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23 Pages

### modulated_transf

Course: MECHANICAL 2.141, Fall 2006
School: MIT
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Word Count: 1303

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MECHANICAL NONLINEAR 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 geometry of motion EXAMPLE: automobile internal combustion engine. reciprocating translational motion of a piston is converted to crankshaft rotation by a crank...

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MECHANICAL NONLINEAR 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 geometry of motion EXAMPLE: automobile internal combustion engine. reciprocating translational motion of a piston is converted to crankshaft rotation by a crank and slider mechanism. x1 x x2 Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 1 Consider torque applied to rotate the crankshaft. Inertia and friction opposing crankshaft rotation depend on position. if 0 or 180 small crankshaft rotation does not move the piston -- inertia is small if 90 crankshaft rotation is nearly proportional to piston translation -- inertia is large How should this phenomenon be modeled? Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 2 CAUTION: A POSITION-MODULATED INERTIA WOULD VIOLATE ENERGY CONSERVATION! Se I SHOULD THIS BE SOME KIND OF MULTIPORT? unclear the angle that "modulates" inertia also modulates stored kinetic energy that change of stored energy should be associated with a power port ... but that angle is already associated with a power port how can a separate port be justified? Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 3 EXAMINE THE MECHANISM KINEMATICS The mechanism imposes a relation between crankshaft angle and piston position x= where l1 = crank length l2 = connecting rod length check: = 0 = 90 = 180 OK x = l2 + l1 x= l22 l12 l22 l12sin2() + l1cos() x = l2 l1 Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 4 MODULATED TRANSFORMER The kinematic relation x = x() may be regarded as a transformation between coordinates. An energetically consistent description of the mechanism inertia includes a modulated transformer. For example, if the inertia of the crank and connecting rod are neglected, and the piston mass is assumed to be the dominant inertia, a bond graph of that model is Se 1 . dt MTF : 1 x . F v I :m Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 5 The angle-dependent transformer modulus is found by differentiating the kinematic relation dx dx d v = dt = dt = j() d j() is often called the Jacobian of the transformation x() For the crank and slider mechanism: dx l12 sin() cos() d = l1 sin() dt dt l22 l12sin2() l12 sin() cos() l1 sin() j() = l22 l12sin2() Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 6 The modulated transformer also relates crankshaft torque to piston force = j() F such that power in = power out = j() F = F j() = F v Thus the Jacobian is an angle-dependent moment arm. check: = 0 = 90 = 180 = 270 OK Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 7 j=0 j = l1 j=0 j = l1 CAUTION: there are subtleties to modulated transformers THE TRANSFORMER RELATIONS MAY BE PROPERLY DEFINED IN ONLY ONE DIRECTION. The map x is well defined -- there is an unique x for every The map x is poorly defined -- for some values of x the relation does not exist When the relation exists it is ambiguous -- many values of correspond to the same x 1. Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 8 NOTE: if the relation between displacements and flows is well defined in one direction then the relation between efforts and momenta* is well defined in the other e.g., v = j() is well defined for all but j() = 0 at = 0 and = 180 thus = j-1() v is not defined at these singular points conversely, = j() F is well defined for all but F = j-1() is not defined at the singular points *discussed later Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 9 2. THE MINIMUM NUMBER OF STATE VARIABLES TO DESCRIBE SYSTEM ENERGY MAY Causal analysis reveals a single independent energy storage element -- this suggests the minimum system order is one but an additional state equation is required to compute the angle -- in fact the minimum system order is two Furthermore, this causal form requires the inverse map (or the inverse jacobian) to find the angle. BE UNCLEAR Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 10 3. ACTIVE ELEMENTS MAY APPEAR AS PASSIVE ELEMENTS A constrained kinematically source may behave like a storage elements -- but significant differences remain. EXAMPLE: SIMPLE PENDULUM r y g x m develop a network model of this system Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 11 A general procedure to find network models for mechanisms: 1. IDENTIFY GENERALIZED COORDINATE(S) -- a set of variables that uniquely define system configuration in this case, angle may be used as a generalized coordinate IDENTIFY THE KINEMATIC RELATION(S) DEFINING OTHER RELEVANT DISPLACEMENTS (COORDINATES) 2. in this example, the vertical position, y, of the mass is relevant kinematics: y = r cos() Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 12 3. DIFFERENTIATE THE KINEMATIC RELATION(S) TO FIND THE TRANSFORMER dy d = r sin() dt dt velocities: dy v = dt d = dt Jacobian: j() = r sin() MODULUS 4. THE BASIC JUNCTION STRUCTURE IS . 1 j( ) : MTF y . :1 Other junction elements may be added as needed. Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 13 5. ADD ENERGY STORAGE, DISSIPATION AND SOURCE ELEMENTS. in this case kinetic energy storage (a rotational inertia) is associated with angular speed gravity may be described as a constant-force source that does work when vertical position changes j() y : :1 :1 Se :mg MTF mr2: I Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 14 Causal analysis shows only one independent energy storage element -- the inertia. -- that suggests the minimum system order is one. From the previous argument we know that an additional state variable is needed to compute the angle. -- the minimum system order is two. But in addition, this system can oscillate. Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 15 The combination of effort source and modulated transformer can behave like a mechanical spring. j() y : :1 Se :mg MTF is equivalent to C :: = mgr sin() The resulting graph indicates the system can oscillate. I :1 C :: = mgr sin() However, this "spring" has a "hidden" causal constraint -- effort (torque) must be the output variable. Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 16 EXAMPLE: SPRING PENDULUM THE PROCEDURE APPLIES TO SYSTEMS WITH MANY GENERALIZED COORDINATES r y g x m Find a network model for this mechanical system Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 17 1. IDENTIFY GENERALIZED COORDINATES polar coordinates, r and , are one suitable choice 2. IDENTIFY THE KINEMATIC RELATIONS DEFINING OTHER RELEVANT DISPLACEMENTS (COORDINATES) Cartesian coordinates, x and y, are relevant x = r sin() y = r cos() Note: By the definition of generalized coordinates, these relations will always be well-defined. Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 18 3. DIFFERENTIATE THE KINEMATIC RELATION(S) TO FIND THE TRANSFORMER Velocities: u = dx/dt v = dy/dt n = dr/dt = d/dt Jacobian: u = n sin() + r cos() v = n cos() - r sin() MODULI Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 19 4. THE BASIC JUNCTION STRUCTURE IS 1 x r MTF MTF MTF MTF 0 1 y 1 0 1 5. ADD ENERGY STORAGE AND SOURCE ELEMENTS. 0:Se 1 x r 1/k: C MTF MTF MTF MTF 0 1 I :m y 1 0 1 Se :mg I:m Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 20 MULTI-BOND NOTATION The basic bond graph notation is cumbersome for mechanisms of even modest complexity. The more compact multibond notation offsets this problem. In multibond notation, the spring pendulum model is C The multibond 1 MTF 1 I depicts a multiple (or vector) of power flows each with an associated multiple (vector) of efforts, flows, momenta, displacements e.g., x y u v Fx Fy px py Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 21 All associated elements are multiports I: multiport inertia e.g., m 0 I= 0 m C: multiport capacitor e.g., 0 0 C= 0 1/k MTF: multiport modulated transformer u sin() r cos() n = v cos() -r sin() Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 22 1 denotes a power-continuous junction where flows associated with each distinct power are identical e.g., utransformer = uinertia vtransformer = vinertia Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 23
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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
MIT - MECHANICAL - 2.160
Department of Mechanical Engineering Massachusetts Institute of Technology2.160 Identification, Estimation, and Learning End-of-Term ExaminationMay 17, 2006 1:00 3:00 pm (12:30 2:30 pm) Close book. Two sheets of notes are allowed. Show how you arrived a
MIT - MECHANICAL - 2.160
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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2.160 Identification, Estimation, and Learning Lecture Notes No. 1February 8, 2006Mathematical models of real-world systems are often too difficult to build based on first principles alone.Figure by MIT OCW. Figure by MIT OCW.System Identification; &quot;L
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2.160 Identification, Estimation, and LearningLecture Notes No. 2February 13, 2006 2. Parameter Estimation for Deterministic Systems 2.1 Least Squares Estimationu1 u2 M umMDeterministic System w/parameter Linearly parameterized modelyInput-output
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2.160 Identification, Estimation, and LearningLecture Notes No. 3February 15, 2006 2.3 Physical Meaning of Matrix PThe Recursive Least Squares (RLS) algorithm updates the parameter vector ^(t - 1) based on new data T (t ), y (t ) in such a way that the
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2.160 Identification, Estimation, and LearningLecture Notes No. 4February 17, 2006 3. Random Variables and Random ProcessesDeterministic System: Input OutputIn realty, the observed output is noisy and does not fit the model perfectly. In the determini
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2.160 System Identification, Estimation, and LearningLecture Notes No. 6February 24, 2006 4.5.1 The Kalman Gain Consider the error of a posteriori estimate xt et xt xt = xt t 1 + K (yt Ht xt t 1 )xt t = xt t 1 +K t (Ht xt +vt H t xt t 1
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2.160 System Identification, Estimation, and LearningLecture Notes No. 9March 8, 2006Part 2 Representation and LearningWe now move on to the second part of the course, Representation and Learning. You will learn various forms of system representation,
MIT - MECHANICAL - 2.160
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2.160 System Identification, Estimation, and LearningLecture Notes No. 13March 22, 20068. Neural Networks8.1 Physiological Background Neuro-physiology A Human brain has approximately 14 billion neurons, 50 different kinds of neurons. . uniform Massive
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