(116c) On Instruction of Statics and Dynamics to a Large Class

This study is based on instruction of Engineering Mechanics to 303 students in 7 semesters at a HBCU, historically black college and university.  The statics section and dynamics section are instructed separately and a 4 credit combined statics and dynamics section is given every semester.  The course was assessed using ABET criteria 3(a) and 3(e) skils.  The course objectives are for the students to: (i) be able to construct free body diagrams; (ii) be able to apply the two laws of mechanical equilibrium, of translation and rotation; (iii) demonstrate the ability to use 10 different supports in equilibrium, problems; (iv) derive effect of distributed loads by using calculus and laws of equilibrium; (v) learn Coulomb's law of friction; (vi) be abe to device simple machines such as wedges, jack screws and belts using principles of friction; (vii) obtain center of gravity, centroid using integral calculus; (viii) obtain volumes and surface areas of objects of rotation; (ix) compute moments of inertia; (x) derive three laws of kinematics governing velocity, acceleration and displacement; (xi) apply to elevator design and projectile motion; (xii) learn Newton's laws of motion and gravitation; (xiii) derive the space flight trajectory in cyliindrical coordinates; (xiv) apply work and energy theorem; (xv) derive and apply three laws of kinematics of rigid bodies.

    A correlation between the students final performances in CVEG 2454 Statics and Dynamics course was sought as a function of their performances in the first test (Bucholtz, 2011).  The trend was fairly linear for 209 students. There was some scatter at the lower end of the graph. This can be accounted for by absences in the first test. The scatter was much less at the higher end. The scatter was less at the average to above average end.

   PowerPoint  slides on FBD, free body diagrams  increased the attention of lot of the students. The number of sets of alternate EOEs, equations of equilibrium  that can be generated can be 10 sets when 3 different moment become significant. 10 = ⁵C₃. This was for the two dimensional case.

  New theory is needed for moment needed to turn a jack-screw at large thread angles and large coefficient of friction. The current theory gives a mathematical “blow-up” and an unrealistic infinite moment.

  The history of how equations of motion and calculus were simultaneously launched with the definition of velocity as rate of change of displacement by Newton and Leibnitz was discussed, More students spent more time with the math skills.

   The units of the coefficient in the Belville springs problem was found to be incorrect. This increased the awareness of the importance of each and every term in any equation.

  Cubic equations were encountered during analysis of force and displacement problems. Vieta's substitution for depressed cubics can be a method of obtaining closed form analytical solution for cubic equations.

   In one problem in projectile motion the student was asked to calculate the angle of the water jet directed at a fire.  Upon setting up the equations of projectile motion the equations were reduced  to one variable described by one equation. This was quadratic. Two roots both real and positive resulted. The two angles meant different points where the water jet will encounter the fire: one before the maximum height and the second after the maximum height.

   Students were required to sketch the s-t, v-t, a-t and v-s graphs.  The equations to a hyperbola, circle, parabola, ellipse etc were not shown in the Appendix. These were available on the Internet throught wikepedia.org. Once these equatiosn were refered to more students got on board.

   In one problem on curvilinear motion, the distance travelled by the car was given as 100 m. The (x,y) coordinates had to be solved for. This needed an lengthier algebraic solution.  ∫dL had to be used as shown in chapter 9..0 on the bent rod problem. The integration also involved several methods such as trigonometric substitution, integration by parts and partial fraction expansion.  The purport of the normal acceleration calculation using the radius of curvature did not come through clearly.  An asymptotic limit can be used, i.e,, the radius of curvature at (100,0) can be assumed for the radius of curvature at (x,y). This leaves the focus of the exercise on radius of curvature and estimate of normal acceleration.

    In another problem at the end of the chapter in the textbook on work and energy theorem on nested springs, the spring compressions of the two springs were positive for one and negative for another or vice-versa.  Two nested springs with both in compression for absorbing the energy from a falling block is conceivable.  It is hard to imagine one of the nested spring in compression and the other in stretched configuration.

