DampedSpring

DampedSpring(
    var_name='damped_spring',
    m=1.0,
    k=1.0,
    c=0.1,
    F=0.0,
    state_variables=None,
    diagnostic_variables=None,
    *args,
    **kwargs,
)

Damped (and optionally driven) spring-mass oscillator.

Parameters

var_name : = 'damped_spring'

Human-readable label for the model.

m : = 1.0

Mass in kg. Must be > 0.

k : = 1.0

Spring constant in N/m. Must be > 0.

c : = 0.1

Linear damping coefficient in N·s/m. c=0 gives undamped SHM; c < 2*sqrt(k*m) gives underdamped (oscillatory) decay.

F : float or callable or cc.Forcing = 0.0

External driving force (N). Default 0.0 (undriven). For a time-varying drive use model.register_forcing('F', forcing_obj).

state_variables : = None

Names for the two integrated state variables (position, velocity).

diagnostic_variables : = None

Names for post-integration diagnostics.

Notes

State variables are x (position) and v (velocity) in that order. Diagnostic variables energy and omega_0 are computed after integration.

Examples

import numpy as np
import matplotlib.pyplot as plt
import climatecritters as cc
from climatecritters.model_critters.damped_spring import DampedSpring

model = DampedSpring(m=1.0, k=4.0, c=0.4)
output = model.integrate(t_span=(0, 30), y0=[1.0, 0.0], method='RK45')
ts = output.to_pyleo(var_names=['x'])
ts.plot()
plt.savefig('docs/reference/figures/DampedSpring_example.png',
            dpi=150, bbox_inches='tight')

DampedSpring example output

With resonant driving (external force at ω₀):

import numpy as np
import matplotlib.pyplot as plt
import climatecritters as cc
from climatecritters.model_critters.damped_spring import DampedSpring

m, k = 1.0, 4.0
omega_0 = np.sqrt(k / m)
model = DampedSpring(m=m, k=k, c=0.0)
model.register_forcing('F', cc.Forcing(lambda t: np.cos(omega_0 * t)))
output = model.integrate(t_span=(0, 30), y0=[0.0, 0.0], method='RK45')
fig, ax = plt.subplots()
ax.plot(output.time, output.state_variables['x'])
ax.set_xlabel('time'); ax.set_ylabel('x')
ax.set_title('DampedSpring — resonant driving at ω₀')
plt.savefig('docs/reference/figures/DampedSpring_resonance_example.png',
            dpi=150, bbox_inches='tight')

DampedSpring_resonance example output

Methods

Name	Description
damping_ratio	Return ζ = c / (2√(km)). ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.
natural_frequency	Return ω₀ = √(k/m) in rad/s (uses current param_values).
natural_period	Return T₀ = 2π/ω₀ in seconds (uses current param_values).

damping_ratio

DampedSpring.damping_ratio()

Return ζ = c / (2√(km)). ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.

natural_frequency

DampedSpring.natural_frequency()

Return ω₀ = √(k/m) in rad/s (uses current param_values).

natural_period

DampedSpring.natural_period()

Return T₀ = 2π/ω₀ in seconds (uses current param_values).