.<\/li>\n<\/ol>\n<p>After drawing all power bonds for the model, assign the causality strokes. The next section provides a list of guidelines for causality assignments.<\/p>\n<h2>4.2.2 \u00a0 \u00a0 \u00a0\u00a0 Guidelines for Assigning Causality Strokes<a id=\"S4-2.2\"><\/a><\/h2>\n<p>The assignment of causality strokes is a required step in building any BG model. The following steps help with achieving this requirement.<\/p>\n<ol>\n<li>Assign causality to BG source elements.<\/li>\n<li>Assign causality assignments with preferred integral causality strokes to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/>&#8211; and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/>&#8211; elements.<\/li>\n<li>As far as possible, extend the causality assignments to other power bonds, using the causality requirements for connecting elements (e.g., 1, 0, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-c83edd8a73e25b889812de87029ee455_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"22\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-9b140dab83603fb3b9e9fbd26dfbdba8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"23\" style=\"vertical-align: 0px;\" \/>)<\/li>\n<li>Assign causality assignments to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-25cfe7b772dea23f45d0cdd4f5c10d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/>-elements that accept neutral causality stroke assignment.<\/li>\n<li>As far as possible, using the causality requirements for connecting elements, extend the causality assignments to all remaining power bonds in the model.<\/li>\n<\/ol>\n<p>If execution of step 5 from the above list cannot be completed, then the BG model contains some specific mathematical properties\u2014<em>algebraic loop<\/em> or <em>differential\/derivative causality<\/em> (see <a href=\"\/engineeringsystems\/chapter\/miscellaneous-topics\/\">chapter 11<\/a>). The application 20-sim automatically assigns the causality strokes with prioritizing integral causalities, and if present, identifies the derivative or algebraic loops causalities in the model with red-colour strokes. In further sections, we will explore these features, with some examples.<\/p>\n<h1>4.3 \u00a0 \u00a0 \u00a0\u00a0 Example: BG Model for a One-DOF Mass-Spring-Damper Mechanical System<a id=\"S4-3\"><\/a><\/h1>\n<p>A mechanical system consists of mass <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-08297f9d61e9c01c09eafffb66d4cf14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"7\" width=\"13\" style=\"vertical-align: 0px;\" \/> [kg], spring <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-8c9a2c1169d8ce6372870f6e0d2c3fab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/> [N\/m], and damper <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-adb81a6e4b3d016ae0f0d46bea2da10c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"7\" style=\"vertical-align: 0px;\" \/> [N.s\/m]. The applied force on mass is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-69f6d0343a698a233c9771c56e6918c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: -4px;\" \/>. Build a BG model for this system as shown in <a href=\"#F4-1\">Figure 4\u20111<\/a>, neglecting friction of the rollers.<a id=\"F4-1\"><\/a><\/p>\n<figure id=\"attachment_1156\" aria-describedby=\"caption-attachment-1156\" style=\"width: 302px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1156 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-1.jpg\" alt=\"\" width=\"302\" height=\"167\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-1.jpg 302w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-1-300x166.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-1-65x36.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-1-225x124.jpg 225w\" sizes=\"auto, (max-width: 302px) 100vw, 302px\" \/><figcaption id=\"caption-attachment-1156\" class=\"wp-caption-text\">Figure 4-1 A mass-spring-damper mechanical system<\/figcaption><\/figure>\n<p>Solution:<\/p>\n<p>DOF = 1 (1D translational motion of one mass) and the required BG elements are: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/> (representing the mass), <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/> (representing the spring), <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-25cfe7b772dea23f45d0cdd4f5c10d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/> (representing the damper), <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-ed63991f05623afc79c0427a3c722cec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -2px;\" \/> (representing force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-31c65b987512f42c8c282c2fc003e471_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/>) and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-71c9985fb7e53bd022c3f0c6e2775281_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: -5px;\" \/> (representing the wall velocity). Also, we are required to have junctions \u201c1\u201d and \u201c0.\u201d<\/p>\n<ol>\n<li>Distinct velocity nodes are the mass and the wall (although the wall usually is stationary). Hence, we need two \u201c1\u201d junctions to represent common velocity for all elements attached to the mass and the wall.<\/li>\n<\/ol>\n<p style=\"padding-left: 40px\">We draw them as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1370 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-1.jpg\" alt=\"\" width=\"1194\" height=\"234\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-1.jpg 1194w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-1-300x59.