Computer Aided Manufacturing, Automation and Production Control

The axis convention is fixed by the spindle, not by the operator. Z always lies along the spindle axis, positive in the direction that increases the distance between tool and work; X is the longest travel perpendicular to Z and lies horizontal where a horizontal surface exists; Y completes a right-handed set. On a turning centre there is no Y, so the slides are X and Z, and X is programmed as a diameter rather than a radius on almost every control — a commanded X40.0 puts the tip 20 mm from the centre line. A vertical machining centre uses X, Y and Z. Rotary axes about those three are named A, B and C. Adding a rotary table gives a fourth axis and lets one setup reach several faces; adding two rotary axes gives five-axis machining, which is built either as table-table, where both rotations are carried by the work, or as head-head, where both are carried by the spindle, with the mixed arrangement in between. The motion-control classes are separate from the axis count: point-to-point control cares only where a move ends, which suits drilling and spot welding; straight-cut control feeds along one axis at a time; contouring control coordinates every axis continuously so the resultant tip path is a specified curve, and that is what milling a profile requires.

An open-loop axis commands a position and hopes; a closed-loop axis measures one. In the open-loop drive a stepping motor is given a train of pulses and advances a fixed angle per pulse, and nothing on the machine ever reports back where the slide actually is. It is cheap, it holds position without power dissipation in a servo amplifier, and it fails in one specific way: if the cutting load exceeds the motor torque the motor loses steps, the control does not know, and every coordinate for the rest of the programme is offset by the lost amount. In the closed-loop drive a rotary encoder on the screw or a linear scale on the slide reports actual position, the control forms the difference against the commanded position and drives the error to zero. The feedback may be indirect, taken from the leadscrew, in which case screw wear and backlash are outside the loop and are not corrected, or direct, taken from a scale on the slide, in which case they are inside the loop and are.

Control resolution, usually called the basic length unit, is the smallest increment of table motion the control can command. It is set by the transmission, not by the electronics alone. Take an axis whose leadscrew has a pitch of 6 mm and is driven through a 2 : 1 reduction, the motor carrying an encoder that issues 500 pulses per motor revolution. Two motor revolutions turn the screw once, so the screw receives 500 × 2 = 1000 pulses per revolution, and one screw revolution advances the table by one pitch. The basic length unit is therefore 6/1000 = 0.006 mm. A commanded move of 30 mm is 30/0.006 = 5000 pulses, and a feed of 360 mm/min is 6 mm/s, which is 6/0.006 = 1000 pulses per second. Every one of those three numbers changes if the gear ratio is misread, which is why the ratio is worth writing down before the arithmetic starts.

Interpolation is how a control makes a curve out of axes that can only be told to move. The interpolator takes the endpoint of a block and issues pulses to each axis at ratios that keep the resultant on the wanted path: a fixed ratio gives a straight line, a continuously varying ratio gives a circle, and a circle combined with a steady third axis gives a helix, which is how a thread or a helical ramp into a pocket is cut. The path a control actually produces is a chain of very short straight chords, and the chord error is what the basic length unit and the block processing rate together decide. One control feature belongs here because it exists only on CNC machines: constant surface speed. Under G96 the programmed S value is a cutting speed in m/min and the control varies the spindle speed to hold it as the diameter changes, using N = 1000V/(πD). Facing a disc at 150 m/min, the spindle runs at 1000 × 150/(π × 80) = 596.83 rev/min at 80 mm diameter and must reach 1000 × 150/(π × 20) = 2387.32 rev/min by the time the tool is at 20 mm — four times the speed, because the diameter has fallen to a quarter. That is also why every G96 programme carries a spindle speed clamp: without one the control would ask for an unbounded speed as the tool reaches the centre.

A part programme is written in word-address format: each block is a line, each word is an address letter followed by a number, and the order of words within a block does not matter to the control. N numbers the block, G gives a preparatory function that sets a mode or a motion type, X Y Z give coordinates, I J K give arc centre offsets, R gives an arc radius, F the feed, S the spindle speed, T the tool and M a miscellaneous machine function. Most G words are modal: once G01 is active every following block with coordinates is a feed move until some other motion code replaces it, so a repeated code is omitted rather than repeated. A few are non-modal and act only in their own block, G04 dwell being the usual example. The same distinction applies to M words, and it matters because the commonest beginner defect is a block that inherits a mode nobody intended.

