Homework #8
Contents
Homework #8¶
Problem 1¶
For the system shown above, block one slides along the x-axis and is attached via a pin to a link, body 2. The system has the following characteristics,
\(m^1 = 0.1~kg\)
\(k = 40~N/m\)
\(F_{spring} = -k(R_x^1-0.2)\)
\(l^2 = 1~m\)
\(m^2 = 1~kg\)
Use the embedding technique to solve for the dynamic response for the system. Separate the generalized coordinates are separated into dependent and independent coordinates as such
\(\mathbf{q} = [\mathbf{q}_d,~\mathbf{q}_i]\).
\(\mathbf{q}_d = [R_y^1,~\theta^1,~R_x^2,~R_y^2]\)
\(\mathbf{q}_i = [R_x^1,~\theta^2]\)
This distinction creates two second order differential equations, where \(n=3\times(\#~bodies)\) and \(n_c=number~of~constraints\). The equations of motion are as such
\(\mathbf{B^TMB +B^TM\gamma-B^TQ_e = 0}\)
where,
\(\mathbf{B}=\left[\begin{array}{c}-\mathbf{C_{q_d}^{-1}C_{q_i}} \\\bar{I}\end{array}\right]\)
\(\gamma = \left[\begin{array}{c} ~\mathbf{-C_{q_d}^{-1}[(C_q\dot{q})_q\dot{q}+2C_{qt}\dot{q}+C_{tt}}] \\ \mathbf{0}\end{array}\right] = \left[\begin{array}{c} ~\mathbf{C_{q_d}^{-1}[Q_d]} \\ \mathbf{0}\end{array}\right]\)
\(\mathbf{C_{q_d}}~and~\mathbf{C_{q_i}}\) are the Jacobian of constraints for dependent and independent coordinates, respectively
You will define the following functions:
C_systhe 4 constraint equations for the systemCq_systhe Jacobian of the systemd C_sys/dqBi_linkthe array \(\mathbf{B}\) that transforms \(\delta \mathbf{q_i}\) to \(\delta \mathbf{q}\)eom_systhe final equation of motion using the embedding technique
Define \(\mathbf{A}\) and \(\mathbf{A}_\theta\)¶
Here, you create two functions to rotate from the body coordinate system at angle \(\theta\) to the global coordinate system. The derivate, \(\mathbf{A}_\theta\) will be used in the Jacobian and equations of motion.
def rotA(theta):
'''This function returns a 2x2 rotation matrix to convert the
rotated coordinate to the global coordinate system
input is angle in radians
Parameters
----------
theta : angle in radians
Returns
-------
A : 2x2 array to rotate a coordinate system at angle theta to global x-y
'''
A=np.zeros((2,2))
A=np.array([[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]])
return A
def A_theta(theta):
'''This function returns a 2x2 rotation matrix derivative
input is angle in radians
Parameters
----------
theta : angle in radians
Returns
-------
dAda : 2x2 array derivative of `rotA`
'''
dAda=np.array([[-np.sin(theta), -np.cos(theta)],
[np.cos(theta), -np.sin(theta)]])
return dAda
Define Jacobian of pin constraint as Cq_pin.
def Cq_pin(qi, qj, ui, uj):
'''Jacobian of a pinned constraint for planar motion
Parameters
----------
qi : generalized coordinates of the first body, i [Rxi, Ryi, thetai]
qj : generalized coordinates of the 2nd body, i [Rxj, Ryj, thetaj]
ui : position of the pin the body-i coordinate system
uj : position of the pin the body-j coordinate system
Returns
-------
Cq_pin : 2 rows x 6 columns Jacobian of pin constraint Cpin
'''
Cq_1=np.block([np.eye(2), A_theta(qi[2])@ui[:,np.newaxis] ])
Cq_2=np.block([-np.eye(2), -A_theta(qj[2])@uj[:,np.newaxis] ])
Cq_pin=np.block([Cq_1, Cq_2])
return Cq_pin
Problem 2¶
Create the system of equations using the augmented technique. Solve these equations numerically using solve_ivp and plot the position and orientation of the links for one revolution of the crankshaft. Assume that \(M^2\) = 10N-m, \(F^4\) = 15N, \(\theta^2\) = 45\(^o\), and \(\dot{\theta}^2\)=150 rad/s.
The augmented technique is a generalized method to solve for the dynamic response in a system of moving parts. The system of differential algebraic equations (DAE) are solve for all generalized coordinates and constraint forces in a system of equations,
\(\left[\begin{array} ~\mathbf{M} & \mathbf{C_q}^T \\ \mathbf{C_q} & \mathbf{0} \end{array}\right] \left[\begin{array} ~\mathbf{\ddot{q}} \\ \mathbf{\lambda}\end{array}\right] = \left[\begin{array} ~\mathbf{Q_e} \\ \mathbf{Q_d}\end{array}\right]\)
Plot the angles of the two connecting links and the position of the sliding block at point B.
Plot the reaction forces at O, A, and B.