كتاب Statics and Dynamics - Andy Ruina

Online electronic version May not be emailed or posted ANYWHERE May not be copied, or printed without express written permission of the authors. Introduction to STATICS and DYNAMICS Andy Ruina and Rudra Pratap Oxford University Press (Preprint) Most recent modifications on August 21, 2010. Filename:Summaryofmechanics Reference Tables: The front and back tables concisely summarize much of the text material. 0) The laws of mechanics apply to any collection of material or ‘body.’ This body could be the overall system of study or any part of it. In the equations below, the forces and moments are those that show on a free body diagram. Interacting bodies cause equal and opposite forces and moments on each other. I) Linear Momentum Balance (LMB)/Force Balance Equation of Motion Fi L The total force on a body is equal to its rate of change of linear momentum. (I) Impulse-momentum (integrating in time) t2 t1 Fi ·dt L Net impulse is equal to the change in momentum. (Ia) Conservation of momentum (if Fi 0 ) L = 0 L = L2 L1 0 When there is no net force the linear momentum does not change. (Ib) Statics (if L is negligible) Fi 0 If the inertial terms are zero the net force on system is zero. (Ic) II) Angular Momentum Balance (AMB)/Moment Balance Equation of motion MC HC The sum of moments is equal to the rate of change of angular momentum. (II) Impulse-momentum (angular) (integrating in time) t2 t1 MCdt HC The net angular impulse is equal to the change in angular momentum. (IIa) Conservation of angular momentum (if MC 0) HC 0 HC HC2 HC1 0 If there is no net moment about point C then the angular momentum about point C does not change. (IIb) Statics (if HC is negligible) MC 0 If the inertial terms are zero then the total moment on the system is zero. (IIc) III) Power Balance (1st law of thermodynamics) Equation of motion Q P EK EP Eint E Heat flow plus mechanical power into a system is equal to its change in energy (kinetic + potential + internal). (III) for finite time t2 t1 Qdt t2 t1 Pdt E The net energy flow going in is equal to the net change in energy. (IIIa) Conservation of Energy (if Q P 0) E 0 E E2 E1 0 If no energy flows into a system, then its energy does not change. (IIIb) Statics (if EK is negligible) Q P EP Eint If there is no change of kinetic energy then the change of potential and internal energy is due to mechanical work and heat flow. (IIIc) Pure Mechanics (if heat flow and dissipation are negligible) P EK EP In a system well modeled as purely mechanical the change of kinetic and potential energy is due to mechanical work on the system. (IIId) Summary of Mechanics Some definitions (Also see the index and back tables) *r or *x Position e.g.,*r i  *r i=O is the position of a point i relative to the origin, O. *v  d*r dt Velocity e.g., *v i  *v i=O is the velocity of a point i relative to O, measured in a non-rotating reference frame. *a  d*v dt D d2*r dt2 Acceleration e.g., *a i *a i=O is the acceleration of a point i relative to O, measured in a Newtonian frame. * F Force e.g., the force on A from B is FA from B. * M or * MC D * M=C Moment or Torque e.g., the moment of a collection of forces about point C. *! Angular velocity A measure of rotational velocity of a rigid object. *! B = angular velocity of rigid object B. *  P *! Angular acceleration A measure of rotational acceleration of a rigid object. * L  8<: P mi *v i discrete R*v dm continuous Linear momentum A measure of a system’s net translational rate (weighted by mass). D mtot *v cm P * L  8<: P mi *a i discrete R*a dm continuous Rate of change of linear momentum The aspect of motion that balances the net force on a system. D mtot *a cm * H=C  8<: P*r i=C  mi *v i discrete R*r =C *v dm continuous Angular momentum about point C A measure of the rotational rate of a system about a point C (weighted by mass and distance from C). P * H=C  8<: P*r i=C  mi *a i discrete R*r =C *a dm continuous Rate of change of angular momentum about point C The aspect of motion that balances the net torque on a system about a point C. EK  8<: 1 2 P mi v2 i discrete 1 2 R v2dm continuous Kinetic energy A scalar measure of net system motion. Eint D (heat-like terms) Internal energy The non-kinetic non-potential part of a system’s total energy. P  P * Fi
*v i CP * Mi
*! i Power of forces and torques The mechanical energy flow into a system. Also, P  WP , rate of work. .Icmچ  26664 I cm xx I cm xy I cm xz I cm xy I cm yy I cm yz I cm xz I cm yz I cm zz 37775 Moment of inertia matrix about center of mass (cm) A measure of the mass distribution in a rigid object.
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