Now read the cutting conditions out of the programme rather than off a chart. The spindle word and the tool together give the cutting speed: V = πDN/1000 = π × 16 × 1200/1000 = 60.32 m/min. The feed word is in mm per minute, so the load on each tooth has to be recovered: with four flutes at 1200 rev/min, the feed per tooth is 240/(1200 × 4) = 0.05 mm. Those two numbers, not S1200 F240, are what decide whether the cut is sensible, and they are the two a control never checks. The geometry gives the time. Block N90 cuts 60 mm; block N100 is a quarter circle of radius 20, so its arc length is 2π × 20/4 = 31.42 mm; block N110 cuts 40 mm. The profile is 60 + 31.42 + 40 = 131.42 mm at 240 mm/min, which is 0.5476 min, or 32.85 s. The plunge of N80 runs 8 mm at 100 mm/min, another 0.08 min or 4.80 s. The cutting time for the part is therefore 37.65 s, and the rapid moves add machine time but no cutting time at all.

• G00 rapid traverse, G01 linear feed, G02 clockwise arc, G03 anticlockwise arc, G04 dwell
• G17 G18 G19 select the XY, ZX and YZ planes, which is what tells an arc block which two axes it lives in
• G20 and G21 select inch and millimetre input; G90 and G91 select absolute and incremental coordinates
• G40 G41 G42 cancel, left and right cutter radius compensation, which lets the programme carry the finished profile rather than a path offset by hand
• G43 applies a tool length offset from a register, so a tool of a different length needs a new offset and not a new programme
• M03 and M04 start the spindle clockwise and anticlockwise, M05 stops it, M06 changes the tool
• M08 and M09 switch coolant on and off, M30 ends the programme and rewinds it

A fluid power actuator converts pressure into force and flow into speed, and those are two independent statements. Pressure is set by the load, not by the pump or the compressor: a cylinder pushing against nothing sits at almost no pressure however hard the supply is trying. Flow is set by the valve and the supply, and it decides how fast the piston travels. A single-acting cylinder is pressurised on one side only and is returned by a spring or by the weight of the load, so it does useful work in one direction and needs only one supply line; a double-acting cylinder is pressurised on both sides in turn and can do work both ways. Two important variants exist. A double-rod cylinder has a rod out of both ends so the two areas are equal and the extend and retract behaviour is symmetric. A telescopic cylinder nests several stages to get a long stroke out of a short body, at the price of a force that falls stage by stage as the effective area shrinks.

Directional control valves are named by two numbers: ports first, then positions. A 3/2 valve has three ports and two positions and is what a single-acting cylinder needs — supply, cylinder, exhaust, switching the cylinder between supply and exhaust. A 5/2 valve has five ports and two positions and is the standard valve for a double-acting cylinder, because each cylinder port needs its own exhaust so that the two sides can be throttled independently. A 4/3 valve has four ports and three positions and appears mostly in hydraulics, where the centre position matters: a closed centre locks the actuator, a tandem centre locks it while unloading the pump to tank, and a float or open centre lets the actuator be moved by hand. Valves are actuated by a push button, a roller lever, a pilot pressure or a solenoid, and it is the solenoid that makes the system electro-pneumatic, because an electrical signal from a relay, a limit switch or a controller output can now command fluid. A single-solenoid valve is spring returned and so has a defined rest state when power fails; a double-solenoid valve is bistable and holds its last position, which is safer for some machines and much less safe for others.

