MPPSC SES Mechanical cheat sheet

Euler-Bernoulli beam bending: EI·d²y/dx² = M(x); curvature is proportional to bending moment and inversely proportional to flexural rigidity EI.

Taylor series: f(x) = f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + …; Maclaurin is the case a = 0, so e^x = 1 + x + x²/2! + …

Laplace transforms: L{e^(at)} = 1/(s−a), L{sin ωt} = ω/(s²+ω²), L{cos ωt} = s/(s²+ω²), L{t^n} = n!/s^(n+1); L{f'(t)} = sF(s) − f(0).

Fourier series of period 2π: f(x) = a₀/2 + Σ(aₙcos nx + bₙsin nx), with a₀ = (1/π)∫f dx, aₙ = (1/π)∫f cos nx dx, bₙ = (1/π)∫f sin nx dx over one period.

For a matrix A, eigenvalues solve det(A − λI) = 0; Cayley-Hamilton states A satisfies its own characteristic equation, so A⁻¹ follows from it when A is nonsingular.

Gradient ∇f points along steepest increase and is normal to level surfaces; divergence measures source density; curl measures rotation; ∇·(∇×F) = 0 identically.

Probability: P(A∪B) = P(A) + P(B) − P(A∩B); Bayes' theorem P(Aᵢ|B) = P(B|Aᵢ)P(Aᵢ)/ΣP(B|Aⱼ)P(Aⱼ).

Standard ODE solution: for y'' + 2ζωₙy' + ωₙ²y = 0, roots are −ζωₙ ± ωₙ√(ζ²−1); ζ < 1 underdamped oscillatory, ζ ≥ 1 non-oscillatory.

Lami's theorem applies to exactly three coplanar concurrent forces in equilibrium: each force is proportional to the sine of the angle between the other two.

Friction: limiting friction F = μN; angle of friction φ = tan⁻¹μ; on an incline the body just slides when tan θ = μ.

Parallel-axis theorem: I = I_G + A·d²; for a rectangle about its base I = bh³/3, about the centroidal axis bh³/12.

Torsion of a circular shaft: τ/r = T/J = Gθ/L, with polar moment J = πd⁴/32 for a solid shaft and J = π(D⁴−d⁴)/32 for a hollow one.

Euler buckling load P_cr = π²EI/(L_e)²; effective length L_e = L (pinned-pinned), 2L (fixed-free), L/2 (fixed-fixed), 0.7L (fixed-pinned).

Principal stresses: σ₁,₂ = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]; maximum shear stress equals the radius of Mohr's circle.

Thin cylindrical shell: hoop stress σ_h = pr/t, longitudinal stress σ_l = pr/2t; for a thin sphere both principal stresses equal pr/2t.

Strain energy in axial loading U = P²L/(2AE); in pure bending U = ∫M²dx/(2EI); proof resilience is the strain energy at the elastic limit.

Grubler's criterion for planar mechanisms: degrees of freedom = 3(n−1) − 2j, where n is the number of links and j the number of lower pairs; a four-bar with four turning pairs has one DOF.

Grashof condition: shortest + longest link ≤ sum of the other two links gives at least one full rotation; equality gives a change-point mechanism.

Simple gear train ratio = product of driven teeth / product of driver teeth; for a compound train the ratio is the product of individual stage ratios.

Governor sensitivity = (N₁ − N₂)/N_mean; a governor is isochronous when equilibrium speed is constant over the working range, i.e. sensitivity zero.

Flywheel energy fluctuation: ΔE = I·ω²·C_s, where C_s = (ω₁ − ω₂)/ω is the coefficient of fluctuation of speed.

Single-degree-of-freedom free vibration: natural frequency ωₙ = √(k/m); damped frequency ω_d = ωₙ√(1−ζ²), and logarithmic decrement δ = 2πζ/√(1−ζ²).

Whirling of shafts: critical speed ω_c = √(g/δ_st), where δ_st is the static deflection; operation near it produces large amplitude resonance.

Dynamic balancing: for a rotating mass m at radius r, the balancing mass m_b at radius r_b satisfies m_b·r_b = m·r, with angular positions set by a force or couple polygon.

Soderberg, Goodman and Gerber criteria relate mean and alternating stresses to yield and endurance limits; the Goodman line is σ_a/S_e + σ_m/S_ut = 1.