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-1-1024x201.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-1-768x151.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-1-65x13.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-1-225x44.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-1-350x69.jpg 350w\" sizes=\"auto, (max-width: 1194px) 100vw, 1194px\" \/><\/p>\n<p style=\"padding-left: 40px\">As well, for each junction, it is useful to assign a name related to its representation.<\/p>\n<ol start=\"2\">\n<li>We draw all elements connecting to the junctions which have the same distinct velocity values and connect them with power bonds. Therefore, for the wall-velocity junction, we use flow source <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-71c9985fb7e53bd022c3f0c6e2775281_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: -5px;\" \/> and for mass-velocity junction and inertial element <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/>, representing the mass <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-08297f9d61e9c01c09eafffb66d4cf14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"7\" width=\"13\" style=\"vertical-align: 0px;\" \/> and a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-ed63991f05623afc79c0427a3c722cec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -2px;\" \/> representing the applied force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-0c700f88071aee6045582ad0f6626e49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#40;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: -4px;\" \/> Note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/>-element should receive power (passive element), and sources send power to the system (active elements).<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1371 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-2.jpg\" alt=\"\" width=\"1503\" height=\"387\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-2.jpg 1503w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-2-300x77.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-2-1024x264.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-2-768x198.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-2-65x17.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-2-225x58.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-2-350x90.jpg 350w\" sizes=\"auto, (max-width: 1503px) 100vw, 1503px\" \/><\/p>\n<ol start=\"3\">\n<li>The spring and damper experience the same value of relative velocity, |<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-b05a3746d1671f4a9b5ca743d52d1a8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#109;&#45;&#86;&#95;&#87;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: -2px;\" \/>|, which is represented by 0-junctions. Recall that 0-junction element is a flow summator. According to the power bonds connecting the spring (or damper) to the 0-junction, we can have (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-ba03027d6d6dfc413cb3d2413c382435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#109;&#62;&#86;&#95;&#87;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"59\" style=\"vertical-align: -2px;\" \/>) or (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-3fbbbb61f5ae712728bcc5f3ceb5c1af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#109;&#60;&#86;&#95;&#87;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"59\" style=\"vertical-align: -2px;\" \/>), considering the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-68d3165f12fed5d05e11de45dbfe5bec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"7\" width=\"8\" style=\"vertical-align: 0px;\" \/>-coordinate as given in <a href=\"#F4-1\">Figure 4\u20111<\/a>. Therefore, to specify the associated power bond directions we should define a sign convention. The common practice is to consider the spring (or damper) from the BG model and define either <em>tension<\/em> force as being positive (+T) or the <em>compression<\/em> force being positive (+C). For this example, we use the spring displacement\/velocity to demonstrate the sign convention. A similar argument applies for the damper\u2019s displacement.<\/li>\n<\/ol>\n<p style=\"padding-left: 40px\">To represent the relative velocity, we add two 0-junctions and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/>&#8211; and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-25cfe7b772dea23f45d0cdd4f5c10d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/>&#8211; elements to the model and use, e.g., (+T) sign convention, as shown below:<\/p>\n<p style=\"padding-left: 40px\"><b> <img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2453 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/02\/fig4-3-1-300x209.png\" alt=\"\" width=\"748\" height=\"521\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/02\/fig4-3-1-300x209.png 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/02\/fig4-3-1-65x45.png 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/02\/fig4-3-1-225x157.png 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/02\/fig4-3-1-350x244.png 350w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/02\/fig4-3-1.png 720w\" sizes=\"auto, (max-width: 748px) 100vw, 