‘w..--9vVv—‘wa-ww.V——.—-V“.fi_-‘ ‘.. q - —_‘..:.>. __. “___ .,-‘-‘-V.V .h.“ ... "‘fl'.“-‘~I‘P.\“"‘"‘““"‘w~d;¢nvv"'-fi “ , ‘ 7 ' A MULTIPGRT APPROACHVTD’MODMG‘ , ' me-mwmsyswms» ~ “mm: “'{mhe-Dez‘m-fims N I mmem STATE uNiVjERSITY; , *‘ r 1 . mn-RMY, ‘ MW" Irv-.1. E”? )W .A\ ’l- ,, l-I .‘ arr" * '- ' ‘if’l’vhe «um». "-17" (.H',’ ’ - p . D .30 $3? _ ‘ f '3"! 'r n ~-' .1.“ 1“? 3:549” to Iglflt 3”}; ,' )4 . ~ .,.. . .3 I —l I r, :45, a ' I I; ' I, . 4,; , w“ . J ' . A‘. ,:.‘ . f . .~,;.;- 51,} 4a; , fi¢ééff ,; . 34%;»; viz-w: 5‘ u j u", r . ’ ~ , ‘12,)!” f. ‘4 ”(ii-3 :' 11W 1 1111111111 11111 31293 0111031212 LIBRARY Michi ganStatc E University {”1 ‘TT-k‘;g WM“ 13»: : 1. , > ' 9 ‘_ , JUN 2 5 1999 ABSTRACT A MULTIPORT APPROACH TO MODELING FLUID POWER SYSTEMS BY Mark R. Ray The ability to predict system dynamics is prOposed as a new tool to overcome problems often encountered in the design of large scale fluid power systems. System ’models are constructed from component models developed by a multiport approach. Standard techniques are used to generate the system state space equations. The component models are compiled into a catalog to be used in construction of system models. A scheme for obtaining system parameters is also introduced. A MULTIPORT APPROACH TO MODELING FLUID POWER SYSTEMS BY . 13 Mark R? Ray A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1974 r A TABLE or CONTENTS LIST OF FIGURES . . . . . . . . Chapter 1. THE PROBLEM . . . . . . . 1.1 New Tools . . . . . 1.2 An Example 2. THE APPROACH . . . . . . 2.1 A Standard Format . . 2.2 The Bond Graph Approach 2.3 Typical Models . . . 2.4 System Synthesis . . . 3 0 THE CATALOG O O O O O O O 3.1 Format . . . . . . 4. CONCLUSIONS AND RECOMMENDATIONS 4.1 Summary . . . . . . 4.2 Future Work . . . . APPENDICES Appendix A. A Definition of the Bond Graph Language. B. Component Catalog . . . . . REFERENCES . . . . . . . . . ii Page iii 10 20 26 26 28 28 29 30 34 48 Figure l. 2. LIST OF FIGURES Hydraulic drive for conveyor system . (a) Hydraulic cylinder; (b) word bond graph; (c) free body diagram . (a) Ideal bond graph model; (b) static bond graph model; (c) dynamic model (a) Positive displacement axial piston pump; (b) definition of variables Stages of development for the bond graph . . . . . . . . . Schematic with components isolated Word bond graph of conveyor belt model Preliminary bond graph of conveyor belt. mOdel O O I O O O O O 0 Bond graph of conveyor belt model Example of a cataloged component . iii Page 11 14 16 19 22 23 24 25 27 CHAPTER 1 THE PROBLEM 1.1 New Tools The design of large scale fluid power systems has become a task of increasing difficulty in recent years due to their rapid expansion in scope and SOphistication. Static analysis has been heavily employed as an aid to design but is proving less and less effective as the systems increase in complexity. Investigation of system dynamics prior to actual construction is a powerful approach to the design problem. The ability to predict the dynamics of a fluid power system is a versatile tool and one well worth em- ploying. The task of predicting system dynamics can be broken down into three phases. The first phase is to develOp or obtain a system model, in this case a mathe- matical model consisting of differential equations describing the system. Next the system parameters, volumes, inertias, etc., used in the system model must be obtained. The last phase is solving the equations. Although these operations are easy to enumerate their execution can be difficult. Trying to obtain the system equations when modeling a large scale fluid power system can be a perplexing task unless one is well armed with experience in work with fluid power systems and mathe- matical descriptions of physical phenomena. Often the parameters can only be obtained by testing in a laboratory. Many methods exist for solving differential equations but choosing the most effective may also require additional effort. Prediction of system dynamics is not easy. But can it be simplified? 1.2 An Example As an example of a design problem involving a large scale fluid power system consider the application detailed here. It may help to illustrate some of the problems previously mentioned. A conveyor belt is to be used to move material down a line composed of individual work stations. A unique operation is performed at each station by an auto- mated machine, so the material must be precisely posi- tioned. The distance between each work station is identi- cal so the belt may be indexed and all functions performed simultaneously. The limiting factor on the rate of production will be how fast the belt can be indexed while still maintaining the position accuracy required. Figure 1 shows the proposed fluid power system used to drive the system. 3 HYDRAUUC CYUNDER HYDRAuuc TEE , MOTOR 5A3? TEE u 9. CHECK é VALVE ACCUMULATOR TEE 3- WAY 3.1—1 [—133 VALVES l—'I \ ‘3‘ I ‘—i S! ’ S! I TEE r" r common MANIFOLD CHECK VALVE “Q: WC HYDRAULIC - . 