Shaft diameter for combined bending moment M and torque T (maximum shear theory): d³ = 16√(M² + T²)/(π·τ_allow).

For a bolted joint, initial preload plus external load sharing depends on bolt and member stiffness: load fraction on bolt = k_b/(k_b + k_m).

Fatigue endurance limit of a steel component = S_e' (rotating beam) multiplied by surface, size, load, temperature and reliability factors; none may exceed 1.

Welded joint throat thickness t = 0.707·h for a fillet weld of leg h; strength = throat area × allowable shear stress × weld length.

Belt drive: ratio T₁/T₂ = e^(μθ) for flat belts, where θ is the angle of lap in radians; centrifugal tension T_c = mv² reduces power transmitted.

Clutch torque capacity for a single plate with uniform pressure: T = (2/3)μP(R_o³ − R_i³)/(R_o² − R_i²); uniform wear gives T = μP(R_o + R_i)/2.

Spring design: Wahl factor K_w = (4C−1)/(4C−4) + 0.615/C, with spring index C = D/d; deflection δ = 8WD³n/(Gd⁴).

Continuity for steady incompressible flow: A₁V₁ = A₂V₂; Bernoulli between two points: p/ρg + V²/2g + z = constant along a streamline for ideal flow.

Reynolds number Re = ρVD/μ; pipe flow is laminar below about 2000 and turbulent above about 4000, with a transitional band between.

Darcy-Weisbach head loss h_f = fLV²/(2gD); Hagen-Poiseuille for laminar pipe flow gives h_f = 32μLV/(ρgD²).

Hydraulic diameter for a non-circular duct D_h = 4A/P; laminar friction factor f = 64/Re, independent of roughness.

Pelton wheel is an impulse turbine suited to high head and low discharge; Francis is mixed flow for medium head; Kaplan is axial flow for low head and large discharge.

Specific speed of a turbine N_s = N√P/H^(5/4); for a pump N_s = N√Q/H^(3/4), with consistent units stated explicitly.

Cavitation occurs where local pressure falls below vapour pressure; NPSH available must exceed NPSH required to avoid it.

Drag on a body F_D = C_D·½ρV²A; lift F_L = C_L·½ρV²A; drag crisis is the sharp C_D drop near critical Reynolds number for bluff bodies.

First law for a steady-flow open system: q − w = (h₂ − h₁) + (V₂² − V₁²)/2 + g(z₂ − z₁), all per unit mass.

Carnot efficiency η = 1 − T_L/T_H with absolute temperatures; no heat engine operating between two reservoirs can exceed it.

Entropy change for an ideal gas: Δs = c_p ln(T₂/T₁) − R ln(p₂/p₁); for a reversible adiabatic process Δs = 0.

Rankine cycle efficiency is raised by increasing boiler pressure, superheating, and lowering condenser pressure; regenerative feed heating raises mean temperature of heat addition.

Fourier conduction: q = −kA dT/dx; for a plane wall q = kA(T₁ − T₂)/L; thermal resistance of the wall is L/(kA).

Newton's law of cooling q = hA(T_s − T_∞); radiation from a black body q = σAT⁴ with σ = 5.67×10⁻⁸ W/(m²·K⁴).

Log mean temperature difference for a heat exchanger: ΔT_lm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂); LMTD correction factor applies for multipass and cross-flow units.

Biot number Bi = hL/k compares internal conduction resistance with surface convection resistance; Bi below 0.1 justifies lumped capacitance analysis.

Merchant's shear angle relation: φ = 45° + α/2 − β/2, where α is rake angle and β is friction angle; cutting force components resolve onto the shear plane.

Taylor's tool life equation VT^n = C; higher n means the tool material tolerates speed increases better, typically n ≈ 0.1–0.15 for HSS and 0.2–0.4 for carbides.

Chip thickness ratio r = t₁/t₂ = sin φ/cos(φ − α); shear strain γ = cot φ + tan(φ − α).

In metal casting, Chvorinov's rule solidification time t = B(V/A)²; risers must freeze after the casting they feed.

Gating design uses the choke area and the ratio rule; aspiration effect is avoided by tapering the sprue so the flow area decreases downward.

Resistance spot welding heat H = I²Rt; weld nugget size grows with current, time and electrode force interplay.

Limits and fits: tolerance is the difference between upper and lower limits; clearance fit has hole minimum above shaft maximum, interference fit the reverse.

