Damped Oscillator Model#

import numpy as np
import matplotlib.pyplot as plt
import uqtestfuns as uqtf

The damped oscillator model is a seven-dimensional scalar-valued function. The model was first proposed in [IK85] and used in the context of reliability analysis in [DKDS91, Dub11].

Note

The reliability analysis variant differs from this base model. Used in the context of reliability analysis, the model also includes additional parameters of a capacity factor and load such that the performance function can be computed. This base model only computes the relative displacement of the spring.

Test function instance#

To create a default instance of the damped oscillator model:

my_testfun = uqtf.DampedOscillator()

Check if it has been correctly instantiated:

print(my_testfun)

Name              : DampedOscillator
Spatial dimension : 7
Description       : Damped oscillator model from Igusa and Der Kiureghian (1985)

Description#

The damped oscillator model is based on a two degree-of-freedom primary-secondary mechanical system characterized by two masses, two springs, and the corresponding damping ratios. Originally, the model computes the mean-square relative displacement of the secondary spring under a white noise base acceleration using the following analytical formula1:

\[ \mathcal{M}(\boldsymbol{x}) = \left( \pi \frac{S_0}{4 \zeta_s \omega_s^3} \frac{\zeta_a \zeta_s}{\zeta_p \zeta_s (4 \zeta_a^2 + \theta^2) + \gamma \zeta_a^2} \frac{(\zeta_p \omega_p^3 + \zeta_s \omega_s^3) \omega_p}{4 \zeta_a \omega_a^4} \right)^{0.5} \]
\[\begin{split} \begin{aligned} \omega_p & = \left( \frac{k_p}{m_p}\right)^{0.5} & \omega_s & = \left(\frac{k_s}{m_s}\right)^{0.5} & \omega_a & = \frac{\omega_p + \omega_s}{2}\\ \gamma & = \frac{m_s}{m_p} & \zeta_a & = \frac{\zeta_p + \zeta_s}{2} & \theta & = \frac{\omega_p - \omega_s}{\omega_a} \\ \end{aligned} \end{split}\]

where \(\boldsymbol{x} = \{ M_p, M_s, K_p, K_s, \zeta_p, \zeta_s, S_0 \}\) is the seven-dimensional vector of input variables further defined below.

Note

In UQTestFuns, this original output is square-rooted to get the relative displacement

Probabilistic input#

Based on [DKDS91], the probabilistic input model for the damped oscillator model consists of seven independent random variables with marginal distributions shown in the table below.

my_testfun.prob_input

Name: DampedOscillator-DerKiureghian1991

Spatial Dimension: 7

Description: Probabilistic input model for the Damped Oscillator model from Der Kiureghian and De Stefano (1991).

Marginals:

No. Name Distribution Parameters Description
1 Mp lognormal [0.40048994 0.09975135] Primary mass
2 Ms lognormal [-4.61014535 0.09975135] Secondary mass
3 Kp lognormal [-0.01961036 0.1980422 ] Primary spring stiffness
4 Ks lognormal [-4.62478054 0.1980422 ] Secondary spring stiffness
5Zeta_p lognormal [-3.06994228 0.38525317] Primary damping ratio
6Zeta_s lognormal [-4.02359478 0.47238073] Secondary damping ratio
7 S0 lognormal [4.60019502 0.09975135] White noise base acceleration

Copulas: None

Reference results#

This section provides several reference results of typical UQ analyses involving the test function.

Sample histogram#

Shown below is the histogram of the output based on \(100'000\) random points:

np.random.seed(42)
xx_test = my_testfun.prob_input.get_sample(100000)
yy_test = my_testfun(xx_test)

plt.hist(yy_test, bins="auto", color="#8da0cb");
plt.grid();
plt.ylabel("Counts [-]");
plt.xlabel("$\mathcal{M}(\mathbf{X})$");
plt.gcf().set_dpi(150);
../_images/damped-oscillator_9_0.png

Moments estimation#

Shown below is the convergence of a direct Monte-Carlo estimation of the output mean and variance with increasing sample sizes.

