- Front tables
- Copyright and acknowledgements
- Preface
- To the student
- Part IMechanics Toolset
- 0 What is mechanics?
Mechanics can predict forces and motions.
The three pillars of mechanics are: I. models of physical behavior, II. geometry, and III. the basic mechanics balance laws. The
laws of mechanics are informally summarized in this introductory chapter. The extreme accuracy of Newtonian mechanics is emphasized. Despite relativity and quantum mechanics
supposedly having ‘overthrown’ seventeenth-century physics, classical mechanics is still incredibly accurate for almost all engineering purposes.
Various uses of the word ‘model’ are described.
- 1 Vectors: position, force and moment
Mechanical analysis uses vectors. We develop vector skills here using the key vectors for statics, namely relative position, force, and moment. Notational clarity is emphasized
because good vector calculation demands distinguishing vectors from
scalars. Vector addition is motivated by the need to add forces and relative
positions.
Dot products are motivated as the tool which reduces vector equations
to scalar equations. And cross products are motivated as the formula which
correctly calculates the heuristically-motivated quantities of moment and moment
about an axis.
- 2 Free-Body Diagrams
On a free-body diagram, systems of forces are often replaced with ‘equivalent’ forces, a special important case of which is a weight force at the center of gravity. The center of gravity is also the Center of Mass, a key concept in Dynamics.
- 3 Computation: Algebra, ODEs, etc.
Here is a quick review of some of the important math and computation for dynamics.
These are things that engineers should just know, like they know that the derivative of is .
- 4 Units and estimation
Similarly, estimating the sizes of things is done by comparison with standards. But, rather than international standards, these ‘standards’ are the sizes of familiar things that you know and remember.
- Part IIStatics
- 5 Statics of one object
One object is in equilibrium if the forces and moments balance. For a particle, force balance tells all. But for an extended object, moment balance is
also essential. There are special shortcuts for an object that has exactly two or exactly three forces
acting on it. If friction forces are relevant the possibility of motion needs to be taken into account. Many real-world problems are not statically determinate and thus yield either only partial solutions, or yield full solutions after you have made extra assumptions.
- 6 Trusses and frames
Here we consider collections of parts designed
to hold something up or in place. Emphasis is on trusses, assemblies
of straight bars connected
by pins at their ends. Trusses are analyzed by
drawing free-body diagrams of the pins (method of joints) or of bigger
parts of the truss (method of sections). Frameworks, built with
other than two-force bodies are also analyzed by
drawing free-body diagrams of parts. Trusses and frames can be rigid or not and
redundant or not, as can be determined by the
equilibrium equations.
- 7 Transmissions and mechanisms
Sometimes solid parts are assembled to
cause force or torque in one place when a different force or torque is applied at another place.
Such assemblies include levers, gear boxes, presses, pliers, clippers, chain drives,
and crank-drives. Besides solid parts connected by pins, a few
special-purpose parts are commonly used, including springs, strings and gears.
Tricks for amplifying force are usually based on principles idealized
by pulleys, levers, wedges and toggles.
Force-analysis of transmissions and mechanisms is done by drawing free-body diagrams
of the parts, writing equilibrium equations for these, and solving the equations
for desired unknowns.
- 8 Tension, shear and bending moment
The ‘internal forces’ (tension, shear and bending moment) can vary from
point to point in long narrow objects. Here we introduce the notion of graphing this
variation and noting the features of these graphs. This graphing is a favorite chore of civil engineers.
- 9 Hydrostatics
Hydrostatics concerns the equivalent force and moment due to distributed pressure on a surface from a still fluid. Pressure increases with
depth. With constant pressure, the equivalent force has magnitude = pressure times area,
acting at the centroid. For linearly-varying pressure on a rectangular plate the equivalent force is the average pressure times the area acting somewhere between 1/2 and 2/3 of the way down. The net force
acting on a totally submerged object in a constant density fluid is the displaced fluid’s weight,
acting at the centroid.
