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Table of Contents

Challenge

In this challenge, we will study how a "limited slip differential" helps a car that is stuck in the mud. Maybe this has happened to you... maybe it hasn't. But getting stuck in the mud looks like this:

Not a great day! And did you know that it's extremely likely that the other wheel, while also stuck in the mud, is not moving at all? This is because the truck's differential is not allowing any torque to be transfered to the wheel with more traction. Why would this be? To understand, we need to understand what a differential in a car actually does.

As the video shows, a standard differential in a car will actually deliver all of the power ($\dot{W} = T\Omega$) from the transmission to the wheel that is slipping more!! This unfortunate reality led to the development of a device called a "limited slip differential," whose operation is explained below.

You should recognize the concept of viscous friction as a catalyst for recovering the lost torque in the axle. You may even consider the effect of the limited slip differential to be similar to that of an idealized rotational damper. Today, you will build a very simplified model of a truck's rear axle when it is equipped with a limited slip differential. You will not have any data with which to validate your model, because your job is to design a limited slip differential with the "right" damping coefficient $b$. You will use your model to determine what damping coefficient will be acceptable.

But before you begin, it is very true that this system may look extremely daunting. How will you know just where the power goes between the left wheel, the right wheel, the differential, and the viscous damping separating the sliding wheels from the mud?

One thing that may help with model construction (and even scoping) in complex problems like this is to explicitly take advantage of the first law of thermodynamics in the form of continuity and compatibility equations. The easiest way to construct your system model in a way that alllows you to take advantage of these concepts is to represent your mechanical system model as a network of circuit-like elements so that the relationships needed in developing compatibility and continuity equations are explicit. To represent our mechanical system as an "equivalent circuit," we need to be able to draw solid analogies between power flow in electrical circuits and power flow in mechanical systems.

Power Variable Types

In all three of the types of systems we've studied, it is possible to write an equation for how power flows between elements. For example:

  1. In a mechanical system, power is $F\cdot v$ for translational systems and $T\cdot \Omega$ for rotational systems
  2. In an electrical system, power is $V \cdot i$
  3. In a fluid system, power is $P \cdot \dot{\mathcal{V}}$.

If the first law of thermodynamics tells us that one of these two variables is conserved at a "node," we interpret this as a statement that it represents a quantity flowing "through" the node. We call this type of variable a "through type variable." In mechanical, electrical, and fluid systems, these are force, current, and flow rate, respectively.

If a variable must be measured with respect to some fixed reference in order to have meaning, and is different from one node in our model to the next, it is called an "across type variable."

For example, velocity must be measured with respect to an inertial reference frame, and varies from one end of an idealized damper to the other. Voltage must be measured with respect to some reference "ground" voltage, and varies from one end of a resistor to another. Pressure must be measured by a gauge which references some environmental (atmospheric) reference pressure, and varies from one end of a valve to the other.

Generalized Idealized Elements

All of the "idealized" system element types we have talked about thus far, by the assumptions applied to their first law equations, either store energy or transfer it to the surrounding environment. This means that all of the elements we've discussed (other than "nodes") have different amounts of power flowing into and out of them. These elements are often visualized as having two "ports," one into which power flows, and one out of which power flows. A drawing is shown below:

image-2.png

The generalized element above will have two distinct values of the A-type variable at each "end" (some systems textbooks call these 'ports'), either because the element is accumulating energy (like a mass, capacitor, or tank does), or because it is transferring energy to outside the system boundaries (like a resistor, damper, or valve). We will typically label the inside of this element with the independent variable that dictates its ability to store energy (e.g. $C$, $m$, $J$) or dissipate energy (e.g. $b$,$R$).

Visualizing elements in this general way allows us to connect elements of our systems together into generalized "equivalent circuits." Doing this provides operational efficiency in using energy conservation (the first law) to build models for our systems.

Assuming no significant net energy transfer via mass flow (as we have in all of our 'standard' elements so far), the first law of thermodynamics for this element could be written:

$$\dot{E} = \dot{W}_{in} - \dot{W}_{out} + \dot{Q}_{in} - \dot{Q}_{out}$$

Then, recognizing that $\dot{W} = A\cdot T$, we could write the power portion of the first law equation as:

$$\dot{E} = A_1T - A_2T + \dot{Q}_{in} - \dot{Q}_{out}$$

note that the visualization of our generalized element does not show whether the element stores energy or transfers it to the element's surroundings as heat. As such this is not a complete energetic diagram. We need to know what the $\dot{Q}$ terms and the $\dot{E}$ term should be for our element. These terms are often based on similar sorts of assumptions, and yield two common types of "generalized elements."

