# Dynamic Analysis

A **modal analysis** is used to determine the fundamental frequencies of
vibration for a structure. This can be useful for isolating vibration problems
due to machinery, human activity or seismic events. A **response analysis**
is a simplified way to estimate seismic or similar dynamic effects using
modal-superposition. The **time-history** response of a structure is
most simply the response (motion or force) of the structure evaluated as a
function of time including inertial effects.

VisualAnalysis advanced provides **design-checks** for dynamic analysis results through Result Superposition Combinations.

## Find Dynamic Mode Shapes

Mode shapes are calculated automatically if enabled through the **Project Settings**, **Analysis** settings or within . Here you can decide whether or not to compute the mode
shapes when an analysis is run, and if so, how many mode shapes to extract. To
run a modal analysis you only need to have a stable and supported model with at
least one element.

You will need to consider how many mode shapes are necessary. Theoretically,
there is one mode shape for each degree of freedom in your model. Generally only
the first few in each direction are really important. Still, you may need to
generate many mode shapes to obtain a few in each direction. *VisualAnalysis*
extracts the lowest frequencies (largest periods) from the frequency spectrum.

In a dynamic analysis you need to model both the stiffness and the mass of the structure correctly. The modal analysis calculates undamped frequencies and mode shapes for the structure. Lumped mass properties are calculated in the process of a modal analysis. Lumped masses come from three sources:

1. Mass associated with the structural members and plates. This is included automatically and calculated based on material density and element sizes. This mass is lumped at the nodes, except for tapered members, where a consistent mass matrix is used.

2. Additional mass you include at each node. You should apply concentrated masses at the nodes to account for any mass not associated with the structural elements themselves. Rotational mass should be applied where there exist nonstructural entities that are affixed to the structure and have appreciable rotational inertia.

3. Additional mass you include through a static load case. The load case and global direction are specified through the

command.The sum of this mass for the structure can be included in a report. You should check this total to make sure that you have accounted for all the mass in the structure.

Reports generated for a dynamic modal analysis provide the nodal
displacements for each mode shape. These displacements are not "real" values but
rather shape displacements resulting from the modal solution. Mode shape values
in *VisualAnalysis*
reports are based on normalization to a unit mass matrix, in other words, the
value S(M_{i} x D_{i}^{2} ) = 1, where:

M_{i} = mass associated with degree of freedom i, D_{i} =
modal displacement associated with degree of freedom i. The sum is carried out for all degrees of freedom in the structure. Frequency, period, and modal participation factors (for each direction) are
shown in the title bar of each mode shape .

*VisualAnalysis* uses a Sparse Lanzcos procedure which has proven to be
very robust.

## Find Dynamic Response

Use the **Load Case Manager** to create a **Dynamic** **Response Case**. The load case does not use any loads, but relies on however many mode shapes have been calculated. You should look at a *Dynamic Analysis Summary* in the reporting system to determine if you have enough mode shapes included.

Modal participation factors are normally checked to see if enough mode shapes have been included in a response spectrum analysis. Building codes usually require a percentage (like 90%) of all the modal weight to be accounted for when performing a modal superposition analysis. Modal participation factors are calculated for all translational directions in the model. For a discussion of the effective modal weight calculation, see the following reference:

Response Spectrum Analysis is based on modal superposition. Results from a modal superposition analysis are all non-negative numbers. This includes displacements, forces, and stresses. You may select from the CQC (recommended) method, or the old SRSS method:

**CQC Method**

This commonly used method allows specification of a uniform modal damping factor. This method and the combination equation are outlined in the following reference:

*Mario Paz & *William Leigh * Structural Dynamics Theory and
Computation, Springer, 5th Ed. 2006ISBN: 978-1402076671.*

Symbol |
Definition |

U | Response (force, moment, translations, etc.) |

U_{IXX} |
Response in I^{th} mode, X
earthquake direction, X spectrum input |

U_{IXY} |
Response in I^{th} mode, X
earthquake direction, Y spectrum input |

… | … |

U_{IZZ} |
Response in I^{th} mode, Z
earthquake direction, Z spectrum input |

**SRSS Method (Square Root of the Sum of the Squares)**

All sums are sums of absolute values.

U_{IIX} = U_{IXX} + U_{IXY} + U_{IXZ}U_{IX}
= Ö (U1_{IX}^{2} + U2_{IX}^{2}
+ U3_{IX}^{2} + . . . + UN_{IX}^{2})N = Number
of Modes