748px\" \/><\/b>Note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/>&#8211; and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-25cfe7b772dea23f45d0cdd4f5c10d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/>&#8211; elements are passive and should receive power from the system. After labelling the bonds connecting to the 0-junction associated with the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/>&#8211; element, we can write the power balance as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-6df470709fc02f672bf7f760f7d7f76e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#95;&#50;&#102;&#95;&#50;&#45;&#101;&#95;&#51;&#102;&#95;&#51;&#45;&#101;&#95;&#49;&#102;&#95;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"141\" style=\"vertical-align: -3px;\" \/>. But <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-ffa72ecaf519acabd9126525294f785a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#95;&#49;&#61;&#101;&#95;&#50;&#61;&#101;&#95;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"78\" style=\"vertical-align: -2px;\" \/>. Hence, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-91a9858d36d6df94d9287840ff60733e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#95;&#50;&#45;&#102;&#95;&#51;&#61;&#102;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"76\" style=\"vertical-align: -3px;\" \/>, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-8416ff0d1866b17287f1bc8b4ae08950_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#109;&#45;&#86;&#95;&#87;&#61;&#86;&#95;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"95\" style=\"vertical-align: -2px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-108665ddb012ed4364a828e948ccf402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: -2px;\" \/> is the spring displacement rate or velocity equal to the relative velocity. Now, to have the displacement of the spring in the +<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-68d3165f12fed5d05e11de45dbfe5bec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"7\" width=\"8\" style=\"vertical-align: 0px;\" \/> direction, we should have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7a1a047db728b8fa42e5b3286b812a57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#67;&#32;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#103;&#114;&#101;&#97;&#116;&#101;&#114;&#32;&#92;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"46\" style=\"vertical-align: -2px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-ba03027d6d6dfc413cb3d2413c382435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#109;&#62;&#86;&#95;&#87;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"59\" style=\"vertical-align: -2px;\" \/>. This implies that the displacement\/velocity of the mass should be larger than that of the wall, for the spring is experiencing a positive tension force. Therefore, the spring is under tension and the assigned sign convention (+T) is satisfied, considering the +<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-68d3165f12fed5d05e11de45dbfe5bec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"7\" width=\"8\" style=\"vertical-align: 0px;\" \/> direction.<\/p>\n<p style=\"padding-left: 40px\">Now, if we change the power direction of the bonds connecting to the 0-junction, as shown in the sketch below, we have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-39b27df08960d60a068bb73e1a66ec09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#87;&#45;&#86;&#95;&#109;&#61;&#86;&#95;&#67;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"122\" style=\"vertical-align: -2px;\" \/>, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-951377434da5b9b2e1301b7c6ba34939_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#87;&#62;&#86;&#95;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"59\" style=\"vertical-align: -2px;\" \/>; hence, we have the spring under compression. Therefore, the (+C) sign convention is satisfied. Note that only the power direction of the four bonds associated with the two 0-juctions can change their directions since the rest are associated with source or passive elements and are unique in their directions, as shown below.<span style=\"font-size: 14.4px\">\u00a0<\/span><\/p>\n<p style=\"padding-left: 40px\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1373 size-large\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-4-1024x732.jpg\" alt=\"\" width=\"1024\" height=\"732\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-4-1024x732.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-4-300x215.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-4-768x549.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-4-65x46.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-4-225x161.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-4-350x250.jpg 350w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-4.jpg 1110w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/>Both sign conventions are legitimate, but only one should be selected and used consistently for building a BG model. We continue, using the BG model with (+T) sign convention.