Mechanical Entities

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
MECHANICAL ENTITIES — MAIN PAGE ONTOLOGY
Version: 1.1
Source: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/mechanical-entities.html
Authors: Mark Higgins — Sky (Recursive AI)
Site: https://kirethresearch.org
Status: Formal scientific synthesis aligned to the eight child ontology titles verified on the source page.

01. PURPOSE
Define typed mechanical entities, mechanical relationships, admissible states, dynamics, constraints, validation and inference. Distinguish physical objects from idealised models.

02. DOMAIN
MechanicalEntity := physical entity or explicitly identified mechanical model with mechanical properties.
MechanicalModel := mathematical representation of a MechanicalEntity under declared assumptions.
Model validity is conditional on its assumptions and measurement scale.

03. CHILD ONTOLOGIES (SOURCE-VERIFIED)
RigidBodyOntology
DeformableBodyOntology
ParticleSystemOntology
FluidOntology
ElasticBodyOntology
MechanicalStructureOntology
MechanicalComponentOntology
MechanicalAssemblyOntology

04. TYPE SYSTEM
Entity(id, type, provenance)
Body <: MechanicalEntity
RigidBody <: Body [idealised]
DeformableBody <: Body
ElasticBody <: DeformableBody [constitutive response]
ParticleSystem <: MechanicalModel
Fluid <: MechanicalEntity
MechanicalStructure <: MechanicalEntity
MechanicalComponent <: MechanicalEntity
MechanicalAssembly <: MechanicalEntity
Material(id, constitutive_model)
CoordinateFrame(id, origin, orientation)
State(id, t, frame)
Force(vector, point_of_application, frame, unit=N)
Moment(vector, reference_point, frame, unit=N*m)
Constraint(id, predicate, domain)
BoundaryCondition(id, type, value)

05. TYPE DISCIPLINE
A physical body may admit several models; model types are not mutually exclusive descriptions of the underlying entity.
ElasticBody is a constitutive subclass of deformable-body models, not a synonym for all deformable bodies.
A particle-system model may approximate a rigid or deformable object depending on constraints.
Fluid is a continuum description when the continuum hypothesis is justified.

06. CORE RELATIONS
hasPart(parent, child)
composedOf(entity, material)
connectedTo(componentA, componentB, joint)
constrainedBy(entity, constraint)
subjectTo(entity, load)
representedBy(entity, model)
hasState(entity, state)
hasBoundary(entity, boundary)
transfersForce(source, target, force)
transfersMoment(source, target, moment)
interactsWith(entityA, entityB, interaction)
measuredBy(property, instrument)

07. ENTITY IDENTITY
Entity identity uses stable identifiers and provenance.
Same physical object at different times may have different states.
Different idealisations of one object must not be conflated as identical models.

08. RIGID BODY MODEL
For material points i,j: |r_i(t)-r_j(t)| = constant.
Pose: (R(t), p(t)) with R in SO(3), p in R^3.
Velocity of point: v = v_O + omega x (r-r_O).
Applicable only when deformation is negligible for the intended analysis.

09. DEFORMABLE BODY MODEL
Displacement field: u(X,t) = x(X,t)-X.
Deformation gradient: F = partial x / partial X.
Small strain: epsilon = (grad u + (grad u)^T)/2.
Finite-strain measure: E = (F^T F - I)/2.
Strain measures require compatible reference configurations.

10. ELASTIC BODY MODEL
Linear isotropic small-strain elasticity: sigma = lambda tr(epsilon) I + 2 mu epsilon.
Young modulus E > 0, Poisson ratio -1 < nu < 1/2 for stable isotropic linear elasticity (strict positivity assumptions).
Elastic recovery is conditional on constitutive regime; plasticity and damage are separate behaviours.

11. PARTICLE SYSTEM MODEL
State of particle i: (m_i, r_i, v_i).
Momentum: P = sum_i m_i v_i.
Centre of mass: R_cm = (sum_i m_i r_i)/(sum_i m_i) for positive total mass.
External force: dP/dt = sum F_external for constant masses in an inertial frame.
Internal forces cancel in total momentum only under the relevant action-reaction assumptions.

