MODEL OF THE FINAL SECTION OF NAVIGATION OF A SELFGUIDED MISSILE STEERED BY A GYROSCOPE


 Christine Cannon
 2 years ago
 Views:
Transcription
1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 2, pp , Warsaw th Anniversary of JTAM MODEL OF THE FINAL SECTION OF NAVIGATION OF A SELFGUIDED MISSILE STEERED BY A GYROSCOPE Edyta ŁadyżyńskaKozdraś Faculty of Mechatronics, Warsaw University of Technology, Warsaw, Poland Zbigniew Koruba Faculty of Mechatronics and Machine Building, Kielce University of Technology, Kielce, Poland This paper presents the modelling of dynamics of a selfguided missile steeredusingagyroscope.insuchkindsofmissiles,themainelementisa selfguiding head, which is operated by a steered gyroscope. The paper presents the dynamics and the method of steering such a missile. Correctness of the developed mathematical model was confirmed by digital simulation conducted for a Maverick missile equipped with a gyroscope being an executive element of the system scanning the earth s surface and following the detected target. Both the dynamics of the gyroscope and the missile during the process of scanning and following the detected target were subject to digital analysis. The results were presented in a graphic form. Key words: selfguiding, dynamics and steering, steered gyroscope, missile 1. Introduction In the case of automatic steering of selfguided missiles, kinematic equations of spins were related to missile equations of dynamics, using the Boltzmann Hamel equations(ładyżyńskakozdraś et al., 2008), which were developed in a relative frame of reference Oxyz, rigidly connected with the missile(żyluk, 2009). Atthemomentofdetectingthetarget,itwasassumehatthemissile automatically passes from the flight on the programmed trajectory to trackingflightofthetargetaccordingtotheassumedalgorithm,inthiscase
2 474 E. ŁadyżyńskaKozdraś, Z. Koruba the method of proportional navigation. Controlling the motion of the missile is carried out by the deflection of control surfaces, i.e. direction steer and heightsteerattheanglesδ V andδ H respectively.thecontrollawsconstitute kinematic relations of deviations of set and realised flight parameters, stabilising the movement of the missile in channels of inclination and deflection. Therealisationofthedesiredflightpathofthemissileiscarriedoutbythe autopilot, which generates control signals based on the compounds derived for the executive system of steering. In the final section of navigation, various types of disturbances may affect the missile, such as wind or shock waves from shells exploding nearby. Therefore, additional stabilisation is necessary, in this case performed by the gyroscope. When searching for a ground target, the gyroscope axis, facing down, strictly outlines defined lines on the earth s surface with its extension. Theopticsystempositionedintheaxisofthegyroscope,withaspecificangle ofview,canthusfinhelightorinfraredsignalemittedbyamovingobject. Therefore, kinematic parameters of reciprocal movement of the missile head andgyroscopeaxisshouldbeselecteodetectthetargetatthehighestprobability possible. After locating the target(receiving the signal by the infrared detector), the gyroscope goes into the tracking mode, i.e. from this point its axis takes a specific position in space, being directed onto the target. 2. The general equations of missile dynamics Dynamic equations of missile motion in flight were derived in quasicoordinates ϕ,θ,ψandquasivelocitiesu,v,w,p,q,r(fig.1)usingtheboltzmann Hamel equations true for mechanical systems in the system associated with the object. The following correlation expresses them in a general form d T T + ω µ π µ k k r=1α=1 γµα r T ω α =Q µ (2.1) ω r where: α,µ,r=1,2,...,k, k numberofdegreesoffreedom, ω µ quasi velocities, T kineticenergyexpressedinquasivelocities, π µ quasi coordinates, Q µ generalisedforces, γ r αµ threeindexboltzmannfactors, expressed by the following correlation γ r αµ = k δ=1λ=1 k ( arδ q λ a rλ q δ ) b δµ b λα (2.2)