'VNTUOF{ ‘pumap. L :flUhAp) Figure 1. Hydraulic drive for conveyor system. The hydraulic motor drives the conveyor belt through a gear arrangement but for sake of clarity they are not shown. The exhaust part of the hydraulic motor is opened and closed by means of a two way valve actuated by the hydraulic cylinder. The cylinder is in turn con- trolled by the two three way solenoid valves on the common manifold. The motor by-pass is also opened and closed by a solenoid valve, in this case a two way valve. The system operates in the following manner. The solenoid valve on the left side of the manifold is nor- mally open to the supply source. The valve on the right is normally closed to the supply source. The normally open and normally closed phases refer to the deenergized state of the valve so with no power to the valves the cylinder is up. The check valve orifice configurations between the valve and cylinder allows free flow from the valve to the cylinder but when flow is reversed the check valve is closed and the flow is metered through the orifice. By changing the orifice size the speed of the cylinder can be controlled. The solenoid valve in the motor by-pass line is normally closed and the orifice in that line is also adjustable. The solenoid valves on the manifold are energized when a signal is generated by a timing board clock. The by-pass solenoid is energized when the material being moved by the conveyor closes a switch. Upon the signal from the timing board the manifold valves switch orientation; the left valve opens to the sump and the right valve opens to the supply source, driving the cylinder down. This action allows the two way valve closing the motor exhaust port to open and the motor begins to drive the conveyor. When the material on the conveyor closes the switch allowing the motor by-pass line to open the motor starts to decelerate. Upon another signal from the timing board the manifold valve switch orientation and begin closing the motor exhaust line. When the material reaches the desired point the conveyor is abruptly halted. Coordination of all these activities into a smooth operation can be extremely difficult, especially when trying to operate stably at high speed. CHAPTER 2 THE APPROACH 2.1 A Standard Format Recognizing the need for better systems analysis, the National Fluid Power Association has formed a simula- tion committee ”to set forth a standard format for mathematical models for fluid power systems and components [1]." This development would be beneficial in several ways. If the format were adopted by the fluid power industry the component manufacturers could develop models for their products and supply the parameters needed in the models to describe their products. The proper format could also simplify the development of system models, making it easier to obtain the system equations. These two advances coupled with the prOper method of solution could enable the designer of fluid power systems to deter- mine the feasibility of a design before taking any action toward construction. The format proposed here will begin by attacking the modeling problem at the component level. Standard models of common fluid power components will be developed and compiled into a catalog. A method of system model construction from component models will also be detailed. A later section on future developments will propose some interesting uses for the catalog. V The standard models for each component are developed using a multiport approach known as bond graphs. (Those unfamiliar with bond graphs are referred to Appendix A.) The bond graphs are then translated into ordinary differential equations describing the component in terms of the so-called "state variables." A set of simple algebraic equations relate the remaining component variables to the state variables. The internal coupling structure of the component can be studied when using bond graph methods as well as the basic nature of the dynamics. The models are acausal in nature so it is unnecessary to make any decision con- cerning input and output at the component level. Deci- sions involving input and output are made when the system model is completed. Once the problems of obtaining the system equa- tions and parameters have been dealt with, the only remaining hurdle is the method of solution. There are a number of analog and digital techniques that can be used to solve the system equations. A digital computer program capable of accepting a system model in bond graph form known as Enport [4] is available. The program is capable of assigning power flow directions and causality but if the system equations are nonlinear the program will not generate the state space equations. At the present time the program is only capable of manipulating small scale, continuous, linear or linearized systems. Expansion of the program capa- bilities is currently being studied with the ability to handle large—scale, linear systems as the next step in development, followed by nonlinear systems. Analog computer schemes would appear to offer the best possibility for solution of large-scale fluid power system state equations. The nonlinearity and discontinuity of the equations can be nicely handled by an analog com- puter. The only drawback is the amount of work in- volved in altering the system equations when parameters are changed. This sometimes makes the use of an analog computer in an iterative manner extremely tedious. This is undesirable since iterative solution techniques are valuable in design work. 2.2 The Bond Graph Approach The modeling of fluid