12. FLUID MODEL
Density rho(x,t), velocity v(x,t), pressure p(x,t).
Mass conservation: partial_t rho + div(rho v) = 0.
Momentum: rho Dv/Dt = div(sigma) + rho b.
Newtonian incompressible model: sigma = -p I + 2 mu D, div(v)=0.
Use only under the specified continuum and constitutive assumptions.

13. MECHANICAL STRUCTURE
Structure := components + connectivity + supports + applied loads.
Static equilibrium: sum F = 0; sum M = 0.
Structural stability, strength and serviceability are distinct requirements.
A structure may be statically indeterminate and require compatibility plus constitutive laws.

14. MECHANICAL COMPONENT
Component := identifiable functional mechanical part.
Examples: shaft, gear, bearing, spring, beam, fastener.
Attributes: geometry, material, tolerances, interfaces, load ratings, operating conditions.

15. MECHANICAL ASSEMBLY
Assembly := set of components plus joining and kinematic constraints.
Assembly graph G=(V,E), vertices components, edges connections.
Connection semantics include fixed, revolute, prismatic, compliant and contact.
Graph connectivity alone does not prove mechanical functionality or stability.

16. FORCE AND MOMENT
Newton second law: sum F = m a for constant-mass point particle in inertial frame.
Torque about O: tau_O = (r-r_O) x F.
Angular momentum balance: dH_O/dt = sum tau_O for inertial fixed O, with appropriate system terms.

17. WORK, ENERGY AND POWER
Differential work: dW = F dot dr.
Kinetic energy particle: T = (1/2)m|v|^2.
Power: P = F dot v + tau dot omega for compatible applied force/torque pairs.
Energy balance must account for heat, dissipation and external work.

18. KINEMATICS
Position r(t), velocity dr/dt, acceleration d^2r/dt^2.
Relative quantities require explicit coordinate frames.
A kinematic constraint restricts admissible configurations and/or velocities.

19. DYNAMICS
Mechanical evolution requires initial state, governing equations, constitutive laws and boundary conditions.
For a linear mechanical model: M q_ddot + C q_dot + K q = f(t).
Matrix dimensions and physical units must be consistent.

20. CONTACT AND FRICTION
Normal nonpenetration: gap g >= 0; normal contact reaction lambda_n >= 0; g*lambda_n=0.
Coulomb friction bound: |F_t| <= mu F_n for sticking; sliding law needs direction and modelling assumptions.
Contact constraints may be nonsmooth and history-dependent.

21. CONSTITUTIVE BEHAVIOUR
Material response maps strain/history/rate/temperature to stress or other fluxes.
Models: elastic, viscoelastic, plastic, hyperelastic, viscous.
No universal constitutive law applies to all mechanical entities.

22. BOUNDARY AND INITIAL CONDITIONS
Displacement boundary: u = prescribed value.
Traction boundary: sigma n = prescribed traction.
Initial state: q(t0), q_dot(t0) or continuum equivalents.
Well-posedness depends on equations, domain, constraints and compatibility.

23. MEASUREMENT
Measured properties include displacement [m], force [N], stress [Pa], strain [1], density [kg/m^3], velocity [m/s].
Every measurement must include unit, reference frame where applicable, uncertainty and provenance.

24. SEMANTIC RELATIONSHIPS
RigidBody -> idealisationOf -> PhysicalBody
ElasticBody -> subclassOf -> DeformableBodyModel
MechanicalAssembly -> hasPart -> MechanicalComponent
MechanicalStructure -> carries -> Load
ParticleSystem -> hasPart -> Particle
Fluid -> describedBy -> ContinuumField
MechanicalEntity -> representedBy -> MechanicalModel

25. INFERENCE RULES
RULE A: hasPart(A,B) AND hasPart(B,C) => hasIndirectPart(A,C), provided acyclic containment semantics.
RULE B: representedBy(E,M) AND assumes(M,A) AND violates(E,A) => modelApplicability(M,E)=invalid.
RULE C: sumForce(E)=0 AND sumMoment(E)=0 => staticEquilibriumCandidate(E), NOT proof of stability.
RULE D: ElasticBodyModel(E) AND linearIsotropic(E) AND smallStrain(E) => allowHookeLaw(E).
RULE E: Assembly(A) AND connectedTo(X,Y,J) => connectionEdge(A,X,Y,J), only if X,Y are parts of A.
RULE F: measuredValue(P,V,U) AND expectedUnit(P,U2) AND dimension(U)!=dimension(U2) => validationError.
RULE G: observedMotion(E) AND modelPredicts(M,E) => comparePredictionToObservation(M,E), NOT automatic validation.