3 Model of the final section of navigation Fig. 1. Assumed reference system and parameters of the missile in the course of guidance Relations between quasivelocities and generalised velocities are k k ω δ = a δα q α q δ = b δµ ω µ (2.3) α=1 µ 1 where: q δ generalised velocities, q k generalised coordinates, a δα = =a δα (q 1,q 2,...,q k )and b δα =b δα (q 1,q 2,...,q k ) coordinatesbeingfunctions of generalised coordinates, while the following matrix correlation exists: [a δµ ]=[b δµ ] 1. The BoltzmannHamel equations, after calculating the values of Boltzmann factors and indicating kinetic energy in quasivelocities, a system of ordinary differential equations of the second order was received which describes the behaviour of the missile on the track during guidance. In the frame of reference associated with the moving object Oxyz, they have the following form M V+KMV=Q (2.4) where:mistheinertiamatrix,k kinematicrelationsmatrix,v velocity vector and
4 476 E. ŁadyżyńskaKozdraś, Z. Koruba m S z S y 0 m 0 S z 0 S x 0 0 m S M= y S x 0 0 S z S y I x I xy I xz S z 0 S x I yx I y I yz S y S x 0 I zx I zy I z 0 R Q R 0 P Q P K= 0 W V 0 R Q W 0 U R 0 P V U 0 Q P 0 V=col[U,V,W,P,Q,R] The vector of forces and moments of external forces Q affecting the moving missileisthesumofinteractionsofthecentre,inwhichitismoving.this vectorconsistsofforces:aerodynamicq a,gravitationalq g,steeringq δ and thrustq T.Theflyingmissileissteeredautomatically.Thesteeringisdonein twochannels:inclinationbytiltingtheheightsteerbyδ H anddeflectionby tiltingthedirectionsteerbyδ V where Q=Q g +Q a +Q δ +Q T =col[x,y,z,l,m,n] (2.5) X= mgsinθ+t 1 2 ρsv2 0(C xa cosβcosα+c ya sinβcosα C za sinα) +X Q Q+X δh δ H +X δv δ V Y=mgcosθsinφ 1 2 ρsv2 0 (C xasinβ C ya sinβ)+y P P+Y R R+Y δv δ V Z=mgcosθcosφ 1 2 ρsv2 0 (C xacosβsinα+c ya sinβsinα+c za cosα) +Z Q Q+Z δh δ H L= 1 2 ρsv2 0 l(c mxacosβcosα+c mya sinβsinα C mza sinα)+l P P +L R R+L δv δ V M= mgx c cosθcosφ 1 2 ρsv2 0 l(c mxasinβ+c mza cosβ)+m Q Q +M W W+M δh δ H N=mgx c cosθsinφ 1 2 ρsv2 0l(C mxa cosβsinα+c mya sinβsinα +C mza cosα)+n P P+N R R+N δv δ V
5 Model of the final section of navigation while:m missilemass,t missileenginethrustvector(fig.2),ρ(h) air density at a given flight altitude, l characteristic dimension(total length of the missile body), S area of reference surface(crosssection of rocket body), V 0 = U 2 +V 2 +W 2 velocityofthemissileflight, C xa,c ya,c za,c mxa, C mya,c mza dimensionlesscoefficientsofaerodynamiccomponentforces, respectively:resistancep xa,lateralp ya andbearingp za aswellasthemoment oftiltingm xa,inclinationm ya anddeflectionm za (Fig.2),X Q,Y P,Y R,Z Q, L P,L R,M Q,N P,N R derivativesofaerodynamicforcesandmomentswith respect to components of linear and angular velocities. Fig.2.Forcesandmomentsofforcesactingonthemissileinflight 3. Layout of gyroscopic selfguidance of missiles Figures 3 presents a simplified diagram of the layout of gyroscopic selfguidance ofmissilesontoagrounargetemittinginfraredradiation(e.g.atankora combat vehicle). Figure4showsthegeneralviewofthemissileusedinthescanningand tracking gyroscope, i.e. one which can perform programmed movements while searching for the target and tracking movements after detecting the ground target through an adequate steering mounted on its frames. The equations expressing dynamics of this kind of gyroscope steered by omitting the moments of inertia of its frames, have the following form dω yg2 J gk ( J go cosϑ g +J gk ω gx2 (ω gz2 +ω gy2 sinϑ g )+M k sinϑ g ω gz2 + dφ g ) ω gx2 cosϑ g +η c dψ g =M c
6 478 E. ŁadyżyńskaKozdraś, Z. Koruba Fig.3.Diagramoftheprocessofselfguidingamissileonatarget where dω ( gx2 J gk J gk ω gy2 ω gz2 +J go ω gz2 + dφ ) g dϑ g ω gy2 +η b d( J go ω gz2 + dφ ) g =M k M rk =M b (3.1)
7 Model of the final section of navigation Fig. 4. General view of the gyroscope and assumed systems of coordinates ω gx2 =Pcosψ g Rsinψ g + dϑ g ( dψg ) ω gy2 =(Pcosψ g +Rsinψ g )sinϑ g + +Q cosϑ g ( dψg ) ω gz2 =(Pcosψ g +Rsinψ g )cosϑ g +Q sinϑ g andj go,j gk momentsofinertiaofthegyroscoperotorintermsofitslongitudinalaxisandprecessionaxis,respectively,ϑ g,ψ g anglesofrotationof internalandexternalframesofthegyroscope,respectively,m k,m rk torques drivingtherotorofthegyroscopeandfrictionforcesintherotorbearingin the frame, respectively. ThesteeringmomentsM b,m c affectingthegyroscopeexpressedbyeqs. (3.1),foundonthePRboard,weshallpresentasfollows M b =Π(t o,t w )M p b (t)+π(t s,t k )M s b M c =Π(t o,t w )M p c (t)+π(t s,t k )M s c (3.2) where: Π( )arefunctionsoftherectangularimpulse, t o timemomentof thebeginningofspatialscanning,t w momentofdetectingthetarget,t s momentofthebeginningoftargettracking, t k momentofcompletingthe process of penetration, tracking and laser lighting of the target.
