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4.1  Input/Output shape matrices

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
• 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, fe_case or fe_load. For sensors see also scell. Understanding what they represent and how they are transformed when model DOFs/states are changed is essential.

Linear mechanical models take the general forms

(1)

in the frequency domain, and

(2)

in the time domain.
N is the number of degrees of freedom in the model, NA is the number of independent actuators, and NS is the number of sensors.

In the model form (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 q_l, the input shape matrix is equal to zero except for DOF l where it takes the value 1

(3)

Since {q_l} = [b_l]^T {q}, the transpose of this Boolean input shape matrix is often called a localization matrix. Boolean input/output shape matrices are easily generated by fe_c Input/output shape matrices become really useful when not Boolean. For applications considered in the SDT they are key to

• distributed FEM loads, see fe_load.
• the description of piezoelectric loads and sensors, see p_piezo.
• the description of sensors that do not directly correspond to DOFs (accelerations in non global directions at positions that do not correspond to finite element nodes).
• model reduction. To allow the changes to the DOFs q while retaining the physical meaning of the I/O relation between {u} and {y}.