Resolve merge.

Author: moorepants

angular.rst

@@ -39,25 +39,25 @@ Learning Objectives
 
 After completing this chapter readers will be able to:
 
-- apply the definition of angular velocity
-- calculate the angular velocity of simple rotations
-- choose Euler angles for a rotating reference frame
-- calculate the angular velocity of reference frames described by successive
-  simple rotations
-- derive the time derivative of a vector in terms of angular velocities
-- calculate the angular acceleration of a reference frame
-- calculate the angular acceleration of reference frames described by
-  successive rotations
+#. apply the definition of angular velocity
+#. calculate the angular velocity of simple rotations
+#. choose Euler angles for a rotating reference frame
+#. calculate the angular velocity of reference frames described by successive
+   simple rotations
+#. derive the time derivative of a vector in terms of angular velocities
+#. calculate the angular acceleration of a reference frame
+#. calculate the angular acceleration of reference frames described by
+   successive rotations
 
 Introduction
 ============
 
 To apply `Euler's Laws of Motion`_ to a multibody system we will need to
 determine how the `angular momentum`_ of each rigid body changes with time.
-This requires that we specify the angular kinematics of each body in the
-system: typically both angular velocity and angular acceleration. Assuming that
-a reference frame is fixed to a rigid body, we will start by finding the
-angular kinematics of a single reference frame and then use the properties of
+This requires specifying the angular kinematics of each body in the system:
+typically both angular velocity and angular acceleration. Assuming that a
+reference frame is fixed to a rigid body, we will start by finding the angular
+kinematics of a single reference frame and then use the properties of
 :ref:`Successive Orientations` to find the angular kinematics of a set of
 related reference frames.
 
@@ -76,6 +76,10 @@ of time.
      <source
        src="https://upload.wikimedia.org/wikipedia/commons/b/be/Dzhanibekov_effect.ogv"
        type="video/ogg">
+     <source
+       src="https://objects-us-east-1.dream.io/mechmotum/dzhanibekow-effect.mp4"
+       type="video/mp4">
+     Your browser does not support the video tag.
    </video>
    <p>Public Domain, NASA</p>
    </center>
@@ -84,8 +88,8 @@ The T-handle exhibits unintuitive motion, reversing back and forth
 periodically. This phenomena is commonly referred to as the "`Dzhanibekov
 effect`_" and Euler's Laws of Motion predict the behavior, which we will
 investigate in later chapters. For now, we will learn how to specify the
-angular kinematics of a reference frame in motion, such as one fixed to this
-T-handle.
+angular kinematics of a reference frame in rotational motion, such as one fixed
+to this T-handle.
 
 .. _Dzhanibekov effect: https://en.wikipedia.org/wiki/Tennis_racket_theorem
 

energy.rst

@@ -164,7 +164,7 @@ of the system.
 Total Energy
 ============
 
- The total energy of the system is:
+The total energy of the system is:
 
 .. math::
 
@@ -573,7 +573,7 @@ are similar to prior chapters, so I leave them unexplained.
              Matplotlib axes for each panel.
 
           """
-          fig, axes = plt.subplots(6, 1, sharex=True)
+          fig, axes = plt.subplots(6, 1, sharex=True, layout='constrained')
 
           fig.set_size_inches((10.0, 6.0))
 
@@ -593,15 +593,13 @@ are similar to prior chapters, so I leave them unexplained.
           axes[4].legend(['$K$', '$V$', '$E$'])
           axes[5].legend(['$T_k$'])
 
-          axes[0].set_ylabel('Distance [m]')
-          axes[1].set_ylabel('Angle [deg]')
-          axes[2].set_ylabel('Speed [m/s]')
-          axes[3].set_ylabel('Angular Rate [deg/s]')
-          axes[4].set_ylabel('Energy [J]')
-          axes[5].set_ylabel('Torque [N-m]')
-          axes[5].set_xlabel('Time [s]')
-
-          fig.tight_layout()
+          axes[0].set_ylabel('Distance\n[m]')
+          axes[1].set_ylabel('Angle\n[deg]')
+          axes[2].set_ylabel('Speed\n[m/s]')
+          axes[3].set_ylabel('Angular Rate\n[deg/s]')
+          axes[4].set_ylabel('Energy\n[J]')
+          axes[5].set_ylabel('Torque\n[N-m]')
+          axes[5].set_xlabel('Time\n[s]')
 
           return axes
 

generalized-forces.rst

@@ -284,6 +284,68 @@ where :math:`\hat{e}_r` is a unit vector in the independent speed
 :math:`\bar{u}_s` vector space, e.g. :math:`\hat{e}_2=\left[0, 1, 0,
 0\right]^T` if :math:`p=4`. See [Kane1985]_ pg. 48 for more explanation.
 
+Revisit the Chaplygin Sleigh
+----------------------------
+
+To see how :math:`\mathbf{A}_n` can be used to formulate a nonholonomic partial
+velocity, recall the :ref:`Chaplygin Sleigh` velocity equations where
+:math:`u_x=\dot{x}` and :math:`u_y=\dot{y}`:
+
+.. jupyter-execute::
+
+   x, y, theta = me.dynamicsymbols('x, y, theta')
+   ux, uy, utheta = me.dynamicsymbols('u_x, u_y, u_theta')
+
+   N = me.ReferenceFrame('N')
+   A = me.ReferenceFrame('A')
+
+   A.orient_axis(N, theta, N.z)
+
+   O = me.Point('O')
+   P = me.Point('P')
+
+   P.set_pos(O, x*N.x + y*N.y)
+
+   O.set_vel(N, 0)
+
+   N_v_P = P.vel(N).express(A).xreplace({x.diff(): ux, y.diff(): uy})
+   N_v_P
+
+and that the nonholonomic constraint is:
+
+.. jupyter-execute::
+
+   fn = N_v_P.dot(A.y)
+   fn
+
+The simplest, but not necessarily computationally efficient, method of finding
+a nonholonomic partial velocity is to substitute all dependent speeds for
+expressions that are functions only of the independent speeds. So, if we select
+:math:`u_x` to be the dependent speed, the nonholonomic partial velocity of
+:math:`u_y` is:
+
+.. jupyter-execute::
+
+   ux_sol = sm.solve(fn, ux)[0]
+   ux_sol
+
+.. jupyter-execute::
+
+   N_v_P.xreplace({ux: ux_sol}).diff(uy, N).simplify()
+
+But, the same result can be found without substitution by using
+:math:`\mathbf{A}_n` from :math:numref:`eq-constraint-linear-form-solve`, which
+for this example is simply the scalar:
+
+.. jupyter-execute::
+
+   An = ux_sol.diff(uy)
+   An
+
+.. jupyter-execute::
+
+   (N_v_P.diff(uy, N) + N_v_P.diff(ux, N)*An).simplify()
+
 Generalized Active Forces
 =========================
 

introduction.rst

@@ -147,24 +147,12 @@ helped improve the text in many ways. We also thank the students of TU Delft's
 Multibody Dynamics course who test the materials while learning.
 
 These are the primary contributors to the SymPy Mechanics software presented in
-the text, in approximate order of first contribution:
-
-- Dr. Luke Peterson, 2009
-- Dr. Gilbert Gede, 2011
-- Dr. Angadh Nanjangud, 2012
-- Tarun Gaba, 2013
-- Oliver Lee, 2013
-- Dr. Chris Dembia, 2013
-- Jim Crist, 2014
-- Sahil Shekhawat, 2015
-- James McMillan, 2016
-- Nikhil Pappu, 2018
-- Sudeep Sidhu, 2020
-- Abhinav Kamath, 2020
-- Timo Stienstra, 2022
-- Dr. Sam Brockie, 2023
-- Hwayeon Kang, 2024
-- Riccardo Di Girolamo, 2024
+the text, in approximate order of first contribution: Dr. Luke Peterson (2009),
+Dr. Gilbert Gede (2011), Dr. Angadh Nanjangud (2012), Tarun Gaba (2013), Oliver
+Lee (2013), Dr. Chris Dembia (2013), Jim Crist (2014), Sahil Shekhawat (2015),
+James McMillan (2016), Nikhil Pappu (2018), Sudeep Sidhu (2020), Abhinav Kamath
+(2020), Timo Stienstra (2022), Dr. Sam Brockie (2023), Hwayeon Kang (2025), and
+Riccardo Di Girolamo (2025).
 