8 480 E. ŁadyżyńskaKozdraś, Z. Koruba Theprogramsteeringmoments M p b (t)and Mp c (t)puttheaxisofthe gyroscopeintherequiredmotionandarefoundbythemethodofsolvingthe inverse problem of dynamics(osiecki and Stefański, 2008) [ M p d 2 b (τ)=π(τ ϑ gz dϑ gz o,τ w ) dτ 2 +b b dτ 1 2 ( dψgz dτ ) 2sin2ϑgz dψ gz dτ cosϑ gz] 1 c b ( d Mc p 2 (τ)=π(τ ψ gz o,τ w ) dτ 2 cos 2 ϑ p gz +b dψ gz c + dψ gzdϑ gz dτ dτ dτ sin2ϑ gz (3.3) where + dϑ gz dτ cosϑ gz) 1 c c τ=tω Ω= J gon g J gk c b =c c = 1 J gk Ω 2 b b =b c = η b J gk Ω ϑ gz =ε ψ gz =σ (3.4) andϑ gz,ψ gz aretheanglesdeterminingthepositionofthegyroscopeaxisin space. For the target tracking status, values of angles determining the given positionofthegyroscopeaxisareequalto where:ε,σaretheanglesdeterminingagivenpositionofthetargetobservation line(tol). The angles ε, σ are defined by the following relationships constituting kinematic equations TOL(Mishin, 1990) where dr e =V pxe V cxe dε r ecosε=v pye V cye (3.5) dσ r e=v pze V cze V pxe =V 0 [cos(ε χ p )cosεcosγ p sinεsinγ p ] V pye = V 0 sin(ε χ p )cosγ p V pze =V 0 [cos(ε χ p )sinεcosγ p cosεsinγ p ] V cxe =V c [cos(ε χ c )cosεcosγ c sinεsinγ c ] V cye = V c sin(ε χ c )cosγ c V cze =V c [cos(ε χ c )sinεcosγ c cosεsinγ c ]
9 Model of the final section of navigation andr e distancebetweenthecentreofgravitymassofpranheground target,v 0,V c velocitiesofpranhegrounarget,γ p =θ α,χ p =ψ β positionanglesoftheprvelocityvector, γ c,χ c positionanglesofthe velocity vector of the ground target. Ifangulardeviationsbetweentherealangles ϑ g and ψ g andrequired anglesϑ gz andψ gz aredenotedasfollows e ψ =ψ g ψ gz e ψ =ψ g ψ gz (3.6) then the tracking steering moments of the gyroscope shall be expressed as follows ( Mb(τ)=Π(τ s de ) ϑ s,τ k ) k b e ϑ k c e ψ +h g dτ ( Mc(τ)=Π(τ s de ) (3.7) ψ s,τ k ) k b e ψ +k c e ϑ +h g dτ where k b = k b J gk Ω 2 k c = k c J gk Ω 2 h g = h g J gk Ω Thus, the steering law for the autopilot, taking into account the dynamics of inclination of steers, shall be expressed as follows d 2 δ m dδ ( m 2 +h mp +k mp δ m =k m (γ p γp )+h dγp ) m dγ p d 2 δ n 2 +h dδ ( n np +k npδ n =k n (χ p χ dχp ) p)+h m dχ p (3.8) where:b m,b n arethecoefficientsofstabilisingsteers,k mp,k np coefficients ofamplificationsofsteerdrives,h mp,h np coefficientsofsuppressionsofsteer drives,k m,k n coefficientsofamplificationsoftheautopilotregulator,h m,h n coefficients of suppressions of the autopilot regulator. Therequiredanglesofpositionγ p,χ p oftheprvelocityvectoraredetermined by the method of proportional navigation(koruba, 2001) dγp =a dϑ g γ dχ p =a χ dψ g (3.9) where:a γ,a χ aretherequiredcoefficientsofproportionalnavigation.
10 482 E. ŁadyżyńskaKozdraś, Z. Koruba 4. Obtained results and final conclusions The tested model of navigation and the steering of the selfguiding missile describes the fully autonomous motion of the Maverick combat vessel, which is to directly attack and destroy a ground target after being detected and identified. Figures 58 show selected results of digital simulation of the dynamics of the missile during selfguidance on a detected ground target. It was assumed that the missile was launched from an aircraftcarrier moving at a speed of 200m/sataheightof400m.Thetargetwasmovingalonganarcofacircleat the speed of 10m/s. The parameters of the steered gyroscope were as follows J gk = kgm 2 J go = kgm 2 n g =600 rad s while the coefficients of its regulator had the values k b = Nm rad k c =2.986 Nm rad η b =η c =0.01 Nms rad h g = Nms rad The coefficients of proportional navigation and parameters of the regulator of PR autopilot were as follows a γ =3.5 a χ =3.5 k m =2.703 Nm rad k n = Nm h m =9.887 Nms rad rad Fig. 5. Spatial trajectory of a selfguiding missile
11 Model of the final section of navigation Fig.6.Changeofanglesofattackandglideanglesinfunctionoftime Fig. 7. Angular position of the missile in function of time during guidance Fig.8.Angularpositionofthegyroscopeaxisinfunctionoftime
12 484 E. ŁadyżyńskaKozdraś, Z. Koruba With the parameters selected as in the above, the target was destroyed in 6softheflight. It should be emphasised that the gyroscope scanning and tracking layout proposed in this paper improves the stability of the system of missile selfguidance and increases resistance to vibrations born from the board of the missile itself. References 1.KorubaZ.,2001,DynamicsandControlofaGyroscopeonBoardofanFlying Vehicle, Monographs, Studies, Dissertations No. 25, Kielce University of Technology, Kielce[in Polish] 2. Koruba Z., ŁadyżyńskaKozdraś E., 2010, The dynamic model of combat target homing system of the unmanned aerial vehicle, Journal of Theoretical and Applied Mechanics, 48, 3, ŁadyżyńskaKozdraś E., 2008, Dynamical analysis of a 3D missile motion under automatic control,[in:] Mechanika w Lotnictwie MLXIII 2008, J. Maryniak(Edit.), PTMTS, Warszawa 4. ŁadyżyńskaKozdraś E., 