Mechanics And Materials

Reference notes on kinematics, rigid-body dynamics, machine elements, materials, and corrosion.

Coordinate Frames And Kinematics

Kinematics describes motion without requiring force balance.

Position, velocity, and acceleration are related by:

$$v=\dot{r}$$$$a=\ddot{r}$$

Coordinate frames define how vectors are represented. A vector can have the same physical meaning but different components in different frames.

A rotation matrix maps vector components between frames:

$$v_A=R_{AB}{v}_B$$

where $v_B$ is the vector represented in frame $B$ and $v_A$ is the same vector represented in frame $A$.

Angular velocity describes the rate of frame rotation. For planar motion:

$$\omega =\dot{\theta }$$

When a vector is differentiated in a rotating frame, the derivative must account for the frame motion:

$${\left(\frac{da}{dt}\right)}_I={\left(\frac{da}{dt}\right)}_B+{\omega }_{B/I}\times a$$

This transport theorem is the source of centripetal, Coriolis, and gyroscopic terms. Those terms are bookkeeping required by the chosen frame, not extra physics.

Generalized coordinates $q$ describe system configuration using the variables that best match the constraints. They may be angles, lengths, modal coordinates, path coordinates, or any independent coordinates sufficient to describe configuration.

Kinematics is a common source of modeling errors. Make sign conventions, frame definitions, derivative frames, and small-angle assumptions explicit.

Rigid-body kinematics also separates translation of a reference point from rotation about that point. For two points $A$ and $B$ on a rigid body:

$$v_B=v_A+\omega \times r_{B/A}$$

For planar vehicle and mechanism models, this relationship is often enough to derive velocity constraints, slip velocities, instantaneous centers, and relative motion at joints or contacts.

Mechanics And Dynamics

Mechanics relates force, momentum, motion, energy, and constraints.

Newton's second law and its angular form are momentum balances. Balance Laws gives the general forms and their reductions to $\sum F=ma$ and $\sum M=I\alpha$.

Common ideal elements:

Element	Translational	Rotational
Inertia	$F=m\dot{v}$	$\tau =J\dot{\omega }$
Spring	$F=kx$	$\tau =k_{\theta }\theta$
Damper	$F=bv$	$\tau =b_{\theta }\omega$

For a mass-spring-damper system:

$$m\ddot{x}+b\dot{x}+kx=F(t)$$

The undamped natural frequency and damping ratio are:

$${\omega }_n=\sqrt{\frac{k}{m}}$$$$\zeta =\frac{b}{2\sqrt{km}}$$

Mechanical systems are often characterized by mass, stiffness, damping, geometry, constraints, friction, and forcing.

Generalized coordinates extend the same idea beyond simple Cartesian coordinates. If $q$ is a generalized coordinate, generalized momentum is often defined from the kinetic energy or Lagrangian as:

$$p_q=\frac{\partial L}{\partial \dot{q}}$$

This makes momentum balance useful even when the system is constrained, rotating, coupled, or described by non-Cartesian coordinates.

Energy methods are also common:

$$T=\frac{1}{2}m{v}^2$$$$V=\frac{1}{2}k{x}^2$$

where $T$ is kinetic energy and $V$ is potential energy. Energy methods are especially useful when constraint forces do no work or when generalized coordinates are clearer than Cartesian force balances.

Lagrange's equations provide a systematic way to form equations of motion:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_i}\right)-\frac{\partial L}{\partial q_i}=Q_i$$

where $L=T-V$ and $Q_i$ is generalized nonconservative force. This form is especially useful for multibody systems because constraints and coordinate choices can be handled more directly than with separate free-body diagrams for every part.

Virtual work states that ideal constraint forces do no work for allowable virtual displacements. This is why generalized coordinates can remove many reaction forces from the equations without ignoring the constraints they represent.

Machine Elements And Failure

Mechanical design connects loads, geometry, material behavior, and failure criteria.

Normal stress and engineering strain are:

$$\sigma =\frac{F}{A}$$$$ϵ=\frac{\mathrm{\Delta }L}{L}$$

Linear elastic behavior is:

$$\sigma =Eϵ$$

For shafts, beams, fasteners, springs, and joints, the key design questions are load path, stress concentration, deformation, fatigue, manufacturability, assembly, and inspection.

Common failure-related quantities:

Quantity	Meaning
Yield strength	stress where plastic deformation begins
Ultimate strength	maximum stress before failure in a tensile test
Endurance limit	stress amplitude below which fatigue failure may not occur for some materials
Factor of safety	ratio between allowable capacity and expected demand
Stress concentration	local stress amplification due to geometry

Von Mises stress is commonly used for ductile materials:

$${\sigma }_{\text{vm}}=\sqrt{\frac{({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_2-{\sigma }_3{)}^2+({\sigma }_3-{\sigma }_1{)}^2}{2}}$$

A design is not fully described by whether the nominal stress is below yield. Deflection, fatigue, buckling, wear, temperature, corrosion, tolerance stackup, and assembly method can control the real design.