Energy Storage Elements

If an element stores energy, but does not transfer a significant amount to its surroundings via heat, it is generally called an "energy storage element." If it stores energy via its across-type variable, it is called an "A-type storage element." Masses, rotational inertias, capacitors, and fluid capacitors are all A-type storage elements. If the element stores energy via its through-type variable, it is called a "T-type storage element." For energy storage elements, zero heat transfer to the element's surroundings is often assumed, which reduces the first law equation to:

$$\dot{E} = A_1 T - A_2 T$$

Dissipative Elements

If an element does not store significant energy, but does transfer significant energy to the element's surroundings via heat, we call that element a "dissipative element." Dissipative elements include electrical resistors, valves, and idealized dampers. The assumption of no energy storage for dissipative elements yields the generalized first law balance of:

$$\dot{Q}_{out} = A_1 T - A_2 T$$

Do all physical things fall into one of these two categories?

No. It may not be reasonable to represent every physical "part" of every system as one of these two things. However, Elements that display significant energy storage and energy dissipative behavior can, under many circumstances, be represented by a combination of a dissipative and a storage element. We call this a "lumped element" approximation, where a "real" element is broken down into a combination of idealized storage and dissipative elements.

Generalized Equivalent Circuits

By conceptualizing each of our idealized elements as having two ports through which power flows, much like an electrical component has, we can arrange the elements into an equivalent circuit that shows us how power flows from one element to another. An example is shown below.

image-2.png

This circuit has a source (which we often model as having a defined, known value for either the T-type or A-type power variable, and thus "infinite" power potential), and 5 idealized elements. Distinct values of the across-type power variable are shown in blue, and distinct values of the through-type variable are shown in red. This circuit-like network allows us to write compatibility relations (loops relating the A type variables) and continuity relations (nodal equations for the T type variables) to aid in model construction.

Note that while this configuration allows for easy use of the first law of thermodynamics for studying how power flows from one element in our system to the others (compatibility and continuity relationships), it does not tell us explicitly about energy storage or dissipation for each element. Individual first law balances for each element are required, along with the construction of loop and node equations, to use the equivalent circuit for model construction.

Note that typically, the blank box used to denote each element would include the element's primary parameter. For example, a damper might be drawn as a box with a "b" in it to represent its damping coefficient. A mass might be drawn as a box with an "m" in it. A capacitor might be labeled "C."

Disciplined Process for using Energy Conservation and Equivalent Circuits to Construct a Model

Now that we have "new tools" (KCL/KVL and equivalent circuits) to help us with model construction, we can update our "step 3" in the general discplined process for Model Construction. The use of equivalent circuits means that we can take explicit advantage of two particular forms of the energy conservation equation: node equations (KCL) and loop equations (KVL). KCL lets us represent power transfer between elements when the T-type power variable splits and the A-type variable is constant. KVL lets us represent power transfer elements when the T-type variable is constant but the A-type variable is not.

  1. Draw an energetic diagram for each element in your model scope, crossing off any terms you are assuming to be insignificant. When you do this, justify each assumption clearly in writing. Making an assumption is always a deliberate risk; the model evaluation step of our overall disciplined process for ES103 will guide you in determining the quality of the assumptions you make.
    1. You may reference the assumptions in the "standard" lumped element table in your justification, but you must still give a reason for assuming that an element acts like one of these "standard" element.
    2. Represent (if possible) each of your elements as a generalized circuit element.
  2. Write the time-derivative of the first law of thermodynamics for each element to describe how the element transfers energy between itself and its surroundings. Take care to substitute what you know about the energy, work, and heat terms for each element into the equation.
    1. You may cross out any "extra" power variables in each final first-law equation for each element (e.g. $V_{12} i = i^2R \rightarrow V_{12} = iR$)
  3. Represent connections between your system's elements by drawing the system as an equivalent circuit. Use physical connections between elements in your system to guide you here. Write equations that indicate how each element in your scope exchanges energy between itself and other elements in the scope, or between itself and the overall system's surroundings. Specifically, power linkges between elements can be written using Krichoff's Current Law (continuity) and Kirchoff's Voltage Law (compatibility):
    1. Construct and label your "equivalent circuit." This may be an iterative process as you identify nodes and loops in your system (see below).
    2. Identify "nodes" in your system and write the principle of continuity for these nodes (e.g. $i_1 - i_2 - i_3 = 0$).
    3. Identify "loops" in your system and write the principle of compatibility for each loop (e.g. $V_{41}+V_{12}+V_{23}+V_{34}=0$.
  4. Using algebraic manipulation, combine your equations from steps 2 and 3 to obtain a final system model that includes only independent variables, dependent variables, and their derivatives.
  5. Check your final system model for internal validity, including the signs and units of each term, and the overall form of the equation.

