Multiple Degree of Freedom Systems (`vibration_toolbox.mdof`)¶

`mdof` – Multiple Degree of Freedom Functions¶

Multiple Degree of Freedom Analysis Tools.

mdof.modes_system(M, K, C=None)[source]¶

Natural frequencies, damping ratios, and mode shapes of MDOF system. This function will return the natural frequencies (wn), the damped natural frequencies (wd), the damping ratios (zeta), the right eigenvectors (X) and the left eigenvectors (Y) for a system defined by M, K and C. If the dampind matrix ‘C’ is none or if the damping is proportional, wd and zeta will be none and X and Y will be equal.

Parameters

M: array: Mass matrix
K: array: Stiffness matrix
C: array: Damping matrix

Returns

wn: array: The natural frequencies of the system
wd: array: The damped natural frequencies of the system
zeta: array: The damping ratios
X: array: The right eigenvectors
Y: array: The left eigenvectors

Examples

>>> # This test has been moved to tests_mdof
>>> M = np.array([[1, 0],
...               [0, 1]])
>>> K = np.array([[2, -1],
...               [-1, 6]])
>>> C = np.array([[0.3, -0.02],
...               [-0.02, 0.1]])
>>> wn, wd, zeta, X, Y = modes_system(M, K, C) # doctest: +SKIP
Damping is non-proportional, eigenvectors are complex.
>>> wn # doctest: +SKIP
array([1.33, 2.5 , 1.33, 2.5 ])
>>> wd # doctest: +SKIP
array([1.32, 2.5 , 1.32, 2.5 ])
>>> zeta # doctest: +SKIP
array([0.11, 0.02, 0.11, 0.02])
>>> X # doctest: +SKIP
array([[-0.06-0.58j, -0.01+0.08j, -0.06+0.58j, -0.01-0.08j],
       [-0.  -0.14j, -0.01-0.36j, -0.  +0.14j, -0.01+0.36j],
       [ 0.78+0.j  , -0.21-0.03j,  0.78-0.j  , -0.21+0.03j],
       [ 0.18+0.01j,  0.9 +0.j  ,  0.18-0.01j,  0.9 -0.j  ]])
>>> Y # doctest: +SKIP
array([[ 0.02+0.82j,  0.01-0.31j,  0.02-0.82j,  0.01+0.31j],
       [-0.05+0.18j,  0.01+1.31j, -0.05-0.18j,  0.01-1.31j],
       [ 0.61+0.06j, -0.12-0.02j,  0.61-0.06j, -0.12+0.02j],
       [ 0.14+0.03j,  0.53+0.j  ,  0.14-0.03j,  0.53-0.j  ]])
>>> C = 0.2*K # with proportional damping
>>> wn, wd, zeta, X, Y = modes_system(M, K, C) # doctest: +SKIP
Damping is proportional or zero, eigenvectors are real
>>> X # doctest: +SKIP
array([[-0.97,  0.23],
       [-0.23, -0.97]])

mdof.modes_system_undamped(M, K)[source]¶

Return eigensolution of multiple DOF system.

Returns the natural frequencies (w), eigenvectors (P), mode shapes (S) and the modal transformation matrix S for an undamped system.

See Notes for explanation of the underlying math.

Parameters

M: float array: Mass matrix
K: float array: Stiffness matrix

Returns

w: float array: The natural frequencies of the system
P: float array: The eigenvectors of the system.
S: float array: The mass-normalized mode shapes of the system.
Sinv: float array: The modal transformation matrix S^-1(takes x -> r(modal coordinates))

Notes

Given \(M\ddot{x}(t)+Kx(t)=0\), with mode shapes \(u\), the matrix of mode shapes \(S=[u_1 u_1 \ldots]\) can be created. If the modal coordinates are the vector \(r(t)\). The modal transformation separates space and time from \(x(t)\) such that \(x(t)=S r(t)\). Substituting into the governing equation:

\(MS\ddot{r}(t)+KSr(t)=0\)

Premultiplying by \(S^T\)

\(S^TMS\ddot{r}(t)+S^TKSr(t)=0\)

The matrices \(S^TMS\) and \(S^TKS\) will be diagonalized by this process (\(u_i\) are the eigenvectors of \(M^{-1}K\)).

If scaled properly (mass normalized so \(u_i^TMu_i=1\)) then \(S^TMS=I\) and \(S^TKS=\Omega^2\) where \(\Omega^2\) is a diagonal matrix of the natural frequencies squared in radians per second.

Further, inverses are unstable so the better way to solve linear equations is with Gauss elimination.

\(AB=C\) given known \(A\) and \(C\) is solved using la.solve(A, C, assume_a=’pos’).