2009, The control laws having a form of kinematic relations between deviations in the automatic control of a flying object, Journal of Theoretical and Applied Mechanics, 47, 2, ŁadyżyńskaKozdraś E., Maryniak J., Żyluk A., Cichoń M., 2008, Modeling and numeric simulation of guided aircraft bomb with preset surface target, Scientific Papers of the Polish Naval Academy, XLIX, 172B, Mishin V.P., 1990, Missile Dynamics, Mashinostroenie, Moskva, pp Osiecki J., Stefański K., 2008, On a method of target detection and tracking used in air defence, Journal of Theoretical and Applied Mechanics, 46, 4, ŻylukA.,2009,Sensitivityofabombtowinurbulence,JournalofTheoretical and Applied Mechanics, 47, 4, Model końcowego odcinka nawigacji samonaprowadzającego pocisku rakietowego sterowanego giroskopem Streszczenie W pracy zaprezentowano modelowanie dynamiki samonaprowadzającego pocisku rakietowego sterowanego przy użyciu giroskopu. W tego rodzaju pociskach rakietowych atakujących samodzielnie wykryte cele głównym elementem jest samonapro
13 Model of the final section of navigation wadzająca głowica, której napęd stanowi giroskop sterowany. W pracy przedstawiona została dynamika i sposób sterowania takiego pocisku. Poprawność opracowanego modelu matematycznego potwierdziła symulacja numeryczna przeprowadzona dla pocisku klasy Maverick wyposażonego w giroskop będący elementem wykonawczym skanowania powierzchni ziemi i śledzenia wykrytego na niej celu. Analizie numerycznej poddana została zarówno dynamika giroskopu, jak i pocisku podczas procesu skanowania i śledzenia wykrytego celu. Wyniki przedstawione zostały w postaci graficznej. Manuscript received July 11, 2011; accepted for print September 12, 2011
CONTROL AND CORRECTION OF A GYROSCOPIC PLATFORM MOUNTED IN A FLYING OBJECT
JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45,1,pp.4151,Warsaw2007 CONTROL AND CORRECTION OF A GYROSCOPIC PLATFORM MOUNTED IN A FLYING OBJECT Zbigniew Koruba Faculty of Mechatronics and Machine Buiding,
More informationDYNAMICS OF A CONTROLLED ANTIAIRCRAFT MISSILE LAUNCHER MOUNTED ON A MOVEABLE BASE
JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 2, pp. 279295, Warsaw 2010 DYNAMICS OF A CONTROLLED ANTIAIRCRAFT MISSILE LAUNCHER MOUNTED ON A MOVEABLE BASE Zbigniew Koruba Zbigniew Dziopa Izabela Krzysztofik
More informationOnboard electronics of UAVs
AARMS Vol. 5, No. 2 (2006) 237 243 TECHNOLOGY Onboard electronics of UAVs ANTAL TURÓCZI, IMRE MAKKAY Department of Electronic Warfare, Miklós Zrínyi National Defence University, Budapest, Hungary Recent
More informationThe impact of launcher turret vibrations control on the rocket launch
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 63, No. 3, 2015 DOI: 10.1515/bpasts20150083 The impact of launcher turret vibrations control on the rocket launch Z. DZIOPA and Z.
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationA B = AB sin(θ) = A B = AB (2) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product
1 Dot Product and Cross Products For two vectors, the dot product is a number A B = AB cos(θ) = A B = AB (1) For two vectors A and B the cross product A B is a vector. The magnitude of the cross product
More informationQuadcopters. Presented by: Andrew Depriest
Quadcopters Presented by: Andrew Depriest What is a quadcopter? Helicopter  uses rotors for lift and propulsion Quadcopter (aka quadrotor)  uses 4 rotors Parrot AR.Drone 2.0 History 1907  BreguetRichet
More informationKINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
More informationPhysics 2A, Sec B00: Mechanics  Winter 2011 Instructor: B. Grinstein Final Exam
Physics 2A, Sec B00: Mechanics  Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry
More informationAdequate Theory of Oscillator: A Prelude to Verification of Classical Mechanics Part 2
International Letters of Chemistry, Physics and Astronomy Online: 213919 ISSN: 22993843, Vol. 3, pp 11 doi:1.1852/www.scipress.com/ilcpa.3.1 212 SciPress Ltd., Switzerland Adequate Theory of Oscillator:
More informationA MONTE CARLO DISPERSION ANALYSIS OF A ROCKET FLIGHT SIMULATION SOFTWARE
A MONTE CARLO DISPERSION ANALYSIS OF A ROCKET FLIGHT SIMULATION SOFTWARE F. SAGHAFI, M. KHALILIDELSHAD Department of Aerospace Engineering Sharif University of Technology Email: saghafi@sharif.edu Tel/Fax:
More informationPrerequisites 20122013
Prerequisites 20122013 Engineering Computation The student should be familiar with basic tools in Mathematics and Physics as learned at the High School level and in the first year of Engineering Schools.
More informationLinear StateSpace Control Systems
Linear StateSpace Control Systems Prof. Kamran Iqbal College of Engineering and Information Technology University of Arkansas at Little Rock kxiqbal@ualr.edu Course Overview State space models of linear
More informationRigid body dynamics using Euler s equations, RungeKutta and quaternions.
