The FE equations of the 20-node option of ROM144 are derived from the system of governing equations of a coupled electrostatic-structural system in modal coordinates (Equation 14–149 and Equation 14–150)

(13–220)

where:

K = stiffness matrix

D = damping matrix

M = mass matrix

F = force

I = electric current

The system of Equation 13–220 is similar to that of the TRANS126 - Electromechanical Transducer element with the difference that the structural DOFs are generalized coordinates (modal amplitudes) and the electrical DOFs are the electrode voltages of the multiple conductors of the electromechanical device.

The contribution to the ROM144 FE matrices and load vectors from the electrostatic domain is calculated based on the electrostatic co-energy W_el (Reduced Order Modeling of Coupled Domains).

The electrostatic forces are the first derivative of the co-energy with respect to the modal coordinates:

(13–221)

where:

Electrode charges are the first derivatives of the co-energy with respect to the conductor voltage:

(13–222)

where:

The corresponding electrode current I_i is calculated as a time-derivative of the electrode charge Q_i. Both, electrostatic forces and the electrode currents are stored in the Newton-Raphson restoring force vector.

The stiffness matrix terms for the electrostatic domain are computed as follows:

(13–223)

(13–224)

(13–225)

(13–226)

where:

The damping matrix terms for the electrostatic domain are calculated as follows:

(13–227)

(13–228)

(13–229)

There is no contribution to the mass matrix from the electrostatic domain.

The contribution to the FE matrices and load vectors from the structural domain is calculated based on the strain energy W_SENE (Reduced Order Modeling of Coupled Domains). The Newton-Raphson restoring force F, stiffness K, mass M, and damping matrix D are computed according to Equation 13–230 to Equation 13–233.

(13–230)

(13–231)

(13–232)

(13–233)

where:

For the 30-node option of ROM144, it is necessary to establish a self-consistent description of both modal coordinates and nodal displacements at master nodes (defined on the RMASTER command defining the generation pass) in order to connect ROM144 to other structural elements UX DOF or to apply nonzero structural displacement constraints or forces.

Modal coordinates q_i describe the amplitude of a global deflection state that affects the entire structure. On the other hand, a nodal displacement u_i is related to a special point of the structure and represents the true local deflection state.

Both modal and nodal descriptions can be transformed into each other. The relationship between modal coordinates q_j and nodal displacements u_i is given by:

(13–234)

where:

Similarly, nodal forces F_i can be transformed into modal forces f_j by:

(13–235)

where:

Both the displacement boundary conditions at master nodes u_i and attached elements create internal nodal forces F_i in the operating direction. The latter are additional unknowns in the total equation system, and can be viewed as Lagrange multipliers λ_i mapped to the UY DOF. Hence each master UX DOF requires two equations in the system FE equations in order to obtain a unique solution. This is illustrated on the example of a FE equation (stiffness matrix only) with 3 modal amplitude DOFs (q₁, q₂, q₃), 2 conductors (V₁, V₂), and 2 master UX DOFs (u₁, u₂):

(13–236)

Rows 6 and 7 of Equation 13–236 correspond to the modal and nodal displacement relationship of Equation 13–234, while column 6 and 7 - to nodal and modal force relationship (Equation 13–235). Rows and columns (8) and (9) correspond to the force-displacement relationship for the UX DOF at master nodes:

(13–237)

(13–238)

where is set to zero by the ROM144 element. These matrix coefficients represent the stiffness caused by other elements attached to the master node UX DOF of ROM144.

In the generation pass of the ROM tool, the ith mode contribution factors for each element load case j (Reduced Order Modeling of Coupled Domains) are calculated and stored in the ROM database file. In the Use Pass, the element loads can be scaled and superimposed in order to define special load situations such as acting gravity, external acceleration or a pressure difference. The corresponding modal forces for the jth load case (Equation 14–149) is:

(13–239)

where: