Dynamic loads applied to a discretized mechanical model can be
decomposed into a product {F}q= [b] {u(t)} where
the input shape matrix [b] is time invariant and
characterizes spatial properties of the applied forces and
the vector of inputs {u} allows the description of the
time/frequency properties.
Similarly it is assumed that the outputs {y}
(displacements but also strains, stresses, etc.) are linearly related
to the model coordinates {q} through the sensor output shape
matrix ({y} = [c] {q}).
Input and output shape matrices are typically generated with fe_c or
fe_load. Understanding what they represent and how they are
transformed when model DOFs/states are changed is essential.
Linear mechanical models take the general forms
[Ms2+Cs+K]N × N {q(s)} = [b]N × NA {u(s)}NA × 1
{y(s)}NS × 1 = [c]NS × N {q(s)}N × 1
(2.1)
in the frequency domain (with Z(s)=Ms2+Cs+K), and
[M] {
..
q
} + [C] {
.
q
} + [K] {q} = [b]{u(t)}
{y(t)}= [c] {q(t)}
(2.2)
in the time domain.
In the model form (2.1), the first set of equations describes
the evolution of {q}. The components of q are called Degrees Of Freedom (DOFs) by
mechanical engineers and states in control theory. The second
observation equation is rarely considered by mechanical
engineers (hopefully the SDT may change this). The purpose of
this distinction is to lead to the block diagram representation of the
structural dynamics
which is very useful for applications in both control and mechanics.
In the simplest case of a point force input at a DOF ql, the input shape
matrix is equal to zero except for DOF l where it takes the value 1
[bl] = [
·
·
·
0
1
0
·
·
·
]
¬ l
(2.3)
Since {ql} = [bl]T {q}, the transpose this Boolean input
shape matrix is often called a localization matrix. Boolean input/output
shape matrices are easily generated by fe_c (see the section on
DOF selection ).
Input/output shape matrices become really useful when not Boolean. For applications considered in the SDT they are key to
test analysis correlation. Since you often have measurements that do not
directly correspond to DOFs (accelerations in non global directions at
positions that do not correspond to finite element nodes, see
section 3.2.1).
model reduction. To allow the changes to the
DOFs q while retaining the physical meaning of the I/O relation
between {u} and {y} (see section 6).
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