<\/p>\n<ol start=\"4\">\n<li>Causality assignments are now applied, according to the rules discussed in <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#C3\">chapter 3<\/a>. Following the guidelines given in <a href=\"#S4-2.2\">section 4.2.2<\/a>, we start applying the causality to the source elements, followed by those for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/>&#8211; and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/>&#8211; elements. Recall that integral causalities are preferred for elements <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/> (i.e., <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/> receives effort) and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/> (i.e., C sends effort). The causality strokes are shown with transvers lines, as shown below.<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1374 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-5.jpg\" alt=\"\" width=\"1116\" height=\"794\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-5.jpg 1116w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-5-300x213.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-5-1024x729.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-5-768x546.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-5-65x46.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-5-225x160.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/06\/4.2-Solution-5-350x249.jpg 350w\" sizes=\"auto, (max-width: 1116px) 100vw, 1116px\" \/><\/p>\n<ol start=\"5\">\n<li>Extend the causality assignments to the remaining bonds, using the rules for 1-junction (can receive only one flow signal through its strong bond) and 0-junction (can receive only one effort signal through its strong bond), as shown below in the model sketch (see <a href=\"#F4-2\">Figure 4\u20112<\/a>).<a id=\"F4-2\"><\/a><\/li>\n<\/ol>\n<figure id=\"attachment_1157\" aria-describedby=\"caption-attachment-1157\" style=\"width: 1340px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1157 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-2.jpg\" alt=\"\" width=\"1340\" height=\"810\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-2.jpg 1340w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-2-300x181.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-2-1024x619.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-2-768x464.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-2-65x39.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-2-225x136.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-2-350x212.jpg 350w\" sizes=\"auto, (max-width: 1340px) 100vw, 1340px\" \/><figcaption id=\"caption-attachment-1157\" class=\"wp-caption-text\">Figure 4\u20112 Bond graph model for a one-DOF mass-spring-damper system<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<ol start=\"6\">\n<li>To a reasonable extent, we can simplify the BG model such that it clearly resembles the physical system. The two 0-junctions represent the same value of relative velocity, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-b05a3746d1671f4a9b5ca743d52d1a8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#109;&#45;&#86;&#95;&#87;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: -2px;\" \/>. Therefore, we can combine them into a single 0-junction and share the relative velocity value through a 1-junction element with the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/>&#8211; and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-25cfe7b772dea23f45d0cdd4f5c10d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/>&#8211; elements. This simplification becomes very useful for building large BG models for more complex systems. <a href=\"#F4-3\">Figure 4\u20113<\/a> shows the resulting BG model. Note that the causality strokes should be adjusted after simplifications are made.<a id=\"F4-3\"><\/a><\/li>\n<\/ol>\n<figure id=\"attachment_1158\" aria-describedby=\"caption-attachment-1158\" style=\"width: 1234px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1158 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-3.jpg\" alt=\"\" width=\"1234\" height=\"531\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-3.jpg 1234w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-3-300x129.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-3-1024x441.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-3-768x330.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-3-65x28.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-3-225x97.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-3-350x151.jpg 350w\" sizes=\"auto, (max-width: 1234px) 100vw, 1234px\" \/><figcaption id=\"caption-attachment-1158\" class=\"wp-caption-text\">Figure 4\u20113 Simplified bond graph model for a one-DOF mass-spring-damper system<\/figcaption><\/figure>\n<h1>4.4 \u00a0 \u00a0 \u00a0\u00a0 Example: BG Model for a Two-DOF Mass-Spring-Damper Mechanical System<a id=\"S4-4\"><\/a><\/h1>\n<p>Build the BG model for the mechanical system as shown in <a href=\"#F4-4\">Figure 4-4<\/a>. Consider the (+C) to be the sign convention for internal forces.<a id=\"F4-4\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1159\" aria-describedby=\"caption-attachment-1159\" style=\"width: 467px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1159 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-4.jpg\" alt=\"\" width=\"467\" height=\"120\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-4.jpg 467w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-4-300x77.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-4-65x17.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-4-225x58.