Overview of the mdof package

The mdof package is designed to provide a convenient interface for computing dynamic analyses of structural vibrations.

Installation

If you are running this notebook locally, install mdof from pypi.

pip install mdof

If you are running this notebook from an external server on JupyterHub/JupyterLab, DataHub, or Binder, uncomment and run the next cell to install the required packages.

# !pip install -Ur requirements.txt

Import

In Python, import the package.

import mdof

Investigate Structural Vibrations

1. Simulate or load a structural response.

Given the modal parameters, what is the structural response data?

from mdof.utilities.config import create_time_vector, gather_system_parameters, generate_input
# time parameters
nt, dt, tf, times = create_time_vector(nt=5000,dt=0.03)
# system parameters
T, m, c, k, Phi = gather_system_parameters(T=[3,2],m=10,c=1)
import numpy as np
zeta = c/(2*np.sqrt(m*k))
print(f"Damping ratios:", zeta)
# input motion
input_motion = generate_input(times)

Periods: [3 2]
Modeshapes:
[[1. 0.]
 [0. 1.]]
Damping ratios: [0.02387324 0.01591549]

# Calculate the response using numerical integration.
import sdof, numpy as np
output_motion = np.array([sdof.integrate(m,c,k[i],input_motion,dt)[0] for i in range(len(T))])

# Plot the input and output motions.
from mdof.utilities.printing import plot_io
plot_io(input_motion, output_motion, times)

2. Modal identification from data.

Given the structural response data, what are the modal parameters?

# With a state-space model:
periods, modeshapes = mdof.modes(input_motion, output_motion, dt=dt, order=2*len(T), method='srim')
print(f"Periods identified from data: {np.round(periods, 3)}")
print(f"Mode shapes identified from data:\n {np.round(np.abs(modeshapes), 3)}")

Periods identified from data: [3.001 2.001]
Mode shapes identified from data:
 [[0.071 0.   ]
 [0.    0.071]]

# With a frequency domain, output-only model:
from mdof.utilities.printing import plot_fdd
periods, modeshapes = plot_fdd(outputs=output_motion, dt=dt)
print(f"Periods identified from data: {np.round(periods, 3)}")
print(f"Mode shapes identified from data:\n {np.round(np.abs(modeshapes), 3)}")

Periods identified from data: [3.    2.027]
Mode shapes identified from data:
 [[1.    0.001]
 [0.001 1.   ]]

3. Response prediction from data.

For any given input motion, can we generate the response of the system?

# Reproduce the response with the state space model
from control import ss, forced_response
from mdof.utilities.printing import plot_pred
realization = mdof.system(method="okid-era", inputs=input_motion, outputs=output_motion,order=2*len(T))
modes = mdof.modal.system_modes(realization,dt)
from mdof.utilities.printing import print_modes
print_modes(modes)
out_pred = forced_response(ss(*realization,dt), U=input_motion, squeeze=False, return_x=False).outputs
plot_pred(ytrue=output_motion, models=out_pred, t=times, title="State Space Model Displacement Response")
print(f"mean absolute error: {np.mean(np.abs(output_motion-out_pred))}")

Spectral quantities:
       T(s)        ζ        EMACO      MPC       EMACO*MPC
      3.001      0.02386    1.0        1.0        1.0
      2.001      0.01589    1.0        1.0        1.0
Mean Period(s): 2.5012359909299615
Standard Dev(s): 0.49975407347712175
mean absolute error: 4.502252724305641e-06

input_motion_2 = np.sin(times)
output_motion_2 = np.array([sdof.integrate(m,c,k[i],input_motion_2,dt)[0] for i in range(len(T))])
out_pred_2 = forced_response(ss(*realization,dt), U=input_motion_2, squeeze=False, return_x=False).outputs
plot_pred(ytrue=output_motion_2, models=out_pred_2, t=times, title="State Space Model Displacement Response on New Input")
print(f"mean absolute error: {np.mean(np.abs(output_motion_2-out_pred_2))}")

mean absolute error: 2.1372453781165088e-07

from mdof.macro import stabilization
fig = stabilization(input_motion, output_motion, dt=dt, orders=(2,50,8), plotly=True)

fig.update_layout({'xaxis': {'autorange': True},
                  'xaxis2':{'range':(0,0.1), 'autorange': False},
                  'yaxis':{'range':(0,10), 'autorange': False}})

Data type cannot be displayed: application/vnd.plotly.v1+json