power systems by employing bond graph techniques can be approached several ways in mnequally valid manner. The approach taken here is by no means unique. The majority of fluid power systems of interest can be characterized as high pressure-low flow rate systems. There are several choices of power variables available to describe fluid power systems; the flow-effort pair of volume flow rate (Q) and dynamic pressure (P) shall be used here. The fluid represented in this manner has no distributed mass, compressibility, or thermal properties. However, a system may be charac- terized by means of lumped models whenever the significant wave lengths of all variables are large compared to the physical dimensions of the system [5]. Although this places some limitations on the models they are not too restrictive. A departure from conventional bond graph notation will be made in that the effort variables will be repre- sented by P (pressure) and F (force) rather than the generalized effort symbol e. The flow variables will be represented by Q (volume flow rate), V (linear velocity) and W (rotational velocity),rather than the generalized flow symbol f. This is done in an effort to make the presentation in terms more familiar to the average engineer. A consistent set of units will be maintained throughout the development which will conflict somewhat with general practice in the fluid power industry. The volume flow rate, for example, is given in cubic inches per second rather than gallons per minute. Other quantities will be defined as required. Several models will now be develOped to illustrate the methods used to obtain the standard 10 models for common fluid power system components in the catalog (Appendix B). 2.3 Typical Models The common hydraulic cylinder shown in Figure 2a has many applications in fluid power systems and can be used to demonstrate the bond graph modeling technique. If the fluid is considered incompressible and the piston and shaft massless, the device will exhibit no dynamics. If the motion of the piston and shaft occurs without dissipation and no leakage occurs the device may be considered ideal. After making these assumptions the next step is to examine the external ports of the device. Figure 2b is a word bond graph of the hydraulic cylinder showing the device as a 3 port element. The power con- vention on the fluid ports is chosen only for convenience and causes no loss of generality to the model. By means of the free body diagram in Figure 2c the basic coupling structure may be examined. The sum of forces on the free body diagram is given as F1 - F2 = F3 2.3.1 Substituting for F1 and F2 PlAl - PZA2 = F3 2.3.2 Equation 2.3.2 shows the relationship between the chamber pressures and the force on the shaft. By ll 01 02 F E—fl‘h qu F l_L V 911 O1 P2 02 HYDRAULIC CYLINDER‘Ti" Figure 2b r—a—g ,___J Figure 2c Figure 2- (a) Hydraulic cylinder; (b) word bond graph; (c) free body diagram. 12 examining the flow equations the relationship between the piston velocity and the inlet and outlet flow is revealed. < u #9 l _ Q2 2.3.3 A = A2 The ideal device is represented by the bond graph model in Figure 3a. The relationships are modeled by two transformers and a l-junction. Now that the ideal model has been deve10ped, the static and dynamic models require only some additions to the basic model. First consider the static model which includes losses but no dynamics. Two mechanical losses occur that are relatively simple to model. A force is needed to overcome sealing friction around the shaft. This force is a function of the velocity of the shaft and is added to the l-junction as a resistance or dissipating element. The other mechanical loss is due to the force needed to shear the fluid in the narrow clearance between the cylinder and piston. This is also a function of the velocity of the piston and is added as a resistance element on the l-junction. Two fluid power losses occur due to leakage. One is internal leakage past the piston, which is dependent on the chamber pressures and the piston velocity. The component of leakage flow due to the piston velocity will be neglected in the model 13 developed here. The leakage flow will be considered to be dependent on chamber pressure only. This effect can be modeled by establishing a O-junction to represent the chamber pressure and inserting a resistance element on a l-junction between them. The flow is shown from P to P2. 1 This is done to establish a convention and causes no loss of generality. The other leakage loss occurs where the shaft passes out of the cylinder body. This leakage is de- pendent on the pressure in the chamber, P2, the velocity of the shaft and the pressure outside the cylinder. By estab— lishing a O-junction for that pressure the leakage past the shaft can be represented by inserting a resistance element on a l-junction between the O-junctions representing the two pressures. The effect of shaft velocity is ignored. Figure 3b shows the static bond graph model. The last model to be developed is the dynamic model. If the inertia of the piston and shaft are lumped together it can be simply added to the l-junction repre- senting their velocity. The compliance of the chamber is dependent on the pressure in the respective chambers and can be modeled by the simple addition of two capacitance elements to the O-junctions referring to these pressures. The inertia of the fluid is assumed negligible when com- pared to the inertia of the piston and shaft and so is not considered to affect the dynamics enough to warrant inclusion. The shaft and cylinder body are l4 w—P: TF l 1 F : d—o—P: TF ‘——J 2 Figure 3a P1 “—‘ol 0 TF L 1/A1\ R F FL“"'1; t'--::‘:; ‘23-0 ‘—"T.F a 2 A2 Figure 3b C .1 4 WAGE—h": i 9.4-1 1—=-: ‘Egoh—TF/IE R 2 P c Figure 3c Figure 3. (a) Ideal bond graph model; (b) static bond graph model; (c) dynamic model. 