Put numbers on one cylinder and every comparison in this section falls out of it. Take a double-acting cylinder of 80 mm bore with a 25 mm rod, supplied at 6 bar gauge. The full piston area is A = π × 80²/4 = 5026.55 mm², and 6 bar is 0.6 N/mm², so the extend force is 0.6 × 5026.55 = 3015.93 N. On the retract stroke the rod occupies part of the face, so the effective annulus is π × (80² − 25²)/4 = 4535.67 mm² and the retract force is 0.6 × 4535.67 = 2721.40 N, which is 4535.67/5026.55 = 90.23 per cent of the extend force. Note what that percentage is a ratio of: areas, not diameters. The rod is 31.25 per cent of the bore in diameter but only 9.77 per cent of it in area, and a candidate who subtracts diameters gets 68.75 per cent and is wrong by a wide margin. Speed comes from the same two areas and the flow. Feeding 20 L/min, which is 333 333 mm³/s, into the full bore gives an extend speed of 333 333/5026.55 = 66.31 mm/s, while the same flow into the annulus gives a retract speed of 333 333/4535.67 = 73.49 mm/s. A rodded cylinder always retracts faster than it extends and always pushes harder than it pulls, by exactly the same area ratio of 1.108.

The choice between air and oil is decided by that last row and by compressibility, and almost never by anything else. The same cylinder at a hydraulic pressure of 150 bar develops 15 × 5026.55 = 75.40 kN, twenty-five times the pneumatic force, because pressure is twenty-five times higher; getting the same force from air would need a bore of 400 mm. Against that, air is compressible, which is why a pneumatic cylinder cannot hold an intermediate position accurately under a varying load and why it lurches when a stiff load suddenly yields — the air behind the piston is a spring that has been storing energy. Air is also cheap to exhaust to atmosphere, needs no return line, tolerates leaks as a nuisance rather than a hazard, and is safe in a wet or explosive area. Oil is stiff, so position and speed are controllable, and it carries away heat, but it needs a return line, a reservoir, filtration and a leak-free installation. Air is not free, though, and the cost is in the compressor. Sizing that compressor means counting free air, not swept volume, because the cylinder is filled with compressed air and the compressor is rated in air drawn from the atmosphere.

Work it through for the same cylinder on a 300 mm stroke running 15 complete cycles a minute. The extend stroke swallows 5026.55 × 300 = 1.508 litres of compressed air and the retract stroke 4535.67 × 300 = 1.361 litres, so one full cycle takes 2.869 litres at working pressure. The absolute pressure is 6 + 1 = 7 bar, so each litre at working pressure came from seven litres of free air: 2.869 × 7 = 20.08 litres of free air per cycle, and at 15 cycles a minute the demand is 301.2 litres of free air per minute. Three slips are worth naming because each gives a plausible wrong number. Forgetting the compression ratio entirely gives 43.0 L/min. Using the gauge pressure 6 instead of the absolute 7 gives 258.2 L/min, an error of one seventh that hides easily. Counting only the extend stroke gives 158.3 L/min, and it is the one that most often reaches a purchase order.

Electro-pneumatics is the layer where the fluid circuit meets the electrical one, and it is worth seeing what that layer actually contains before a controller is introduced. The power side stays as it was: compressor, receiver, a filter-regulator-lubricator unit, a 5/2 valve, a cylinder. The signal side becomes electrical: reed switches clamped to the cylinder barrel sense the magnetic piston at each end of stroke, push buttons and limit switches supply operator and position signals, and relays with normally-open and normally-closed contacts combine those signals and energise the valve solenoids. A two-cylinder sequence such as A+ B+ A- B- is then built by making each step trigger the next: the reed switch at the extended end of A energises the relay that fires B, and so on around the cycle, with a latching relay holding the start condition and a normally-closed emergency-stop contact breaking the whole control supply. That circuit works, and it is exactly the circuit whose weaknesses a programmable controller was invented to remove, since a change of sequence means rewiring a panel rather than editing a file.

A programmable logic controller is a computer with its input and output hardened for a factory and its operating system reduced to one repeated loop. Five parts make it up. The processor holds the user program and executes it. Memory is divided in a way worth being precise about: program memory holds the rungs, while data memory holds the input image table, the output image table, and the timer, counter and internal bit files that the rungs read and write. The power supply feeds the rack and often the input field devices. The input and output modules are the whole reason the machine survives a factory: every channel is optically isolated, so a 240 V surge on a field wire does not reach the processor, and each module converts between field levels and logic levels. Finally a communications port and programming terminal load the program, force bits during commissioning and read the live image tables, which is how a PLC is debugged without stopping the plant. Physically these come as a modular rack, where a backplane accepts whatever mix of modules a machine needs, or as a compact brick with a fixed I/O count for small machines.

