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linearized_therm

Course: MECHANICAL 2.141, Fall 2006
School: MIT
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Word Count: 1255

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THERMAL EXAMPLE: DAMPING work in air sealed outlet A BICYCLE PUMP WITH THE OUTLET SEALED. When the piston is depressed, a fixed mass of air is compressed. --mechanical work is done. The mechanical work done on the air is converted to heat. --the air temperature rises A temperature difference between the air and its surroundings induces heat flow. --entropy is produced The original work done is not recovered when...

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THERMAL EXAMPLE: DAMPING work in air sealed outlet A BICYCLE PUMP WITH THE OUTLET SEALED. When the piston is depressed, a fixed mass of air is compressed. --mechanical work is done. The mechanical work done on the air is converted to heat. --the air temperature rises A temperature difference between the air and its surroundings induces heat flow. --entropy is produced The original work done is not recovered when the piston is withdrawn to the original piston. --available energy is lost Mod. Sim. Dyn. Syst. Thermal damping example page 1 MODEL THIS SYSTEM GOAL: the simplest model that can describe thermal damping (the loss of available energy) ELEMENTS: TWO KEY PHENOMENA work-to-heat transduction a two port capacitor represents thermo-mechanical transduction entropy production a two port resistor represents heat transfer and entropy production BOUNDARY CONDITIONS: For simplicity assume a flow source on the (fluid-)mechanical side a constant temperature heat sink on the thermal side Mod. Sim. Dyn. Syst. Thermal damping example page 2 A BOND GRAPH IS AS SHOWN. P Tgas dSgas/dt thermal domain To dSo/dt Q(t): Sf dV/dt (fluid) mechanical domain C 0 R Se :To CAUSAL ANALYSIS: The integral causal form for the two-port capacitor (pressure and temperature outputs) is consistent with the boundary conditions and with the preferred causal form for the resistor Mod. Sim. Dyn. Syst. Thermal damping example page 3 CONSTITUTIVE EQUATIONS: Assume air is an ideal gas and use the constitutive equations derived above. T T o P P o = V c V v o R V c V v o R S S o exp mc v = + 1 S S o exp mc v Assume Fourier's law describes the heat transfer process. kA = Q l (T1 - T2) Mod. Sim. Dyn. Syst. Thermal damping example page 4 ANALYSIS: For simplicity, linearize the capacitor equations about a nominal operating point defined by So and Vo T T = o S o mcv P P = o S o mc v T TR = o (- V ) o Vo c v P P R = o + 1 (- V ) o Vo c v Po To mc mc v v -1 Inverse capacitance: C = Po Po R + 1 mc v Vo c v equality of the off-diagonal terms (the crossed partial derivatives) is established using Po Vo = mRTo Linearized constitutive equations To T = mc v Po P mc v where S Po R + 1 (- V ) Vo c v Po mcv S = S - So, V = V - Vo, T = T - To(So,Vo), P = P - Po(So,Vo) Mod. Sim. Dyn. Syst. Thermal damping example page 5 NETWORK REPRESENTATION The linearized model may be represented using the following bond graph P . -V T This representation shows that in the isothermal case ( T = 0 ) the fluid capacitance is Cfluid = Vo Po & in the constant-volume case ( V = 0 ) the thermal capacitance is C thermal = mc v To the strength of thermo-fluid coupling is To Po This uses the convention that the transformer coefficient is for the flow equation with output flow on the output power bond T & T & V = o S and hence T = o P Po Po though causal considerations may require the inverse equations. Mod. Sim. Dyn. Syst. Thermal damping example page 6 : : 1 C : Vo/Po TF To/Po 0 . S : C: mcv/To ALTERNATIVELY: It may be useful to express the parameters in term of easily-measured reference variables To and Vo as follows P . -V T : This representation shows that the strength of the coupling is Vo mR proportional to the (nominal) gas volume inversely proportional to the mass of gas V & V & V = o S and T = o P mR mR : 1 TF Vo/mR : 0 . S C : Vo2/mRTo C: mcv/To Mod. Sim. Dyn. Syst. Thermal damping example page 7 RESISTOR EQUATIONS The two-port resistor constitutive equations are & & 1 = Q = kA T1 - T2 S T1 l T1 & & 2 = Q = kA T1 - T2 S T T2 l 2 Linearize the resistor constitutive equations about a nominal operating point defined by T1o and T2o T2o & S1 2 kA = T1o l 1 & S2 T 2o 1 T T1o 1 T1o - 2 T2 T2o - This is in conductance form, f = Ge Note that this conductance matrix is singular: G =- T1oT2o 1 - =0 2 T12 T2o T1o T2o o this is because both entropy flows are associated with the same heat flow Mod. Sim. Dyn. Syst. Thermal damping example page 8 LINEARIZE ABOUT ZERO HEAT FLOW If the two nominal operating temperatures are equal, T1o = T2o = To , the linearized constitutive equations are 1 & S1 = kA To l 1 & S2 T o 1 T To 1 1 - T2 To - This simple form can be represented by an equally simple bond graph T1 . S1 1 T2 . S2 R : Tol/kA This follows the usual convention of writing the resistor parameter in resistance form Mod. Sim. Dyn. Syst. Thermal damping example page 9 ASSEMBLE THE PIECES... LINEARIZED BOND GRAPH Pgas . -Vgas : Tgas : . So : Sf 1 C TF Po/To : 0 C 1 R Se: To : Vo/Po mcv/To : Tol/kA Note the sign change on the capacitor thermal port (to avoid a superfluous 0-junction) Causal assignment indicates a first-order system Time-constant is determined by thermal (conduction) resistance and thermal capacitance Gas pressure is determined by fluid capacitance and (reflected) thermal capacitance and resistance Mod. Sim. Dyn. Syst. Thermal damping example : page 10 INCLUDE PISTON INERTIA BOND GRAPH . Vgas Tgas : . -Sout_of_gas : : : mpiston 1/Apiston : I TF : 1 C TF To/Po : 0 C 1 R Se Tambient : Vo/Po mcv/To : Tol/kA Causal analysis indicates a third-order system capable of resonant oscillation In this model the only damping is in the thermal domain heat transfer, entropy flow Mod. Sim. Dyn. Syst. Thermal damping example : page 11 SUMMARIZING THE GAS STORES ENERGY. It also acts as a transducer because there are two ways to store or retrieve this energy --two interaction ports energy can be added or removed as work or heat. The "energy-storing transducer" behavior is modeled as a two-port capacitor. --just like the energy-storing transducers we examined earlier. Mod. Sim. Dyn. Syst. Thermal damping example page 12 IF POWER FLOWS VIA THE THERMAL PORT, AVAILABLE ENERGY IS REDUCED --the system also behaves as a dissipator. The dissipative behavior is due to heat transfer. Gas temperature change due to compression and expansion does not dissipate available energy. If the walls were perfectly insulated, no available energy would be lost, but then, no heat would flow either. Without perfect insulation temperature gradients induce heat flow Heat flow results in entropy generation. Entropy generation means a loss of available energy. THE SECOND LAW. Mod. Sim. Dyn. Syst. Thermal damping example page 13 DISCUSSION ALL MODELS ARE FALSE. It is essential to understand what errors our models make, and when the errors should not be ignored. It is commonly assumed that modeling errors become significant at higher frequencies. --not so! Compression and expansion of gases is common in mechanical systems. Hydraulic systems typically include accumulators (to prevent over-pressure during flow transients). The most common design uses a compressible gas. Compression and expansion of the gas can dissipate (available) energy. Mod. Sim. Dyn. Syst. Thermal damping example page 14 This dissipation requires heat flow, but heat flow takes time. For sufficiently rapid compression and expansion, little or no heat will flow, and little or no dissipation will occur. The simplest model of a gas-charged accumulator may justifiably ignore "thermal damping". That is an eminently reasonable modeling decision but that model will be in error at low frequencies not high frequencies. THIS IS A GENERAL CHARACTERISTIC OF PHENOMENA DUE TO THERMODYNAMIC IRREVERSIBILITIES. Mod. Sim. Dyn. Syst. Thermal damping example page 15
