Merge pull request #56 from TTK4130/assignment-7

Converted assignment 7 to rst.

docs/assignments/assignment-7.rst

@@ -0,0 +1,160 @@
+==========================
+Assignment 7 - Bond Graphs
+==========================
+
+.. note::
+
+    Submit your assignment as a single PDF, including plots and source code (if any).
+    We expect academic honesty. Collaboration is encouraged, but must be declared. Any use of AI must be declared along with any other sources used.
+    This is not an exam. Do your best and show that you put in effort and the assignment will be approved.
+
+In this assignment we will study how we can use power bonds to connect submodel into a system model.
+
+Problem 1 - Steering mechanism
+------------------------------
+
+.. figure:: ./figures/assignment_7/Steeringmechanism.png
+    :width: 100%
+    :align: center
+    :name: steering
+
+    Simple model of a steering mechanism.
+
+.. figure:: ./figures/assignment_7/word_bg_steering_mech.png
+    :width: 100%
+    :align: center
+    :name: bond_graph_steering
+
+    Word bond graph for the steering mechanism.
+
+Figure :numref:`steering` shows a simple model of a steering mechanism for a vehicle,
+while figure :numref:`bond_graph_steering` shows a *word bond graph* for the same
+steering mechanism.
+
+.. admonition:: Tasks
+
+    a) Draw a block diagram of the system based on the word bond graph and the causality assignment given to it.
+
+    .. hint::
+        :class: dropdown
+
+        Recall that the causal stroke is on the submodel where the effort is 
+        input and the flow is output.
+
+    b) The following information is given about each submodel:
+
+    - **Batter**:
+        Gives a constant voltage.
+    - **DC motor**:
+        A DC-motor can be described by the following equations
+
+        .. math::
+            L_a \frac{di_a}{dt} = -R_a i_a - K_t \omega_m + u_a
+            \\
+            J \dot{\omega}_m = K_T i_a - \tau
+        
+        where :math:`L_a` is the armature inductance, 
+        :math:`i_a` is the armature current, 
+        :math:`R_a` is the resistance in the armature circuit, 
+        :math:`K_T` is a constant, 
+        :math:`\omega_m` is the speed of the motor, 
+        :math:`u_a` is the armature voltage,
+        and the torque :math:`\tau` is the load.
+    - **Flexible shaft**: 
+        A flexible shaft may have a slight difference in the angular velocity :math:`\omega_1`
+        and :math: `\omega_2` of each side, resulting also in a slight difference
+        :math:`\Delta \theta = \int_{t_0}^t (\omega_2 - \omega_1) dt` in the angular
+        displacement on each side.
+        We can think of the flexible shaft as an angular spring with a
+        linear relation between the angular displacement and the torque, such that :math:`\tau = k_s \Delta \theta`.
+        The moment of inertia for the shaft is small compared to the moment of inertia of the DC-motor
+        and the rack, so we may consider it as massless.
+    - **Gear (or pinion)**:
+        The gear is modelled as massless and loss-less
+        (i.e. it does not remove energy from the system).
+        Its function in this system is to transform between the angular velocity
+        and torque on one port, and linear velocity and a force on the other port.
+        The relationship between the linear velocity and the angular velocity is :math:`v = r\omega`. 
+        Since it is loss-less, we also have that the power on each port is identical.
+        In equation form this can be stated as :math:`\omega \tau = v F`.
+    - **Rack**:
+        The rack can be modelled as mass m that can move with one degree of freedom.
+        This mass is governed by the equation: :math:`ma = \sum F` .
+    - **Spring**: 
+        The spring gives a linear relation between the force and the displacement, such that
+        the spring force :math:`F_s`  is given as :math:`F_s = kx` , where :math:`x = \int_{t_0}^t v dt`.
+    - **Damper**:
+        The damper (or dashpot) is governed by the law: :math:`F_d = k_d v`.
+
+    Use the block diagram you made together with the information given above and the word bond
+    graph to show that the system equations for the steering mechanism can be written as
+
+    .. math::
+        L_a \frac{di_a}{dt} &= -R_a i_a - K_t \omega_1 + u_a
+        \\
+        J \dot{\omega}_m &= K_T i_a - k_s \Delta \theta
+        \\
+        \dot{\Delta \theta} &= \frac{1}{r} v - \omega_1
+        \\
+        \dot{x} &= v
+        \\
+        m \dot{v} &= - kx - dv + \frac1{r} k_s \Delta \theta
+    
+    .. hint::
+        :class: dropdown
+
+        Recall that the direction of the half arrows in the word bond graph
+        defines which direction of positive power (energy flow).
+        This means that the half arrows will be useful in defining the sign of
+        efforts and flows in the final set of differential equations.
+
+Problem 2 - Two rotating shafts
+-------------------------------
+
+.. figure:: ./figures/assignment_7/two_rotating_masses.png
+    :width: 100%
+    :align: center
+    :name: two_rotating_masses
+
+    Two rotating shafts connected by a gearbox
+
+.. figure:: ./figures/assignment_7/word_bg_flywheels.png
+    :width: 100%
+    :align: center
+    :name: bond_graph_two_rotating_masses
+
+    Word bond graph of the system
+
+Figure :numref:`two_rotating_masses` shows two rotating flywheels connected by a gearbox. 
+The gearbox is friction-less. 
+There are two torques :math:`T_1` and :math:`T_2` applied to the two flywheels. 
+The left flywheel rotates with the angular speed :math:`\omega_1` and has moment of inertia :math:`J_1`,
+while the right-hand flywheel rotates with an angular speed :math:`\omega_2` and has moment of inertia :math:`J_2`. 
+A system model is shown in the form of a word bond graph in figure :numref:`bond_graph_two_rotating_masses`.
+
+
+.. admonition:: Tasks
+
+    a)
+        Draw a block diagram for the system based on the word bond graph and the causality defined by
+        the causal strokes.
+
+    b)
+        While it is tempting to use two differential equations on the form
+
+        .. math::
+            J_i \dot{\omega}_i = \sum T
+
+        this is not viable because the two flywheels are not able to rotate independently of each other.
+        If we know :math:`\omega_1`, we also know :math:`\omega_2`
+        (and if we know :math:`\dot{\omega_1}` we know :math:`\dot{\omega_2}`).
+        In particular, :math:`r \omega_1 = \omega_2` and :math:`\tau_1 = r \tau_2`.
+        Therefore we only need a single differential equation to describe both of them.
+        Show that the equation of motion for the two flywheels can be written as
+
+        .. math::
+            \left( J_1 + r^2 J_2 \right) \dot{\omega}_1 = T_1 + r T_2
+
+    c)
+        Derive the same expression using Lagrange mechanics with your generalized coordinate
+        :math:`q = \theta_1` and :math:`\dot{q} = \omega_1`.

docs/assignments/index.rst

@@ -13,3 +13,4 @@ Assignments
     assignment-4
     assignment-5
     assignment-6
+    assignment-7