Translational and Rotational 
                                                                          Dynamics!
                                                                          Robert Stengel!
                                        Robotics and Intelligent Systems MAE 345, 
                                                            Princeton University, 2017
                                                  Copyright 2017 by Robert Stengel.  All rights reserved.  For educational use only.          1
                                                                  http://www.princeton.edu/~stengel/MAE345.html
                                                                          Reference Frame 
                                  •!   Newtonian (Inertial) Frame of 
                                       Reference
                                        –! Unaccelerated Cartesian frame 
                                               •! Origin referenced to inertial 
                                                  (non-moving) frame
                                        –! Right-hand rule
                                        –! Origin can translate at 
                                            constant linear velocity
                                        –! Frame cannot  rotate with 
                                            respect to inertial origin                                 ! x $
                                   •!   Position: 3 dimensions                                  r = # y &
                                          –! What is a non-moving frame?                               #        &
                                                                                                       # z &
                                                                                                       "        %
                                                                 •!   Translation = Linear motion                                             2
                                                   Velocity and Momentum 
                                                                      of a Particle
                                       •!   Velocity of a particle
                                                                      ! ! $           ! vx $
                                                       dr             # x &           #         &
                                                                           !              v
                                                                !          y      =
                                                v = dt = r = #                 &      #     y   &
                                                                      # ! &           #         &
                                                                      " z %           # vz &
                                                                                      "         %
                                       •!   Linear momentum of a particle
                                                                                      ! vx $
                                                                  p=mv=m# v &
                                                                                      #     y   &
                                                                                      # v &
                                                                                      #     z   &
                                                                                      "         %
                                                                                                                                         3
                                    Newtons Laws of Motion: !
                                         Dynamics of a Particle 
                                                                            First Law
                                                           . If no force acts on a particle, 
                                                it remains at rest or continues to move in 
                                                        straight line at constant velocity, 
                                                                . Inertial reference frame 
                                                                . Momentum is conserved
                                                    d (mv)= 0 ; mv =mv
                                                                                                      t                 t
                                                   dt                                                 1                   2
                                                                                                                                         4
                                                 Newtons Laws of 
                                                                Motion: !
                                           Dynamics of a Particle 
                                                                          Second Law
                                          •!   Particle acted upon by force 
                                          •!   Acceleration proportional to and in direction 
                                               of force 
                                           •!  Inertial reference frame 
                                           •!  Ratio of force to acceleration is particle mass
                                       d (mv)= mdv = ma=Force                             ! dv= 1Force= 1 I Force
                                      dt                  dt                                     dt      m               m 3
                                                   ! fx $                                     " 1/m          0         0      %" fx %
                                     Force=# f &=force vector                              =$ 0            1/m         0      '$ f      '
                                                   #    y   &                                 $                               '$     y  '
                                                   #        &                                 $    0         0       1/m '$ f '
                                                       fz                                     #                               &$     z  '
                                                   #        &                                                                  #        &     5
                                                   "        %
                                                Newtons Laws of Motion: !
                                                     Dynamics of a Particle 
                                                                           Third Law
                                  For every action, there is an equal and opposite reaction
                                                Force on rocket motor = –Force on exhaust gas
                                                                               F =!F
                                                                                 R          E
                                                                                                                                         6
                               One-Degree-of-Freedom Example 
                                         of Newton
                                                                 s Second Law
                                             nd
                                           2 -order, linear, time-invariant 
                                           ordinary differential equation
                                                  d2x(t)                       fx(t)          !  "Defined as"
                                                             ""       "                       
                                                    dt2    ! x(t)= vx(t)=       m
                                                   
                                                                           st
                                       Corresponding set of 1 -order equations
                                                   (State-Space Model)
                              dx (t)
                                 1        "
                                       ! x (t)! x (t)! v (t)                     x (t) ! x(t),  Displacement
                                 dt        1        2        x                    1
                              dx (t)                                 f (t)      x (t)! dx(t),     Rate
                                 2        ""       "        "         x           2
                                       ! x (t)= x (t)= v (t)=                             dt
                                 dt        1        2        x        m
                                                                                                             7
                                                                                                       st
                               State-Space Model is a Set of 1 -
                         Order Ordinary Differential Equations
                             State, control, and output vectors for the example
                                         ! x (t) $                                           ! x (t) $
                                x(t)= #      1      &;    u(t)=u(t)= f (t);          y(t)= #      1     &
                                         # x (t) &                          x                # x (t) &
                                         "   2      %                                        "    2     %
                                          Stability and control-effect matrices
                                                 F=! 0 1 $; G=! 0 $
                                                       # 0 0 &                # 1/m &
                                                       "          %           "         %
                                                     Dynamic equation
                                                     !
                                                     x(t) = Fx(t)+Gu(t)
                                                                                                             8