RecallWeave
FIELD LAB 02
ORIGINAL · OFFLINE

Oscillations / normal coordinates

Two masses.
Two independent modes.

Choose a motion, inspect an instant, and separate what each mass does from the two modes that describe it. Every reading comes from the analytical solution.

Equal massesIdeal springsNo damping or drive

Set the experiment

01 / inputs

A preset applies all eight values and returns the inspection time to zero.

System & time window
kg · 0.1–10
N/m · 1–200
N/m · 0–200
s · 0.1–20
Initial displacement & velocity
m · −1 to +1
m · −1 to +1
m/s · −2 to +2
m/s · −2 to +2

Preparing the applied experiment…

The applied motion

Preparing

Two equal masses connected to wall springs and one coupling springDisplacements are scaled for this schematic. Dashed lines mark the two equilibrium positions.

Schematic displacements are scaled to fit. Wall spacing and spring lengths do not define a geometric contact model. Positive x points right.

0 s
0 s8 s

The two modal clocks

Angular frequency

+ / in-phase basis vector (1, 1)

—rad/s · ω₊ = √(k/m)

− / opposite basis vector (1, −1)

—rad/s · ω₋ = √((k + 2c)/m)

Each mode evolves independently. A mass generally combines both frequencies; a generic initial condition need not transfer all its energy to the other mass.

Physical coordinates

x₁ · solidx₂ · dashed

Displacement (m) / time (s). Marker: inspected instant.

q₊ · solidq₋ · dashed

Same displacement and time scales as the physical plot.

Instantaneous state

02 / readings
Applied state at the inspection time; values displayed to eight significant digits
QuantityValueUnit
Time t—s
Position x₁—m
Position x₂—m
Velocity v₁—m/s
Velocity v₂—m/s
Acceleration a₁—m/s²
Acceleration a₂—m/s²
Net force F₁—N
Net force F₂—N
Normal position q₊—m
Normal position q₋—m
Normal velocity u₊—m/s
Normal velocity u₋—m/s

One conserved total

Energy / joules
Physical and modal energy decompositions of the same conserved total
Kinetic T—J
Wall springs Vwall—J
Coupling spring Vc—J
Physical total T + V—J
Mode energy E₊—J
Mode energy E₋—J
Signed drift E(t) − E(0)—J
T / kineticVwall / wallsVc / coupling

E₊ and E₋ are individually constant. The physical kinetic and spring energies can vary. The two decompositions describe the same total; do not add them together.

Readings use eight significant digits. The observation JSON retains the model’s full numeric precision and sampled trajectory.

Coordinate convention

A half-sum, a half-difference.

q₊ = (x₁ + x₂)/2
q₋ = (x₁ − x₂)/2
x₁ = q₊ + q₋ · x₂ = q₊ − q₋
u₊ = (v₁ + v₂)/2
u₋ = (v₁ − v₂)/2

Each modal coordinate has effective mass 2m in this convention. Displacements are measured from equilibrium. The ideal springs are unstrained at equilibrium.

Equations of motion

Coupling cancels in the sum.

m ẍ₁ = −k x₁ + c(x₂ − x₁)
m ẍ₂ = −k x₂ + c(x₁ − x₂)
q̈₊ + (k/m)q₊ = 0
q̈₋ + ((k + 2c)/m)q₋ = 0

The + mode leaves the coupling spring unextended. The − mode changes its extension and has the larger frequency when c > 0.

Energy bookkeeping

Two descriptions, one E.

T = ½m(v₁² + v₂²)
V = ½k(x₁² + x₂²) + ½c(x₂ − x₁)²
E₊ = m u₊² + k q₊²
E₋ = m u₋² + (k + 2c)q₋²
E = T + V = E₊ + E₋

The ideal solution has no damping or external drive. Small displayed drift reflects floating-point evaluation, not a dissipative model.

Keep the reasoning

Take the course, guide, or observation.

The course and guide are always available. An observation contains only a successfully applied experiment and its inspected time. All files are created in this page; nothing is sent over the network and there is no automatic saving.

In RecallWeave, choose Course JSON under Bring your own lesson, preview, then Start this deck.