15 modeled with no compliance though this need not be the case. The dynamic model developed above is only one of many which may include some of the effects which may be important in predicting the response of the component. Figure 3c shows the dynamic bond graph model. The hydraulic cylinder was a relatively simple device to illustrate the bond graph modeling technique. Figure 4a is an axial piston positive displacement pump which will be modeled using a slightly different approach. Initially the pump will be considered as an ideal device having no mass or compliance. The fluid will also be considered ideal. No fluid leakage will occur nor any mechanical losses of any kind. The shaft shall be rotated causing the piston to both move in the cylinder and rotate. When the piston is at bottom dead center the volume of the cylinder shall be considered minimum. When the piston is at top dead center the volume shall be maximum. Both these positions shall be centered between the high and low pressure ports. The volume of the piston cylinder is given given by equation 2.3.4. v = A R 1- o e SIN 9 2.3.4 pp(°38) y These terms are defined in Figure 4b by taking the derivative of this equation with respect to time and expression for the volume flow rate is obtained 16 O M 4 . 11‘ Figure 4a 9% |..e. Figure 4b = Yoke Angle (Radians) O 9 = Shaft Angle (Radians) R = Radius of Piston Circle (Inches) A = Surface Area of Piston (Inchesz) Figure 4. (a) Positive displacement axial piston pump; (b) definition of variables. O = A R SING SINO O + A R l-cosO C O O 2. . v (pp y s)s (ppl. slosy)y 36 Since in the case of this particular motor 6y = 0 V = A R SING SINO 9 2.3.7 ( P P Y S) S Neglecting the discontinuity at 68 = 0° and 180° the piston and cylinder charge when O°—4 : ‘r’ AccumuLAToR ; 1 I O+OALA~SEU "“\‘\ ll 4”\\ 'c-tz“ , ’I‘ ’ 5“”'“’ ELE!SF+l-1F-h1 l—bkn g r Momz‘f/ V K / gt'. - ’CHecK VALVE ' 05-33:: aim;- :SUMP Figure 8. Preliminary bond graph of conveyor belt model. 1———L1——— r‘i a i if c>°é IF“ 1—‘1/ J /° 1 K C F 1 3‘1 1‘3 \/“ “‘1 L1 1——_ /)\z 55241 is: 1 1 1 is C K‘i )1255 9 R I O 1—* \x' /\ t SE R :yaAL1v_arup.a¢1 1W“FL EL ’V ’\ I K K 04-5! Figure 9. Bond graph of conveyor belt model. CHAPTER 3 THE CATALOG 3.1 Format The catalog, though not extensive, contains a representative collection of common fluid power system components. The models themselves do not represent all possible configurations but hopefully some of the most useful. Instructions for the use of the catalog are relatively simple. The components are grouped by main function only. For example, all motors are under a common listing, whether hydraulic or electric. There is no other organization other than the general groups at present; as the catalog grows more formal organization may be necessary. An attempt was made at consistency but in some instances it was difficult to maintain. A typical catalog page is shown in Figure 10. It should provide some idea of how the material in the catalog is presented. Three models are usually presented for each component representing an ideal, static and dynamic level of analysis. Occasionally more than one model is presented at certain levels to detail a special case. Appendix B contains the catalog. 26 2'7 P. I G . P4611 HYDRAUUC F. CYLINDER v A.- suntan: was” me 2. runs MGR Poms A" m ”.mr FAG. \ mama-1mm". Post Moons. (DEAL. * i ___EEA-STFF a}. ;. ’ /A. L r: L ——.- 4 V a ,__| 4—— TF at x ‘ ELEMENT: 1' Funcnou/Rsunonsms TF“ NEWER P. ' ¢1 lA. 1.ng mum Pg. “IA; NMIIHJ1“*NKL"1»F, r— ‘ ‘V 410.11,; at. p ' " R , ELEMENT °. mm/mnmmo Rs me SHEAR es“ 3““ R, mm”: LIAKMG. 13' 93/12: K“ Mums. Llama-I '1'.“ 2.. [2“ Figure 10. Example of a cataloged component. CHAPTER 4 CONCLUSIONS AND RECOMMENDATIONS 4.1 Summary In a presentation of the National Fluid Power Association Simulation Committee's proposed standard format to the National Conference on Fluid Power [1] in September of 1972 the following was stated. The Simulation Committee avoided the enticing avenue of writing a separate general equation, with variable parameters which each manufacturer would supply, for each type of valve, pump, cylinder, etc., commercially available. Such an approach would merely temporarily side-step the more general problem of guaranteeing compatibility of mathematical equations of two components which can be physically connected in the laboratory. It has been demonstrated that bond graph tech- niques can be used to effectively model fluid power system components not only in a manner that allows standard models of common components to be developed by also guarantees mathematical compatibility of the equations obtained when the component models are connected. A 28 29 method was also introduced to simplify the system synthesis from the component models. A beginning was made at collecting the component models into a catalog useful to the manufacturer and designer for exchanging information. The use of bond graph techniques also simplifies the task of obtaining the system equations. 