Lab Goals and Instructions

In this lab we will investigate the motion of a pendulum.  We will consider damped and undamped, linear and non-linear models; in particular, the similarities and differences between them.  We will also be introduced to the notion of solutions in phase space.  These solutions will be interpreted and related to their physical properties.  There are topics at the end of the lab which need to be addressed in the lab write-up.

Follow these guidelines (points may be deducted for non-compliance):

• three (3) students is the maximum group size
• use a cover page
• for each student in the group, on the cover page, list the
• name
• recitation number
• I.D. number (last 4 digits)
• TA name
• Instructor's name
• submit the report in class; do not put the reports in mailboxes or they will be returned
• see here the lab report guidelines
• make sure to include in the appendix all steps of a derivation. Even when the desired result or equation is given you must show all necessary steps to obtain it.
• four decimal place accuracy will suffice for this lab; for example, e is approximately 2.7183

Each group is required to turn in one lab report in class on Thursday, 3 December, 2009. Late labs will NOT be accepted under any circumstances.

Model

We wish to study the motion of an ideal pendulum.  The derivation of the equations of motion is given here:

• Derivation in PDF format

Note that there are two cases being considered in this model.

1. The pendulum hangs from a nail and can swing all the way around; in a complete circle.
2. The pendulum hangs from a ceiling; the swinging is restricted, i.e. the pendulum can't swing up through the ceiling!

Read through the derivation to get an idea of the model.  Your report should give a description of the physical configuration, defining
the important terms; however, you do not need to give details of the derivation of equation (1).  An appropriate place for this description
is in the introduction.

We will be working with the following equation (stated in the derivation):

Now let u = θ giving us the following system:

When answering the questions, do not assign values to the model parameters except when plotting.  For generating the plots, let L = 1, m = 0.5, g = 9.8, and b = 0 (unless noted otherwise as, for example, in questions 6 and 12). All units in S.I. 
Note the notation

The first step in understanding this model is to consider the phase-plane portrait for (2); the set of variables (u, v) is called the phase plane.  If b = 0, the vector field is a plot of u' versus u (or v versus u in the notation of (2)), namely (v,-(g/l)sin u).

We can create phase-plane portraits using the Mathematics Visualization Tool (MVT) or by using Mathematica, Matlab, or Maple. You might also consider using "pplane7.m" for Matlab that can be found online. The input will look similar to (2). The slope field is a plot of vectors that determines the path of the curves. Take -4π < u < 4π, and -10 < v < 10 which offers a good start for viewing the vector field and phase portraits.  These inequalities can be tweaked once we gain a better understanding of the plots.

Linearized Model

Although we can extract a fair amount of information from the equation using phase planes (1), the sine term still makes the equation a little unwieldy.  In general, non-linear equations yield little useful analytic information without a sizable amount of effort.  For this reason, we want to find a linear equation which approximates equation (1).

Using Taylor's theorem, we can rewrite the sine term in equation (1) as a series expanded about the lower equilibrium point.  Recall Taylor's theorem, in this case, states that

We can then obtain an approximation to equation (1), under certain conditions to be determined later, by truncating this series. 

Questions and Issues to Address:

1. Classify (1) (e.g. linear? homogeneous? order? autonomous?) and interpret each term of the model
2. From the physical configuration, θ = θ₀ is the same point as θ = -θ₀. Why?  The previous statement is not valid for v = dθ/dt, why not?
3. Produce a phase portrait for (2).  Overlay some solution curves onto this vector field.
4. Find the equilibria of (2) analytically (by hand) and identify (draw) them on the phase portrait.  Where do these equilibria occur physically? Make sure this explanation is in layman's terms.
5. On the phase portrait, identify the distinct types of trajectories and interpret these different behaviors in terms of the physical motion of the pendulum.  Take care to consider those that, while technically possible, are unlikely to be observed physically. 
6. Notice that the notion of time is absent in the phase-plane portrait.  However, while there are some trajectories that can take only a short time, there are others that can take infinite time!  For example, it can be shown (although not easily...you do not need to verify this fact) that, in the undamped (b=0) case of the pendulum nailed to the wall, equation (1) is satisfied by: 
    
   Create a plot of this solution and describe the motion of this trajectory.  (Think about the behavior of this curve as time is extended from -∞ to +∞?  The curve has two asymptotes: θ = π and θ = -π.  What do they indicate?  Note where these points are located on your phase portrait!)
7. Include the following derivation of the energy formula in the body of your lab; other calculations are to be included in an appendix.  Now that we have some idea of how our model behaves, we will do some analysis to verify the geometric (qualitative) results.  Write the model as 
    

   Multiply this equation by the velocity dθ/dt and notice that both terms can be rewritten as expressions like a total derivative.  By integrating the (total) derivatives, obtain the result

   
   where c is some constant.  Show that, by an appropriate choice of the constant of integration c, we may rewrite this result as:
    

   where C₂ is a new constant.  (Note: The first term in this equation is an expression for the kinetic energy of the pendulum.  The second term is an expression for the potential energy of the pendulum.  Hence (3) is the "total energy" of the pendulum.)

8. Equation (3) makes a statement about the total energy of the system.  What is that statement?  (hint: Set θ = π)
9. Write the Taylor series expansion for the sine term in equation (1).  Approximate equation (1) by truncating the series after the first term to obtain a very simple approximation.  Under what conditions should this simplified equation give a good approximation to equation (1)?

10. Solve this new equation, by hand, with the initial conditions 

11. Create a phase portrait of the linear equation that you derived in number 10. What are the similarities and differences between the phase portrait in number 4 and the phase portrait for equation (1)?
12. For all of the plots, we have been considering the somewhat unrealistic situation where there is no damping on the pendulum's movement.  In other words, we set b = 0.  Now, produce a phase-plane portrait as well as solution curves for the damped (non-linear) pendulum with b = 0.7 and b = 1.2.  Do not overlay these plots.  Describe the results of each case.
13. Briefly comment on the similarities and differences between the damped non-linear case and the undamped non-linear case. You might want to consider how the physical system and the phase portraits are related.
14. What are the shortcomings of the model?  In what ways can the model be improved?

15. We will now walk through the steps required to solve the nonlinear model. Refer back to step 7, where we had, after shuffling constants:

    Multiply through by dθ/dt and integrate with respect to t, arriving at:

    

    How would you solve for the constant? Why do you need more than just θ(0) to obtain a value for the constant?

16. Suppose that you have solved for the constant, using the method that you just described, and that its value is α. Using this notation, separation of variables, and the equation above, show that the solution is: