Mechanical engineering basics: 50 core concepts

All 50 cards

1. 1
   Moment of a force about a point
   $M = Fd$

   $d$ is the perpendicular distance from the point to the force’s line of action. Measured in $\mathrm{N\,m}$.
2. 2
   Couple
   Two equal, opposite, parallel forces whose lines of action don’t coincide; they produce a pure moment with no net force

   Its moment $M = Fd$ is the same about every point, where $d$ is the distance between the forces.
3. 3
   Free-body diagram
   A sketch of a body isolated from its surroundings, showing every external force and moment acting on it

   Supports are replaced by the reactions they exert.
4. 4
   Equations of static equilibrium in 2D
   $\sum F_x = 0,\ \sum F_y = 0,\ \sum M = 0$

   Three independent equations, so a planar rigid body can be solved for at most three unknown reactions.
5. 5
   Two-force member
   A member loaded at only two points, with no applied moments; in equilibrium the two forces are equal, opposite and act along the line joining the points

   Ideal truss members are two-force members, in pure tension or compression.
6. 6
   Reactions at a pin, a roller and a fixed support (2D)
   Pin: two force components. Roller: one force perpendicular to the surface. Fixed: two force components and a moment
7. 7
   Truss
   A structure of straight, slender members joined at their ends and loaded at the joints, so each member carries only axial tension or compression

   Solved by the method of joints or the method of sections.
8. 8
   Centroid of an area
   $\bar{x} = \dfrac{\int x\,dA}{A}$, $\bar{y} = \dfrac{\int y\,dA}{A}$

   Its geometric center. For a uniform plate it coincides with the center of gravity.
9. 9
   Normal stress
   $\sigma = \dfrac{F}{A}$

   $A$ is the cross-sectional area perpendicular to the force. $1\ \mathrm{MPa} = 1\ \mathrm{N/mm^2}$.
10. 10
    Engineering strain
    $\varepsilon = \dfrac{\Delta L}{L_0}$

    Dimensionless, often given in percent or in microstrain ($10^{-6}$).
11. 11
    Average shear stress
    $\tau = \dfrac{V}{A}$

    $V$ is the shear force and $A$ the area parallel to it, such as the cross-section of a bolt in single shear.
12. 12
    Young’s modulus
    $E = \dfrac{\sigma}{\varepsilon}$ in the linear-elastic region

    About $200\ \mathrm{GPa}$ for steel and $70\ \mathrm{GPa}$ for aluminium.
13. 13
    Change in length of a bar under an axial load
    $\delta = \dfrac{FL}{AE}$

    For a uniform bar in the linear-elastic range.
14. 14
    Poisson’s ratio
    $\nu = -\dfrac{\varepsilon_{\text{lateral}}}{\varepsilon_{\text{axial}}}$

    About 0.3 for most metals; rubber is close to 0.5.
15. 15
    Shear modulus
    $G = \dfrac{\tau}{\gamma}$

    $\gamma$ is the shear strain in radians. For an isotropic material $G = \dfrac{E}{2(1 + \nu)}$, about $80\ \mathrm{GPa}$ for steel.
16. 16
    Elastic vs plastic deformation
    Elastic deformation disappears when the load is removed; plastic deformation is permanent
17. 17
    Yield strength
    The stress at which a material starts to deform plastically

    For metals without a sharp yield point it is found with the 0.2% offset method.
18. 18
    Ultimate tensile strength
    The maximum engineering stress on the stress–strain curve

    After it, a ductile specimen necks and the engineering stress falls until fracture.
19. 19
    Ductile vs brittle materials
    Ductile materials deform plastically a lot before they fracture (mild steel, copper); brittle ones fracture with little plastic deformation (glass, ceramics, gray cast iron)
20. 20
    Hardness
    Resistance to localized plastic deformation, such as indentation or scratching

    Measured with the Brinell, Rockwell and Vickers indentation tests.
21. 21
    Toughness
    The energy a material can absorb before it fractures

    The area under the stress–strain curve up to fracture; impact toughness is measured with the Charpy test.
22. 22
    Stiffness vs strength
    Stiffness is resistance to elastic deformation; strength is the stress a material can carry before it yields or breaks

    All steels have about the same Young’s modulus, however strong they are.
23. 23
    Linear thermal expansion
    $\Delta L = \alpha L_0\,\Delta T$

    $\alpha$ is about $12 \times 10^{-6}\ \mathrm{K^{-1}}$ for steel and $23 \times 10^{-6}\ \mathrm{K^{-1}}$ for aluminium.
24. 24
    Steel vs cast iron
    Steel is iron with up to about 2% carbon; cast iron has roughly 2 to 4%

    The extra carbon lowers the melting point and makes cast iron easy to cast, but gray cast iron is brittle.
25. 25
    Efficiency
    $\eta = \dfrac{\text{useful output}}{\text{input}}$

    Of power or energy. Real machines always have $\eta < 1$ because of friction and other losses.
26. 26
    Power transmitted by a rotating shaft
    $P = T\omega$