4.2 Future Work If the models developed in this thesis using bond graph techniques prove unsatisfactory, further development must be done to develop models which can be standardized throughout the fluid power industry. Once these goals are accomplished manufacturers could provide the com- ponent parameters specified by the standard model. A computer library of models could be developed for use with a simulation program. Ideally the component models could be called out and assembled according to designer specification. The program would generate and solve the system equations and simulate system response. The designer may be able to use the design program without any specific knowledge about the component models, other than choosing appropriate levels of complexity. APPENDICES APPENDIX A A DEFINITION OF THE BOND GRAPH LANGUAGE RAW Associate Professor. Department of Mechanical Engineering. Michigan State University. East Lansing. Mich. I). C. KARIOI’I' Professor. Department of Mechanical Engineering. University of California. Davis. Calif. language Introduction The purpose of this paper is to present the basic definitions of the bond graph language in a compact but general form. The language presented herein is a formal mathematical system of definitions and symbolism. The descriptive names are stated in terms related to energy and power, because that is the historical basis of the multiport concept. It is important that the fundamental definitions of the lan- guage be standardized because an increasing number of people around the world are using and developing the bond graph language as a modeling tool in relation to multiport systems. A common set of reference definitions will be an aid to all in promoting ease of communication. Some care has been taken from the start to construct defini- tions and notation which are helpful in communicating with digital computers through special programs, such as ENPORT [5].l It is hoped that any subsequent modifications and exten- sions to the language will give due consideration to this goal. Principal sources of extended descriptions of the language and physical applications and interpretations will be found in Paynter [l], Karnopp and Rosenberg [2, 3], and Takahashi, et al. [4]. This paper is the most highly codified version of language definition, drawing as it does upon all previous efl'orts. Basic Definitions fluttlpert mutant. Ports. and lends. Multiport elements are the nodes of the graph, and are designated by alphanumeric characters. They are referred to as elements, for convenience. For example, in Fig. 1(a) two multiport elements, 1 and R, are shown. Ports of a multiport element are designated by line members is braehts ddgnats Muses at and of papa. Contributed by the Automatic Control Division for publication (without prusotatioa) in the Joanna or Dnuunc Brenna. MIMUIIIIN‘I. awn goers-ass. Wm reedved at ABME Headquarters. May 9. 1972. Paper 0. 72-bit- . mmumumwnm. A llefinition of the Bond Graph segments incident on the element at one and. Ports are places where the element can interact with its environment. For example, in Fig. 1(b) the 1 element has three ports and the R element has one port. We say that the 1 element is a 3- port, and the R element is a l-port. Bonds are formed when pairs of ports are joined. Thus bonds are connections between pairs of multiport elements. For example, in Fig. 1(c) two ports have been joined, forming a bond between the l and the R. Bead Graphs. A bond graph is a collection of multiport elements bonded together. In the general sense it is a linear graph whose nodes are multiport elements and whose branches are bonds. A bond graph may have one part or several parts, may have no loops or several loops, and in general has the characteristics of any linear graph. An example of a bond graph is given in Fig. 2. In part (a) a bond graph with seven elements and six bonds is shown. In part (b) the same graph has had its powers directed and bonds labeled. A bond graph fragment is a bond graph not all of whose ports have been paired as bonds. An example of a bond graph fragment is given in Fig. 1(a), which has one bond and two open, or unconnected, ports. PertVariahles. Associated with a given port are three direct and three integral quantities. Eflori, e(i), and flow, fa), are directly associated with a given port, and are called the port power variables. They are assumed to be scalar functions of an independent variable (1). Power, P0), is found directly from the scalar product of efiort and flow, as PG) - ¢(¢)'f(¢)- The direction of positive power is indicated by a half-arrow on the bond. Momentum, 32(1), and displacement, q(i), are related to the effort and flow at a port by integral relations. That is, Discussion on this paper will be accepted at ASME Headquarters until January 2. 