Rigid body dynamics using Euler s equations, RungeKutta and quaternions. Indrek Mandre http://www.mare.ee/indrek/ February 26, 2008 1 Motivation I became interested in the angular dynamics
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationANALYSIS OF THE DYNAMICS AND CONTROL OF THE MODIFIED OPTICAL TARGET SEEKER USED IN ANTIAIRCRAFT ROCKET MISSILES
JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 3, pp. 629639, Warsaw 2014 ANALYSIS OF THE DYNAMICS AND CONTROL OF THE MODIFIED OPTICAL TARGET SEEKER USED IN ANTIAIRCRAFT ROCKET MISSILES Daniel Gapiński,
More informationDESCRIPTION OF MOTION OF A MOBILE ROBOT BY MAGGIE S EQUATIONS
JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 3, pp. 511521, Warsaw 2005 DESCRIPTION OF MOTION OF A MOBILE ROBOT BY MAGGIE S EQUATIONS Józef Giergiel Wiesław Żylski Rzeszow University of Technology
More informationPhysics 1A Lecture 10C
Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. Oprah Winfrey Static Equilibrium
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationDesign of Retraction Mechanism of Aircraft Landing Gear
Mechanics and Mechanical Engineering Vol. 12, No. 4 (2008) 357 373 c Technical University of Lodz Design of Retraction Mechanism of Aircraft Landing Gear Micha l Hać Warsaw University of Technology, Institute
More informationChapter 24 Physical Pendulum
Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...
More informationTRACKING ERRORS DECREASING IN CNC SYSTEM OF MACHINE TOOLS
Andrzej WAWRZAK Kazimierz KARWOWSKI Krzysztof KARWOWSKI Sławomir MANDRA Mirosław MIZAN TRACKING ERRORS DECREASING IN CNC SYSTEM OF MACHINE TOOLS ABSTRACT In the paper the procedures and practical methods
More informationSingle and Double plane pendulum
Single and Double plane pendulum Gabriela González 1 Introduction We will write down equations of motion for a single and a double plane pendulum, following Newton s equations, and using Lagrange s equations.
More informationChapter 9 Rotation of Rigid Bodies
Chapter 9 Rotation of Rigid Bodies 1 Angular Velocity and Acceleration θ = s r (angular displacement) The natural units of θ is radians. Angular Velocity 1 rad = 360o 2π = 57.3o Usually we pick the zaxis
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationA R C H I V E S O F M E T A L L U R G Y A N D M A T E R I A L S Volume 52 2007 Issue 3
A R C H I V E S O F M E T A L L U R G Y A N D M A T E R I A L S Volume 52 2007 Issue 3 Z. JĘDRZYKIEWICZ MODELING AND SIMULATION TESTS ON LIFT VALVES OPERATING IN THE HYDRAULIC SYSTEM OF A PRESSURE CASTING
More informationMODELLING A SATELLITE CONTROL SYSTEM SIMULATOR
National nstitute for Space Research NPE Space Mechanics and Control Division DMC São José dos Campos, SP, Brasil MODELLNG A SATELLTE CONTROL SYSTEM SMULATOR Luiz C Gadelha Souza gadelha@dem.inpe.br rd
More informationLesson 5 Rotational and Projectile Motion
Lesson 5 Rotational and Projectile Motion Introduction: Connecting Your Learning The previous lesson discussed momentum and energy. This lesson explores rotational and circular motion as well as the particular
More informationPhysics1 Recitation7
Physics1 Recitation7 Rotation of a Rigid Object About a Fixed Axis 1. The angular position of a point on a wheel is described by. a) Determine angular position, angular speed, and angular acceleration
More informationRotation: Kinematics
Rotation: Kinematics Rotation refers to the turning of an object about a fixed axis and is very commonly encountered in day to day life. The motion of a fan, the turning of a door knob and the opening
More informationResponse to Harmonic Excitation Part 2: Damped Systems
Response to Harmonic Excitation Part 2: Damped Systems Part 1 covered the response of a single degree of freedom system to harmonic excitation without considering the effects of damping. However, almost
More informationProblem of the gyroscopic stabilizer damping
Applied and Computational Mechanics 3 (2009) 205 212 Problem of the gyroscopic stabilizer damping J. Šklíba a, a Faculty of Mechanical Engineering, Technical University in Liberec, Studentská 2, 461 17,
More informationSIX DEGREEOFFREEDOM MODELING OF AN UNINHABITED AERIAL VEHICLE. A thesis presented to. the faculty of
SIX DEGREEOFFREEDOM MODELING OF AN UNINHABITED AERIAL VEHICLE A thesis presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirement
More informationPeople s Physics book 3e Ch 251
The Big Idea: In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate
More informationBead moving along a thin, rigid, wire.
Bead moving along a thin, rigid, wire. odolfo. osales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing
More informationOUTCOME 2 KINEMATICS AND DYNAMICS TUTORIAL 2 PLANE MECHANISMS. You should judge your progress by completing the self assessment exercises.
Unit 60: Dynamics of Machines Unit code: H/601/1411 QCF Level:4 Credit value:15 OUTCOME 2 KINEMATICS AND DYNAMICS TUTORIAL 2 PLANE MECHANISMS 2 Be able to determine the kinetic and dynamic parameters of
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME  TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationKepler s Laws of Planetary Motion and Newton s Law of Universal Gravitation
Kepler s Laws of lanetary Motion and Newton s Law of Universal Gravitation Abstract These notes were written with those students in mind having taken (or are taking) A Calculus and A hysics Newton s law
More informationChapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.
Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems
More informationAngular Velocity. The purpose of this verification is to confirm that the angular velocity algorithm used by the program is working correctly.
E R I I C A T I O # 5 Angular elocity The purpose of this verification is to confirm that the angular velocity algorithm used by the program is working correctly. The example consists of a slope with two
More informationRotational Dynamics. Luis Anchordoqui
Rotational Dynamics Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation ( O ). The radius of the circle is r. All points on a straight line
More informationOrbital Mechanics. Angular Momentum
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
More informationCh.8 Rotational Equilibrium and Rotational Dynamics.
Ch.8 Rotational Equilibrium and Rotational Dynamics. Conceptual question # 1, 2, 9 Problems# 1, 3, 7, 9, 11, 13, 17, 19, 25, 28, 33, 39, 43, 47, 51, 55, 61, 79 83 Torque Force causes acceleration. Torque
More informationFirst Semester Learning Targets
First Semester Learning Targets 1.1.Can define major components of the scientific method 1.2.Can accurately carry out conversions using dimensional analysis 1.3.Can utilize and convert metric prefixes
More informationChapter 8 Rotational Motion
Chapter 8 Rotational Motion Textbook (Giancoli, 6 th edition): Assignment 9 Due on Thursday, November 26. 1. On page 131 of Giancoli, problem 18. 2. On page 220 of Giancoli, problem 24. 3. On page 221
More informationThe quest to find how x(t) and y(t) depend on t is greatly simplified by the following facts, first discovered by Galileo:
Team: Projectile Motion So far you have focused on motion in one dimension: x(t). In this lab, you will study motion in two dimensions: x(t), y(t). This 2D motion, called projectile motion, consists of
More informationChapt. 10 Rotation of a Rigid Object About a Fixed Axis
Chapt. otation of a igid Object About a Fixed Axis. Angular Position, Velocity, and Acceleration Translation: motion along a straight line (circular motion?) otation: surround itself, spins rigid body:
More informationGyroscope Angular Rate Sensor Three main types
Gyroscopes Gyroscope Angular Rate Sensor Three main types Spinning Mass Optical Ring Laser Gyros Fiber Optic Gyros Vibratory Coriolis Effect devices MEMS 4 March 2011 EE 570: Location and Navigation: Theory
More informationPath Tracking for a Miniature Robot
Path Tracking for a Miniature Robot By Martin Lundgren Excerpt from Master s thesis 003 Supervisor: Thomas Hellström Department of Computing Science Umeå University Sweden 1 Path Tracking Path tracking
More information5.2 Rotational Kinematics, Moment of Inertia
5 ANGULAR MOTION 5.2 Rotational Kinematics, Moment of Inertia Name: 5.2 Rotational Kinematics, Moment of Inertia 5.2.1 Rotational Kinematics In (translational) kinematics, we started out with the position
More informationProjectile Motion  Worksheet
Projectile Motion  Worksheet From the given picture; you can see a skateboarder jumping off his board when he encounters a rod. He manages to land on his board after he passes over the rod. 1. What is
More informationAircraft Flight Dynamics!
Aircraft Flight Dynamics Robert Stengel, Princeton University, 2014 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2014 by Robert Stengel. All rights reserved. For educational
More informationBasic Principles of Inertial Navigation. Seminar on inertial navigation systems Tampere University of Technology
Basic Principles of Inertial Navigation Seminar on inertial navigation systems Tampere University of Technology 1 The five basic forms of navigation Pilotage, which essentially relies on recognizing landmarks
More informationIMU Components An IMU is typically composed of the following components:
APN064 IMU Errors and Their Effects Rev A Introduction An Inertial Navigation System (INS) uses the output from an Inertial Measurement Unit (IMU), and combines the information on acceleration and rotation
More informationVersion PREVIEW Practice 8 carroll (11108) 1
Version PREVIEW Practice 8 carroll 11108 1 This printout should have 12 questions. Multiplechoice questions may continue on the net column or page find all choices before answering. Inertia of Solids
More informationA Simulation Study on Joint Velocities and End Effector Deflection of a Flexible Two Degree Freedom Composite Robotic Arm
International Journal of Advanced Mechatronics and Robotics (IJAMR) Vol. 3, No. 1, JanuaryJune 011; pp. 90; International Science Press, ISSN: 09756108 A Simulation Study on Joint Velocities and End
More informationSIGNAAL MODULAR COMBAT SYSTEM TACTICOS CAPABILITIES DURING HELICOPTER TASKS REALIZATION
ZESZYTY NAUKOWE AKADEMII MARYNARKI WOJENNEJ ROK XLIX NR 4 (175) 2008 Arkadiusz Panasiuk Polish Naval Academy SIGNAAL MODULAR COMBAT SYSTEM TACTICOS CAPABILITIES DURING HELICOPTER TASKS REALIZATION ABSTRACT
More informationSOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
More informationPresentation of problem T1 (9 points): The Maribo Meteorite
Presentation of problem T1 (9 points): The Maribo Meteorite Definitions Meteoroid. A small particle (typically smaller than 1 m) from a comet or an asteroid. Meteorite: A meteoroid that impacts the ground
More informationCoordinated 6DOF Control of Dual Spacecraft Formation
Coordinated 6DOF Control of Dual Spacecraft Formation M.S.Siva, Scientist, ISRO Satellite Centre, Bangalore, INDIA R.Pandiyan, Scientist, ISRO Satellite Centre, Bangalore, INDIA Debasish Ghose, Professor,
More informationLecture L293D Rigid Body Dynamics
J. Peraire, S. Widnall 16.07 Dynamics Fall 2009 Version 2.0 Lecture L293D Rigid Body Dynamics 3D Rigid Body Dynamics: Euler Angles The difficulty of describing the positions of the bodyfixed axis of
More informationWe have just introduced a first kind of specifying change of orientation. Let s call it AxisAngle.