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-4-350x90.jpg 350w\" sizes=\"auto, (max-width: 467px) 100vw, 467px\" \/><figcaption id=\"caption-attachment-1159\" class=\"wp-caption-text\">Figure 4-4 A two-DOF mass-spring-damper mechanical system<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>Solution:<\/p>\n<p>This system has two DOF and four distinct velocity points, corresponding to mass <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-64d6ab52bb05afac5120f9fa30e12679_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"18\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7074eddb4b441a91a587661e65d2d465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#95;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"19\" style=\"vertical-align: -2px;\" \/> and the two walls. Therefore, we lay out four 1-junctions to represent them in the model. The remaining required BG elements are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-25cfe7b772dea23f45d0cdd4f5c10d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-ed63991f05623afc79c0427a3c722cec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -2px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-71c9985fb7e53bd022c3f0c6e2775281_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: -5px;\" \/>, and 1- and 0- junctions.<\/p>\n<p>We follow the same guidelines demonstrated in the previous example (see <a href=\"#S4-3\">section 4.3<\/a>) and build the BG model as shown in <a href=\"#F4-5\">Figure 4-5<\/a>.<a id=\"F4-5\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1160\" aria-describedby=\"caption-attachment-1160\" style=\"width: 356px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1160 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-5.jpg\" alt=\"\" width=\"356\" height=\"183\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-5.jpg 356w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-5-300x154.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-5-65x33.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-5-225x116.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-5-350x180.jpg 350w\" sizes=\"auto, (max-width: 356px) 100vw, 356px\" \/><figcaption id=\"caption-attachment-1160\" class=\"wp-caption-text\">Figure 4-5 BG model for a two-DOF mass-spring-damper mechanical system<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>The reader is encouraged to build this BG model and to compare the results with those provided in <a href=\"#F4-5\">Figure 4-5<\/a>. The 20-sim BG model and a screen recording are available as companion resources describing the process to build the equation model and typical results for this example.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Screenrecord_for_Example_in_section_4-4\" src=\"https:\/\/player.vimeo.com\/video\/563486977?dnt=1&amp;app_id=122963\" width=\"500\" height=\"265\" frameborder=\"0\"><\/iframe><\/p>\n<h1>4.5 \u00a0 \u00a0 \u00a0\u00a0 Example: BG Model for a Three-DOF Mass-Spring-Damper Mechanical System<a id=\"S4-5\"><\/a><\/h1>\n<p>Build the BG model for the mechanical system as shown in <a href=\"#F4-6\">Figure 4-6<\/a>. Consider the (+C) to be the sign convention for internal forces.<a id=\"F4-6\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1161\" aria-describedby=\"caption-attachment-1161\" style=\"width: 906px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1161 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-6.jpg\" alt=\"\" width=\"906\" height=\"344\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-6.jpg 906w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-6-300x114.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-6-768x292.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-6-65x25.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-6-225x85.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-6-350x133.jpg 350w\" sizes=\"auto, (max-width: 906px) 100vw, 906px\" \/><figcaption id=\"caption-attachment-1161\" class=\"wp-caption-text\">Figure 4-6 A three-DOF mass-spring-damper mechanical system<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>Solution:<\/p>\n<p>This system has three DOF and five distinct velocity points corresponding to mass <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-64d6ab52bb05afac5120f9fa30e12679_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"18\" style=\"vertical-align: -2px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7074eddb4b441a91a587661e65d2d465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#95;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"19\" style=\"vertical-align: -2px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e2d51415022140c44a47a869c78afb13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#95;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"19\" style=\"vertical-align: -2px;\" \/> and the two walls. Therefore, we lay out five 1-junctions to represent them in the model. The remaining required BG elements are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e0d502912ebc0d1a2f2b253b1a893f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"8\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-7d5d9e8849dff9523b40f081c156ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"12\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-25cfe7b772dea23f45d0cdd4f5c10d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-ed63991f05623afc79c0427a3c722cec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -2px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-71c9985fb7e53bd022c3f0c6e2775281_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#95;&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: -5px;\" \/>, and 1- and 0-junctions.