Response spectrum analysis

Response spectrum analysis, also called shock spectrum analysis, is an approximate method for predicting the peak response of a structure due to base excitations (enforced motion). Instead of performing a time-consuming transient analysis, you can use response spectrum analysis to quickly approximate the peak response (for example, von Mises stresses) of a structure in a long time history. Response spectrum analysis can thus be used as a design tool.

Using a series of single degree of freedom (SDOF) oscillators, a response spectrum of the base motion can be generated from the transient motion for each given damping ratio:

	Equation 1

where:

	= the displacement spectrum functions of the peak values
	= the velocity spectrum functions of the peak values
	= the acceleration spectrum functions of the peak values
	= a given damping ratio
ω	=
f	= a given frequency
	= the transient base excitation

In Response Dynamics, you can apply the response spectrum generated from SDOF oscillators to the structure as an enforced motion excitation to predict the peak responses using the mode participation factors (MPF), which are defined as:

	Equation 2

where:

	= normal mode
	= constraint mode
	= partitions of the modal mass matrix
	= modal mass
	= associated with normal mode and excitation DOF, respectively

Let

be the value of the s-th excitation (input) response spectrum at the natural frequency of the

mode. The modal peak of each excitation can be calculated by:

	Equation 3

where:

p	= peak value
s	= excitation DOF
	= modal DOF

The physical peak response from the

excitation can then be calculated by:

	Equation 4

Response Dynamics offers five commonly used rules to compute the peak values by combining the peak response (Equation 4) of each mode and each excitation:

• ABS

• SRSS

• NRL

• CQA

• NRC

Absolute Value rule (ABS)

The Absolute Value rule (ABS) assumes a worst case scenario in which all modal peak responses occur at the same time with all modes in phase.

	Equation 5a

Square Root of the Sum of the Squares rule (SRSS)

The Square Root of the Sum of the Squares rule (SRSS) assumes that the modal responses are uncorrelated and that the peak response for each mode will occur at different times. The SRSS results are therefore optimistic and determine a lower bound of the peak dynamic response.

	Equation 5b

Naval Research Lab method (NRL)

The Naval Research Lab (NRL) method is a combination of the ABS and SRSS methods, and is therefore a compromise between a worst case and an optimistic analysis.

	Equation 5c

When you select the NRL method, the software first identifies the largest single model peak of the response spectrum at

(the mode).

Then, using a set of closely spaced modes

inside the frequency range

the software identifies neighbor modes where:

	= the natural frequency of the -th mode, which is at the peak response
	= NRL Neighboring factor, which you define
	= the natural frequency of the 1-th mode, which is the lower end of the neighboring frequency range
	= the natural frequency of the 2-th mode, which is the upper end of the neighboring frequency range

If you set the NRL Neighboring factor (C) to 1.0, no neighboring modes are used to compute the peak modal response.

If you set the NRL Neighboring factor to 1.1, and a peak modal response is at 100 Hz, modes in the range of 90.9 to 110 Hz (that is, +/- 10 percent of the peak modal frequency) are included in the peak modal response calculation.

The larger the NRL Neighboring factor, the larger the range from which neighboring modes are selected. The lower the factor, the smaller the range from which neighboring modes are selected.

For more information, see Theoretical Background of Shock and Response Spectrum Analysis in the Simcenter Nastran Advanced Dynamics User's Guide.

Complete Quadratic Combination rule (CQC)

To combine responses to multiple excitations using the CQC rule, you can select either ABS or SRSS as the combination method:

ABS Combination Method

SRSS Combination Method

where:

Nuclear Regulatory Commission rule (NRC)

The NRC rule is used for modal models with closely spaced natural frequencies. Response Dynamics uses the double-sum method for NRC. With the NRC rule, you can further select ABS or SRSS as the combination method for multiple excitations:

ABS Combination Method

SRSS Combination Method

where:

t = duration of excitation

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