Why springs, and how every part is one
Treating structure as compliance rather than as rigid geometry.
Every part in a machine is a spring, whether or not it was built to look like one. Anything solid deflects a real, if often tiny, amount under load and springs back once the load is removed, and treating a supposedly rigid bracket, shaft or frame as a spring with an unusually high stiffness explains its behaviour far more completely than treating it as perfectly solid.
Rigid is always a matter of degree
Every solid material obeys the same elastic relationship a coiled spring does, deflecting in proportion to the load applied and returning to shape once the load is removed, right up until the load is large enough to cause permanent damage. What separates an object everyone calls a spring from one everyone calls rigid is how much load it takes to produce a deflection large enough to notice. Steel's material stiffness is roughly ten thousand times that of soft rubber, so a steel bracket moves far less than a rubber mount under the same push, but the same proportional law governs both, and the mathematics used to size a coiled spring applies to the bracket with a much larger stiffness entered into it.
A spring designer coils thinner wire into more turns to make a softer spring, or thicker wire into fewer turns to make a stiffer one. A bracket's designer makes the same kind of choice every time they decide how thick to cast a rib or how far apart to space two mounting bosses.
A skyscraper looks like the most unmoving structure imaginable, yet its top sways a measurable amount in a strong wind, enough that engineers design for it and some towers carry a heavy moving mass near the top to damp the sway. A small steel bracket in a machine behaves in the same way, on a scale no eye could catch without instruments.
Asking how far a part will move
A designer who treats a bracket as rigid is assuming its deflection is exactly zero. That is convenient for rough calculations and wrong once the bracket's small movement matters to something else: a precisely aligned bearing, a sensitive measurement, a mechanism that depends on staying where it was placed. Treating each part as a spring with its own calculable stiffness lets the designer ask how much it will move under its expected load, and whether that movement matters. Mechanisms that lose precision under load, instruments that read differently depending on how they are mounted, and assemblies that resonate at unpredicted frequencies all trace back to a rigid part's real compliance never having been accounted for.
The harder skill is judging which compliances matter. A machine analysed with equal rigour for every part's tiny deflection would spend enormous effort on distinctions that change nothing, and the judgement rests on knowing what the rest of the machine is sensitive to. The bolted joint covered earlier in this era is the clearest case, since the bolt's own stretch was the mechanism holding the joint together. The next two articles follow the same habit into how compliance limits a machine's speed and which speeds a compliant structure has to avoid.