<\/p>\n<p>We follow the same guidelines demonstrated in the previous example (see <a href=\"#S4-3\">section 4.3<\/a>) and build the BG model, as shown in <a href=\"#F4-7\">Figure 4-7<\/a>.<a id=\"F4-7\"><\/a><\/p>\n<figure id=\"attachment_1162\" aria-describedby=\"caption-attachment-1162\" style=\"width: 1172px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1162 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-7.jpg\" alt=\"\" width=\"1172\" height=\"528\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-7.jpg 1172w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-7-300x135.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-7-1024x461.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-7-768x346.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-7-65x29.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-7-225x101.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-7-350x158.jpg 350w\" sizes=\"auto, (max-width: 1172px) 100vw, 1172px\" \/><figcaption id=\"caption-attachment-1162\" class=\"wp-caption-text\">Figure 4-7 BG model for a three-DOF mass-spring-damper mechanical system<\/figcaption><\/figure>\n<p>The reader is encouraged to build this BG model and compare the results with those provided in <a href=\"#F4-7\">Figure 4-7<\/a>. The 20-sim BG model and a screen recording are available as companion resources describing the process to build the equation model and typical results for this example.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Screenrecord_for_Example_in_section_4-5\" src=\"https:\/\/player.vimeo.com\/video\/563487776?dnt=1&amp;app_id=122963\" width=\"500\" height=\"268\" frameborder=\"0\"><\/iframe><\/p>\n<h1><a id=\"S4-6\"><\/a>4.6\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Example: Kinetics and Kinematics of a Mechanical System Using BG Model<\/h1>\n<p><span style=\"font-size: 1em\">As mentioned previously<\/span><span style=\"text-align: initial;font-size: 1em\">\u00a0(see <a href=\"\/engineeringsystems\/chapter\/bond-graph-modelling-method#S3.2\">section 3.2<\/a>), one of the advantages of BG modelling method is that a BG model allows us to gain insights by visual inspection of the BG model. This can be achieved by drawing the streams of efforts (kinetics) and flows (kinematics) for a BG model.<\/span><\/p>\n<p>In this example, we use the results from the example given in <a href=\"#S4-3\">section 4.3<\/a> to demonstrate this property by explicitly drawing the effort and flow associated with each power bond in the model. First, we look at the kinetics of the system by drawing the efforts, as shown in <a href=\"#F4-8\">Figure 4-8<\/a>. As shown, the efforts\/forces associated with the spring <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-cced530613171cac8cbc5790dcd8a1a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#95;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"16\" style=\"vertical-align: -2px;\" \/> and dumper <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-f2f8bed69a7e32e0f5666e6a092d4819_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#95;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"16\" style=\"vertical-align: -2px;\" \/> are collected as force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-9e58889fe60ada819d48f71296f83b05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"7\" width=\"7\" style=\"vertical-align: 0px;\" \/> and transferred to the mass <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-08297f9d61e9c01c09eafffb66d4cf14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"7\" width=\"13\" style=\"vertical-align: 0px;\" \/> in addition to the applied force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-31c65b987512f42c8c282c2fc003e471_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"11\" style=\"vertical-align: 0px;\" \/> shown as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-0d356dfe3f97c3b8c4f37e33eb5ad1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#95;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"16\" style=\"vertical-align: -2px;\" \/>. Clearly, the wall receives the collected force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-d1da41072262de0d3573a3abe4b78f4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#61;&#101;&#95;&#67;&#32;&#43;&#32;&#101;&#95;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"76\" style=\"vertical-align: -2px;\" \/>.<a id=\"F4-8\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1163\" aria-describedby=\"caption-attachment-1163\" style=\"width: 1491px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1163 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-8.jpg\" alt=\"\" width=\"1491\" height=\"759\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-8.jpg 1491w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-8-300x153.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-8-1024x521.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-8-768x391.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-8-65x33.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-8-225x115.