Assignment

Design Requirements

You work at a car company as an engineer. The company saw the viral video of last year's truck getting stuck in the mud on YouTube, which is pasted at the top of this notebook. You've been tasked with equipping the 2021 version of the truck with a limited slip differential to decrease the likelihood of the new model getting stuck.

To complete your initial, rough model, you assume that the truck has whatever torque is necessary to keep the right wheel spinning at a known, constant angular velocity while the left wheel, which is initially stationary, starts to spin due to torque from the limited slip differential. No other source of torque powers (has the potential to increase the energy of) the left wheel.

Using the concept of an equivalent circuit, scope and construct a model for the axle of the truck that is stuck in the mud in the first video. Then, assuming that you can design a limited slip differential with any damping coefficient $b_d$, use a simulation of the truck's right wheel spinning at a constant $\Omega_r = 60 \frac{rad}{s}$ (10 revs/second) to choose a value of $b_d$ that will satisfy the following design requirements:

  1. The left wheel must come up to a constant steady-state speed no sooner than 2 seconds of when the right wheel begins to spin at its constant, input angular velocity of $\Omega_r = 60 \frac{rad}{s}$.
  2. The left wheel must have an eventual angular velocity of no faster than $1 \%$ of the right wheel's angular velocity. While this will hurt the truck's ability to climb out of the mud, it is required to prevent unnecessary wear during normal driving around turns at low speeds, where the two wheels must turn at different rates.

What you know

Your department at your engineering firm doesn't have all that many details about the internal workings of limited slip differentials. You only know that if you specify a particular effective damping coefficient $b_d$, the company that you outsource the differential manufacturing to can make it. You also know that:

  • The truck's wheels with tires have a mass of $15kg$ each and a radius of $0.31m$.
  • The axle and differential have negligible rotational inertia, but the differential will have a significant rotational damping coefficient $b_d$ that transfers torque between the right and left wheels.
  • Based on very rudimentary initial testing, you estimate that the average viscous damping coefficient representing the friction between the stationary wet mud on the ground and your company's wheels/tires is approximately $b_g=2 \frac{Nms}{rad}$.
  • For the purposes of this initial model, you can neglect the possibility that the truck starts to move during the test (including this possibility would complicate the model substantially).

Deliverable: Model Scoping

Present a model scope with a well-thought-out explanation of each modeling choice below.

YOUR ANSWER HERE

Deliverable: Model Construction

Using an equivalent circuit to represent the system of two wheels connected to each other via the limited slip differential, and to the ground via viscous friction to the mud, construct a model that relates the (known, constant) right wheel's angular velocity, $\Omega_r$ to the left wheel's angular velocity, $\Omega_l$. You may assume that the right wheel is powered by an angular velocity "source" for the purposes of your model.

YOUR ANSWER HERE

Deliverable: Model Evaluation

You do not have data with which to validate your model (yet). However, you do have more than one way to build a model of a mechanical system. Below, show the following:

  1. Show that you can obtain the same model as you developed using an equivalent circuit by using Newton's laws and free body diagrams of each system component.
  2. Explicitly show a check of the units governing each term in your model.

YOUR ANSWER HERE

Deliverable: Use your model for design

Construct a simulation that allows you to tune the parameter $b_d$ for the differential, and create a simulation that, with the correct value for $b_d$, meets the design specifications above.

In [4]:
% YOUR CODE HERE
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Deliverable: Discussion

Using your model, modify it (slightly) to also show the torque that is applied to the ground by the left wheel with your design when the left wheel reaches its constant speed. Compare this to the torque applied by the right wheel to the ground. Is this limited slip differential design, under the design constraints you have, going to make a significant improvement to the truck's ability to climb out of the mud? Why or why not?

YOUR ANSWER HERE