Class Schedule

Text: Applied Mechanics - Dynamics, by Housner and Hudson

Week 1. June 6 - 12

Ch. 1.  The General Principles of Dynamics

Secs. 1.1 - 1.6

Problems: 1.1, 1.2, 1.4, 1.5*

* Seek F in the form F = CA^av^br^c, with C dimensionless.  Express F, A, v, and r in terms of fundamental units, then find a, b, and c by matching dimensions of both sides of equation.

Sec. 2.1: Displacement, Velocity, and Acceleration

Problems: 1, 3, 4, 7, 8

Sec. 2.2: Angular Velocity

Problems: 9, 10, 12

Sec. 2.3: Motion Referred to a Moving Coordinate System

Problems: 13, 14, 15, 18, 19, 20

Sec. 3.1: Integration of the Equation of Motion for Particular Problems

Problems: 2, 3, 4, 5

Notes

a) For an equation of motion of the form y'' = f(y), where t is the independent variable, the transformation y'' = dy'/dt = (dy'/dy)(dy/dt) = y'(dy'/dy) transforms the equation into the form y'dy' = f(y)dy, which can be integrated. This is a separation-of-variables type of transformation.

b) For an equation of the form y'' = f(y'), or dy'/dt = f(y'), take y' as dependent variable, and rewrite the equation as dy'/f(y') = dt. This is also a separation of variables method.  Integration yields t as function of y'.

Another separation of variables method:  Using  y'' =  y'(dy'/dy),  as shown in a), the equation y'' = f(y') is transformed into y'dy'/f(y') = dy.  Integration yields y as function of y'.

Sec. 3.2: The Equation of Impulse and Momentum

Problems: 8, 10, 11, 14, 15

Note: The equation of impulse and momentum does not imply that the impulse is of a very short duration. It can always be used if the impulse ∫Fdt can be evaluated. In that case it is the same as integrating mx'' = F with respect to time.

Sec. 3.3: The Equation of Work and Energy

Problems: 16, 17, 19, 21 - 25

Sec. 3.4, 3.5, 3.6: Potential, Potential Energy, Conservation of Energy

Problems: 26 - 30

Sec. 3.7: The Solution of Problems in Dynamics

Problems: 31, 32, 34, 35, 36, 37, 39, 40, 42, 43, 45, 46, 47

Notes

a) The equation of work and energy is sometimes called a 'first integral' of the equations of motion, because it goes from acceleration to velocity. Notes (a) and (c) on Sec. 3.1 can be seen as leading to that equation, and some of the problems of that section can be solved by the Work-Energy approach.

b) When the forces are conservative, i.e. when they have a potential, their work between two positions depends only on those positions, and not on the path between them. That's why it is then possible to have a first integral of the equations of motion. So, think of the work-energy equation whenever you encounter a problem where the forces are conservative.

c) The equation of work and energy can be written for any force system, conservative or not. Forces that do zero work, such as the normal reaction on a surface of contact, and the force in an inextensible string, do not affect the equation. If friction forces are present, their work must be included in the equation.

Sec. 4.4: Impact

Problems: 24, (27, 28), 30, 31, 32, 33, 34

Note: In problem 27, assume that e is defined with respect to the direction normal to the surface of rebound, and that the ball is on the verge of sliding during impact.

Sec. 4.7: Variable Mass Systems

Problems: 39, 42, 43

Ch. 5.  Dynamics of Vibrating Systems

Sec. 5.1 to 5.4:  The Vibration Problem, Characteristics of Forces, Differential Equation, Free Undamped Vibrations.

Problems: 5.2, 5.4, 5.5, 5.7, 5.11, 5.13, 5.14, 5.15, 5,16, 5.17

Sec. 5.5: Damped Vibrations

Problems: 5.18 (try also by Laplace), 5.21, 5.22, 5.23, 5.25

Sec. 5.6: Forced Vibrations

Problems: 5.28, 5.29, 5.32, 5.33, 5.35, 5.37(try also by Laplace), 5.38

Sec. 5.7, 5.8, 5.9, 5.10

Problems on applications of Duhamel's integral and resonance: 

5.56: Resonance of an undamped system. Solution by Duhamel's integral.  What form of particular solution would you seek if solving the diff. equ. by usual means?  Review phenomenon of resonance in sec. 5.6

5.57: Impulse as a limit. What do you expect the answer to part (b) to be?

5.59: The right hand side of the equation of motion is a second degree polynomial in t.  What form of particular solution to seek?

5.60: Duhamel's integral for a damped system.  To form integral, obtain the response of a damped system to a unit impulse.

5.61, 5.62: Of mathematical interest.

Weeks 6, 7: July 11 - July 19

Ch. 6.  Principles of Dynamics for Systems of Particles

Sec. 6.1 to 6.3: Equations of Motion, Motion of Center of mass, Kinetic Energy.

Problems: 6.3, 6.4, 6.6, 6.8, 6.10, 6.11, 6.12

Sec. 6.4: Moment of Momentum

Problems: 6.14, 6.15, 6.16, 6.17, 6.18, 6.19, 6.20, 6.21, 6.22

(6.17, 6.18: See also notes: Dynamics of a Particle, Equations of motion, Example 3)

Week 7 (Cont.): July 20 - July 24

Ch. 7. The Dynamics of Rigid Bodies

Read web notes "Dynamic Equations for a Rigid Body", Part 1, Sec. 1.