Everything surprising about a PLC follows from the scan, so learn the order before the instructions. One scan runs four stages and then starts again. First housekeeping: the processor checks memory, the watchdog timer and the I/O modules. Second the input scan: every input channel is read at that instant and copied into the input image table, and it is that frozen copy — not the field terminal — which the rungs will examine for the rest of the scan. Third the program scan: the rungs are solved top to bottom and, within a rung, left to right, and output instructions write to the output image table rather than to the terminals. Fourth the output scan: the whole output image is transferred to the output modules at once. Then communications are serviced and the loop repeats. The frozen images are the design decision that makes a PLC deterministic: a rung solved late in the scan sees exactly the same inputs as one solved early, so the logic cannot depend on how long the processor took.

Read the last row of that table carefully, because it is the number a machine designer needs and the number a beginner never computes. An input that changes state one microsecond after the input scan has passed it will not be seen at all this scan; it is read at the start of the next scan, acted on during that scan, and written to the terminal at the end of it. The delay from the physical event to the physical output is therefore up to two full scans, 5.8 ms here, plus the hardware filter delay of the input module, which is typically several milliseconds more and is there deliberately to reject contact bounce. The same arithmetic sets the shortest event a PLC can catch on an ordinary input: a pulse narrower than one scan may fall entirely between two input scans and be missed. Consider a proximity sensor watching a shaft at 1500 rev/min with one 30-degree target on it. The shaft turns once every 40 ms, and the target covers 30/360 of that, so the sensor output is high for 40 × 30/360 = 3.33 ms. Against a 2.90 ms scan that pulse is seen, but with no margin worth having, and at 2000 rev/min it would be 2.5 ms and would be missed intermittently — which is the worst possible failure, because it works on the bench. The fix is never a faster program; it is a high-speed counter module or a latching input, which catches the edge in hardware and holds it until the processor asks.

• Discrete DC input modules are wired as sinking or sourcing, which is a statement about the direction of conventional current at the module terminal and must match the sensor that drives it
• Relay outputs switch AC or DC and tolerate a wide load range but wear out mechanically; transistor outputs switch DC only, are silent and fast, and survive millions of operations
• Analogue input modules digitise 4 to 20 mA or 0 to 10 V signals; the current loop is preferred on a long run because a broken wire reads as zero current and is detectable, while a broken voltage wire reads as zero volts and looks like a real measurement
• A watchdog timer resets the processor if one scan overruns its limit, which is what stops a program fault from leaving the outputs frozen in a dangerous state
• An emergency stop is wired to drop power to the output devices directly and is never left to the program alone, because a processor fault must not be able to defeat it

Ladder logic is drawn as a relay schematic because it was meant to be read by people who already maintained relay panels. Two vertical rails carry notional power from left to right through horizontal rungs. A rung is a series and parallel arrangement of contacts ending in one or more coils: if a continuous path of true contacts exists from the left rail to the coil, the coil is energised. The one idea that must be right before anything else works is that a contact is a question about a bit in memory, not a picture of the field device. The symbol --] [--, called examine-if-closed, is true when its bit is a one; --]/[--, examine-if-open, is true when its bit is a zero. Whether the switch out in the plant is normally open or normally closed is a separate, electrical fact, and the two must be reasoned about together: a stop button wired as a normally-closed field contact holds its input bit at one while all is well, so the rung examines it with --] [--, and breaking the wire drops the bit to zero and stops the machine. That is a fail-safe design, and writing --]/[-- there instead would give a machine that runs correctly until the day its stop wire is cut.

Rung one is the latch, and the whole of it is in one parallel branch. I0.0 is a momentary start button, so it is true only while a finger is on it; Q0.0 is the output itself, examined as a contact in parallel with the button. Press the button and the rung is true, so Q0.0 goes to one; from the next scan onwards the parallel branch through Q0.0 is true on its own and the output holds even though the button has been released. That is the seal-in or holding contact, and it is how a momentary signal becomes a maintained state without any special instruction. I0.1 is the stop input in series with the whole branch, so it can break the path from either route, which is why the stop must be in series and not in parallel with the start. Some controllers offer set and reset coils instead, which latch without the seal-in branch; the behavioural difference is worth knowing, because a set coil is retentive across a power failure on most processors while a seal-in latch is not, and a machine that restarts itself when the supply returns is a different machine from one that waits to be told.