\(BA=C\) given known \(A\) and \(C\) is solved by first transposing the equation to \(A^TB^T=C^T\), then solving for \(C^T\). The resulting command is la.solve(A.T, C.T, assume_a=’pos’).T

Examples

>>> M = np.array([[4, 0, 0],
...               [0, 4, 0],
...               [0, 0, 4]])
>>> K = np.array([[8, -4, 0],
...               [-4, 8, -4],
...               [0, -4, 4]])
>>> w, P, S, Sinv = modes_system_undamped(M, K)
>>> w # doctest: +SKIP
array([0.45, 1.25, 1.8 ])
>>> S
array([[ 0.16, -0.37, -0.3 ],
       [ 0.3 , -0.16,  0.37],
       [ 0.37,  0.3 , -0.16]])

mdof.response_system(M, C, K, F, x0, v0, t)[source]¶

Returns system response given the initial displacement vector ‘X0’, initial velocity vector ‘V0’, the mass matrix ‘M’, the stiffness matrix ‘M’, and the damping matrix ‘C’ and force ‘F’. T is a row vector of evenly spaced times. F is a matrix of forces over time, each column corresponding to the corresponding column of T, each row corresponding to the same numbered DOF.

Parameters

M: array: Mass matrix
K: array: Stiffness matrix
C: array: Damping matrix
x0: array: Array with displacement initial conditions
v0: array: Array with velocity initial conditions
t: array: Array withe evenly spaced times

Returns

Tarray: Time values for the output.
youtarray: System response.
xoutarray: Time evolution of the state vector.

Examples

>>> M = np.array([[9, 0],
...               [0, 1]])
>>> K = np.array([[27, -3],
...               [-3, 3]])
>>> C = K/10
>>> x0 = np.array([0, 1])
>>> v0 = np.array([1, 0])
>>> t = np.linspace(0, 10, 100)
>>> F = np.vstack([0*t,
...                3*np.cos(2*t)])
>>> tou, yout, xout = response_system(M, C, K, F, x0, v0, t)
>>> tou[:10] # doctest: +SKIP
array([0.  , 0.1 , 0.2 , 0.3 , 0.4 , 0.51, 0.61, 0.71, 0.81, 0.91])

>>> yout[:10]
array([[ 0.  ,  1.  ,  1.  ,  0.  ],
       [ 0.1 ,  1.  ,  0.99,  0.04],
       [ 0.2 ,  1.01,  0.95,  0.1 ],
       [ 0.29,  1.02,  0.88,  0.15],
       [ 0.38,  1.04,  0.79,  0.19],
       [ 0.45,  1.06,  0.68,  0.2 ],
       [ 0.51,  1.08,  0.55,  0.17],
       [ 0.56,  1.09,  0.41,  0.09],
       [ 0.59,  1.09,  0.26, -0.04],
       [ 0.61,  1.08,  0.11, -0.22]])

>>> xout[:10]
array([[ 0.  ,  1.  ,  1.  ,  0.  ],
       [ 0.1 ,  1.  ,  0.99,  0.04],
       [ 0.2 ,  1.01,  0.95,  0.1 ],
       [ 0.29,  1.02,  0.88,  0.15],
       [ 0.38,  1.04,  0.79,  0.19],
       [ 0.45,  1.06,  0.68,  0.2 ],
       [ 0.51,  1.08,  0.55,  0.17],
       [ 0.56,  1.09,  0.41,  0.09],
       [ 0.59,  1.09,  0.26, -0.04],
       [ 0.61,  1.08,  0.11, -0.22]])

mdof.response_system_undamped(M, K, x0, v0, max_time)[source]¶

This function calculates the time response for an undamped system and returns the vector (state-space) X. The n first rows contain the displacement (x) and the n last rows contain velocity (v) for each coordinate. Each column is related to a time-step. The time array is also returned.

Parameters

M: array: Mass matrix
K: array: Stiffness matrix
x0: array: Array with displacement initial conditions
v0: array: Array with velocity initial conditions
max_time: float: End time

Returns

t: array: Array with the time
X: array: The state-space vector for each time

Examples

>>> M = np.array([[1, 0],
...               [0, 4]])
>>> K = np.array([[12, -2],
...               [-2, 12]])
>>> x0 = np.array([1, 1])
>>> v0 = np.array([0, 0])
>>> max_time = 10
>>> t, X = response_system_undamped(M, K, x0, v0, max_time)
>>> # first column is the initial conditions [x1, x2, v1, v2]
>>> X[:, 0] # doctest: +SKIP
array([1., 1., 0., 0.])
>>> X[:, 1] # displacement and velocities after delta t
array([ 1.  ,  1.  , -0.04, -0.01])

Multiple Degree of Freedom Systems (`vibration_toolbox.mdof`)¶

`mdof` – Multiple Degree of Freedom Functions¶

Vibration Toolbox

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Multiple Degree of Freedom Systems (vibration_toolbox.mdof)¶

mdof – Multiple Degree of Freedom Functions¶

Multiple Degree of Freedom Systems (`vibration_toolbox.mdof`)¶

`mdof` – Multiple Degree of Freedom Functions¶