Read H & H,  Sec. 7.1

Problems: 7.1, 7.5, 7.6, 7.10

                7.2, 7.3, 7.7, 7.9

Read web notes "Indicial and Matrix Notations for Vector and Tensor components"

Read web notes "Dynamic Equations for a Rigid Body", Part 1, Secs. 2, 3, 4

Read H & H,  Secs. 7.2 to 7.7

Week 8: July 25 - July 31

Ch. 7. The Dynamics of Rigid Bodies

Problems (bold face results are good to memorize)

1.  Show that the second moment of area of a circle about a diameter is pr⁴/4.

2.  Show that the second moment of area of a circle about an axis perpendicular to its plane (polar moment of area) and passing through its center is pr⁴/2.  Show in general that the polar moment of area about an axis perpendicular to the area is equal to the sum of the moments of area about two orthogonal axes in the plane of the area.

3.  Use 2. to do Problem 7.14.

4.  Use 1. and 2. to do Problem 7.16.

5.  Show that the second moment of area of a rectangle of sides b and h about a central axis parallel to the b side is bh³/12.

6.  Use 5.  to do Problem 7.15 (a).

7.  In 5. let the axis coincide with a b side.  Use the parallel axes theorem to show that the second moment of area is bh³/3.

8. Use 7. to do problem 7.15 (b).

9. Show that the second moment of a line of length l about a perpendicular axis at its mid-point is l³/12. Show that the second moment at an end point of the line is l³/3.  Show this directly and by the parallel axes theorem. 

10. Use 9. to do Problem 7.18, and find J = ml²/12.  Show that if the axis is at one end of the rod J = ml²/3

11. Obtain the second moments of area I_xx, I_yy, I_xy for the sections of problems 7.19 and 7.21.  Hint: break a section into rectangular areas, then use known inertia properties of these areas about their centroidal axes and the parallel axes theorems.  Note that I_xy is zero if either axis is an axis of symmetry.

Determine the principal axes of the z-section of Problem 7.19.

12. Indicate how the results of 11. are used to do problems 7.19 and 7.21.  For numerical calculations use MATLAB or similar software.

13. Do Problems 7.17, 7.20, 7.22

14. Problem 7.27:  Compute I_xy using the definition, and also using transformation formulas from principal axes.

15. Problems 7.28, 7.31, 7.29, 7.34: Determine the matrices l and J so that the transformation l^TJl yields the answer to the problem.

16. Problem 7.33

Week 9: Aug 1 - Aug 7

Ch. 7. The Dynamics of Rigid Bodies (Cont.)

Read H & H, Secs. 7.8, 7.9, Web Notes, Secs. 5 to 8.1

Problems: 7.35, 7.36, 7.37, 7.38, 7.39, 7.42

Read H & H, Sec. 7.10,  Web Notes, Sec. 8.2

Problems: 7.43, 7.50, 7.51, 7.53, 7.54, 7.56, 7.57, 7.58, 7.59, 7.61, 7.62, 7.64, 7.65, 7.66, 7.67, 7.69, 7.73, 7.76 , 7.77, 7.79

Week 10: Aug 8 - Aug 14

Ch. 7. The Dynamics of Rigid Bodies (Cont.)                                   

H & H

Sec. 7.11: Plane Motion. 

Problems:  7.80, 7.81, 7.82, 7.84, 7.86, 7.87, 7.88, 7.89, 7.92, 7.93, 7.94, 7.95, 7.96

Secs. 7.12 to 7.14: Rotation about a Fixed Point. Symmetric Top and Gyroscope. Gyroscopic Compass.

Problems:  7.97, 7.99, 7.100, 7.101, 7.102

Sec. 7.15, 7.16: General Motion in Space.  Rolling of a Disk. Stability of Rigid Body Motion.  The Rolling Disk.

Problems:  7.103, 7.104

Sec. 7.17:  D'Alembert's Principle

Problems: 7.105, 7.106, 7.107, 7.108, 7.110, 7.112, 7.113, 7.114

Week 11: Aug 15 - Aug 21

Ch. 9.  Lagrange's Equations.  Vibration of Multi Degree of Freedom Systems. Calculus of Variations.  Hamilton's Principle.

Secs. 9.1, 9.2:  Lagranges' Equations for a Particle

Problems: 9.1, 9.2, 9.3, 9.5

Sec. 9.3:  Lagranges' Equations for a System of Particles

Problems: 9.7, 9.8, 9.10, 9.11, 9.14, 9.15

Secs. 9.4, 9.5:  Oscillations of Two Degree of Freedom Systems.  Principal Modes of Vibrations.

Problems:  9.17, 9.18, 9.19, 9.20, 9.21, 9.22, 9.24, 9.25, 9.26, 9.27, 9.29

Secs. 9.6 to 9.11:  Small Oscillations of Conservative Systems. Natural Frequencies and Mode Shapes.

Problems:  9.33, 9.34, 9.35

Sec. 9.12:  Forced Oscillations

Problem 9.38

Secs. 9.13, 9.14:  The Calculus of Variations

Problems:  9.39, 9.41

Sec. 9.15:  Hamilton's Principle

Problems:  1) below, 9.43

1)  Show directly that the Euler equations of the variational equation (9.40) are Lagrange's Equations.  Do the same for the equation derived in Example 1.