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Beam and discrete oscillator

Verification method · assumptions, essential equations, full derivation, algorithm, code and verification references.

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Verification methodANALYTICAL ORACLE · exact boundary conditions declared

A cantilever and one-degree-of-freedom spring–mass case check stiffness, mass, modal response, and frame conventions.

01 Model and scope

Cantilever responses exercise beam stiffness and local-frame mapping; an SDOF oscillator gives the analytical frequency √(k/m)/(2π). Agreement with the oscillator checks the assembled stiffness–mass pair but does not validate unrelated beam modes or support conditions.

Route assumptions and calculation checks

  1. The beam oracle is chosen slender so transverse-shear error is bounded against Euler–Bernoulli theory.
  2. The oscillator checks one DOF; it does not validate general beam assemblies.
  3. Report reaction, displacement, frequency, and analytic relative error together.

Useful foundations and prerequisites

Units, signs and error measuresInterpolation, mapping and assembly

02 Essential equations and symbols

v(L)=PL3/(3EI)+PL/(κGA)v(L) = PL^3/(3EI) + PL/(\kappa{}GA)f1=(1/2π)(k/m)f_{1} = (1/2π)\sqrt{}(k/m)

Definitions and conventions

E,A,I
Elastic modulus, area and bending inertia.
k,m
Oscillator stiffness and mass.
f1f_{1}
First undamped natural frequency.
TT
Beam local/global transformation.
Beam and discrete oscillator · method mapA cantilever and a one-DOF oscillator test separate element and point-entity quantities against closed-form references.
3 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Use a cantilever oracle for the beam terms

The Timoshenko cantilever tip displacement contains both bending and transverse-shear contributions. For a slender beam, the shear term is small and the result approaches Euler–Bernoulli theory; a shorter/thicker beam makes the section shear correction observable.

v(L)=PL33EI+PLκGAv(L)=\frac{PL^3}{3EI}+\frac{PL}{\kappa GA}

2 · Use the spring–mass oscillator for the discrete pair

A single free translational DOF with spring k and mass m has a closed-form undamped frequency. This checks the assembled spring sign, mass value, support, and eigenvalue-to-frequency conversion with a minimal model.

ω1=km,f1=12πkm\omega_1=\sqrt{\frac{k}{m}},\qquad f_1=\frac{1}{2\pi}\sqrt{\frac{k}{m}}k=1000 N/m,  m=10 kg⇒f1=1.5915494309 Hzk=1000\,\mathrm{N/m},\;m=10\,\mathrm{kg}\Rightarrow f_1=1.5915494309\,\mathrm{Hz}

3 · Compare the right outputs and limits

Report reactions, tip displacement, mode frequency, and relative analytical error together. The beam check exercises its local frame, section and mass assumptions; the oscillator checks one DOF only. Neither result validates arbitrary beam networks, eccentric masses, or every support convention.

erel=∣yFE−yref∣max⁡(∣yref∣,s0)e_{rel}=\frac{|y_{FE}-y_{ref}|}{\max(|y_{ref}|,s_0)}

04 Algorithm and implementation

This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.

05 Verification and limitations

ANALYTICAL ORACLE · exact boundary conditions declared

Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.

Beam V&V
Capability matrix →Recorded cases and original sources →Run a maintained example →

06 Sources and related sheets

Beam inputOscillator inputBeam V&V

Continue the reasoning

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