1973 30 SE--'0‘--'F‘--i l 1 ‘ I 3 a. | ‘Zl J6 fi-u-s (a) Anesampieetahnndg with d (b) : (a)ahendgrapn:(h)tnehand this! n. Daniels 2 re 0'!" :nd pm - pa.) + 11. com sad «to - w.) + f. mm. respectively. Momentum and displaament 'are sometimes referred to as energy vu-iables. Energy, 8(1), is related to the power at a port by Ira) - Ea.) + f. Pmdx. The quantity E(1) — Ea.) represents the net energy transferred through the port in the dimotion of the half-arrow (i.e., positive power) over the interval (1., 1). In common bond graph usage the effort and the flow are often shown explicitly next to the port (or bond). The power, d'u- placemeot, momentum, and energy quantities are all implied. Basic Initipert Elements. There are nine basic multiport elements, grouped into four categories according to their energy characteristics. These elements and their definitions are sum- marised in Fig. 3. W Source of alert, written SE3, is defined by e :- 9(1). 80w offices, written SFZ, is defined by f- 1(1). Shapes. Capacitance, written ’5, C, is defined by a - etc) and «(o - at.) + fl: max. That is, the effort is a static function of the displacement and the displacement is the time integral of the flow. ;I,isdefinedby I - on) ma pa) - pa.) + f. «Max. That is, the flow is a static function of the momentum and the momentum is the time integral of the efiort. I nerionee, written ”plan. Residence, written-.4 13"“ defined by "m swims. an: SE—Lp ‘ _ e I nit) source of effort SF—'-p f I Ht) scarce of flow 6+ a I .(q) capacitance air) - ditch [Mt i‘+—— f I “9) inertance pin-pitch {e-dt it+- Me.” I 0 resistance l 2 \ fi‘rF— c' I .eez traHSformr lzm ”If! . ‘2 I 2 -—-’GY—--—y e' I r f2 gyrator r .2 - refl ._.'._,o_}_.' 9' - e2 3 e3 con-non effort 2 junction f| 4» f2 - 3 0 +l3q fI-fz-f commn flow 2 3 junction eI + a2 - e3 I 0 Fig. I Definitiens at to basic multiport ebmants Th’t is, a static relation ex'mts between the effort and flow at the port. . Junctions: Z-M Transformer, written :7: T? ;-: , is a linear 2-port element de- fined by e. - rn-e. and rn-f; - ’3, where m is the modulus. Gyrator, written 2 G!" 2, is a linear 2-port element defined by Cr ' ”ft and £3 -' "II: where r is the modulus. Both the transformer and grater preserve power (i.e., P. = P. in each case shown), and they must each have two ports, so they are called essential z-port junctions. Junctions: i-Part. . 1 3 Common wort junction, wntten —-7 O —-7 21 is a linear 3-port element defined by e. - e. - as (common efi'ort) and In +13 - I. I 0. (flow summation) 32 1 3 “'71 ““7. 2r zero junction. Common flow junction, written is a linear 3-port element defined by fl =f2=f1 l'i +8: '— l'. ='- 0. (common flow) and (effort summation) Other names for this element. are the eflort junction and the our junction. Both the common effort junction and the common flow junc- tion preserve power (i.e.. the rut power in is zero at all times). so they are called junctions. If the reference power directions are changed the signs on the summation relation must change accordingly. Extended Definitions lultlpert Fields . Storage Fields. M ultt'port capacitances, or C-fields, are written I n ' _.7 C T" — , and characterized by 2’] °° '1' = ¢6(qt, (I!) °°' (10),; = [tons and q;(1) = (Idle) + j: frOUdh, i = l to n. I n /I\-—-. 21' Multiport incrtancrs, or I-firlds, are written - and characterized by ft = 4M“. Pa. C and pa(1) -= p410) + 1:. ea(>\)d)\,s' = l to n. ...p.),i= lton, If a C-field or l-field is to have an associated “energy” state function then certain integrability conditions must be met by the 4% functions. In multiport terms the relations given in the foregoing are sufficient to define a C-field and l-field, respectively. Mixed multiport storage fields can arise when both C and I- type storage efl'ects are present simultaneously. The symbol for such an element consists of a set of 0’s and I ’s with appropriate ports indicated. For example, -—1 7 10] V33“- indicates the existence of a set ’12 of relations I: = 4MP» (It. 10:). 02 = 4’20", qt. Pa). ft = “(Pb 0:» Pa). and M0 = M.) + «mat. «hm = (We) + flaunt. Mt) = m.) + g «out. Multiport dzssz'pators, or lit-fields, are written )7 R, A! 2 P. and are characterized by ¢I'(¢’h fl) 53’ fir ° - If the R-field is to represent pure dissipation, then the power function associated with the R-field must be positive definite. M ultiport junctions include 0 junctions and l - junctions with n ports, n 2 2. The general case for each junction is given in the following. .eg,ffi)=0,i=lt¢0n. l n l "7 ” l . / 1*" 6|=¢g=....-‘=fl. f|=f3=....=fn 5'1 6-1 Modulated z-Port Junctions. The modulated tromformrr, or j m(n) M TF written 1 M TF' 2 implies the realtions .7 7 Ci "‘ "“01: moo-s - fa. where m(x) is a function of a set of variables, a. The modulated transformer preserves power; i.e., P1“) - P.