In structural analysis, enforced motion is a common excitation. Examples of this behavior include the response of a building to an earthquake, the vibration of a device carried by a vehicle, etc.
The equations of motion in terms of absolute displacements can be expressed by:
 | (14–370) |
where:
= structural mass matrix
= structural damping matrix
= structural stiffness matrix
= nodal acceleration vector
= nodal velocity vector
= nodal displacement vector
= applied load vector
Partition the degrees of freedom into two sets:
 | (14–371) |
where:
= displacements remaining free
= displacements with enforced motion
Assuming that the only excitation source is the enforced motion, the load vector applied on degree of freedom {u1} is zero. Equation 14–370 can be expanded to:
 | (14–372) |
Where {F2} = the reaction force between the structure and its supports.
In transient and harmonic analyses using the full method (TRNOPT,FULL and HROPT,FULL), the upper part of Equation 14–372 is rearranged and solved as:
 | (14–373) |
where
 is the
enforced acceleration. In a full transient analysis, it is input with
the D command and Lab = ACCX, ACCY, ACCZ, DMGX, DMGY, or DMGZ (not supported in harmonic). |
 is the
enforced velocity. In a full transient analysis, it is input with
the D command and Lab = VELX, VELY, VELZ, OMGX, OMGY, or OMGZ (not supported in harmonic). |
 is the enforced displacement. In full transient
and harmonic analyses, it is input with the D command
and Lab = UX, UY, UZ, ROTX, ROTY, or ROTZ. |
The solution of Equation 14–373 is the
absolute displacement vector
.
The nodal displacement due to enforced motion can be separated into the quasi-static response and the dynamic response, also called relative motion (see Paultre [418]):
 | (14–374) |
where:
{y} is the dynamic response, representing the relative motion of the structure with respect to the base motion.
is the quasi-static
response.
By introducing Equation 14–374 into Equation 14–372, and neglecting time-derivative terms, the following equation is obtained:
 | (14–375) |
From the upper part of Equation 14–375,
is derived as:
 | (14–376) |
The absolute displacement can be written as:
 | (14–377) |
Substituting Equation 14–376 and Equation 14–377 into Equation 14–373 gives:
 | (14–378) |
Neglecting the enforced velocity term on the right hand side, the equation reduces to:
 | (14–379) |
This equation is exact in the case of stiffness-based proportional damping (BETAD) and results will match those obtained with a full method using Equation 14–373. It is approximate for all other cases of damping.
When a structure is subjected to excitations from different
supports and/or along different axes, Equation 14–379 can be solved using the mode-superposition method (TRNOPT,MSUP and HROPT,MSUP). In that case, the degrees of freedom
are first
fixed for the modal analysis. The natural frequencies and mode shapes
of the conservative system without damping, derived from Equation 14–379 are used to obtain the uncoupled equations,
as explained in Mode-Superposition Method in this guide.
If different supports are excited, the vector of displacements
from enforced
motion and the associated accelerations
are composed of subsets
with degrees of freedom corresponding to the different supports. Each
subset, also called base, is identified using the Value argument on the D command during the modal analysis,
and for all the degrees of freedom of a subset, the common value of
enforced displacement
or enforced acceleration
is input using the DVAL command in the mode-superposition transient and harmonic
analyses.
In a harmonic analysis, depending on whether displacements or accelerations are input with the DVAL command, the following relationship is used internally:
where Ω is the current forced circular frequency.
This equation defines the acceleration of supports in Equation 14–379 when displacements are specified and defines displacements of supports in Equation 14–376 when accelerations are specified.
In a transient analysis, the Newmark time integration method described in Transient Analysis is internally used to calculate the enforced displacements when enforced accelerations are specified. Conversely, the same method is used to calculate the enforced accelerations when enforced displacements are specified.
When KeyCal is ON in the DVAL command, the final displacement vector
is calculated
with Equation 14–374 and displacements are absolute.
When KeyCal is OFF in the DVAL command, the final displacement vector is {y} and
displacements are relative.
Stresses from the quasi-static solution are not zero and may not be negligible in either of the following scenarios:
If excitations act in different directions and/or intensity.
If the quasi-static solution is calculated in the second phase of a linear perturbation modal analysis, even if only one support is excited, because prestressing a structure modifies the rigid body motions.
Therefore, stresses obtained with KeyCal set to ON are
different from those obtained with KeyCal set to OFF.
For more information, see Enforced Motion Method for Mode-Superposition Transient and Harmonic Analyses in the Structural Analysis Guide.
When a structure is subjected to global motion of a rigid support (see Geradin and Rixen [365]), reaction force vector {F2} is zero and the quasi-static response defined in Equation 14–376 can be rewritten as:
 | (14–380) |
where:
| [r1] represents the rigid-body modes of the part of the structure remaining free |
 = the support motion, the acceleration of which
is: |
 | (14–381) |
where:
| γ = the global acceleration of the support, input on the ACEL command. |
In this case, the coupled mass between the free and constrained degrees of freedom ([M12]) is ignored in Equation 14–379 to obtain:
 | (14–382) |
To solve Equation 14–382, degrees of freedom
must be
fixed and the full method transient and harmonic analyses are used.
The solution is the relative displacement vector {y}.
As stresses of the quasi-static solution are zero, the stresses obtained with a full method working with absolute displacements (D) or relative displacements (ACEL) are identical.
The relative solution obtained with Equation 14–382 matches the relative results obtained with the enforced motion method (DVAL,,ACC,,OFF). However, the static element nodal forces and the static reaction forces do not match as there is no external inertia load in the enforced motion method.
The large mass method is an approximate technique that treats the response to acceleration excitation as a response to external forces.
Assuming that the masses associated with subsystem are augmented
so that [M22 ] becomes
, then Equation 14–372 becomes:
 | (14–383) |
Solve the equations of
:
 | (14–384) |
Define
and substitute Equation 14–384 into
the upper part of Equation 14–383 to get:
 | (14–385) |
where:
 | (14–386) |
If
is large enough,
tends to zero, and the following
approximation is verified:
 | (14–387) |
Equation 14–385 will be in the same form as Equation 14–373, and Equation 14–383 is equivalent to Equation 14–372.
A ratio of the large mass to the mass of the entire structure in the range of 104 to 108 keeps the modeling error small (see Léger et al. [419]).
Large lumped masses can be implemented by using element type MASS21.
The solution of Equation 14–385 is the absolute
displacement vector
.