SECTION 8 RESPONSE METHOD
The two types of response analysis in linear dynamics are Transient response analysis (response in the time domain) Frequency response analysis (response in the forcing frequency domain) Transient analysis is intuitively the dynamic response that may be thought of in many typical situations, i.e. impulsive loading on a satellite during launch, response of an automobile to pot holes, earthquake response of a building. An alternative is to vary the loading as a function of frequency. This can simulate a shaker table or exciter where the input frequency is controlled and the response across a frequency domain is investigated. Input Input Output Output P(t) u(t) t (sec) P(f) u(f) f (Hz) t (sec) f (Hz) RESPONSE TYPES
Look at the simple spring mass system from the previous section on normal modes Run this model as a transient analysis and as a frequency response analysis for an overview of the two methods. It was previously found that the natural frequencies and eigenvectors are SPRING MASS SYSTEM
Problem 1 P(t)=1.0 sin 5.629 t Problem 2 P(t)=1.0 sin 10.875 t TRANSIENT ANALYSIS For the first problem, apply a force, P(t), to Grid Point 3. It varies sinusoidally, has a magnitude of 1.0, and a frequency of 0.896 Hz (an input at the first resonant frequency). For the second problem, again apply a force, P(t), to the same point. It varies sinusoidally, has a magnitude of 1.0, but this time a frequency of 1.731 Hz (an input at the second resonant frequency). Damping in both cases is 2% of critical
TRANSIENT ANALYSIS (Cont.) For the first problem, notice it takes around 33 secs of load input for the input load to reach an energy level where it balances the energy loss due to damping. From this point on the response is at steady state, and it can be seen the amplitude of Grid Point 2 and 3 varies over +/- .015 and +/- .011 units.
TRANSIENT ANALYSIS (Cont.) In the second case, notice a similar response, taking around 15 secs of load input for the input load to reach an energy level where it balances the energy loss due to damping From this point on the response is at steady state, and it can be seen the amplitude of Grid Point 2 and 3 varies over +/- .0012 and +/- .003 units.
The two sets of responses can be plotted as amplitude versus the driving frequency of loaded grid point The response of the structure at the two natural frequencies has converged, but it may be important to know what happens below 0.896 Hz, above 1.731 Hz and between these values. It is possible to do more transient analyses to obtain this information, or a frequency response analysis can be used as shown below. Displacement Frequency (Hz) 0.896 1.731 Grid 2 Grid 2 Grid 3 Grid 3 TRANSIENT ANALYSIS (Cont.)
f1 = 0.896 Hz f2 = 1.731 Hz u3 = 0.015 u2 = 0.011 u2 = 0.003 u3 = 0.001 FREQUENCY RESPONSE ANALYSIS Frequency Response analysis The applied load is a continuous function of frequency Calculates the response of the two grid/node points as a function of frequency at defined frequency response points The displacement values from the transient response analysis match those from the frequency response analysis Grid 3 Grid 2
It may be that one or other type of loading is defined in a specification and it may be required to carry out the respective response analysis However, both types of response may be useful in assessing a structure’s behavior in a loading environment For a transient analysis, it is necessary to assess the frequency content of the response, either by simple methods such as peak to peak estimate of the time period, or carrying out a Fourier analysis Whichever technique is used it is essential that the normal modes of the structure have been investigated first. This is a theme that will be discussed in later case studies and workshops. RESPONSE TYPES
In the previous sections it was seen that the response in the time domain can be defined using physical coordinates, xj(t), or in terms of modal coordinates, xi(t). It was also seen that the number of modal coordinates needed to define a response is typically a small fraction of the number of physical coordinates This motivates us to have an option to solve dynamics problems in the modal coordinate system in the interest of saving CPU time Extend the technique to responses as a function of frequency Thus all MSC.Nastran linear dynamic solutions have two versions. Direct - the solution is solved in terms of physical coordinates. Modal - the solution is solved in terms of modal coordinates. The steps in the modal solution sequences are Transform from physical to modal coordinates. Solve in modal coordinates Transform back to physical coordinates MODAL AND DIRECT METHODS
The Solution Sequences are Direct Transient response analysis - SOL 109 Frequency response analysis - SOL 108 Modal Transient response analysis - SOL 112 Frequency response analysis - SOL 111 MODAL AND DIRECT METHODS (Cont.)
MODAL AND DIRECT METHODS Further details on transient response and frequency response, using the modal or direct techniques, are covered in subsequent sections.