Fatigue connects stress amplitude, mean stress, surface condition, size, notches, residual stress, and load history. An $S$-$N$ curve relates stress amplitude to cycles to failure:

$${\sigma }_a=f(N)$$

Mean stress corrections such as Goodman-style relations are used when cyclic loads are not fully reversed:

$$\frac{{\sigma }_a}{S_e}+\frac{{\sigma }_m}{S_{\text{ut}}}\le \frac{1}{n}$$

Buckling can control slender members before material yield. For an ideal pinned column:

$$P_{\text{cr}}=\frac{{\pi }^{2}EI}{(KL{)}^2}$$

Connections need the same level of attention as primary members. Bolts, bearings, welds, bonded joints, press fits, keys, splines, and pins all create local load paths, stiffness discontinuities, and inspection requirements.

Materials And Microstructure

Materials engineering relates structure, processing, properties, and performance.

Stress, strain, and linear elastic behavior are defined in Machine Elements And Failure.

Material behavior may include elastic deformation, plastic deformation, yield, fracture, fatigue, creep, wear, corrosion, thermal expansion, anisotropy, and manufacturing defects.

Microstructure affects macroscopic behavior through grain size, phases, defects, inclusions, porosity, texture, and processing history.

Processing affects microstructure, and microstructure affects properties. Heat treatment, cold work, casting, welding, additive manufacturing, and machining can change grain size, residual stress, phase distribution, hardness, ductility, and fatigue behavior.

Fatigue describes failure under repeated loading, often at stresses below static yield. Creep describes time-dependent deformation under sustained load, especially at elevated temperature.

Thermal expansion is commonly approximated by:

$$\mathrm{\Delta }L=\alpha L_0\mathrm{\Delta }T$$

Microscopy and materials characterization connect observed structure to manufacturing route and mechanical performance. Failure theories such as maximum shear stress and von Mises criteria are models for predicting failure under multiaxial stress states.

Scanning electron microscopy, optical microscopy, spectroscopy, and related methods are evidence tools. They can reveal fracture surfaces, inclusions, porosity, grain morphology, corrosion products, coating defects, and wear features. The observation is only useful when connected to the loading, environment, manufacturing route, and failure hypothesis.

Fracture mechanics relates crack size, stress, and material toughness. A common mode-I stress intensity form is:

$$K_I=Y\sigma \sqrt{\pi a}$$

Fracture becomes critical when $K_{\text{I}}$ approaches the fracture toughness $K_{\text{IC}}$.

Hardness, tensile tests, microscopy, spectroscopy, and fractography answer different material questions. A useful material investigation connects the test method to the suspected mechanism instead of treating characterization as a checklist.

Corrosion

Corrosion is material degradation driven by chemical or electrochemical reactions.

An electrochemical corrosion process includes anodic and cathodic reactions. The anodic reaction removes metal atoms into solution. The cathodic reaction consumes electrons.

Important concepts include electrode potential, galvanic coupling, reference electrodes, polarization, passivation, corrosion current density, concentration effects, environment chemistry, and surface condition.

A corrosion cell requires an anode, cathode, electrolyte, and conductive path. Changing any one of these can reduce or stop the corrosion mechanism.

The Nernst equation relates electrode potential to reaction conditions:

$$E=E^{\circ }-\frac{RT}{nF}\ln Q$$

Faraday's law relates electric charge transfer to material consumption:

$$m=\frac{QM}{nF}$$

where $m$ is mass, $Q$ is charge, $M$ is molar mass, $n$ is electrons transferred, and $F$ is Faraday's constant.

Observed degradation depends on material, environment, geometry, time, and electrochemical compatibility with nearby materials.

Common corrosion forms include uniform corrosion, galvanic corrosion, pitting, crevice corrosion, stress-corrosion cracking, intergranular corrosion, and erosion-corrosion.

Corrosion control can involve material selection, coatings, cathodic protection, inhibitors, geometry changes, isolation of dissimilar metals, and environment control.

Galvanic corrosion occurs when dissimilar materials are electrically connected in an electrolyte. The less noble material tends to corrode faster, especially when a small anodic area is coupled to a large cathodic area.

Passivation forms a protective surface film. It can greatly reduce corrosion rate, but local chemistry, chloride concentration, mechanical damage, or crevices can break down the passive layer.

Corrosion design is geometric as well as chemical. Crevices, stagnant water, poor drainage, trapped contaminants, inaccessible inspection surfaces, and coating defects can dominate material selection.