Lecture 26 of 16.346 Astrodynamics, Fall 2008. Pages 1–2 derive the linear relative-motion equations with radial coordinate xi and along-track coordinate eta; page 3 gives their solution using a different axis convention. Institutional attribution follows the PDF masthead.

MIT teaching module dated 14 June 2001. By David Roylance.

Verified primary passage: printed pp. 8–11, equation (5) and specimen-geometry discussion: critical stress intensity, crack size and small-plastic-zone validity. The numerical example in the wiki is independently calculated.

Official Weather in a Tank laboratory guide. No separate personal byline or publication date established.

Inspected Tank experiment, equations 1–6, and Atmosphere example, equations 7–11: rotating camera, free-surface pressure gradient and Rossby/geostrophic balance. Existing Massachusetts Institute of Technology/work records are reused.

By George Gabriel Stokes.

1851 memoir, read 9 December 1850. Opening pendulum motivation and Part I section IV, articles 39–42, especially equation (126) for steady sphere resistance. Original notation distinguishes density from viscosity; citations use modern dynamic viscosity.

By Heinrich Blasius. Zeitschrift für Mathematik und Physik 56, 1–37.

Original text or historical translation: ntrs.nasa.gov/api/citations/20050028493/downloads/20050028493.pdf

Primary 1950 NACA-TM-1256 translation/reprint inspected: section I flat plate, no-slip boundary conditions and similarity transformation. Keep 1908 original and 1 February 1950 translation dates distinct.

NIST-hosted primary paper, Cement, Concrete, and Aggregates 20(2), pp. 241–247. The abstract and apparatus description distinguish final slump from the additional time measurement used to estimate plastic viscosity.

Manufacturer training slides, particularly slides 8–14 on stress, strain, strain rate and measured rotational signals.

DOE Office of Science, BES Discoveries, section Harnessing the Thermoacoustic Effect. Identifies Greg Swift and explains acoustic heat engines and refrigerators. Historical account, not a current device-performance benchmark.

Institutional physics demonstration using coils, magnets and cones or cups.

Conference abstract describing laser vibrometry, proximity sensing and high-speed video during the onset of free-reed oscillation.

Institutional description of standing water oscillations, their basin setting, and wind or pressure disturbances. The rectangular shallow-water frequency calculation in the wiki is a separate idealized derivation.

University fluid-mechanics lecture notes, §3.11. Derives the force on an accelerating sphere in unbounded ideal fluid and introduces the added-mass matrix.

By Daisuke Takahashi and Junkichi Satsuma. Journal of the Physical Society of Japan 59, 3514–3519, published 15 October 1990. Publisher abstract describes solitary-wave interactions preserving identities and infinitely many conserved quantities. Full text access is restricted; no claim to inspection of the full paper is made. Publisher affiliations identify Takahashi with applied mathematics/informatics and Satsuma with applied physics.

By Bruno Andreotti, Philippe Claudin and Olivier Pouliquen. Geomorphology 123, 343–348. Author-laboratory copy of the published paper, available online 7 August 2010. This six-page publication is distinct from the 2008 four-page arXiv preprint of the same title. Inspected §1, equation (1), and §2, experimental method.

Physical Review E 50, R647–R650. Original research article and author-hosted abstract.

Cited passage: abstract, systematic search for simple quadratic three-dimensional flows and characterization by fixed points, Lyapunov exponents and fractal dimensions.

CompuTiX project documentation, hosted on Inria GitLab Pages. The Hertz theory section gives the sphere-plane contact radius, load, reduced modulus and pressure profile; the associated verification tests concern sphere-plane collision geometry. No personal author or publication date is inferred from the hosting service.

Original experimental mechanics treatise. Digitized edition: www.e-rara.ch/zut/content/titleinfo/1227245

Caltech apparatus-plate record: digital.archives.caltech.edu/collections/Images/RB-GA1784-1/

Cited passage: original apparatus plate and Caltech description of weights, pulleys and pendulum-clock timing.

A passive mechanical network can store and dissipate energy, but cannot supply more net work than was previously stored or delivered through its terminals. For a two-terminal mechanical element, choose signed force $F$ and relative velocity $v$ so that $p=Fv$ is power entering the network. With a nonnegative mechanical energy storage function $E$, passivity requiresIn particular, the net extracted energy over that interval cannot exceed $E(t_0)$. This permits temporary outward power: a stretched spring can release stored elastic potential energy. A requirement that $Fv$ be nonnegative at every instant would incorrectly exclude such a spring.^[ref]

For a spring, dashpot and inerter connected between the same two terminals, let $x$ be relative displacement and $v=\dot{x}$. Their constitutive equations giveFor $k,b,c\ge 0$, integration proves the passivity inequality: the extra term $\int c{v}^{2}dt$ is dissipated energy. The inerter stores kinetic energy associated with relative motion, so a passive mechanical network need not be a device that merely resists displacement or converts all input work into heat. This is central to mechanical network synthesis using springs, dashpots and inerters.^[ref]

For example, a spring with $k=1000N/m$ initially extended by $0.10\mathrm{m}$ stores $5\mathrm{J}$. It can deliver up to $5\mathrm{J}$ while relaxing, despite being passive. A purely viscous element with $c=-10\mathrm{N}s/m$ would instead have input power $-10v^2$: maintaining $v=1m/s$ for one second would extract $10\mathrm{J}$ even from an initially unenergized model. An apparatus producing that sustained behavior needs another energy source.