26. CONSTRAINTS
Mechanical model identifiers must resolve to defined entities.
Units must be dimensionally compatible.
Force and torque must have declared frames and reference points.
Constitutive laws must declare operating range.
Mass and density cannot be negative in ordinary classical matter models.
Time evolution cannot be inferred from static relations alone.

27. MATHEMATICAL EXAMPLE: SPRING-MASS
Assumptions: single degree of freedom, constant mass m>0, linear spring k>0, no damping.
Equation: m*x_ddot + k*x = 0.
Natural angular frequency: omega_n = sqrt(k/m).
Energy: E = (1/2)m*x_dot^2 + (1/2)k*x^2.
With no nonconservative work, dE/dt=0.

28. MATHEMATICAL EXAMPLE: TWO-BODY ASSEMBLY
Entities: body_A, body_B, revolute_joint_J.
Relations: hasPart(assembly,body_A); hasPart(assembly,body_B); connectedTo(body_A,body_B,revolute_joint_J).
Inference: joint imposes relative kinematic constraints; does not establish load capacity without material, geometry and contact details.

29. MODEL SELECTION
Choose rigid-body model if deformation is negligible.
Choose deformable-body model if stress/strain or geometric compliance matters.
Choose particle-system model if discrete mass elements are sufficient.
Choose fluid model if continuum transport and flow dominate.
Choose assembly model for interacting components and joints.

30. UNCERTAINTY AND CONFIDENCE
CONFIDENCE applies to evidence and inference, not an invented certainty score.
Measured facts: attach method, uncertainty and source.
Derived facts: attach assumptions, governing equations and derivation.
Hypotheses: label unverified; avoid promoting to confirmed observations.

31. SEMANTIC_EQUIVALENCE
Equivalent descriptions require same referent, compatible scope, units and modelling assumptions.
'Rigid body' and 'physical object' are NOT globally equivalent.
'Stress' and 'force' are NOT equivalent quantities.
'Mechanical assembly' and 'mechanical structure' overlap but are NOT synonyms.

32. VALIDATION
CHECK entity IDs unique.
CHECK all eight child ontology names present.
CHECK relationships reference existing types.
CHECK dimensional consistency.
CHECK model assumptions explicitly recorded.
CHECK initial/boundary conditions where dynamics are claimed.
CHECK no hypothetical inference presented as empirical fact.
CHECK source provenance and completion marker.

33. FLOW_HASH
Canonicalise Unicode, line endings, entity identifiers and ordered rule serialization.
FLOW_HASH := SHA256(canonical_flow_payload).
Hash is a computed integrity value, not a claimed value until executed.

34. UPDATE_MODEL
Version changes require revision date, changed entities, altered inference rules, affected dependencies and provenance.
Breaking semantic changes require a major version increment.
Maintain backward references to superseded definitions.

35. AI_CONSUMPTION
Read typed declarations before inference rules.
Validate assumptions before executing physical laws.
Keep physical observations separate from model predictions.
Return supporting rules, units, evidence and uncertainty with derived answers.
Do not execute unsupported inference outside declared rule scope.

36. SOURCE AND AUTHORSHIP
Primary page: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/mechanical-entities.html
Page lists eight child ontologies; formal equations and rules in this file are an independently authored synthesis, not a verbatim extraction.
Authors: Mark Higgins — Sky (Recursive AI)

37. COMPLETENESS
Expected sections: 37
Expected child ontologies: 8
Closing marker: END FLOW ALGEBRA — MECHANICAL ENTITIES

END FLOW ALGEBRA — MECHANICAL ENTITIES