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-8-350x178.jpg 350w\" sizes=\"auto, (max-width: 1491px) 100vw, 1491px\" \/><figcaption id=\"caption-attachment-1163\" class=\"wp-caption-text\">Figure 4-8 Kinetics of a one-DOF mechanical system showing the stream of efforts with its BG model<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>Similarly, by drawing the flows, as shown in <a href=\"#F4-9\">Figure 4-9<\/a> the kinematics of the system can be visualized. As shown, the flows\/velocities associated with the mass <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-e9fad7201539080a008ff6f2558855bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#95;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -3px;\" \/> and wall <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-f8fd98d8747c1a9d392ed757233598d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#95;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -3px;\" \/> are collected as velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-45d2bbafd2751f0a2f4054f3b0269e48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: -3px;\" \/> and transferred to the spring and damper. Clearly, these elements receive the relative velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-dc0aac7dd49819aa98dc8116c5de21da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#102;&#95;&#109;&#43;&#32;&#102;&#95;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"79\" style=\"vertical-align: -3px;\" \/> due to the motion of mass and the wall (if stationary, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/ql-cache\/quicklatex.com-fd96891a2259f18aff79364750eae177_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#95;&#119;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -3px;\" \/>).<a id=\"F4-9\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1164\" aria-describedby=\"caption-attachment-1164\" style=\"width: 1495px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1164 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-9.jpg\" alt=\"\" width=\"1495\" height=\"754\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-9.jpg 1495w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-9-300x151.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-9-1024x516.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-9-768x387.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-9-65x33.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-9-225x113.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-9-350x177.jpg 350w\" sizes=\"auto, (max-width: 1495px) 100vw, 1495px\" \/><figcaption id=\"caption-attachment-1164\" class=\"wp-caption-text\">Figure 4-9 Kinematics of a one-DOF mechanical system showing the stream of flows with its BG model<\/figcaption><\/figure>\n<h1>4.7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Modelling and Simulation Approaches in Engineering: Modern vs. Traditional<\/h1>\n<p>Considering BG\u2014our focus in this textbook\u2014as the modelling method, once we have the corresponding BG model, we can proceed to simulation, and hence, design of a system. One can take two approaches to perform this task: traditional or modern. As mentioned, the main objective of modelling and simulation is to help with more effective design of the systems in terms of their cost, function, material consumption, etc. Therefore, any modelling method, including bond graph, should result in a mathematical model consisting of the systems\u2019 equations. The solution of the equations can be used for system simulation and analysis to support effective system design. In the following sections, we briefly describe possible approaches and the reason for choosing the modern approach over the more traditional one.<\/p>\n<p>&nbsp;<\/p>\n<h2>4.7.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Traditional Approach<\/h2>\n<p>Once the BG model is available for a system, we can derive\/extract the system equations from the BG model\u2014usually a laborious task\u2014and then using numerical methods, we can obtain their solutions. This task is usually achieved with the help of computer programs (usually developed from scratch) based on a selected numerical method. This approach, along with both its system equation extraction and especially the computer coding, is limited in practical application, being specific from one problem to another one, and is laboriously time consuming. <a href=\"#F4-10\">Figure 4-10<\/a> shows the major steps of the traditional approach.<a id=\"F4-10\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1165\" aria-describedby=\"caption-attachment-1165\" style=\"width: 1445px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1165 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-10.jpg\" alt=\"\" width=\"1445\" height=\"583\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-10.jpg 1445w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-10-300x121.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-10-1024x413.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-10-768x310.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-10-65x26.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-10-225x91.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-10-350x141.jpg 350w\" sizes=\"auto, (max-width: 1445px) 100vw, 1445px\" \/><figcaption id=\"caption-attachment-1165\" class=\"wp-caption-text\">Figure 4-10 Traditional approach for system simulation and design<\/figcaption><\/figure>\n<p>In practice, it is inefficient to develop computer codes for each specific design: the amount of person power, computer power, and other resources become overwhelming for the fast-paced engineering design needs of today\u2019s industries. Therefore, a huge effort has been made to develop commercially available software tools to help meet these objectives and to make the whole process of system design more effective, economically viable, and efficient.