(t). and '(x) l The modulated gyrator, or MGY, written 1 MGY 2 7 '7 implies the relations (I = “this sad CI "' r(‘).flr where r(s) is a function of set of variables, it. The modulated gyrator preserves power; i.e., P;(t) a P,(1). Junction Structure. The junction structure of a bond graph is the set of all 0, l, CY, and TF elements and their bonds and ports. The junction structure is an n-port that preserves power (i.e., the net power in is zero). The junction structure may be modulated (if it contains any M G Y’s or MTF's) or unmodulated. For example, the junction structure of the graph in Fig. 2(b) is a 4-port element with ports 1, 2, 5, and 6 and bonds 3 and 4. It contains the elements 0, TF, and 1. Physical Interpretations The physical interpretations given in this section are very succinctly stated. References [l], [2], and [3] contain extensive descriptions of physical applications and the interested reader is encouraged to consult them. Mechanical Translation. To represent mechanical translational phenomena we may make the following variable associations: 1 efl‘ort, e, is interpreted as force; 2 flow, I, is interpreted as velocity; 3 momentum, p, is interpreted as impulse-momentum; 4 displacement, q, is interpreted as mechanical displacement. Then the basic bond graph elements have the following in- terpretations: 1 source of effort, SE, is a force source; 2 source of flow, SP, is a velocity source (or may be thought nf An I mmntrin runners-mint}- .33 3 ran-n.8,reprmantsfrictionandothermedlanial lo- madafins; . 4 capaétance, 0, represents potential or elastic energy storage fleets (or spring-like behavior); 5 instance, I, reprusnta kinetic energ storm (or mass elects); 0 Warmer, TI, represents linear lever or linkage action (motion rutricted to small angles); 7 grater, 0?, represents gryational coupling or interaction between two ports; 8 0-iunetion represents a common force coupling among the mveral ineirhnt ports (or among the ports of the system bonded to the Gimfion); and 9 l-iunction represents a common velocity constraint among tbecva'alinddentportshramongtheportscfthesystem bonded to the l-junction). The extension of the interpretation to rotational mechanim is a natural one. It is based on the following mociations: 1 do“, e, is associated with torque; and 2 flow, I, is associated with angular velocity. Becamethedsvelopmentisaoeimilartotheonefor translational mechanic it will not be repeated here. mm Inelectricalnetworksthekeystepieto intupretaportasaterminal—pair. Thenvariable amociatiom may be made as follows: 1 eta-t, e, is interpreted as "flags; 2 1",], isinterpretedascurrenl; 3 momentum, p, 'w interpreted as flux linkage; 4 diplacement, q, is interpreted as charge. The basic bond graph elements have the following interpreta- tions: 1 some of eflort, SE, is a voltage source; 2 aouru of flow, SF, is a current source; 3 rm'utance, R, represent: electrical resistance; 4 «pianos, C, represents capacitance efiect (stored electric mg); 5 inertance, I, repraenta inductance (stored magnetic my); 6 transformer, TF, represents ideal transformer coupling; 7 gyrator, 0?, represents gntional coupling; 8 O-jnnction represents a parallel connection of ports (com- mon voltage across the terminal pairs); and . 9 l-iunction represents a series connection of ports (common current through the terminal pairs). Hydraulic Clea-Its. For fluid systems in which the significant fluid powu' is given as the product of pressure times volume flow, tin following variable associations are useful: 1 start, e, is interpreted as pressure; 2 flow, I, is interpreted as volumeflow. 3 momentum, p, is interpreted as pressure-momenturn; 4 displacement, q, is interpreted as volume. The basic bond graph elements have the following interpreta- tion: l source of effort, SE, is a pressure source; 2 return of flow, 8!, is a volume flow source; 3 resistance, 8, rqreaanta lo- deeta (e.g., due to leakage, valvm, crises, ate); 4 capacitance, C, reprmsnh accumulation or tank-like effects (had nor-so); 6 inertance, 1, represents slug-flow inertia eflecta; 6 o-junction represents a at of ports having a common Dre—m (as. a ripe m); 7 l-junction represents a set of ports having a common volume flow (i.e., series). Other Interpretations. This brief listing of physical interpreta- tions of bond graph elements is restricted to the simplest, most direct, applications. Such applications came first by virtue of historical development, and they are a natural point of de- parture for most classically trained scientists and engineers. As references [1—4] and the special issue collection in the Jomar. or Drmurrc Srsrsus, Msasonsusm', urn Cos- mos, Tame. ASME, Sept. 1972, indicate, bond graph elements can be used to describe an amazingly rich variety of complex dynamic systems. The limits of applicability are not bound by energy and power in the sense of physics; they include any areas in which there exist useful analogous quantities to energy. Concluding Remarks In this brief definition of the bond graph language two im- portant concepts have been omitted. The first is the concept of bond adiaatr’on, in which one of the two power variables is sup- premed, producing a pure signal coupling in place of the bond. Th'n is very useful modeling device in active systems. Further d'ncusion of activation will be found in reference [3], section 2.4, as well as in references [1] and [2]. Another concept omitted from discussion in this definitional paper is that of operational causality. It is by means of causality operations applied to bond graphs that the algebraic and dif- ferditial relations implied by the graph and its elements may be organised and reduced to state-space form in a systematic manner. Extensive discussion of causality will be found in reference [3], section 3.4 and diapter 5. Systematic formulation of relations is presented in reference [6]. References 1 Pa ter, H. M., Analysis and Design of Engineering Systems. . .I.T. Press. 