   Theory behind the  formulae used in projectile motion in a x-y coordiantye system was not explained in detail in the book. This was done in detail in the blackboard. The students learnt the significance of v².  Although the force of gravity changes in direction from negative to positive during the projectile motion the same mathematical expression was shown to be valid for all the phases of the motion in a chosen cartesisn coordinate system.

   Power Flight trajectory of a satellite was derived from first principles. The cylindrical coordinates conversion from Cartesian coordinates was shown. This motivated more students.  Use of phase angle eased the interpretation of the integration constants.

   Many book mistakes were found . This got the students’ morale up. For example in a problem in Chapter 13.0 the mass of the train was not needed to complete the analysis of the problem.

  Solutions to quadratic equation with two roots both positive could not be interpreted in the spring-mass inclined plane problem. The second root has to be negative with a compressed position interpretation and the positive root the stretched position interpretation.

  In one problem, a car was driving up the incline under rain.  The stoppage distance under wet and dry conditions were required to be calculated.  In one case the stoppage distance was the solution to a quadratic equation.  This lead to two positive roots. It was not clear which root is the correct answer.  The work and energy theorem was needed to resolve the extraneous root.

  The word “erratic motion” can be changed to “domain restricted” solutions in rectilinear kinematics: erratic motion (Chapter 12.0).

    Calculation of bank angle for race car track motivated the students further.

   In one problem Ball A was released from a tall building and Ball B was thrown vertically upward. The students were asked to calculate the height from the ground and time at which Ball A & Ball B pass each other.  It turns out for the numbers given in their textbook, the meeting takes place only when the Ball B descends again. Both A and B move vertically downward! This can be misleading to the novice. The numbers have to be changed so that the Ball A and Ball B cross each other.

   Industrial visits to NASA and planetariums are planned.

   Students were encouraged to derive equations that they apply to real situations themselves rather referring to formulae derived and kept in books. This improved the fundamental grasp of the course.

  The role played by Sir Isaac Newton in the history of the subject was discussed. The story of Galileo was mentioned. How Kepler’s equations were integrated into Newton’s laws by use of cylindrical coordinates got more students tend to the subject. The planetary laws were around before Newton observed the fall of apple from the tree.

References

REQUIRED TEXTBOOK:         

Engineering Mechanics - Statics and Dynamics,  R. C. Hibbeler,  Pearson Prentice Hall, 12^th Edition, (2010), Upper Saddle River, NJ. ISBN-13:  978-0138149291          

REFERENCES,  SOURCES AND FURTHER READING

 

Philosophiæ Naturalis Principia Mathematica, Sir Isaac Newton, Vol I-III, (1687), Latin, English Translation by R. Bentley.

 

Engineering Mechanics - Statics and Dynamics, I. H. Shames, Prentice Hall, 4^thEdition, (1997), Upper Saddle River, NJ. ISBN 0133569241.

Engineering Mechanics Vol. I-Statics,  J. L. Meriam and L. G. Kraige, John Wiley and Sons, (1986),  New York, NY.

Engineering Mechanics Vol II – Dynamics, J. L. Meriam and L. G. Kraige, John Wiley and Sons, (1986), New York, NY.

Schaum’s Outline of Lagrangian Dynamics, Wells, McGraw Hill Professional, (1967), New York, NY.

Vector Mechanics for Engineers, F. P. Beer and E. R. Johnston, McGraw Hill Professional,  8^th Edition, (2007), New York, NY.

Engineering Mechanics : Dynamics, A. Pytel and J. Kiusalaas,  CENGAGE Learning, Third Edition,  (2010),  Stamford, CT.

Engineering Mechanics: Statics and Dynamics, F. Constanzo,  M. E. Plesha and G. L. Gray,  McGraw Hill Professional, (2010), New York, NY.