The interlock is the NOT Q0.1 in rung one and the NOT Q0.0 in rung two, and the sense of those contacts is the whole point. Each output is examined for the off state in the other output's rung, so energising either one immediately makes the other rung false and it cannot be energised until the first drops out. Putting an examine-if-closed contact there instead would produce the opposite machine, one in which neither contactor can start until the other is already running, and it would simply never start. Two cautions belong with this. A software interlock is not a safety function. A short across a contactor coil, a welded main contact or a processor fault defeats it, so a forward-reverse starter also carries a mechanical interlock on the contactors and, where the risk warrants it, a hardwired safety relay in the coil circuit. And the output image table is written only at the end of the scan, so if two different rungs write the same output coil, the one solved last silently decides the result and the earlier rung appears to have no effect at all — the commonest logic bug in a large program, and the reason a coil address is written in exactly one place.

A timer is set by two numbers and the product of them is the delay: the time base is the tick, and the preset is how many ticks. Rung three above uses a 10 ms base with a preset of 750, so the done bit sets 750 × 0.010 = 7.5 s after the rung became true. Choosing a coarse base and a small preset saves nothing and loses resolution, while an unnecessarily fine base can overflow a sixteen-bit accumulator on a long delay, so the base is chosen to suit the delay rather than by habit. Counters are simpler but carry one trap of their own: a counter increments on the transition of its rung, not on the condition being true. A rung that is true for two hundred consecutive scans increments the counter once, which is what allows a slow sensor to be counted correctly and what makes a counter useless for measuring how long something lasted.

Productivity is a ratio of output to input, and production is only the numerator. A plant that doubles its output by doubling everything it consumes has raised production and left productivity where it was. Because there are several inputs, there are several ratios: partial productivity divides output by one input, so labour productivity is pieces per labour hour and machine productivity is pieces per machine hour; total factor productivity divides output by labour and capital together; total productivity divides it by every input valued in money. Each answers a different question, and they can move in opposite directions — installing a robot raises labour productivity and may lower total productivity if the machine is idle half the shift. Work study is the pair of techniques that improves the ratio without buying anything: method study attacks the numerator and the input together by changing how the job is done, and work measurement establishes how long the job should take once it is done that way.

Method study runs in a fixed order, and the order is the method. Select the job worth studying — usually a bottleneck, a high-cost operation or an unsafe one. Record what is done now, using a chart chosen to match the question: an operation process chart for the sequence of operations and inspections on a whole product, a flow process chart for one item through operations, transports, delays, inspections and storages, a string diagram or flow diagram for distance travelled on the floor, a two-handed process chart for the movements of one operator's hands, and a man-machine chart for the idle time of each when they work in turn. Examine the record by challenging every element in the order purpose, place, sequence, person, means. Develop the improved method, define it in writing, install it, and maintain it, since a method left unaudited reverts. Underneath sit the principles of motion economy — symmetrical two-handed motions, materials pre-positioned within a normal working area, gravity used to deliver and remove, hands relieved of holding by a jig — and beneath those the therbligs, the elementary motions such as reach, grasp, move and release into which any manual task can be decomposed.

Work measurement turns a stopwatch reading into a standard, and it takes two corrections to get there. The analyst times an element over many cycles and takes the mean, which is the observed time. That figure describes the operator who was watched, not a qualified average operator, so it is corrected by a rating factor: a rating above 100 means the observer judged the worker faster than standard pace, below 100 slower. Observed time times rating gives the normal time, sometimes called basic time. The normal time still assumes a person who never rests, so allowances are added — relaxation allowance covering personal needs and fatigue, contingency allowance for small irregular delays, and where applicable a policy or interference allowance. Normal time plus allowances is the standard time, and it is the number that feeds capacity planning, costing, line balancing and incentive schemes.

Time study is not the only way to measure work, and the alternatives exist because it is expensive and intrusive. Work sampling takes a large number of instantaneous random observations and estimates the proportion of time spent in each activity rather than the duration of a cycle; it suits long, irregular and non-repetitive work such as maintenance, it needs no stopwatch, and its precision improves only as the square root of the number of observations, so halving the error costs four times the observations. Predetermined motion time systems such as MTM dispense with observation altogether by summing published times for elementary motions, which allows a standard to be set for a job that does not exist yet — the decisive advantage when a line is being designed. Standard data reuses element times already established for similar work. The choice is a trade between cost and directness, and the honest ordering is that a synthetic standard is cheaper and quicker while a timed standard is more defensible in front of the people it will be applied to.

Six sigma is a programme for reducing variation, and its name is a target expressed in standard deviations. The improvement cycle is DMAIC and each phase has a deliverable rather than an attitude. Define fixes the problem, the customer requirement it violates and the boundaries of the process, and leaves a charter. Measure establishes how the characteristic is measured, proves the measurement system itself is capable, and records the current performance as a baseline. Analyse finds the few causes that actually drive the variation, using the data rather than opinion. Improve changes the process and proves with a trial that the change worked. Control puts the new setting under a control chart and a revised procedure so the gain survives the departure of the team. A separate cycle, DMADV, applies when the process does not exist yet and has to be designed. The roles are borrowed from martial arts — green belts run projects part time, black belts full time, master black belts train and coach, champions remove obstacles — and the point of naming them is that the method is a resourcing commitment, not a technique.

The sigma level of a process is how many standard deviations fit between the process mean and the nearer specification limit, and the published table that converts it to defects assumes the mean has drifted 1.5 standard deviations off centre. That drift allowance is a convention, not a law of nature; it exists because a process observed over a long period wanders more than a short capability study suggests, and it is why six sigma corresponds to 3.4 defects per million rather than to the two parts per billion a perfectly centred normal distribution would give. Defects are counted as defects per million opportunities, which normalises for complexity: a unit with many chances to go wrong is not being compared unfairly with a simple one. If 5 defects are found in 400 units inspected and each unit offers 12 opportunities, then DPMO = 5/(400 × 12) × 10⁶ = 1041.7, which sits between the four and five sigma rows of the table, at about 4.6 sigma.

Process capability compares the voice of the process with the voice of the customer, and it takes two indices because a process can fail in two different ways. C_p = (USL − LSL)/6σ compares the specification width with the spread and says nothing whatever about where the process is sitting. C_pk = min[(USL − μ)/3σ, (μ − LSL)/3σ] takes the nearer limit and so folds the centring in. Work a case. A diameter is specified as 50.00 ± 0.15 mm, so USL = 50.15, LSL = 49.85 and the tolerance is 0.30 mm. The process has σ = 0.04 mm and a mean of 50.04 mm. Then C_p = 0.30/(6 × 0.04) = 1.25, the upper index is (50.15 − 50.04)/(3 × 0.04) = 0.9167 and the lower is (50.04 − 49.85)/0.12 = 1.5833, so C_pk = 0.9167. Read the pair together: the spread is comfortably inside the tolerance, and the process is still producing scrap, because it is sitting off centre and the upper limit is close. Recentring the mean on 50.00 with no reduction in variation at all would lift C_pk to 1.25, the value of C_p, which is the ceiling — C_pk can never exceed C_p, and the two are equal exactly when the process is centred.

Lean and six sigma are often bolted together and they are not the same idea. Six sigma attacks variation and its instruments are statistical — capability studies, control charts, designed experiments. Lean attacks waste, meaning any activity the customer would not pay for, and its instruments are physical — pull scheduling with kanban cards, single-minute exchange of dies to make small batches economic, standard work, levelled production, and a stop-the-line rule so a defect is fixed rather than passed on. The waste categories usually listed are overproduction, waiting, transport, over-processing, inventory, motion and defects, with unused human ingenuity added later. They complement each other because they fail differently: a lean line with an unstable process starves the moment a defect appears, since there is no buffer to hide behind, and a statistically capable process inside a batch-and-queue factory still has a long lead time and a warehouse full of work in progress. Lean makes the problems visible by removing the inventory that was concealing them; six sigma is what fixes the problems lean exposes.