2.1.5 Rotations in 3D around the origin; Axis of rotation In threedimensional space, it will not be sufficient just to indicate a center of rotation, as we did for plane kinematics. Any change of orientation
More informationLinear and Rotational Kinematics
Linear and Rotational Kinematics Starting from rest, a disk takes 10 revolutions to reach an angular velocity. If the angular acceleration is constant throughout, how many additional revolutions are required
More informationTRANSCOMP XIV INTERNATIONAL CONFERENCE COMPUTER SYSTEMS AIDED SCIENCE, INDUSTRY AND TRANSPORT
TRANSCOMP XIV INTERNATIONAL CONFERENCE COMPUTER SYSTEMS AIDED SCIENCE, INDUSTRY AND TRANSPORT Tadeusz STEFAŃSKI 1 Łukasz ZAWARCZYŃSKI 2 Induction motor, DTC control method, angular velocity and torque
More information9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration
Ch 9 Rotation 9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration Q: What is angular velocity? Angular speed? What symbols are used to denote each? What units are used? Q: What is linear
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA FURTHER MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 11  NQF LEVEL 3 OUTCOME 3  ROTATING SYSTEMS
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA FURTHER MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 11  NQF LEVEL 3 OUTCOME 3  ROTATING SYSTEMS TUTORIAL 1  ANGULAR MOTION CONTENT Be able to determine the characteristics
More informationPROPORTIONAL FLOW CONTROLLER MODEL IN MATLABSIMULINK MODEL PROPORCJONALNEGO REGULATORA PRZEPŁYWU W MATLABSIMULINK
TECHNICAL TRANSACTIONS MECHANICS CZASOPISMO TECHNICZNE MECHANIKA 1M/2013 MARIUSZ KRAWCZYK* PROPORTIONAL FLOW CONTROLLER MODEL IN MATLABSIMULINK MODEL PROPORCJONALNEGO REGULATORA PRZEPŁYWU W MATLABSIMULINK
More informationExample (1): Motion of a block on a frictionless incline plane
Firm knowledge of vector analysis and kinematics is essential to describe the dynamics of physical systems chosen for analysis through ewton s second law. Following problem solving strategy will allow
More informationStandard Terminology for Vehicle Dynamics Simulations
Standard Terminology for Vehicle Dynamics Simulations Y v Z v X v Michael Sayers The University of Michigan Transportation Research Institute (UMTRI) February 22, 1996 Table of Contents 1. Introduction...1
More informationRotation, Angular Momentum
This test covers rotational motion, rotational kinematics, rotational energy, moments of inertia, torque, crossproducts, angular momentum and conservation of angular momentum, with some problems requiring
More informationPhysics 211 Week 12. Simple Harmonic Motion: Equation of Motion
Physics 11 Week 1 Simple Harmonic Motion: Equation of Motion A mass M rests on a frictionless table and is connected to a spring of spring constant k. The other end of the spring is fixed to a vertical
More informationIntroduction to Engineering System Dynamics
CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are
More informationDevelopment of models for high precision simulation of the space mission MICROSCOPE
Development of models for high precision simulation of the space mission MICROSCOPE Stefanie Bremer Meike List Hanns Selig Claus Lämmerzahl Outline The space mission MICROSCOPE High Performance Satellite
More informationEstimation of TireRoad Friction by Tire Rotational Vibration Model
53 Research Report Estimation of TireRoad Friction by Tire Rotational Vibration Model Takaji Umeno Abstract Tireroad friction is the most important piece of information used by active safety systems.
More informationAP1 Dynamics. Answer: (D) foot applies 200 newton force to nose; nose applies an equal force to the foot. Basic application of Newton s 3rd Law.
1. A mixed martial artist kicks his opponent in the nose with a force of 200 newtons. Identify the actionreaction force pairs in this interchange. (A) foot applies 200 newton force to nose; nose applies
More informationLesson 03: Kinematics
www.scimsacademy.com PHYSICS Lesson 3: Kinematics Translational motion (Part ) If you are not familiar with the basics of calculus and vectors, please read our freely available lessons on these topics,
More informationA1. An object of mass m is projected vertically from the surface of a planet of radius R p and mass M p with an initial speed v i.