Grinding wheel marking gives abrasive type, grit size, grade, structure and bond; finer grit and softer grade suit hard materials and small contact areas.

Economic order quantity EOQ = √(2DS/H), where D is annual demand, S ordering cost per order, H holding cost per unit per year.

Reorder level = lead-time demand + safety stock; safety stock rises with service level and demand variability.

PERT expected time t_e = (t_o + 4t_m + t_p)/6; variance = ((t_p − t_o)/6)²; critical path has zero total float.

Line balancing: cycle time = available time/output required; theoretical minimum number of workstations = Σ task times / cycle time, rounded up.

Work study: standard time = observed time × rating factor × (1 + allowance fraction); allowances cover personal, fatigue and delay needs.

Learning curve: time for the nth unit T_n = T₁·n^b, where b = log(learning rate)/log 2; an 80% curve gives b ≈ −0.322.

Break-even quantity = fixed cost/(selling price − variable cost per unit); margin of safety = (actual sales − break-even sales)/actual sales.

Six Sigma targets 3.4 defects per million opportunities; C_p = (USL − LSL)/(6σ) measures potential capability, C_pk measures actual centring.

Iron-carbon diagram: eutectoid at 0.76% C and 723 °C gives pearlite; eutectic at 4.3% C and 1147 °C gives ledeburite; austenite is stable above the A₃ line.

Martensite forms by diffusionless shear from austenite; its hardness rises with carbon content, and tempering restores toughness at some strength cost.

Hall-Petch relation σ_y = σ₀ + k/√d: yield strength increases as grain size d decreases; fine grains also improve toughness.

Ttt diagrams: fast cooling above critical rate produces martensite; isothermal holding produces pearlite or bainite depending on temperature.

Case hardening processes include carburising, nitriding and cyaniding; carburising raises surface carbon, nitriding forms hard nitrides without a quench.

Stainless steel resists corrosion through a passive chromium oxide film, requiring roughly 10.5% or more chromium; 18-8 is austenitic stainless.

Fatigue failure starts at stress concentrations; surface finish, residual compressive stress by shot peening, and inclusion control all raise fatigue life.

Non-destructive tests: dye penetrant and magnetic particle find surface flaws; ultrasonic and radiography detect internal defects, with radiography best for volumetric ones.

Madhya Pradesh was formed on 1 November 1956 by merging Bhopal, Madhya Bharat, Vindhya Pradesh and Mahakoshal; Bhopal is the capital and Indore the largest city.

The Narmada is the principal west-flowing river of MP, rising at Amarkantak; the Tapti also rises in MP and flows west through a rift valley.

MP national parks include Kanha, Bandhavgarh, Satpura, Pench, Panna and Madhav; Kanha and Satpura are among India's earliest tiger reserves.

Panna is famous for diamond mining and Khajuraho for Chandela-era temples; Sanchi's Buddhist stupa is a UNESCO World Heritage Site.

The Constitution of India came into force on 26 January 1950; the Fundamental Duties were added by the 42nd Amendment in 1976.

Directive Principles of State Policy are in Part IV, non-justiciable, and inspired by the Irish constitution; Fundamental Rights are in Part III and enforceable.

Basic sciences refresher: unit of force is the newton, pressure the pascal, power the watt; 1 bar = 10⁵ Pa and 1 kgf ≈ 9.81 N.

Environmental conventions: the Montreal Protocol targets ozone-depleting substances and the Paris Agreement targets greenhouse gas mitigation.

Time and work: if A does a job in a days and B in b days, together they take ab/(a+b) days; work rates add.

Compound interest amount A = P(1 + r/n)^(nt); simple interest is P·r·t/100 with the rate as a percentage per year.

Percentage change after successive changes of x% and y% is x + y + xy/100 percent; a rise then equal fall always leaves a net decrease.

Average speed for equal distances at speeds u and v is 2uv/(u+v), not the arithmetic mean; for equal times it is the arithmetic mean.

Permutations and combinations: nPr = n!/(n−r)! and nCr = n!/(r!(n−r)!); nCr = nC(n−r) simplifies counting problems.

Ethics: a professional engineer's first duty is public safety, health and welfare; conflicts of interest must be disclosed, not merely avoided.

Whistleblowing is justified when there is serious harm, evidence is documented, internal channels have failed, and proportionate disclosure is used.

Professional competence and honest reporting of test data are core obligations; signing for work not personally verified is misconduct.