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MIT - MECHANICAL - 2.141
NONLINEAR MECHANICAL SYSTEMS (MECHANISMS) The analogy between dynamic behavior in different energy domains can be useful. Closer inspection reveals that the analogy is not complete. One key distinction of mechanical systems is the role of kinematics - the
MIT - MECHANICAL - 2.141
ENERGY-STORING COUPLING BETWEEN DOMAINS MULTI-PORT ENERGY STORAGE ELEMENTS Context: examine limitations of some basic model elements. EXAMPLE: open fluid container with deformable walls P=gh h=AV V = Cf P where Cf = A g-fluid capacitor But when squeezed,
MIT - MECHANICAL - 2.141
NODICITY One of the important ways that physical system behavior differs between domains is the way elements may be connected. Electric circuit elements may be connected in series or in parallel - networks of arbitrary structure may be assembled This no
MIT - MECHANICAL - 2.141
Convection bonds and &quot;pseudo&quot; bondsEven in the simplest case of matter transport, power has two components, one due to the rate of work done, the other due to transported internal energy of the material. &quot;Pseudo&quot; bond graphs depict two distinct bonds. On
MIT - MECHANICAL - 2.141
MULTI-DOMAIN MODELING WHAT'S THE ISSUE? Why not just &quot;write down the equations&quot;? - standard formulations in different domains are often incompatible usually due to incompatible boundary conditions (choice of &quot;inputs&quot;) EXAMPLE: SIMPLE FLUID SYSTEM Scenario
MIT - MECHANICAL - 2.141
CAUSAL ANALYSISThings should be made as simple as possible - but no simpler. Albert Einstein How simple is &quot;as simple as possible&quot;? Causal assignment provides considerable insight.EXAMPLE: AQUARIUM AIR PUMPoscillatory motion in this direction coil leve
MIT - MECHANICAL - 2.141
EXAMPLE: ELECTROMAGNETIC SOLENOID A common electromechanical actuator for linear (translational) motion is a solenoid.Current in the coil sets up a magnetic field that tends to center the movable armature.Electromagnetic Solenoidpage 1 Neville HoganO
MIT - MECHANICAL - 2.141
CO-ENERGY (AGAIN) In the linear case, energy and co-energy are numerically equal. -the value of distinguishing between them may not be obvious. Why bother with co-energy at all? EXAMPLE: SOLENOID WITH MAGNETIC SATURATION. Previous solenoid constitutive eq
MIT - MECHANICAL - 2.141
LINEARIZED ENERGY-STORING TRANSDUCER MODELS Energy transduction in an electro-mechanical solenoid may be modeled by an energy-storing multiport.e= i..ICF . xEnergy transduction in an electric motor may be modeled by a gyrator.e= iGYF . xBut the
MIT - MECHANICAL - 2.141
Massachusetts Institute of Technology Department of Mechanical Engineering2.141 Modeling and Simulation of Dynamic SystemsINTRODUCTIONGOAL OF THE SUBJECT Methods for mathematical modeling of engineering systems Computational approaches are ubiquitous i
MIT - MECHANICAL - 2.141
Junction elements in network models. Classify by number of ports and examine the possible structures that result. Using only one-port elements, no more than two elements can be assembled.Combining two two-ports yields another two-port.At most two one-po
MIT - MECHANICAL - 2.141
EXAMPLE: THERMAL DAMPINGwork in air sealed outletA BICYCLE PUMP WITH THE OUTLET SEALED. When the piston is depressed, a fixed mass of air is compressed. -mechanical work is done. The mechanical work done on the air is converted to heat. -the air tempera
MIT - MECHANICAL - 2.141
NONLINEAR MECHANICAL SYSTEMS CANONICAL TRANSFORMATION S AND NUMERICAL INTEGRATION Jacobi Canonical Transformations A Jacobi canonical transformations yields a Hamiltonian that depends on only one of the conjugate variable sets. Assume dependence on new mo
MIT - MECHANICAL - 2.141
NETWORK MODELS OF TRANSMISSION LINES AND WAVE BEHAVIOR MOTIVATION: Ideal junction elements are power-continuous. Power out = power out instantaneously In reality, power transmission takes finite time. Power out power in Consider a lossless, continuous uni
