Note
You can download this example as a Python script: :jupyter-download:script:`generalized-forces` or Jupyter Notebook: :jupyter-download📓`generalized-forces`.
.. jupyter-execute:: import sympy as sm import sympy.physics.mechanics as me me.init_vprinting(use_latex='mathjax')
At this point we have developed the three primary ingredients to formulate the equations of motion of a multibody system:
- Angular and Translational Kinematics
- Mass and Mass Distribution
- Forces, Moments, and Torques
For a single rigid body B with mass m_B, mass center B_o, and central inertia dyadic \breve{I}^{B/B_o} having a resultant force \bar{F} at B_o and moment \bar{M} about B_o the Newton-Euler Equations of Motion in the inertial reference frame N can be written as follows:
\bar{F} = & \frac{{}^N d \bar{p}}{dt} \quad \textrm{ where } \bar{p} = m_B{}^N\bar{v}^{B_o} \\
\bar{M} = & \frac{{}^N d\bar{H}}{dt} \quad \textrm{ where }
\bar{H} = \breve{I}^{B/B_o} \cdot {}^N\bar{\omega}^{B}
The left hand side contains the forces, moments, and torques (3.) acting on the rigid body and the right hand side contains the kinematics (1.) and the mass distribution (2.).
For a set of particles and rigid bodies that make up a multibody system, defined with generalized coordinates, generalized speeds, and constraints, the generalized speeds characterize completely the motion of the system. The velocities and angular velocities of every particle and rigid body in the system are a function of these generalized speeds. The time rate of change of the generalized speeds \frac{du}{dt} will then play a critical role in the formulation of the right hand side of the multibody system equations of motion.
Take for example the multibody system shown in :numref:`fig-generalized-forces-multi-pendulum`. A force \bar{F} applied at point Q will cause all three of the lower particles to move. The motion of the particles are described by the velocities, which are functions of the generalized speeds. Thus \bar{F} will, in general, cause all of the generalized speeds to change. But how much does each generalized speed change? The so called partial velocites of Q in N will provide the answer to this question.
Four particles attached by massless links making up a 3 link planar simple pendulum. The top particle is fixed in N. If the generalized coordinates q_1,q_2,q_3 represent the angles of the three pendulums then three generalized speeds could be defined as u_i=\dot{q}_i for i=1,\ldots,3.
Recall that all translational and angular velocities of a multibody system can be written in terms of the generalized speeds. By definition (Eq. :math:numref:`eq-generalized-speeds`), these velocities can be expressed uniquely as linear functions of the generalized speeds. For a holonomic system with n degrees of freedom in a single reference frame any translational velocity or angular velocity can be written as ([Kane1985]_, pg. 45):
\bar{v} = \sum_{r=1}^n \bar{v}_r u_r + \bar{v}_t \\
\bar{\omega} = \sum_{r=1}^n \bar{\omega}_r u_r + \bar{\omega}_t
We call \bar{v}_r and \bar{\omega}_r the rth holonomic partial velocity and angular velocity in the single reference frame, respectively. \bar{v}_t and \bar{\omega}_t are the remainder terms that are not linear in a generalized speed. Since the velocities are linear in the generalized speeds, the partial velocities are equal to the partial derivatives with respect to the generalized speeds:
\bar{v}_r = \frac{\partial \bar{v}}{\partial u_r} \quad
\bar{v}_t = \frac{\partial \bar{v}}{\partial t} \\
\bar{\omega}_r = \frac{\partial \bar{\omega}}{\partial u_r} \quad
\bar{\omega}_t = \frac{\partial \bar{\omega}}{\partial t}
Note
The reference frame these partials are taken with respect to should match that which the velocities are with respect to.
Given that the partial velocities are partial derivative, means that we may interpret the partial velocities as the sensitivities of translational and angular velocities to changes in u_r. The partial velocities give an idea of how any given velocity or angular velocity will change if one of the generalized speeds changes. Figure :numref:`fig-generalized-forces-partial-velocities` gives a graphical interpretation of how a velocity of P in N is made up of partial velocities and a remainder.
Velocity vector {}^N\bar{v}^P of point P shown expressed as a sum of linear combinations of generalized speeds and partial velocity vectors and a remainder vector. In this case there are two generalized speeds.
Partial velocities can be determined by inspection of velocity vector expressions or calculated by taking the appropriate partial derivatives. Take, for example, the single body system shown in :numref:`fig-generalized-forces-par-vel-rod`. What are the partial velocities for {}^N\bar{v}^A, {}^N\bar{v}^B, and {}^N\bar{\omega}^R?