An experimental check records terminal force and relative velocity simultaneously and integrates their product, accounting for initial energy and every power-carrying connection. A brief interval of negative measured power is compatible with stored-energy release. For a linear zero-initial-state network, the mechanical driving-point impedance provides a complementary frequency description; a nonnegative real part at measured sinusoidal frequencies checks mean absorbed power there, but a finite frequency sweep alone does not establish the full time-domain passivity inequality.

Mechanical driving-point impedance describes the relation between a driving force and motion at the same mechanical port. Here the convention iswhere $F,V$ are Laplace transforms with zero initial conditions, and $v$ is the signed relative velocity of the two terminals. Choose signs so $Fv$ is entering power. The units of this mechanical impedance are $\mathrm{N}s/m$. Its inverse is the mechanical driving-point mobility. Conventions must be checked explicitly: in The Inerter: A Retrospective by Malcolm C. Smith, the force–current analogy calls $V/F$ the mechanical impedance and $F/V$ the admittance.^[ref]

For a spring, dashpot and inerter sharing the same terminals, relative velocities agree and forces add. HenceAt a sinusoidal angular frequency $\omega$, using complex amplitudes proportional to $e^{i\omega t}$,The real part determines mean absorbed power, while the imaginary part records the out-of-phase reactive response. With peak velocity amplitude $|V|$, the mean entering power is $\frac{1}{2}\mathrm{Re}(Z)|V{|}^2$.

For an original example, choose $b=10\mathrm{kg}$, $c=30\mathrm{N}s/m$ and $k=1000N/m$. At $\omega =10rad/s$, the inertial and elastic terms cancel, giving $Z=30\mathrm{N}s/m$. A relative velocity amplitude $0.020m/s$ requires force amplitude $0.60\mathrm{N}$, and mean absorbed power is $0.006\mathrm{W}$. The individual spring and inerter still store and exchange energy.

A mechanical impedance measurement drives the chosen port and records synchronized force and motion there. A load cell measures force; displacement or an accelerometer can supply motion. For an acceleration complex amplitude $A$, $V=A/(i\omega )$, so the phase as well as magnitude matters. Fixture inertia, sensor measurement calibration and the second terminal's motion belong in the measurement model. Using a response at a different point instead defines a transfer relation, not this driving-point relation. Initial stored motion also prevents interpreting an arbitrary transient quotient as a zero-initial-state transfer function.

Rotating shaft bending fatigue occurs when a shaft rotates while carrying a bending moment whose direction is approximately fixed in space. A material point on the outer surface moves alternately through the tensile and compressive sides of the bent section. A constant external bending load can therefore produce cyclic local stress.

For an ideal circular shaft in elastic pure bending, with diameter $d$, second moment of area $I=\pi d^4/64$ and steady moment magnitude $M$, the nominal surface stress amplitude isA rotating surface point has $\sigma (t)={\sigma }_a\cos (\omega t+\varphi )$ in this model. The mean is zero, ${\sigma }_{\max }={\sigma }_a$, ${\sigma }_{\min }=-{\sigma }_a$ and stress ratio $R={\sigma }_{\min }/{\sigma }_{\max }=-1$. One shaft revolution produces one full stress cycle at that point. A bending moment rotating with the shaft would require a different analysis; shaft rotation alone does not guarantee this cycle.

For an original example, $M=20\mathrm{N}\mathrm{m}$ and $d=20\mathrm{mm}$ give ${\sigma }_a\approx 25.5\mathrm{MPa}$. At $1800$ revolutions per minute, one hour supplies $108000$ cycles. These values specify a nominal loading example, not a predicted life or a safe design stress. Stress concentration, surface finish, material condition and other loads affect the local response and mechanical fatigue behavior.

A motivating apparatus is four-point rotating-bending testing. A NIST/NASA benchmark used machined and polished titanium-alloy specimens, a $100\mathrm{Hz}$ test frequency and fully reversed loading with $R=-1$. Comparing differently treated specimens under a controlled arrangement helps connect material state to fatigue response.^[ref] A lifetime claim needs appropriate experimental fatigue data and treatment of scatter; the elastic stress formula does not supply one.