1961. 2 Kai-no & D. C.. and Rosenberg. R. 0.. Analysis and Simulation of ultfport Systems. M.I.T. Press, 1968. 3. Karnopp D. C.. and Rosenberg. R. 0., “System Dy- namics: A nified Approach,” Division of Engineering Re- search. Coll? of Engineering, Michigan State Umversity, East Lansinfi‘ Mi ., 1971. 4 akahaslri. Y.. Rabins. M., and Auslander. D.. Control. Afidfif-Wmley. Reading, Ma, 1970 (see chapter 6 in par- ticu . E 5 Roses. “£1,062!leng User’sGuiriefthivisign of neenng c . o ngrneerrng, i igan tate Ugtpversity. East Lansin . Mich.. 1972. 6 Rosenberg. R. .. “StateSpace Formulation for Bond Graph Models of Multiport Systems.” Jousnar. or Drmmc Smears. Museums-r. ann Corn-non. TRANS. ASME. Series G. Vol. 93. No. 1. Mar. 1971. pp. 35-40. APPENDIX B COMPONENT CATALOG 34 LlNE - STRAlGl-lT P. 8 P, —’ -’ Q, Q, MODELZIDEAL M P. P‘ —-—7 1 -—-7 Q , Q1 Mum-mo H P. 1 p, ___._, t G. 3 a, R ELEMENT: Re cerium Fum Moses: Dynamo * I P. ~ P... T “NE—a: 2. FUND NWER PORTS FUNCTION] mammals fs' ‘3 I as I 1. i. l: C R ELEMENI". WNcnoN/nunmw 14 mile "Etna f4 . PVT-4 Cg Funo 'MW\CAL e.‘ 0 Ch“ ’6“ meme. ma 35 l“- P P1Q‘ P Q,—-> f P '—-I- Q3 Q. '—'Q3 3 FLUID NWERWRI'S MODEL: IDEAL. #l P P G... p -—;O—-» Q. 3 MocEL: Dvauwuc #l .1... P 0‘ P Q. I L‘, G3 C ELEMENT -. Fvucnon [mmmmp C4 Commueo MID-MEWICAL eq' (be/C4 COMPLIANCE. 36 ACCUMULATOR-SPRING LOADED P. ’ Pa. -—Q-"- ACCUMUtATOR -a’- P. _—J l: P. ‘ 1 --> -> Qt Q1 Armaments ZFLUID POWERRJRTS Moose: DYNAMK. * I C p. 13 [)1 _._h. 0 -—-h Q. t I. z ELEMENT: FUNCTION] mom C, cam e3. Chg/c, Moon. '. DVNNMC. * 7. C 21a. C h 1.. o-rt-ro-r'rr-r-t-fl 9J0, P l‘ It R. I/A t I1. K on FICE FLOW Ftuin ammuw LAMINAR. teams We: swam: PNDN manna FLUID SHEAR. Funniest/Enema») ‘fcr" Tau/R4 et‘ $b/c-b f1 ' ‘1 / 7-1 Cb. eg/A etc' $te/C-Ie flu ‘ Pin/tn e n. = Raf". 37 ORIF'ICE - P. m a P, 3". 1—1 —" s L _~ 2 Qt MODEL: STATIC. 3| P. P. ___,1_., Q. l, 1. R ELEMENT IE, ounce PM P. p, T 0R1 FICE. ‘23: 2 FLUID POWERPOZTS FUNCTlON / RELATlDNSl-“P fa'fi/Z‘s 38 VALVE- TWO WAY,SPR1NG LOADED CHECK P. k . P1- 9. P, ——-> -—b- -- CHECK VALVE —-5 0.. -____4 Q‘ Q. Q; Av EFFECTIVE sums NEEA 7- FW'D MER F0375 0N can. MooeL; STATIc “ I E 1. P. P. _.__, ____., Q0 Qt ELEMENT '. F'uNchON / Remnonsma 2., MODE Pg P, as 6),, P2“): Q1: 0 MODEL: DYNAMIC. 1"’I , at» VT 1--:-o-+-TF—9—1-‘~C .. 5 Pl th VAe l 1 I ELEMENT FVNCTION/KELATIONSNP 23 , MWMTED RESISTANCE f3 = ¢3 (83 (X) W“ “ANSWE{ e5 3 er, [AS I1 Nani/KO: BALL {'1 : P1/I) CB SPKth ee‘ ‘be/Ce 39 HYDRAULIC CYLINDER- DOUBLE ACTING *Gt *3!- H A.-mcEsasA,LErrm Agent‘s am,mrm MODEL: STATIC. * l P: TF 0. P. ‘Q. 9462. HYDRAULlC f CYLINDER v 2 mm PDWEK Poms I MEGLPOWEK M ‘lh. ' | F P: Q, A: ‘ ELEMENT WNCTtON/WMSWP ‘I'F.3 wasmuu P.‘ 2, IA. TF1, masseuse: P, = Cg IN. MocEL151’A1'ic. * I P. O ___...11: Q. ‘ . a ‘ l. /A. Fe K T i 1 —\T: ’ \ 41° ‘ "ht! ‘ K at [0 A‘ 2 ELEMENT Merton/Ware 25 mun m e5 - 15$; 25 Unmet Lanes. {-5 - eggs 3» mm W ‘fi.’ e../z., MODEL: DVNAMK. it I P. G. t n' 4 I X I K ‘7 1 1 Q... 133?“: 4 ' K I. ELEMENT Mum/Gunman I; Dim-am mama fg' Pelt; 40 PUMP- HYDRAULIC , no. AXIAL PISTON .4 Q. t. “9 +— Q2. 9. Oysuxe ANGLE D..- m amateur MteEL'. IDEAL *l Pa 7.71 Tr I-IYD DUMP ‘1)" .1... Z FLUID POWER. M13 I ”£01. MEL MT .1... . a .. w, P. In. °'“ ELEMENT Mam [alumnus 1F” m fs' PM”; MoocL: State. ‘21 P 4.1:), K G‘ I:\' In x, R‘: 1 1 ‘7" ‘a' 1 ‘1; {./" 4‘ Is ’ P‘ o R R Q. I’ R. easiest Wren/«Liam's 2., wmeeu. U68 2.3' Kn. Fla 2.. Fur": sweat 2.. r In?“ R5 SEAL menus en ' girl's 25 MM LIAKAeE ' (all: 21 MIN“ W " 61", k, mamas mace ’- 6912: MODEL: IIIIIMMKé 1* I .131. c I1. I g G. Irv. I\ ’04:, a <- 1 1‘: m.- 1‘13. t W" '3" .1. 0 -;§ c K 9. w r. ELEMENt Funcmluumme C“ Flute WBIUW ‘rL' “bit/Cu. c.. we mum's-um ¢n ' $nl¢n In ”MAL mm 'Fn ' P" [In 41 NOTOR- HYDRAULIC - PD. AXIAL PISTON ’1. Qt 1' mo. moms—.3:- .1... 99m mu; 7. FLUID FINE: Plus o..- More summer I m. mm MODEL: IDEAL *1 P‘1G‘ to 1 '—*TF '—* 4 |.. w, P. I a. /°~ ELEMENT Memes/MW! TF3. mat G.‘ Draws MosEL: Stem ‘I I a 1x9 n1 .1‘ E l;- 1 1 jam-- 1‘ 1 f, P. ’1 I, it V. A O -—> K K E a. ' ELEMENT mam/semis K, was W {3 ' e; Ills g. Mug: W fir cells It“ MW f.‘ $.11!» Ru wmaoe cos! c!" Kai-3‘ I‘M sumac» auto 9" ' Kai-n R's Sen Recess ct: ' 2n! 9.: MODEL: DYNAMK. * 1. r. I» . "—-O A K as 1 \., re K h \ 0 't‘ K ‘T’l 1'7? T’ '7‘ 1 .a-N LIZ." 4v ‘ 6, is R’ ‘ K C. ELEMENT Mmlmmmw C. Plano muss-saute 2.... angle... Cn Reno Maximum 2.. - Cb“ It... I. Wanna mm f.‘ 8 Pie/Ire REFERENCES REFERENCES Unruh, D. R. “A Standard Format for Mathematical Models of Fluid Power Systems." A paper pre- F sented to the National Conference on Fluid Power, ‘1 1972. . Karnopp, D. C. and Rosenberg, R. C. Analy§is and :*1 Simulation of Multiport Systems. M.I.T. Press, 1968. Karnopp, D. C. and Rosenberg, R. C. System Dynamics: A Unified Approach. East Lansing: Michigan State University, College of Engineering, Division of Engineering Research, 1972. Rosenberg, R. C. A Users Guide to Enport-4. East Lansing: Michigan State university, College of Engineering, Division of Engineering Research, 1972. Blackburn, J. F.; Reethof, G.; Shearer, J. L. Fluid Power Control. M.I.T. Press, 1960. Merritt, H. E. Hydraulic Control Systems. New York: Wiley & Sons, 1967. 42 "‘Willi!W