PEC EPE Mechatronics cheat sheet

Laplace pairs: L{1}=1/s, L{e^(-at)}=1/(s+a), L{sin(wt)}=w/(s^2+w^2); derivative rule L{f'}=sF(s)-f(0).

Final value theorem: limit as t tends to infinity of f(t) equals limit as s tends to zero of sF(s), valid only when all poles of sF(s) lie strictly in the left half-plane.

Nyquist criterion: sample at fs > 2fmax; a component above fs/2 folds to |f - kfs| and cannot be recovered by filtering afterwards.

Continuous-time stability requires every eigenvalue of A (roots of det(lambda*I - A)=0) to have negative real part; discrete-time stability requires all poles |z| < 1.

First-order step response y(t)=K(1-e^(-t/tau)): 63.2% at t=tau, 95% at 3tau, 98% at 4tau.

Second-order standard form s^2+2zetawns+wn^2; overshoot Mp=exp(-pizeta/sqrt(1-zeta^2)); 2% settling time about 4/(zeta*wn).

Solid mechanics: sigma=F/A, epsilon=dL/L, sigma=Eepsilon; bending stress sigma=My/I and shear tau=VQ/(It).

Thermodynamics: ideal gas PV=mR*T; first law dU=Q-W; Carnot efficiency eta=1-Tc/Th for reversible operation between two reservoirs.

Uncertainty propagation: for products and quotients relative errors combine in quadrature, (dz/z)^2=(dx/x)^2+(dy/y)^2; for sums absolute errors combine in quadrature.

KVL and KCL give series/parallel reduction; voltage divider Vout=Vin*R2/(R1+R2); current divider splits inversely with branch resistance.

Thevenin equivalent: Voc measured at open terminals, Rth found by killing independent sources; maximum power transfer to a load occurs at RL=Rth with 50% efficiency.

Diode drops: silicon about 0.7 V, germanium about 0.3 V, Schottky about 0.3 V; full-wave bridge ripple frequency is twice line frequency and C=I/(2fVripple).

Ideal op-amp with negative feedback: inverting gain -Rf/Rin, non-inverting gain 1+Rf/Rin, integrator -1/(sRC); the virtual short holds only while feedback is intact.

BJT: IC=beta*IB, IE=(beta+1)*IB, VBE about 0.7 V in active mode, VCE(sat) about 0.2 V; MOSFET gate current is essentially zero in static operation.

ADC: LSB=VFS/2^n, quantization error +/-0.5 LSB, SNR=6.02n+1.76 dB; a sample-and-hold and at least n+1 settling time constants are needed for full accuracy.

PWM: average voltage = DVsupply where D is duty cycle; buck inductor ripple dI=VsD*(1-D)/(L*f).

I2C uses 7-bit addressing, open-drain lines with pull-ups, START/STOP conditions and clock stretching; SPI is 4-wire full duplex with CPOL/CPHA selecting modes 0-3.

Watchdog timer resets the MCU if firmware fails to service it; interrupt latency equals completion of the current instruction plus vector fetch and context save.

Gear geometry: module m=d/N, circular pitch p=pi*m, centre distance (d1+d2)/2, ratio N2/N1; standard involute pressure angle is 20 degrees.

Compound gear train ratio equals product of driven teeth divided by product of driver teeth; idler gears change direction only.

Bearing life: L10=(C/P)^p in millions of revolutions, p=3 for ball and 10/3 for roller; L10h = 10^6L10/(60n).

Fatigue: endurance limit Se=kakbkckdke*Se'; Goodman line sigma_a/Se + sigma_m/Sut = 1/n at failure.

Bolt tightening torque T=KFd with K about 0.2 for dry steel; preload is typically 0.75 of proof load for a reusable joint.

Helical spring rate k=Gd^4/(8D^3Na); wire shear stress tau=8FD/(pid^3) multiplied by the Wahl factor.

Shaft design uses von Mises sigma'=sqrt(sigma^2+3tau^2); lateral critical speed varies as sqrt(EI/(m*L^4)).

Preferred fits: H7/g6 clearance, H7/k6 transition, H7/p6 interference; position tolerance zone diameter relates to coordinate deviation by 2*sqrt(dx^2+dy^2).

Belt drives transmit by friction, so slip and creep reduce the ideal ratio d2/d1; chain drives need lubrication and controlled slack.

Negative feedback closed-loop transfer function is G/(1+GH); closed-loop poles are the roots of the characteristic equation 1+GH=0.

Steady-state error: type 0 with step 1/(1+Kp); type 1 with ramp 1/Kv; type 2 with parabola 1/Ka; increasing system type removes one order of error.

Routh array: number of sign changes in the first column equals right-half-plane poles; a zero first element is replaced by epsilon and a zero row by the derivative of the auxiliary polynomial.

Bode margins: gain margin = -20*log|GH| at the 180 degree phase crossover, phase margin = 180 + angle(GH) at the 0 dB crossover; PM near 60 degrees gives zeta near 0.6.

Root locus: branches start at open-loop poles and end at zeros or infinity; a real-axis segment is part of the locus when the count of poles plus zeros to its right is odd; asymptote angles (2k+1)*180/(n-m).

Ziegler-Nichols ultimate-gain tuning for PID: Kp=0.6*Ku, Ti=Pu/2, Td=Pu/8, based on the gain that produces sustained oscillation.

State space xdot=Ax+Bu: controllability requires rank[B AB ... A^(n-1)B]=n; observability requires rank of the stacked [C; CA; ...; CA^(n-1)] to equal n.

Digital control mapping z=e^(sT); stability inside the unit circle; zero-order hold adds about half a sample of effective delay, reducing phase margin.

