# Modal Time History Analysis

Use modal time history analysis when the variation of each load with time is known explicitly, and you are interested in the response as a function of time.

• General time-varying loads (periodic or non-periodic)
• Uniform base motion (displacement, velocity, or acceleration applied to all supports)
• Support motions (displacement, velocity, or acceleration applied to selected supports non-uniformly)
• Initial conditions (a finite displacement, velocity, or acceleration applied to a part or the whole model at time t =0)

The solution of the equations of motion for multi degree-of-freedom systems incorporates Modal analysis techniques.

The solution's accuracy can improve by using a smaller time step.

After running the study, you can view displacements, stresses, strains, reaction forces, etc. at different time steps, or you can graph results at specified locations versus time. If no locations are specified in Result Options, results at all nodes are saved.

Modal, Rayleigh, composite modal, and concentrated dampers are available for modal time history analysis.

## Analysis Procedure - Modal Time History

The system of equations of motion of a linear n-degree-of-freedom system excited by a time varying force is:

(Equation 1)

where:

[M] = n x n symmetric inertia matrix

[C] = n x n symmetric damping matrix

[K] = n x n symmetric stiffness matrix

{f(t)} = n-dimensional force vector

{u}, , and are the displacement, velocity, and acceleration n-dimensional vectors, respectively.

(Equation 1) is a system of n simultaneous ordinary differential equations with constant coefficients. The equations of motion are coupled through the mass, stiffness, and damping terms. Coupling depends on the coordinate system used to describe the equations of motion mathematically.

The basic idea behind modal analysis is to transform the coupled system of (Equation 1) into a set of independent equations by using the modal matrix [Φ] as a transformation matrix. [Φ] contains the normal modes {f}i for i = 1, ....,n arranged as:

(Equation 2)

The normal modes and eigenvalues of the system are derived from the solution of the eigenvalue problem:

(Equation 3)

where [ω2] is a diagonal matrix of the natural frequencies squared.

For linear systems, the system of n equations of motion can be de-coupled into n single-degree-of-freedom equations in terms of the modal displacement vector {x}:

(Equation 4)

Substituting vector {u} from (Eq.4) and pre-multiplying it by [Φ]T (Equation 1) yields:

(Equation 5)

The normal modes satisfy the orthogonality property, and the modal matrix [Φ] is normalized to satisfy the following equations:

(Equation 6)

(Equation 7), and

(Equation 8).

By substituting (Equations.6--8), (Equation 5) becomes a system of n independent SDOF second-order differential equations:

for i =1, ..., n (Equation 9)

(Equation 9) is solved by using step-by-step integration methods like Wilson-Theta, and Newmark.

The integration is performed in the time domain, where the results of the last step are used to predict those of the next one.

The system's displacement vector (u) is derived from (Equation 4).

## Modal Time History Analysis - Advanced Options

The Advanced tab in the Modal Time History dialog box sets the numerical integration method and its parameters.

 Newmark The uncoupled equations of motion are solved with the Newmark time-stepping method. For a linear variation of acceleration between time-steps, select: First integration parameter a =0.5 Second integration parameter beta = 1/6 For a constant acceleration between time-steps, select: a = 0.5 and beta =0.25. Wilson-Theta The Wilson-Theta method of integration is used to solve the uncoupled equations of motion. Theta. The value of theta controls the numerical stability For theta = 1, the solution formulation is similar to the Newmark's linear acceleration method. For theta greater or equal to 1.37, the Wilson's method is unconditionally stable.