# --- Compute the mean and variance estimate
np.random.seed(42)
sample_sizes = np.array([1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7], dtype=int)
mean_estimates = np.empty(len(sample_sizes))
var_estimates = np.empty(len(sample_sizes))

for i, sample_size in enumerate(sample_sizes):
    xx_test = my_testfun.prob_input.get_sample(sample_size)
    yy_test = my_testfun(xx_test)
    mean_estimates[i] = np.mean(yy_test)
    var_estimates[i] = np.var(yy_test)

# --- Compute the error associated with the estimates
mean_estimates_errors = np.sqrt(var_estimates) / np.sqrt(np.array(sample_sizes))
var_estimates_errors = var_estimates * np.sqrt(2 / (np.array(sample_sizes) - 1))

# --- Do the plot
fig, ax_1 = plt.subplots(figsize=(6,4))

ax_1.errorbar(
    sample_sizes,
    mean_estimates,
    yerr=mean_estimates_errors,
    marker="o",
    color="#66c2a5",
    label="Mean",
)
ax_1.set_xlabel("Sample size")
ax_1.set_ylabel("Output mean estimate")
ax_1.set_xscale("log");
ax_2 = ax_1.twinx()
ax_2.errorbar(
    sample_sizes + 1,
    var_estimates,
    yerr=var_estimates_errors,
    marker="o",
    color="#fc8d62",
    label="Variance",
)
ax_2.set_ylabel("Output variance estimate")

# Add the two plots together to have a common legend
ln_1, labels_1 = ax_1.get_legend_handles_labels()
ln_2, labels_2 = ax_2.get_legend_handles_labels()
ax_2.legend(ln_1 + ln_2, labels_1 + labels_2, loc=0)

plt.grid()
fig.set_dpi(150)
../_images/damped-oscillator_11_0.png

The tabulated results for each sample size is shown below.

from tabulate import tabulate

# --- Compile data row-wise
outputs = []
for (
    sample_size,
    mean_estimate,
    mean_estimate_error,
    var_estimate,
    var_estimate_error,
) in zip(
    sample_sizes,
    mean_estimates,
    mean_estimates_errors,
    var_estimates,
    var_estimates_errors,
):
    outputs += [
        [
            sample_size,
            mean_estimate,
            mean_estimate_error,
            var_estimate,
            var_estimate_error,
            "Monte-Carlo",
        ],
    ]

header_names = [
    "Sample size",
    "Mean",
    "Mean error",
    "Variance",
    "Variance error",
    "Remark",
]

tabulate(
    outputs,
    headers=header_names,
    floatfmt=(".1e", ".4e", ".4e", ".4e", ".4e", "s"),
    tablefmt="html",
    stralign="center",
    numalign="center",
)
Sample size Mean Mean error Variance Variance error Remark
1.0e+01 1.8183e+02 2.8732e+01 8.2554e+03 3.8916e+03 Monte-Carlo
1.0e+02 1.8413e+02 1.1151e+01 1.2434e+04 1.7673e+03 Monte-Carlo
1.0e+03 1.8705e+02 3.9042e+00 1.5243e+04 6.8201e+02 Monte-Carlo
1.0e+04 1.8924e+02 1.2149e+00 1.4759e+04 2.0874e+02 Monte-Carlo
1.0e+05 1.8869e+02 3.8241e-01 1.4624e+04 6.5401e+01 Monte-Carlo
1.0e+06 1.8850e+02 1.2086e-01 1.4607e+04 2.0658e+01 Monte-Carlo
1.0e+07 1.8852e+02 3.8246e-02 1.4628e+04 6.5418e+00 Monte-Carlo

References#

IK85

Takeru Igusa and Armen Der Kiureghian. Dynamic characterization of two-degree-of-freedom equipment-structure systems. Journal of Engineering Mechanics, 111(1):1–19, 1985. doi:10.1061/(asce)0733-9399(1985)111:1(1).

DKDS91(1,2)

Armen Der Kiureghian and Mario De Stefano. Efficient algorithm for second-order reliability analysis. Journal of Engineering Mechanics, 117(12):2904–2923, 1991. doi:10.1061/(asce)0733-9399(1991)117:12(2904).

Dub11(1,2)

Vincent Dubourg. Adaptive surrogate models for reliability analysis and reliability-based design optimization. phdthesis, Université Blaise Pascal - Clermont II, Clermont-Ferrand, France, 2011.

1

see, for instance, Eqs. (5.5), pp. 184 in [Dub11].