 SymPy Mechanics is built on top of SymPy, whose `1000+ contributors`_ have also
 greatly helped SymPy Mechanics be what it is. Furthermore, the software sits on

motion.rst

@@ -720,8 +720,8 @@ nonholonomic constraints:
 .. math::
    :label: eq-snakeboard-constraints
 
-   {}^A\bar{v}^{Bo} \cdot \hat{b}_y = 0 \\
-   {}^A\bar{v}^{Co} \cdot \hat{c}_y = 0
+   {}^N\bar{v}^{Bo} \cdot \hat{b}_y = 0 \\
+   {}^N\bar{v}^{Co} \cdot \hat{c}_y = 0
 
 .. jupyter-execute::
 

notation.rst

@@ -90,8 +90,8 @@ Angular and Translational Kinematics
 Constraints
 ===========
 
-.. todo:: Add Y_k, kin diff eqs.
-
+:math:`\bar{u} = \mathbf{Y}_k \dot{\bar{q}} + \bar{z}_k`
+   Generalized speeds defined as the kinematical differential equations.
 :math:`N,M,n,m,p`
    :math:`N` coordinates, :math:`M` holonomic constraints, :math:`n`
    generalized coordinates and generalized speeds, :math:`m` nonholonomic
@@ -134,8 +134,8 @@ Mass Distribution
    Central inertia dyadic of body :math:`B` or set of particles :math:`B` with respect
    to mass center :math:`B_o`.
 :math:`{}^A \bar{H}^{B/O}`, ``A_H_B_O``
-   Angular momentum of rigid body :math:`B` with respect to point :math:`O` in
-   reference frame :math:`A`.
+   Angular momentum of rigid body :math:`B` with respect to point :math:`O`
+   when viewed from reference frame :math:`A`.
 
 Force, Moment, and Torque
 =========================
@@ -145,6 +145,8 @@ Force, Moment, and Torque
 :math:`\bar{R}^{S/Q}`, ``R_S_Q``
    Resultant of the vector set :math:`S` bound to a line of action through
    point :math:`Q`.
+:math:`\bar{F}^{Q}`, ``F_Q``
+   Force vector bound to a line of action through point :math:`Q`.
 :math:`\bar{M}^{S/P}`, ``M_S_P``
    Moment of the resultant of the vector set :math:`S` about point :math:`P`.
 :math:`\bar{T}^{B}`, ``T_B``
@@ -187,16 +189,19 @@ Generalized Forces
 Unconstrained Equations of Motion
 =================================
 
-:math:`\bar{f}_k(\dot{\bar{q}}, \bar{u}, \bar{q}, t)  = 0`
-   Kinematical differential equations.
+:math:`\bar{f}_k(\dot{\bar{q}}, \bar{u}, \bar{q}, t) = 0`
+   Kinematical differential equations that take the form: :math:`\bar{f}_k =
+   \mathbf{Y}_k \dot{\bar{q}} + \bar{z}_k - \bar{u} = \mathbf{M}_k
+   \dot{\bar{q}} + \bar{g}_k = 0`.
 :math:`\mathbf{M}_k`
    Linear coefficient matrix for :math:`\dot{\bar{q}}` in the kinematical
    differential equations.
 :math:`\bar{g}_k`
    Terms not linear in :math:`\dot{\bar{q}}` in the kinematical differential
    equations.
 :math:`\bar{f}_d(\dot{\bar{u}}, \bar{u}, \bar{q}, t) = 0`
-   Dynamical differential equations.
+   Dynamical differential equations that take the form: :math:`\bar{f}_d =
+   \mathbf{M}_d \dot{\bar{u}} + \bar{g}_d = 0`.
 :math:`\mathbf{M}_d`
    Linear coefficient matrix for :math:`\dot{\bar{u}}` in the dynamical
    differential equations, often called the "mass matrix".
@@ -210,6 +215,8 @@ Unconstrained Equations of Motion
    motion.
 :math:`\bar{g}_m`
    Terms not linear in :math:`\dot{\bar{x}}` in the equations of motion.
+:math:`\dot{\bar{x}} = \bar{f}_m(\bar{x}, t) = -\mathbf{M}_m^{-1} \bar{g}_m`
+   Equations of motion in first order explicit form.
 
 Equations of Motion with Nonholonomic Constraints
 =================================================

orientation.rst

@@ -564,6 +564,10 @@ explicitly:
 
    N.orient_explicit(A, A_C_N)
 
+.. todo::
+
+   When we change to SymPy 1.13, change ``orient_explicit`` to ``orient_dcm``.
+
 .. warning::
 
    Note very carefully what version of the direction cosine matrix you pass to