Analysis of covariance (structural analysis)

The analysis of covariance is a method in structural mechanics for the investigation of structures that are stressed by a stochastic dynamic load . It is used to determine statistical parameters ( variances and covariances ) in order to assess the load on the structure.

description

The analysis of covariance uses a filter representation of the load: the load time series is analyzed and a shape filter is identified. This is based on the relatives of the spectral power density in the time domain , the correlation functions . The result of the analysis of covariance are the variances and covariances of the structural response of all structural degrees of freedom .

Filter display

The basis of the analysis of covariance is the equivalent representation of load and structure as a filter . A shape filter is identified for the load , and the structural data is transformed into a structural filter in the state space representation . The two filter models can then be combined to form an overall filter that has Gaussian white noise as the system input. ${\displaystyle \mathbf {S} _{1}\left({\mathbf {A} _{1},\mathbf {B} _{1},\mathbf {C} _{1},\mathbf {D} _{1}}\right)}$ ${\displaystyle \mathbf {S} _{2}\left({\mathbf {A} _{2},\mathbf {B} _{2},\mathbf {C} _{2},\mathbf {D} _{2}}\right)}$ ${\displaystyle \mathbf {S} _{G}\left({\mathbf {A} _{G},\mathbf {B} _{G},\mathbf {C} _{G},\mathbf {D} _{G}}\right)}$

Lyapunov equation

From a mathematical point of view, the analysis of covariance corresponds to a solution to the continuous Lyapunov equation :

${\displaystyle \mathbf {A} _{G}\cdot \mathbf {P} _{z_{G}z_{G}}+\mathbf {P} _{z_{G}z_{G}}\cdot \mathbf {A} _{G}^{T}+\mathbf {B} _{G}\cdot \mathbf {B} _{G}^{T}=0}$

Where and are the system matrices of the overall filter and is the one with the state vector of the system ${\displaystyle \mathbf {A} _{G}}$ ${\displaystyle \mathbf {B} _{G}}$ ${\displaystyle \mathbf {P} _{z_{G}z_{G}}}$

${\displaystyle \mathbf {z} _{G}(t)={\begin{pmatrix}\mathbf {x} (t)\\{\dot {\mathbf {x} }}(t)\\\mathbf {z} _{1}(t)\end{pmatrix}}}$

corresponding covariance matrix. and the displacement vector or the velocity vector of the structure. is the system vector of the load filter. The covariance matrix contains, among other things, the variances and covariances of the structural displacements and structural velocities . ${\displaystyle \mathbf {x} (t)}$ ${\displaystyle {\dot {\mathbf {x} }}(t)}$ ${\displaystyle \mathbf {z} _{1}(t)}$ ${\displaystyle \mathbf {P} _{xx}}$ ${\displaystyle \mathbf {P} _{{\dot {x}}{\dot {x}}}}$

${\displaystyle \mathbf {P} _{z_{G}z_{G}}={\begin{pmatrix}{\mathbf {P} _{xx}}&{\mathbf {P} _{x{\dot {x}}}}&{\mathbf {P} _{xz_{1}}}\\{\mathbf {P} _{{\dot {x}}x}}&{\mathbf {P} _{{\dot {x}}{\dot {x}}}}&{\mathbf {P} _{{\dot {x}}z_{1}}}\\{\mathbf {P} _{z_{1}x}}&{\mathbf {P} _{z_{1}{\dot {x}}}}&{\mathbf {P} _{z_{1}z_{1}}}\end{pmatrix}}}$

Calculation of higher spectral moments

By forming the expected value, the matrix can be obtained from the covariance matrix , which contains the variances and covariances of the structural accelerations: ${\displaystyle \mathbf {P} _{z_{G}z_{G}}}$ ${\displaystyle \mathbf {P} _{{\dot {z}}_{G}{\dot {z}}_{G}}}$ ${\displaystyle \mathbf {P} _{{\ddot {x}}{\ddot {x}}}}$

${\displaystyle {\begin{aligned}P_{{\dot {z}}_{G}{\dot {z}}_{G}}&=E\left[{{\dot {\mathbf {z} }}_{G}\cdot {\dot {\mathbf {z} }}_{G}^{T}}\right]\\&=\mathbf {A} _{G}\cdot E\left[{\mathbf {z} _{G}\cdot \mathbf {z} _{G}^{T}}\right]\cdot \mathbf {A} _{G}^{T}+\mathbf {A} _{G}\cdot E\left[{\mathbf {z} _{G}\cdot \mathbf {w} ^{T}}\right]\cdot \mathbf {B} _{G}^{T}\\&+\mathbf {B} _{G}\cdot E\left[{\mathbf {w} \cdot \mathbf {z} _{G}^{T}}\right]\cdot \mathbf {A} _{G}^{T}+\mathbf {B} _{G}\cdot E\left[{\mathbf {w} \cdot \mathbf {w} ^{T}}\right]\cdot \mathbf {B} _{G}^{T}\\&=\mathbf {A} _{G}\cdot \mathbf {P} _{z_{G}z_{G}}\cdot \mathbf {A} _{G}^{T}+\mathbf {B} _{G}\cdot \mathbf {B} _{G}^{T}\end{aligned}}}$

The matrix is structured as follows: ${\displaystyle P_{{\dot {z}}_{G}{\dot {z}}_{G}}}$

${\displaystyle \mathbf {P} _{{\dot {z}}_{G}{\dot {z}}_{G}}={\begin{pmatrix}{\mathbf {P} _{{\dot {x}}{\dot {x}}}}&{\mathbf {P} _{{\dot {x}}{\ddot {x}}}}&{\mathbf {P} _{{\dot {x}}{\dot {z}}_{1}}}\\{\mathbf {P} _{{\ddot {x}}{\dot {x}}}}&{\mathbf {P} _{{\ddot {x}}{\ddot {x}}}}&{\mathbf {P} _{{\ddot {x}}{\dot {z}}_{1}}}\\{\mathbf {P} _{{\dot {z}}_{1}{\dot {x}}}}&{\mathbf {P} _{{\dot {z}}_{1}{\ddot {x}}}}&{\mathbf {P} _{{\dot {z}}_{1}{\dot {z}}_{1}}}\end{pmatrix}}}$

The variances for structural displacements, structural speeds and structural accelerations are the spectral moments of the structural displacements. Even higher spectral moments can be calculated to determine the internal forces and stresses of the structure. The spectral moments form the basis for various verification methods of the stochastic structure analysis, for example for the calculation of the damage during the fatigue verification. Many analytical methods for determining the number of cycles (Rice, Dirlik) use the spectral moments.

Further methods of stochastic dynamic structure analysis

Time domain integration

This method (also called time history method after the English name ) is based on an integration of the equation of motion . Like any dynamic process, stochastic loads can also be calculated in this way. In order to obtain statistically meaningful results, however, a long time series must be analyzed, which makes this method computationally intensive and time-consuming.

PSD method

This method works in the frequency domain. The basis is the spectral power density or power spectral density (PSD) of the load. The load time series must therefore be converted beforehand. The power spectral density of the structural response (displacement, internal force, stress) is calculated using the frequency response of the system that describes the structure. This method is significantly faster and statistically more meaningful than a time domain integration. However, the result is only the spectrum of the structural response of one degree of freedom. If there are several answer functions, the computational effort increases accordingly; you have to know in advance which point on the structure is important.

See also

• Covariance matrix