OBAFMI AWOLOWO UNIVRSITY, ILIF, IF, NIGRIA. FACULTY OF SCINC DPARTMNT OF PHYSICS B.Sc. (Physics) Degree xamination PHY GNRAL PHYSICS I TUTORIAL QUSTIONS IN GRAVITATION, FLUIDS AND OSCILLATIONS SCTION
More informationPrecise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility
Precise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility Renuka V. S. & Abraham T Mathew Electrical Engineering Department, NIT Calicut Email : renuka_mee@nitc.ac.in,
More informationPhysics 271 FINAL EXAMSOLUTIONS Friday Dec 23, 2005 Prof. Amitabh Lath
Physics 271 FINAL EXAMSOLUTIONS Friday Dec 23, 2005 Prof. Amitabh Lath 1. The exam will last from 8:00 am to 11:00 am. Use a # 2 pencil to make entries on the answer sheet. Enter the following id information
More informationPhysics 403: Relativity Takehome Final Examination Due 07 May 2007
Physics 403: Relativity Takehome Final Examination Due 07 May 2007 1. Suppose that a beam of protons (rest mass m = 938 MeV/c 2 of total energy E = 300 GeV strikes a proton target at rest. Determine the
More informationSlide 10.1. Basic system Models
Slide 10.1 Basic system Models Objectives: Devise Models from basic building blocks of mechanical, electrical, fluid and thermal systems Recognize analogies between mechanical, electrical, fluid and thermal
More informationRigid Body Dynamics (I)
Rigid Body Dynamics (I) COMP768: October 4, 2007 Nico Galoppo From Particles to Rigid Bodies Particles No rotations Linear velocity v only 3N DoFs Rigid bodies 6 DoFs (translation + rotation)
More informationRotational inertia (moment of inertia)
Rotational inertia (moment of inertia) Define rotational inertia (moment of inertia) to be I = Σ m i r i 2 or r i : the perpendicular distance between m i and the given rotation axis m 1 m 2 x 1 x 2 Moment
More informationHW 7 Q 14,20,20,23 P 3,4,8,6,8. Chapter 7. Rotational Motion of the Object. Dr. Armen Kocharian
HW 7 Q 14,20,20,23 P 3,4,8,6,8 Chapter 7 Rotational Motion of the Object Dr. Armen Kocharian Axis of Rotation The radian is a unit of angular measure The radian can be defined as the arc length s along
More informationAN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 2001 APPLIED MATHEMATICS HIGHER LEVEL
M3 AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 00 APPLIED MATHEMATICS HIGHER LEVEL FRIDAY, JUNE AFTERNOON,.00 to 4.30 Six questions to be answered. All questions carry equal marks.
More informations r or equivalently sr linear velocity vr Rotation its description and what causes it? Consider a disk rotating at constant angular velocity.
Rotation its description and what causes it? Consider a disk rotating at constant angular velocity. Rotation involves turning. Turning implies change of angle. Turning is about an axis of rotation. All
More informationPhysics 53. Rotational Motion 1. We're going to turn this team around 360 degrees. Jason Kidd
Physics 53 Rotational Motion 1 We're going to turn this team around 360 degrees. Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid body, in which each particle
More informationROTATIONS AND THE MOMENT OF INERTIA TENSOR
ROTATIONS AND THE MOMENT OF INERTIA TENSOR JACOB HUDIS Abstract. This is my talk for second year physics seminar taught by Dr. Henry. In it I will explain the moment of inertia tenssor. I explain what
More information1 Theoretical Background of PELTON Turbine
c [m/s] linear velocity of water jet u [m/s) runner speed at PCD P jet [W] power in the jet de kin P T [W] power of turbine F [N] force F = dj Q [m 3 /s] discharge, volume flow ρ [kg/m 3 ] density of water
More informationG U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M
G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More informationLecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014
Lecture 07: Work and Kinetic Energy Physics 2210 Fall Semester 2014 Announcements Schedule next few weeks: 9/08 Unit 3 9/10 Unit 4 9/15 Unit 5 (guest lecturer) 9/17 Unit 6 (guest lecturer) 9/22 Unit 7,
More informationSpacecraft Dynamics and Control. An Introduction
Brochure More information from http://www.researchandmarkets.com/reports/2328050/ Spacecraft Dynamics and Control. An Introduction Description: Provides the basics of spacecraft orbital dynamics plus attitude
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Exam Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) A lawn roller in the form of a uniform solid cylinder is being pulled horizontally by a horizontal
More informationCORE STANDARDS, OBJECTIVES, AND INDICATORS
Aerospace Engineering  PLtW Levels: 1112 Units of Credit: 1.0 CIP Code: 14.0201 Core Code: 38010000150 Prerequisite: Principles of Engineering, Introduction to Engineering Design Test: #967 Course
More informationA Simple Model for Ski Jump Flight Mechanics Used as a Tool for Teaching Aircraft Gliding Flight
eπεριοδικό Επιστήμης & Τεχνολογίας 33 A Simple Model for Ski Jump Flight Mechanics Used as a Tool for Teaching Aircraft Gliding Flight Vassilios McInnes Spathopoulos Department of Aircraft Technology
More informationAssignment Work (Physics) Class :Xi Chapter :04: Motion In PLANE
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Assignment Work (Physics) Class :Xi Chapter :04: Motion In PLANE State law of parallelogram of vector addition and derive expression for resultant of two vectors
More informationAircraft Design. Lecture 9: Stability and Control. G. Dimitriadis. Introduction to Aircraft Design
Aircraft Design Lecture 9: Stability and Control G. Dimitriadis Stability and Control Aircraft stability deals with the ability to keep an aircraft in the air in the chosen flight attitude. Aircraft control
More informationResponse to Harmonic Excitation
Response to Harmonic Excitation Part 1 : Undamped Systems Harmonic excitation refers to a sinusoidal external force of a certain frequency applied to a system. The response of a system to harmonic excitation
More informationMatrices in Statics and Mechanics
Matrices in Statics and Mechanics Casey Pearson 3/19/2012 Abstract The goal of this project is to show how linear algebra can be used to solve complex, multivariable statics problems as well as illustrate
More information