MIT - MECHANICAL - 2.141
WORK-TO-HEAT TRANSDUCTION IN THERMO-FLUID SYSTEMS ENERGY-BASED MODELING IS BUILT ON THERMODYNAMICS - the fundamental science of physical processes. THERMODYNAMICS IS TO PHYSICAL SYSTEM DYNAMICS WHAT GEOMETRY IS TO MECHANICS. WHY SHOULD WE CARE ABOUT THERM
MIT - MECHANICAL - 2.154
MANEUVERING AND CONTROL OF MARINE VEHICLESMichael S. Triantafyllou Franz S. HoverDepartment of Ocean Engineering Massachusetts Institute of Technology Cambridge, Massachusetts USAManeuvering and Control of Marine Vehicles Latest Revision: November 5, 2
MIT - MECHANICAL - 2.154
111.1KINEMATICS OF MOVING FRAMESRotation of Reference FramesWe denote through a subscript the specific reference system of a vector. Let a vector ex pressed in the inertial frame be denoted as , and in a body-reference frame b . For the x x moment, w
MIT - MECHANICAL - 2.154
2VESSEL INERTIAL DYNAMICSWe consider the rigid body dynamics with a coordinate system affixed on the body. A common frame for ships, submarines, and other marine vehicles has the body-referenced xaxis forward, y-axis to port (left), and z-axis up. This
MIT - MECHANICAL - 2.154
3NONLINEAR COEFFICIENTS IN DETAILThe method of hydrodynamic coefficients is a somewhat blind series expansion of the fluid force in an attempt to provide a framework on which to base experiments and calculations to evaluate these terms. The basic dificu
MIT - MECHANICAL - 2.154
44.1VESSEL DYNAMICS: LINEAR CASESurface Vessel Linear ModelWe rst discuss some of the hydrodynamic parameters which govern a ship maneuvering in the horizontal plane. The b ody x-axis is forward and the y -axis is to port, so positive r has the vessel
MIT - MECHANICAL - 2.154
55.1SIMILITUDEUse of Nondimensional GroupsFor a consistent description of physical processes, we require that all terms in an equation must have the same units. On the basis of physical laws, some quantities are dependent on other, independent quantit
MIT - MECHANICAL - 2.154
6CAPTIVE MEASUREMENTSBefore making the decision to measure hydrodynamic derivatives, a preliminary search of the literature may turn up useful estimates. For example, test results for many hull-forms have already been published, and the basic lifting su
MIT - MECHANICAL - 2.154
7STANDARD MANEUVERING TESTSThis section describes some of the typical maneuvering tests which are performed on full-scale vessels, to assess stability and performance.7.1Dieudonn Spiral e1. Achieve steady speed and direction for one minute. No change
MIT - MECHANICAL - 2.154
88.1STREAMLINED BODIESNominal Drag ForceA symmetric streamlined body at zero angle of attack experiences only a drag force, which has the form 1 (109) FA = - CA Ao U 2 . 2 The drag coefficient CA has both pressure and skin friction components, and hen
MIT - MECHANICAL - 2.154
99.1SLENDER-BODY THEORYIntroductionConsider a slender body with d &lt; L, that is mostly straight. The body could be asymmetric in cross-section, or even flexible, but we require that the lateral variations are small and(Continued on next page)409 SLE
MIT - MECHANICAL - 2.154
1010.1PRACTICAL LIFT CALCULATIONSCharacteristics of Lift-Producing MechanismsAt a small angle of attack, a slender body experiences transverse force due to: helical body vortices, the blunt trailing end, and fins. The helical body vortices are stable
MIT - MECHANICAL - 2.154
11FINS AND LIFTING SURFACESVessels traveling at significant speed typically use rudders, elevators, and other streamlined control surfaces to maneuver. Their utility arises mainly from the high lift forces they can develop, with little drag penalty. Lif
MIT - MECHANICAL - 2.154
1212.1PROPELLERS AND PROPULSIONIntroductionWe discuss in this section the nature of steady and unsteady propulsion. In many marine vessels and vehicles, an engine (diesel or gas turbine, say) or an electric motor drives the(Continued on next page)12