- Part IIIDynamics
- 10 Dynamics in 1D
The scalar equation introduces the concepts of motion
and time derivatives to mechanics. In particular the equations of dynamics are seen to reduce to ordinary differential equations, the simplest of which have memorable analytic solutions. The harder differential equations need be solved on a computer. We explore various concepts and applications involving momentum, power, work, kinetic and potential energies, oscillations, collisions and multi-particle systems.
- 11 Vibrations
The ideal harmonic oscillator (previous chapter) oscillates with
simple sinusoidal motion forever. In the real world
there is friction so oscillations decay more or less quickly, there are forces that
cause the oscillations, and there are multiple parts that move multiple
ways. Understanding something about these realities is the topic of this chapter.
Some key words are damping, resonance, frequency response and normal modes.
- 12 Particles in space
This chapter is about the vector equation for one particle.
Concepts and applications include ballistics and planetary motion.
The differential equations of motion are set up in cartesian coordinates and integrated
either numerically, or for special simple cases,
by hand. Constraints, forces from ropes, rods, chains, floors, rails and guides that can
only be found once one knows the acceleration, are not considered.
- 13 Many particles in space
This more advanced chapter concerns the motion of two or
more particles in space. We will use for each
particle. We will use Cartesian coordinates only. The start is the set up of “two-body” type
problems which are easily generalized to 3 or more particles. The first section concerns smooth motions due to forces from gravity, springs, smoothly applied forces and friction.
The second section concerns the sudden change in velocities when impulsive forces are applied.
- 14 Straight line motion
Here is an introduction to
kinematic constraint in its simplest context, systems that are
constrained to move without rotation in a straight line. In one dimension,
pulley problems provide the main example. Two- and three-dimensional
problems are covered, such as finding structural support forces in
accelerating vehicles and the slowing or incipient capsize of a braking
car or bicycle. Angular momentum balance is introduced as a needed tool but
without the complexities of rotational kinematics.
- 15 Circular motion of a particle
After movement on straight lines the next important special
motion is rotation on a circular path. Polar coordinates and base vectors are
introduced in this simplest possible context. The key new idea is that not just coordinates,
but base vectors, can change with time.
- 16 Circular motion of a rigid object
Here we extend the idea of circular motion from particles to objects.
A particle can go in circles. On a rigid object in 2D, all pairs of particles go in circles
around each other. The key theoretical idea is of rigid-object rotation. The primary applications
are pendulums, gear trains, and rotationally accelerating motors or
brakes.
- 17 Planar motion of an object
The main goal here is to generate equations of motion for general planar motion
of a (planar) rigid object that may roll, slide or be in free flight. Multi-object
systems are also considered so long as they do not involve
other kinematic constraints between the bodies. Features of the solution that can
be obtained from analysis are discussed, as are numerical solutions.
- 18 Time-varying basis vectors
Here is a second approach to the kinematics of particle motion. Now, instead of using constant base vectors, we use time-varying
base vectors. The discussion of polar coordinates
started in Chapter 15 is completed here. Path coordinates, where one base vector is parallel to the velocity and the others orthogonal to that, are introduced. The
challenging kinematics topic of relative motion is introduced in two stages: first using rotating base vectors connected to a moving rigid object and then using the more abstract notation associated with frame-dependent differentiation and the famous “five term acceleration formula.”
- 19 Constrained particles and rigid objects
The dynamics of particles and rigid objects
is studied using the relative-motion kinematics ideas
from chapter chapter 18. This is the capstone chapter for a two-dimensional
dynamics course. After this chapter a good student should be able to navigate
through and use most of the skills in the concept map inside the back cover.
- 20 Fixed-axis rotation in 3D
The ideas of chapter 10 on 2D circular motion are
extended to fixed axis rotation in
3D. The key difference here is the non-trivial use of the
cross product for calculating velocities and accelerations. Fixed axis rotation
is the simplest motion with which one can introduce the full moment of inertia
matrix, where the diagonal terms are analogous to the scalar 2D moment of inertia
and the off-diagonal terms have a “centripetal” interpretation. The main new
application is dynamic balance.
- 21 Elementary introduction to 3D rigid-object dynamics
We begin more advanced 3D dynamics here. First we discuss general
motion of a rigid body in 3D. Then we discuss a special simple class of problems, instantaneous dynamics in 3D.
- Back tables