A rod R pinned at A on the horizontal line. A's horizontal translation is described by with the generalized coordinate q_1 and the angle of the rod relative to the horizontal is described by the generalized coordinate q_2.
First calculate the velocities and ensure they are only in terms of the generalized speeds and generalized coordinates. In this case, we have chosen u_1=\dot{q}_1 and u2=\dot{q}_2.
.. jupyter-execute::
L = sm.symbols('L')
q1, q2, u1, u2 = me.dynamicsymbols('q1, q2, u1, u2')
N = me.ReferenceFrame('N')
R = me.ReferenceFrame('R')
R.orient_axis(N, q2, N.z)
.. jupyter-execute:: N_v_A = u1*N.x N_v_A
.. jupyter-execute:: N_w_R = u2*N.z N_w_R
.. jupyter-execute:: r_A_B = -L*R.x N_v_B = N_v_A + me.cross(N_w_R, r_A_B) N_v_B.express(N)
Now, take the partial derivatives with respect to the generalized speeds to find the six partial velocities:
.. jupyter-execute:: v_A_1 = N_v_A.diff(u1, N) v_A_2 = N_v_A.diff(u2, N) v_A_1, v_A_2
.. jupyter-execute:: v_B_1 = N_v_B.diff(u1, N) v_B_2 = N_v_B.diff(u2, N) v_B_1, v_B_2
.. jupyter-execute:: w_R_1 = N_w_R.diff(u1, N) w_R_2 = N_w_R.diff(u2, N) w_R_1, w_R_2
SymPy Mechanics provides a convenience function :external:py:func:`~sympy.physics.vector.functions.partial_velocity` to calculate a set of partial velocities for a set of generalized speeds:
.. jupyter-execute:: me.partial_velocity((N_v_A, N_v_B, N_w_R), (u1, u2), N)
If a system is nonholonomic, it is also true that every translational and angular velocity can be expressed uniquely in terms of the p independent generalized speeds (see Eq. :math:numref:`eq-contraint-linear-form-solve`). Thus, we can also define the nonholonomic partial velocities \tilde{v}_r and nonholonomic partial angular velocities \tilde{\omega}_r as per ([Kane1985]_, pg. 46):
\bar{v} = \sum_{r=1}^p \tilde{v}_r u_r + \tilde{v}_t \\
\bar{\omega} = \sum_{r=1}^p \tilde{\omega}_r u_r + \tilde{\omega}_t
See [Kane1985]_ pg. 48 for the relationship between holonomic and nonholonomic partial velocities.
Suppose we have a holonomic multibody system made up of \nu particles with n degrees of freedom in a reference frame A that are described by generalized speeds u_1,\ldots,u_n. Each particle may have a resultant force \bar{R} applied to it. By projecting each of the forces onto the partial velocity of its associated particle and summing the projections, we arrive at the total scalar force contribution associated with changes in that generalized speed. We call these scalar values, one for each generalized speed, the generalized active forces. The rth holonomic generalized active force for this system in A is defined as ([Kane1985]_, pg. 99):
F_r := \sum_{i=1}^\nu {}^A\bar{v}^{P_i}_r \cdot \bar{R}_i
where i represents the ith particle.
Notice that the rth generalized active force is:
- a scalar value
- has contributions from all particles except if {}^A\bar{v}^{P_i} \perp \bar{R}_i
- associated with the rth generalized speed
We will typically collect all of the generalized active forces in a column vector to allow for matrix operations with these values:
\bar{F}_r = \begin{bmatrix}
\sum_{i=1}^\nu {}^A\bar{v}_1^{P_i} \cdot \bar{R}_i \\
\vdots \\
\sum_{i=1}^\nu {}^A\bar{v}_r^{P_i} \cdot \bar{R}_i \\
\vdots \\
\sum_{i=1}^\nu {}^A\bar{v}_n^{P_i} \cdot \bar{R}_i
\end{bmatrix}
Eq. :math:numref:`eq-rth-gaf` shows that the partial velocities transform the forces applied to the multibody system from their Cartesian vector space to a new generalized speed vector space.
Now let us calculate the generalized active forces for a simple multibody system made up of only particles. :numref:`fig-generalized-forces-double-pendulum` shows a double simple pendulum made up of two particles P_1 and P_2 with masses m_1 and m_2 respectively.
Double simple pendulum a) kinematic schematic, b) free body diagram of P_1, c) free body diagram of P_2.
To calculate the generalized active forces we first find the velocities of each particle and write them in terms of the generalized speeds which we define as u_1=\dot{q}_1,u_2=\dot{q}_2.
.. jupyter-execute::
l = sm.symbols('l')
q1, q2, u1, u2 = me.dynamicsymbols('q1, q2, u1, u2')
N = me.ReferenceFrame('N')
A = me.ReferenceFrame('A')
B = me.ReferenceFrame('B')
A.orient_axis(N, q1, N.z)
B.orient_axis(N, q2, N.z)
O = me.Point('O')
P1 = me.Point('P1')
P2 = me.Point('P2')
O.set_vel(N, 0)
P1.set_pos(O, -l*A.y)
P2.set_pos(P1, -l*B.y)
P1.v2pt_theory(O, N, A)
P2.v2pt_theory(P1, N, B)
P1.vel(N), P2.vel(N)
.. jupyter-execute::
repl = {q1.diff(): u1, q2.diff(): u2}
N_v_P1 = P1.vel(N).xreplace(repl)
N_v_P2 = P2.vel(N).xreplace(repl)
N_v_P1, N_v_P2
We will need the partial velocities of each particle with respect to the two generalized speeds, giving four partial velocities:
.. jupyter-execute:: v_P1_1 = N_v_P1.diff(u1, N) v_P1_2 = N_v_P1.diff(u2, N) v_P2_1 = N_v_P2.diff(u1, N) v_P2_2 = N_v_P2.diff(u2, N) v_P1_1, v_P1_2, v_P2_1, v_P2_2
To determine the resultant forces acting on each particle we isolate each particle from the system and draw a free body diagram with all of the forces acting on the particle. Each particle has a gravitational force as well as distance, or tension, forces that ensure the particle stays connected to the massless rod. The resultant forces on each particle are then:
.. jupyter-execute::
T1, T2 = me.dynamicsymbols('T1, T2')
m1, m2, g = sm.symbols('m1, m2, g')
R1 = -m1*g*N.y + T1*A.y - T2*B.y
R1
.. jupyter-execute:: R2 = -m2*g*N.y + T2*B.y R2
With the resultants and the partial velocities defined, the two generalized active forces can then be found:
.. jupyter-execute:: F1 = me.dot(v_P1_1, R1) + me.dot(v_P2_1, R2) F1
.. jupyter-execute:: F2 = me.dot(v_P1_2, R1) + me.dot(v_P2_2, R2) F2
Notice that the distance forces T_1,T_2 are not present in the generalized active forces F_1 or F_2. This is not by coincidence, but will always be true for noncontributing forces. They are in fact named "noncontributing" because they do not contribute to the generalized active forces (nor the full equations of motion we eventually arrive at). Noncontributing forces need not be considered in the resultants, in general, and we will not include them in further examples.
Notice also that the generalized forces have units of \textrm{force} \times \textrm{length}. This is because our generalized speeds are angular rates. If our generalized speeds were linear rates, the generalized forces would have units of \textrm{force}.
If a holonomic multibody system with n degrees of freedom in reference frame A includes a rigid body B then the loads acting on B can be described by a resultant force \bar{R} bound to an arbitrary point Q in B and a couple with torque \bar{T}. The generalized active force then for a single rigid body in a multibody system is defined as ([Kane1985]_, pg. 106):
(F_r)_B := {}^A\bar{v}^Q_r \cdot \bar{R} + {}^A\bar{\omega}^B_r \cdot \bar{T}
A generalized active force for each rigid body and particle in a system must be summed to obtain the total generalized active force.
To demonstrate finding the generalized active forces for a multibody system with two rigid bodies consider :numref:`fig-generalized-forces-3d-rods` which shows two thin rods of length l that are connected at points O and B_o.
A multibody system comprised of two uniformly dense thin rods of length L and mass m. Rod A is pinned at O and can rotate about \hat{n}_z through q_1. Rod B is pinned to A and can rotate relative to A about \hat{a}_x through q_2. Linear torisional springs of stiffnes k with a free length of zero resists each relative rotation. Gravitational forces are in the \hat{n}_x direction.
The first step is to define the necessary velocities we'll need: translational velocities of the two mass centers and the angular velocities of each body. We use the simple definition of the generalized speeds u_i=\dot{q}_i.
.. jupyter-execute::
m, g, k, l = sm.symbols('m, g, k, l')
q1, q2, u1, u2 = me.dynamicsymbols('q1, q2, u1, u2')
N = me.ReferenceFrame('N')
A = me.ReferenceFrame('A')
B = me.ReferenceFrame('B')
A.orient_axis(N, q1, N.z)
B.orient_axis(A, q2, A.x)
A.set_ang_vel(N, u1*N.z)
B.set_ang_vel(A, u2*A.x)
O = me.Point('O')
Ao = me.Point('A_O')
Bo = me.Point('B_O')
Ao.set_pos(O, l/2*A.x)
Bo.set_pos(O, l*A.x)
O.set_vel(N, 0)
Ao.v2pt_theory(O, N, A)
Bo.v2pt_theory(O, N, A)
Ao.vel(N), Bo.vel(N), A.ang_vel_in(N), B.ang_vel_in(N)
Now determine the holonomic partial velocities in N:
.. jupyter-execute:: v_Ao_1 = Ao.vel(N).diff(u1, N) v_Ao_2 = Ao.vel(N).diff(u2, N) v_Bo_1 = Bo.vel(N).diff(u1, N) v_Bo_2 = Bo.vel(N).diff(u2, N) v_Ao_1, v_Ao_2, v_Bo_1, v_Bo_2
and the holonomic partial angular velocities in N:
.. jupyter-execute:: w_A_1 = A.ang_vel_in(N).diff(u1, N) w_A_2 = A.ang_vel_in(N).diff(u2, N) w_B_1 = B.ang_vel_in(N).diff(u1, N) w_B_2 = B.ang_vel_in(N).diff(u2, N) w_A_1, w_A_2, w_B_1, w_B_2
The resultant forces on the two bodies are simply the gravitational forces that act at each mass center (we ignore the noncontributing pin joint contact forces):
.. jupyter-execute:: R_Ao = m*g*N.x R_Bo = m*g*N.x R_Ao, R_Bo
With linear torsion springs between frames A and N and frames A and B the torques acting on each body are:
.. jupyter-execute:: T_A = -k*q1*N.z + k*q2*A.x T_B = -k*q2*A.x T_A, T_B
Note that k q_2\hat{a}_x in \bar{T}_A is the reaction torque of of body B on A via the torsional spring.
Now, a generalized active force component can be found for each body and each generalized speed using :math:numref:`eq-gaf-rigid-body`:
.. jupyter-execute:: F1_A = v_Ao_1.dot(R_Ao) + w_A_1.dot(T_A) F1_B = v_Bo_1.dot(R_Bo) + w_B_1.dot(T_B) F2_A = v_Ao_2.dot(R_Ao) + w_A_2.dot(T_A) F2_B = v_Bo_2.dot(R_Bo) + w_B_2.dot(T_B) F1_A, F1_B, F2_A, F2_B
Summing for each generalized speed and then stacking the two scalars in a column vector gives the generalized active forces for the system:
.. jupyter-execute:: F1 = F1_A + F1_B F2 = F2_A + F2_B Fr = sm.Matrix([F1, F2]) Fr
For a nonholonomic system with p degrees of freedom in reference frame A, the p generalized active forces can be formed instead. The nonholonomic generalized active force contributions from a particle P and rigid body B are:
(\tilde{F}_r)_P = {}^A\tilde{v}^{P} \cdot \bar{R} \\
(\tilde{F}_r)_B = {}^A\tilde{v}^Q \cdot \bar{R} + {}^A\tilde{\omega}^B \cdot \bar{T}
See [Kane1985]_ pg. 99 for the relationship between holonomic and nonholonomic generalized active forces.