    $T$ is the torque and $\omega$ the speed in rad/s; from $n$ in rpm, $\omega = \dfrac{2\pi n}{60}$.
27. 27
    Kinetic energy stored in a flywheel
    $E = \tfrac{1}{2}I\omega^2$

    $I$ is the mass moment of inertia about the axis of rotation and $\omega$ the speed in rad/s.
28. 28
    One mechanical horsepower in watts
    About $746\ \mathrm{W}$

    Defined as $550\ \mathrm{ft\,lbf/s}$. Metric horsepower is about $735.5\ \mathrm{W}$.
29. 29
    Mechanical advantage
    $\mathrm{MA} = \dfrac{\text{load}}{\text{effort}}$

    A ratio of forces. A machine with an MA above 1 trades distance for force.
30. 30
    Velocity ratio of a machine
    $\mathrm{VR} = \dfrac{\text{distance moved by effort}}{\text{distance moved by load}}$

    Also called the ideal mechanical advantage. The efficiency is $\eta = \dfrac{\mathrm{MA}}{\mathrm{VR}}$.
31. 31
    Law of the lever
    $F_1 d_1 = F_2 d_2$

    The moments about the fulcrum balance; each $d$ is a force’s perpendicular distance from the fulcrum.
32. 32
    Three classes of lever
    First class: fulcrum between effort and load (seesaw, scissors). Second class: load in the middle (wheelbarrow). Third class: effort in the middle (tweezers, the forearm)
33. 33
    Ideal mechanical advantage of a pulley system
    The number of rope segments supporting the moving block

    Assuming frictionless pulleys and a weightless rope.
34. 34
    Gear ratio of a gear pair
    $\dfrac{N_{\text{driven}}}{N_{\text{driver}}} = \dfrac{\omega_{\text{driver}}}{\omega_{\text{driven}}}$

    $N$ is the number of teeth. Output torque rises by the same ratio, less friction losses.
35. 35
    Spur vs helical gears
    Both commonly connect parallel shafts. Spur gears have straight teeth; helical gears have angled teeth that engage gradually, so they run more quietly but produce an axial thrust load
36. 36
    Worm gear
    A screw-like worm driving a worm wheel on a perpendicular, non-intersecting shaft, giving a large speed reduction in one stage

    With a small lead angle the drive is self-locking: the wheel can’t turn the worm.
37. 37
    Lead vs pitch of a screw thread
    Pitch is the distance between adjacent threads; lead is the distance the screw advances in one turn

    $\text{lead} = \text{pitch} \times \text{number of starts}$, so they are equal for a single-start thread.
38. 38
    Closed, open and isolated systems
    Closed: energy crosses the boundary but mass doesn’t. Open (a control volume): both mass and energy cross. Isolated: neither crosses
39. 39
    Gauge vs absolute pressure
    $p_{\text{abs}} = p_{\text{gauge}} + p_{\text{atm}}$

    Standard atmospheric pressure is $101.325\ \mathrm{kPa}$, about $1\ \mathrm{bar}$.
40. 40
    Ideal gas law with the specific gas constant
    $pV = mRT$

    For air $R \approx 287\ \mathrm{J\,kg^{-1}\,K^{-1}}$. $T$ must be an absolute temperature, in kelvin.
41. 41
    Enthalpy
    $H = U + pV$

    Handy for flowing fluids, because it includes the flow work needed to push fluid across a boundary.
42. 42
    Three modes of heat transfer
    Conduction (through a material, without bulk motion), convection (carried by a moving fluid) and radiation (electromagnetic waves, needing no medium)
43. 43
    Casting
    Pouring molten metal into a mold and letting it solidify in the mold’s shape

    Sand casting, die casting and investment casting are common types.
44. 44
    Forging
    Shaping metal with compressive forces, by hammering or pressing, often while it is hot

    The grain flow follows the part’s shape, which makes forgings strong and tough.
45. 45
    Turning vs milling
    Turning: the workpiece rotates against a stationary single-point tool on a lathe. Milling: a rotating multi-point cutter removes material as the workpiece is fed past it
46. 46
    Welding vs brazing vs soldering
    Welding melts the base metals to fuse them. Brazing and soldering melt only a filler metal: above 450 °C for brazing, below it for soldering
47. 47
    Additive manufacturing
    Building a part layer by layer from a digital model

    Also called 3D printing; common processes include material extrusion and laser powder bed fusion.
48. 48
    Annealing
    Heating a metal and cooling it slowly to soften it, relieve internal stresses and restore ductility
49. 49
    Quenching and tempering
    Quenching cools steel rapidly from a high temperature to harden it; tempering then reheats it to a lower temperature to reduce brittleness

    Quenching forms hard, brittle martensite; tempering trades some hardness for toughness.
50. 50
    Tolerance
    The permitted variation in a dimension: the difference between its upper and lower limits

    Tighter tolerances cost more to make and to inspect.