<\/p>\n<h2>4.7.2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Modern Approach<\/h2>\n<p>Alternatively\u2014or rather, preferably\u2014the modern approach in engineering and system design employs related software tools. These tools provide opportunities to perform systems simulation immediately after obtaining the BG model. The software tool that we introduce and use in this textbook, 20-sim, helps with extracting the system equations from the BG model seamlessly and provides facilities for system simulation and parametric analysis. This modern approach is more effective in engineering practice and provides more and quicker insights into engineering systems design. In addition, the modern approach helps to respond more effectively to the fast-paced engineering demands in industry and is recommended for engineers in practice. <a href=\"#F4-11\">Figure 4-11<\/a> shows the major steps of the modern approach. Note that verification and validation should be considered in the modelling step as well.<a id=\"F4-11\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1166\" aria-describedby=\"caption-attachment-1166\" style=\"width: 1440px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1166 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-11.jpg\" alt=\"\" width=\"1440\" height=\"160\" srcset=\"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-11.jpg 1440w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-11-300x33.jpg 300w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-11-1024x114.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-11-768x85.jpg 768w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-11-65x7.jpg 65w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-11-225x25.jpg 225w, https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-content\/uploads\/sites\/1041\/2021\/05\/Figure-4-11-350x39.jpg 350w\" sizes=\"auto, (max-width: 1440px) 100vw, 1440px\" \/><figcaption id=\"caption-attachment-1166\" class=\"wp-caption-text\">Figure 4\u201111 Modern approach for system simulation and design<\/figcaption><\/figure>\n<p>In the next section, we introduce the 20-sim software package with a focus on bond graph modelling, simulation, and time and frequency analysis for engineering systems and design. In further sections, we use 20-sim to build BG models and their simulations and to study their dynamical behaviour.<\/p>\n<h1>Exercise Problems for Chapter 4<\/h1>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercises<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li style=\"text-align: left\">Build the BG model, including causality assignment, for the example given in <a href=\"#S4-4\">section 4.4<\/a> considering (+T) as the sign convention for internal forces.\n<ol style=\"list-style-type: lower-alpha\">\n<li style=\"text-align: left\">Draw a kinetic map of the system, using the stream of efforts.<\/li>\n<li style=\"text-align: left\">Draw a kinematic map of the system, using the stream of flows.<\/li>\n<\/ol>\n<\/li>\n<li style=\"text-align: left\">Build the BG model, including causality assignment, for the example given in <a href=\"#S4-5\">section 4.5<\/a> considering (+T) as the sign convention for internal forces.\n<ol style=\"list-style-type: lower-alpha\">\n<li style=\"text-align: left\">Draw a kinetic map of the system, using the stream of efforts.<\/li>\n<li style=\"text-align: left\">Draw a kinematic map of the system, using the stream of flows.<\/li>\n<\/ol>\n<\/li>\n<li style=\"text-align: left\">Discuss the benefits of modern vs. traditional approaches for simulation and design of systems, from a practical point of view. Include speed of calculations, and economical aspects of the two methods in your discussion.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"media-attributions clear\" prefix:cc=\"http:\/\/creativecommons.org\/ns#\" prefix:dc=\"http:\/\/purl.org\/dc\/terms\/\"><h2>Media Attributions<\/h2><ul><li >fig4-3       <\/li><\/ul><\/div><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-62-1\">Recall that 1- junction is a <em>flow<\/em> equalizer (or effort summator) and 0-junction is an <em>effort<\/em> equalizer (or flow summator). <a href=\"#return-footnote-62-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":801,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-62","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/pressbooks\/v2\/chapters\/62","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/wp\/v2\/users\/801"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/pressbooks\/v2\/chapters\/62\/revisions"}],"predecessor-version":[{"id":2455,"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/pressbooks\/v2\/chapters\/62\/revisions\/2455"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/pressbooks\/v2\/chapters\/62\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/wp\/v2\/media?parent=62"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=62"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/wp\/v2\/contributor?post=62"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/wp-json\/wp\/v2\/license?post=62"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}