Filter (structural analysis)

In structural dynamics, a model is referred to as a filter that is represented in the state space. This representation of a filter as a system is also common in electrical engineering for describing vibration systems.

State space representation

The state space representation consists of a control equation

${\displaystyle {\dot {z}}(t)=A\cdot z(t)+B\cdot u(t)}$

and an observation equation

${\displaystyle y(t)=C\cdot z(t)+D\cdot u(t)}$

Here is the state vector of the system , corresponds to the input vector and the output vector. The matrices and are the matrices of the filter. The system matrix describes the dynamics of the process, it is square; the other matrices usually do not. is called the input or control matrix; is the output or observation matrix and is the passage matrix. ${\displaystyle z(t)}$ ${\displaystyle S\left({A,B,C,D}\right)}$ ${\displaystyle u(t)}$ ${\displaystyle y(t)}$ ${\displaystyle A,B,C}$ ${\displaystyle D}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle D}$

Shape filter (load filter)

A shape filter is the system representation of a stochastic dynamic load. Gaussian white noise w (t) is assumed at the system input . The required load vector f (t) results at the system output :

${\displaystyle {\dot {\mathbf {z} }}_{1}(t)=\mathbf {A} _{1}\cdot \mathbf {z} _{1}(t)+\mathbf {B} _{1}\cdot \mathbf {w} (t)}$

${\displaystyle \mathbf {f} (t)=\mathbf {C} _{1}\cdot \mathbf {z} _{1}(t)+\mathbf {D} _{1}\cdot \mathbf {w} (t)}$

The system status variable has no physical meaning. The system matrices are unknown and must be determined. This process is known as identification . ${\displaystyle \mathbf {z} _{1}(t)}$ ${\displaystyle \mathbf {A} _{1},\mathbf {B} _{1},\mathbf {C} _{1},\mathbf {D} _{1}}$

The shape filter shown above is a continuous filter made up of differential equations. To identify the system matrices, it is better to choose a discrete filter display:

${\displaystyle \mathbf {z} _{1d<k+1>}=\mathbf {A} _{1d}\cdot \mathbf {z} _{1d<k>}+\mathbf {B} _{1d}\cdot \mathbf {w} _{d<k>}}$

${\displaystyle \mathbf {f} _{d<k>}=\mathbf {C} _{1d}\cdot \mathbf {z} _{1d<k>}+\mathbf {D} _{1d}\cdot \mathbf {w} _{d<k>}}$

The matrices of the discrete system are different from those of the continuous system. However, they can be converted using a discrete-continuous transformation.

The order of the system is denoted by n, the number of load channels (output channels) by p and the number of input channels by q. Then has the size (nxn), the size (nxq), the size (pxn) and the size (pxq). ${\displaystyle \mathbf {A} _{1}}$ ${\displaystyle \mathbf {B} _{1}}$ ${\displaystyle \mathbf {C} _{1}}$ ${\displaystyle \mathbf {D} _{1}}$

Structural filter (structure filter)

If you apply the system representation to the structural calculation, the equation of motion of a dynamically loaded structure is the basis:

${\displaystyle \mathbf {M} \cdot {\ddot {\mathbf {x} }}(t)+\mathbf {D} \cdot {\dot {\mathbf {x} }}(t)+\mathbf {K} \cdot \mathbf {x} (t)=\mathbf {T} \cdot \mathbf {f} (t)}$

Here T is a transformation matrix that connects the structural degrees of freedom with the channels of the load f (t). M , D and K are the mass, damping and stiffness matrix of the structure. The system state variable x (t) corresponds to the displacement quantities of the structure.

By simply transforming and adding the identity equation

${\displaystyle {\dot {\mathbf {x} }}\left(t\right)={\dot {\mathbf {x} }}\left(t\right)}$

one obtains the state space representation and thus a filter for the mechanical system of the structural reaction:

${\displaystyle {\begin{pmatrix}\mathbf {\dot {x}} (t)\\{\ddot {\mathbf {x} }}(t)\end{pmatrix}}={\begin{pmatrix}\mathbf {0} &\mathbf {I} \\{-\mathbf {M} ^{-1}\mathbf {K} }&{-\mathbf {M} ^{-1}\mathbf {D} }\end{pmatrix}}\cdot {\begin{pmatrix}\mathbf {x} (t)\\{\dot {\mathbf {x} }}(t)\end{pmatrix}}+{\begin{pmatrix}\mathbf {0} \\{-\mathbf {M} ^{-1}\mathbf {T} }\end{pmatrix}}\cdot \mathbf {f} (t)}$