Analogous to the generalized active forces and their relationship to the left hand side of the Newtwon-Euler equations (Eq. :math:numref:`eq-newton-euler`, generalized inertia forces map the right hand side of the Newton-Euler equations, time derivatives of linear and angular momentum, to the vector space of the generalized speeds for a multibody system. For a holonomic multibody system in A made up of a set of \nu particles the rth generalized inertia force is defined as ([Kane1985]_, pg. 124):
F_r^* := \sum_{i=1}^\nu {}^A\bar{v}^{P_i}_r \cdot \bar{R}^*_i
where the resultant inertia force on the ith particle is:
\bar{R}^*_i := -m_i {}^A\bar{a}^{P_i}_i
The generalized inertia force for a single rigid body B with mass m_B, mass center B_o, and central inertia dyadic \breve{I}^{B/Bo} is defined as:
(F_r^*)_B := {}^A\bar{v}^{B_o}_r \cdot \bar{R}^* + {}^A\bar{\omega}^B_r \cdot \bar{T}^*
where the inertia force on the body is:
\bar{R}^* := -m_{B} {}^A\bar{a}^{B_o}
and the inertia torque on the body are
\bar{T}^* := -\left(
{}^A\bar{\alpha}^B \cdot \breve{I}^{B/Bo} +
{}^A\bar{\omega}^B \times \breve{I}^{B/Bo} \cdot {}^A\bar{\omega}^B
\right)
Coming back to the system in :numref:`fig-generalized-forces-3d-rods` we can now calculate the generalized inertia forces for the two rigid body system. First, the velocities and partial velocities are found as before:
.. jupyter-execute::
m, g, k, l = sm.symbols('m, g, k, l')
q1, q2, u1, u2 = me.dynamicsymbols('q1, q2, u1, u2')
N = me.ReferenceFrame('N')
A = me.ReferenceFrame('A')
B = me.ReferenceFrame('B')
A.orient_axis(N, q1, N.z)
B.orient_axis(A, q2, A.x)
A.set_ang_vel(N, u1*N.z)
B.set_ang_vel(A, u2*A.x)
O = me.Point('O')
Ao = me.Point('A_O')
Bo = me.Point('B_O')
Ao.set_pos(O, l/2*A.x)
Bo.set_pos(O, l*A.x)
O.set_vel(N, 0)
Ao.v2pt_theory(O, N, A)
Bo.v2pt_theory(O, N, A)
v_Ao_1 = Ao.vel(N).diff(u1, N)
v_Ao_2 = Ao.vel(N).diff(u2, N)
v_Bo_1 = Bo.vel(N).diff(u1, N)
v_Bo_2 = Bo.vel(N).diff(u2, N)
w_A_1 = A.ang_vel_in(N).diff(u1, N)
w_A_2 = A.ang_vel_in(N).diff(u2, N)
w_B_1 = B.ang_vel_in(N).diff(u1, N)
w_B_2 = B.ang_vel_in(N).diff(u2, N)
We will need the translational accelerations of the mass centers and the angular accelerations of each body expressed in terms of the generalized speeds, their derivatives, and the generalized coordinates:
.. jupyter-execute:: Ao.acc(N), Bo.acc(N)
.. jupyter-execute:: A.ang_acc_in(N), B.ang_acc_in(N)
The central moment of inertia of a thin uniformly dense rod of mass m and length L about any axis normal to its length is:
.. jupyter-execute:: I = m*l**2/12 I
This can be used to formulate the central inertia dyadics of each rod:
.. jupyter-execute:: I_A_Ao = I*me.outer(A.y, A.y) + I*me.outer(A.z, A.z) I_B_Bo = I*me.outer(B.x, B.x) + I*me.outer(B.z, B.z) I_A_Ao, I_B_Bo
The resultant inertia forces acting at the mass center of each body are:
.. jupyter-execute:: Rs_Ao = -m*Ao.acc(N) Rs_Bo = -m*Bo.acc(N) Rs_Ao, Rs_Bo
And the inertia torques acting on each body are:
.. jupyter-execute::
Ts_A = -(A.ang_acc_in(N).dot(I_A_Ao) +
me.cross(A.ang_vel_in(N), I_A_Ao).dot(A.ang_vel_in(N)))
Ts_A
.. jupyter-execute::
Ts_B = -(B.ang_acc_in(N).dot(I_B_Bo) +
me.cross(B.ang_vel_in(N), I_B_Bo).dot(B.ang_vel_in(N)))
Ts_B
Now the generalized inertia forces can be formed by projecting the inertia force and inertia torque onto the partial velocities:
.. jupyter-execute:: F1s_A = v_Ao_1.dot(Rs_Ao) + w_A_1.dot(Ts_A) F1s_B = v_Bo_1.dot(Rs_Bo) + w_B_1.dot(Ts_B) F2s_A = v_Ao_2.dot(Rs_Ao) + w_A_2.dot(Ts_A) F2s_B = v_Bo_2.dot(Rs_Bo) + w_B_2.dot(Ts_B)
We then sum for each generalized speed and then stack them in a column vector \bar{F}_r^*:
.. jupyter-execute:: F1s = F1s_A + F1s_B F2s = F2s_A + F2s_B Frs = sm.Matrix([F1s, F2s]) Frs
For a nonholonomic system with p degrees of freedom in reference frame A, the p generalized active forces can be formed instead. The nonholonomic generalized active force contributions from a particle P and rigid body B are:
(\tilde{F}^*_r)_P = {}^A\tilde{v}^{P} \cdot \bar{R} \\
(\tilde{F}^*_r)_B = {}^A\tilde{v}^Q \cdot \bar{R} + {}^A\tilde{\omega}^B \cdot \bar{T}
The relationship between the nonholonomic and holonomic generalized inertia forces is give in [Kane1985]_ pg. 124.