Reliability is a probability with a mission attached: the probability that an item performs its function, under stated conditions, for a stated period. Drop the period and the number means nothing. The failure rate λ is the proportion of the surviving population failing per unit time, and when it is constant the reliability is R(t) = e^(−λt). Constant failure rate is the one case in which the arithmetic is easy, because the mean time between failures is then simply 1/λ. Take a pump with λ = 2 × 10⁻⁴ failures per hour. Its MTBF is 1/2 × 10⁻⁴ = 5000 hours, and its reliability over a 1000 hour campaign is e^(−0.2) = 0.8187. Two readings of that pair are worth fixing. The MTBF is not a life: at t = MTBF the reliability is e^(−1) = 0.368, so only 37 per cent of a population survives to its own mean time between failures. And a constant failure rate means the item is memoryless — a unit that has run 4000 hours is exactly as likely to fail in the next hour as a new one, which is why replacing a good component early buys nothing at all in this regime.

The constant failure rate is only the middle of the story, and the bathtub curve is the whole of it. Plot failure rate against age and three regimes appear. Infant mortality comes first, with a failure rate that falls with age, caused by manufacturing defects, bad assembly and weak components; it is removed by burn-in, that is by running the population under stress and shipping the survivors, and it is the reason a burn-in improves a delivered batch without improving any individual unit. The useful life follows, with a roughly constant failure rate dominated by random overloads rather than by age, and this is the only region where the exponential model and the MTBF arithmetic apply. Wear-out comes last, with a failure rate that rises with age through fatigue, abrasion, corrosion and creep; here age does predict failure, so scheduled preventive replacement pays, and it is exactly the region a memoryless model would advise against replacement in. The Weibull distribution parameterises all three with one shape factor: β < 1 is infant mortality, β = 1 reduces to the exponential, and β > 1 is wear-out.

Systems are built out of components, and two configurations cover most of the arithmetic. In series every unit must work, so reliabilities multiply: R_s = R₁R₂R₃…. That has an unpleasant consequence worth feeling — a series system is always less reliable than its worst component, and the more components there are the worse it gets, which is why adding a monitoring device can reduce the reliability of the thing it monitors. In parallel, or active redundancy, the system works if any one unit works, so it is the failures that multiply: R_p = 1 − (1 − R₁)(1 − R₂)…. Take three units in series at 0.95, 0.90 and 0.98 over the mission. The system is 0.95 × 0.90 × 0.98 = 0.8379. Duplicate the weakest unit so that the pair is in parallel: the pair is now 1 − 0.10² = 0.99, and the system becomes 0.95 × 0.99 × 0.98 = 0.9217. One duplicated component moved the system from 0.838 to 0.922, and it did so because redundancy pays where the failures are, not where the components are cheapest. When the failure rates are constant there is a second shortcut for series systems: the rates add, so units at 2 × 10⁻⁴, 3 × 10⁻⁴ and 5 × 10⁻⁴ per hour give λ_s = 10⁻³ and a system MTBF of 1000 hours — half the 2000 hours of the worst unit alone.

Reliability alone does not tell an owner what fraction of the year a machine will actually be producing, and availability does. Availability is uptime divided by total time, and in its steady-state form it is A = MTBF/(MTBF + MTTR), where the mean time to repair measures maintainability — spares held, access designed in, diagnostics fitted, crew trained. The pump above with an MTBF of 5000 hours and an MTTR of 50 hours has A = 5000/5050 = 0.9901, so across a 8760 hour year it is down for about 86.7 hours. Now notice which lever is cheaper. Doubling the MTBF to 10 000 hours takes the availability to 0.9950; halving the MTTR to 25 hours takes it to 0.9950 as well — identical benefit, and the second is usually a stores decision rather than a redesign. That is the practical content of the whole topic, and it is what links it to lifecycle analysis: over a product life the owner pays for acquisition, energy, spares, repair labour and disposal, and the reliability decisions made at design time fix most of the cost that will be incurred after the sale. The design-side tools follow from that — FMEA, which walks each failure mode and scores severity, occurrence and detection into a risk priority number so that attention goes to the worst rather than to the most recent, and fault tree analysis, which works backwards from one undesired event to the combinations of causes that could produce it.