By introducing filter matrices and one obtains ${\displaystyle \mathbf {A} _{2}}$ ${\displaystyle \mathbf {B} _{2}}$

${\displaystyle {\dot {\mathbf {z} }}_{2}\left(t\right)=\mathbf {A} _{2}\cdot \mathbf {z} _{2}\left(t\right)+\mathbf {B} _{2}\cdot \mathbf {f} \left(t\right)}$

With

${\displaystyle \mathbf {z} _{2}(t)={\begin{pmatrix}\mathbf {x} (t)\\{\dot {\mathbf {x} }}(t)\end{pmatrix}}}$

These filter matrices and are given directly by the supporting structure and can be determined using a finite element program. If you add an observation equation ${\displaystyle \mathbf {A} _{2}}$ ${\displaystyle \mathbf {B} _{2}}$

${\displaystyle \mathbf {z} _{2}\left(t\right)=\mathbf {C} _{2}\cdot \mathbf {z} _{2}\left(t\right)+\mathbf {D} _{2}\cdot \mathbf {f} \left(t\right)}$

with and , you get a system for the structural filter. This filter is a continuous filter made up of differential equations. ${\displaystyle \mathbf {C} _{2}=\mathbf {1} }$ ${\displaystyle \mathbf {D} _{2}=\mathbf {0} }$ ${\displaystyle \mathbf {S} _{2}\left({\mathbf {A} _{2},\mathbf {B} _{2},\mathbf {C} _{2},\mathbf {D} _{2}}\right)}$

Combination of load filter and structure filter to form an overall filter

The filter equations of the load filter and structure filter can be combined to form an overall filter . The link is the load vector . This results in a filter that describes both the load and the structure and has Gaussian white noise as an input signal: ${\displaystyle \mathbf {S} _{1}\left({\mathbf {A} _{1},\mathbf {B} _{1},\mathbf {C} _{1},\mathbf {D} _{1}}\right)}$ ${\displaystyle \mathbf {S} _{2}\left({\mathbf {A} _{2},\mathbf {B} _{2},\mathbf {C} _{2},\mathbf {D} _{2}}\right)}$ ${\displaystyle \mathbf {f} (t)}$ ${\displaystyle \mathbf {w} (t)}$

${\displaystyle {\begin{pmatrix}{\dot {\mathbf {x} }}(t)\\{\ddot {\mathbf {x} }}(t)\\{\dot {\mathbf {z} }}_{1}(t)\end{pmatrix}}={\begin{pmatrix}\mathbf {0} &\mathbf {I} &\mathbf {0} \\{-\mathbf {M} ^{-1}\mathbf {K} }&{-\mathbf {M} ^{-1}\mathbf {D} }&{\mathbf {M} ^{-1}\mathbf {T} \mathbf {C} _{1}}\\\mathbf {0} &\mathbf {0} &{\mathbf {A} _{1}}\end{pmatrix}}\cdot {\begin{pmatrix}\mathbf {x} (t)\\{\dot {\mathbf {x} }}(t)\\\mathbf {z} _{1}(t)\end{pmatrix}}+{\begin{pmatrix}\mathbf {0} \\\mathbf {0} \\{\mathbf {B} _{1}}\end{pmatrix}}\cdot \mathbf {w} (t)}$

${\displaystyle \Leftrightarrow {\begin{pmatrix}{\dot {\mathbf {z} }}_{2}(t)\\{\dot {\mathbf {z} }}_{1}(t)\end{pmatrix}}={\begin{pmatrix}\mathbf {A} _{2}&{\mathbf {B} _{2}\mathbf {C} _{1}}\\\mathbf {0} &\mathbf {A} _{1}\end{pmatrix}}\cdot {\begin{pmatrix}\mathbf {z} _{2}(t)\\\mathbf {z} _{1}(t)\end{pmatrix}}+{\begin{pmatrix}\mathbf {0} \\\mathbf {B} _{1}\end{pmatrix}}\cdot \mathbf {f} (t)}$

${\displaystyle \Leftrightarrow {\dot {\mathbf {z} }}_{G}(t)=\mathbf {A} _{G}\cdot \mathbf {z} _{G}(t)+\mathbf {B} _{G}\cdot \mathbf {w} (t)}$

This filter is used in the analysis of covariance in structural dynamics to investigate problems with stochastic loads in order to obtain statistical parameters (variances and covariances) of the structural response.