MIT - MECHANICAL - 2.154
13ELECTRIC MOTORSModern underwater vehicles and surface vessels are making increased use of electrical ac tuators, for all range of tasks including weaponry, control surfaces, and main propulsion. This section gives a very brief introduction to the most
MIT - MECHANICAL - 2.154
14TOWING OF VEHICLESVehicles which are towed have some similarities to the vehicles that have been discussed so far. For example, towed vehicles are often streamlined, and usually need good directional stability. Some towed vehicles might have active li
MIT - MECHANICAL - 2.154
15TRANSFER FUNCTIONS &amp; STABILITYThe reader is referred to Laplace Transforms in the section MATH FACTS for preliminary material on the Laplace transform. Partial fractions are presented here, in the context of control systems, as the fundamental link be
MIT - MECHANICAL - 2.154
1616.116.1.1CONTROL FUNDAMENTALSIntroductionPlants, Inputs, and OutputsController design is about creating dynamic systems that behave in useful ways. Many target systems are physical; we employ controllers to steer ships, fly jets, position electri
MIT - MECHANICAL - 2.154
1717.1MODAL ANALYSISIntroductionThe evolution of states in a linear system occurs through independent modes, which can be driven by external inputs, and observed through plant output. This section provides the basis for modal analysis of systems. Thro
MIT - MECHANICAL - 2.154
(Continued on next page)18.2 Roots of Stability Nyquist Criterion87S(s) =e(s) 1 = , r(s) 1 + P (s)C(s)where P (s) represents the plant transfer function, and C(s) the compensator. The closedloop characteristic equation, whose roots are the poles of t
MIT - MECHANICAL - 2.154
1919.1LINEAR QUADRATIC REGULATORIntroductionThe simple form of loopshaping in scalar systems does not extend directly to multivariable (MIMO) plants, which are characterized by transfer matrices instead of transfer functions. The notion of optimality
MIT - MECHANICAL - 2.154
2020.1KALMAN FILTERIntroductionIn the previous section, we derived the linear quadratic regulator as an optimal solution for the full-state feedback control problem. The inherent assumption was that each state was known perfectly. In real applications
MIT - MECHANICAL - 2.154
2121.1LOOP TRANSFER RECOVERYIntroductionThe Linear Quadratic Regulator(LQR) and Kalman Filter (KF) provide practical solutions to the full-state feedback and state estimation problems, respectively. If the sensor noise and disturbance properties of th
MIT - MECHANICAL - 2.154
2222.122.1.1APPENDIX 1: MATH FACTSVectorsDefinitionA vector has a dual definition: It is a segment of a a line with direction, or it consists of its projection on a reference system 0xyz, usually orthogonal and right handed. The first form is indepe
MIT - MECHANICAL - 2.154
23 APPENDIX 2: ADDED MASS VIA LAGRANGIAN DYNAMICSThe development of rigid body inertial dynamics presented in a previous section depends on the rates of change of vectors expressed in a moving frame, specifically that of the vehicle. An alternative appro
MIT - MECHANICAL - 2.154
24APPENDIX 3: LQR VIA DYNAMIC PROGRAM MINGThere are at least two conventional derivations for the LQR; we present here one based on dynamic programming, due to R. Bellman. The key observation is best given through a loose example.(Continued on next pag
MIT - MECHANICAL - 2.154
25Further Robustness of the LQRThe most common robustness measures attributed to the LQR are a one-half gain reduction in any input channel, an infinite gain amplification in any input channel, or a phase error of plus or minus sixty degrees in any inpu
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Identification, Estimation, and Learning3-0-9 H-Level Graduate Credit Prerequisite: 2.151 or similar subject2.160Reference Books BooksLennart Ljung, &quot;System Identification: Theory for the User, Second Edition&quot;, Prentice-Hall 1999 Graham Goodwin and
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