ASEN 5022 - Spring 2005Dynamics of Aerospace StructuresLecture 01: 11 January 2005IntroductionDynamics according toGalilei Galileo (Two New Sciences, 1636):A subject of never-ending interest.Isaac Newton (The Principia, 1687):We offer this work as the mathematical principles of philos-ophy; for all the difficulty of philosophy seems to consist inthis – from phenomena of motions to investigate the forces ofnature, and then from these forces to demonstrate the otherphenomena; and to this end the general propositions in thefirst and second book are directed.What we will learn/acquire in this course:Kinematics which are essential for the description of motionsof masses.Ability to derive the equations of motion for complex dynam-ical systems.Understanding of vibration phenomena for structural ele-ments.Abilitytomodelcomplexvibrationproblemsbyreducednum-ber of equations.Applications of dynamical principles for engineering designapplications.Transition from Statics to DynamicsStatics DynamicsForce equilibrium :fora” free body” ijfij= 0jfij− miai= 0§Moment equilibrium :around point ”P”jMPj= 0MPj− rPj× (mjaj) = 0§§Observation: The crucial aspect of dynamics is the need to com-pute the acceleration vector for every mass in the system.Landau’s Uniqueness Theorem∗If all the position vectors {x1, x2, ..., xn} and the velocity vec-tors {˙x1,˙x2, ...,˙xn} for n particles are given at some instant,the accelerations {¨x1,¨x2, ...,¨xn} at that instant are uniquelydefined.The relations between the accelerations, velocities and posi-tion vectors are called the equations of motion.∗L. D. Landau and E. M. Lifshits, Mechanics (3rd ed.), PergamonPress, 1959.Reference Frame and Position VectorConsider two frames, K and K, where V is constant in bothmagnitude and direction, and in which the properties of spaceand time are the same.rr'VtOO'Frame KFrame K'r=Vt+r'Example of Two Inertial FramesReference Frame and Position Vector-Cont’dThe kinematic relations between frames K and Kare:r = Vt + r˙r = V +˙r⇒ v = V + v¨r =¨r⇒ a = aConclusion: Theaccelerationvectorsarethesameinallinertialframes!Hence, Galileo’s relativity principle holds.In all inertial frames, the laws of mechanics are the same, which is re-ferred to Galileo’s relativity principle, one of the most important prin-ciples of mechanics.Newton’s Eight DefinitionsDEFINITION I: The quantity of matter[mass] is the measure of the same, arisingfrom its density and bulk conjointly.DEFINITION II: The quantity of motion[linear momentum] is the measure of thesame, arising from the velocity and quantity of matter conjointly.DEFINITION III: The vis insita, or the innate force of matter[inertia force], is a powerof resisting, by which every body, as much as in it lies, continues in its present state,whether it be rest, or of moving uniformly forwards in a right[straight] line.DEFINITION IV: An impressed force is an action exerted upon a body, in order tochange its state, either of rest, or of uniform motion in a right line.Newton’s Eight Definitions - cont’dDEFINITION V: A centripetal force is that by which bodies are drawn or impelled,or any way tend, towards a point as to a centre.DEFINITION VI: An absolute centripetal force is the measure of the same, propor-tional to the efficacy of the cause that propagates it from the centre, through the spacesround about.DEFINITION VII: The accelerative quantity of a centripetal force is the measure ofthe same, proportional to the velocity which it generates in a given time.DEFINITION VIII: The motive[motion-causing] quantity of a centripetal force is themeasure of the same, proportional to the motion which it generates in a given time.Newton’s Three LawsLaw I: Every body continues in its state of rest, or of uniformmotion in a right line, unless it is compelled to change that stateby forces impressed upon it.Law II: The change of motion is proportional to the motive forceimpressed; and is made in the direction of the right line in whichthe force is impressed.Law III: To every action there is always opposed an equal re-action; or, the mutual actions of two bodies upon each other arealways equal, and directed to contrary parts.Test of Galileo’s Relativity PrinciplemkfFree-freestateFrame Koo'(mass center)Frame K'StaticdisplacementAssumeddynamic stateXX'dx'xhmgk(x-d)= kx'maTest of Galileo’s Relativity Principle-cont’dPosition vector in frame K : X = h − d +xPosition vector in frame K: X= xRelation between the two frames : x= x − dEquation of motion in frame K : − mg −k(x − d) − m¨x = 0Equation of motion in frame K: − mg −kx− m¨x= 0Test of Galileo’s Relativity Principle-cont’dEOM in frame K : m¨x + kx = (−mg + kd)(static eq.!) = 0EOM in frame K: m¨x+ kx=−mgRemark 1: The mass-point based coordinate system (frame K) mustinclude the dead weight mg in the equations of motion.Remark 2: The ground based coordinate system (frame K) does notrequire to account for the static equilibrium.Remark 3: When the ground itself moves, i.e., buildings subjectedearthquakes and automobile riding on wavy roads, one has to mod-ifyhtoX= xg+h −d +x where xgis the motion of the ground. Asimilar modification must be made for X.Test of Galileo’s Relativity Principle-cont’dWhat happens to the mass point-based coordinate system when theground