Cascade control uses a fast inner loop for the disturbance-prone variable; anti-windup limits integrator accumulation while the actuator is saturated.

Wheatstone bridge balances when R1/R2=R3/R4; a quarter-bridge strain gauge gives relative sensitivity GFepsilon/4, half bridge GFepsilon/2, full bridge GF*epsilon.

Metal foil strain gauge factor GF=(dR/R)/epsilon is about 2; apparent strain from thermal expansion is cancelled by a dummy gauge or self-temperature-compensating foil.

Pt100 follows R(T)=100*(1+3.9083e-3T-5.775e-7T^2); near 0 degrees Celsius the slope is about 0.385 ohm per degree.

Thermocouple output E=alpha*dT (Seebeck effect); type K is about 41 microvolt per degree Celsius and always needs cold-junction compensation.

4-20 mA current loop with a 250 ohm sense resistor gives 1-5 V; the live zero at 4 mA distinguishes a failed loop from a true zero measurement.

Total measurement uncertainty combines quantization (+/-0.5 LSB), amplifier noise, reference drift and sensor error in root-sum-square; expanded uncertainty at k=2 is about 95% confidence.

Orifice and venturi flow: Q proportional to sqrt(dP); differential-pressure level measurement P=rhogh; Coriolis meters measure mass flow directly.

LVDT is an AC-excited differential transformer; phase-sensitive demodulation gives both magnitude and direction of core displacement with essentially infinite resolution.

Static specifications: accuracy versus precision, linearity, hysteresis, repeatability, dead band and drift; calibration traces the reading to a reference standard.

Homogeneous transform T=[R p; 0 1] maps a point from one frame to the next; composition is 0T2 = 0T1 * 1T2, and the inverse uses R transpose and -R transpose*p.

Denavit-Hartenberg convention: T = Rot_z(theta)*Trans_z(d)*Trans_x(a)*Rot_x(alpha), four parameters per joint.

Mobility: planar M=3(L-1)-2*J1-J2; spatial M=6(L-1)-sum(6-fi) over joints, giving the number of independent inputs.

Differential kinematics v=Jqdot; at a singularity det(J)=0, some Cartesian directions are unreachable and joint rates can diverge; manipulability w=sqrt(det(JJ transpose)).

Common architectures: SCARA for fast planar assembly, 6R articulated for general pose, delta parallel for high-speed pick and place, Cartesian for simple gantry motion.

Hybrid stepper step angle: a 1.8 degree motor gives 200 full steps per revolution, and microstepping divides that angle further but reduces torque per microstep.

Encoder resolution: quadrature counting multiplies counts per revolution by 4; linear resolution = screw lead/(CPR*4) for a direct-coupled rotary encoder.

Trapezoidal motion profile: acceleration time v/a, cruise time from total distance minus accel and decel distances, giving minimum-time motion under acceleration limits.

Safety standards: ISO 10218 for industrial robots, ISO/TS 15066 for collaborative operation; teach pendant use in manual reduced-speed mode is a standard requirement.

PLC scan cycle: read input image, solve ladder top to bottom, write output image; scan time determines the shortest reliably detected field pulse.

Ladder instructions: XIC normally open, XIO normally closed, OTE output coil, OTL/OTU latch/unlatch; TON on-delay, TOF off-delay, RTO retentive timer, CTU/CTD counters.

Modbus RTU over RS-485: slave addresses 1-247, function 03 reads holding registers, 06 writes a single register, 16 writes multiple, with CRC checking.

Fieldbus facts: Profibus DP runs up to 12 Mbit/s with up to 126 nodes; EtherNet/IP uses CIP over TCP/UDP; OPC UA listens on port 4840.

Functional safety per IEC 61508: SIL 1 PFDavg 0.01-0.1, SIL 2 0.001-0.01, SIL 3 0.0001-0.001, SIL 4 0.00001-0.0001 for low-demand mode.

VFD control: constant V/f keeps air-gap flux constant below base speed, PWM carrier is typically 2-16 kHz, and a braking resistor is required for fast deceleration of high-inertia loads.

Control valve selection: Cv sizes flow capacity, equal-percentage trim suits wide range, linear trim suits constant pressure drop; fail-safe action is specified as fail-closed or fail-open.

SCADA/HMI functions: data acquisition, alarm management, trending and historian; alarm rationalisation is described by ISA 18.2.

Takt time = available production time / customer demand, and it sets the design pace of an automated line.

CNC codes: G00 rapid, G01 linear feed, G02/G03 circular interpolation, G90/G91 absolute/incremental, M03 spindle clockwise, M05 spindle stop, M08 coolant on.

Cutting parameters: vc=piDN/1000 in m/min, feed per tooth fz, table feed vf=fzzN, material removal rate aeapvf.

Quality capability: Cp=(USL-LSL)/(6sigma), Cpk=min(USL-mean, mean-LSL)/(3sigma); 1.33 is a common minimum and 1.67 is world class.

Six Sigma uses DMAIC (define, measure, analyse, improve, control); SPC control charts separate common-cause from special-cause variation.

CPM: the critical path is the longest network path and has zero total float; float = LS-ES = LF-EF. PERT expected time=(O+4M+P)/6 with sigma=(P-O)/6.

Costing: breakeven quantity Q=F/(P-v); payback period = capital/annual net cash flow; depreciation is straight-line or accelerated.

Professional ethics: hold public safety paramount, work only within competence, disclose conflicts of interest, reject bribery, and report unsafe conditions through proper channels.

Risk control hierarchy: eliminate, substitute, engineering controls, administrative controls, then PPE; risk = likelihood times severity before and after controls.

Contract types: lump sum, unit price, cost plus; variations, retention and liquidated damages are standard commercial clauses in engineering projects.