9.2 Types of Analysis

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If the applied loading on the structure is to change with time, the designer should make a decision on

whether or not a dynamic analysis is required. To be able to make this decision, information about the

loading and the natural frequencies of the structure are required. From the loading point of view, we

classify a general loading on a structure into one of four categories: steady, cyclic, transient, and random.

Figure 9.21 shows a schematic presentation of the four loading categories. Figure 9.21b shows two

types of cyclic loading, one is a simple harmonic loading with amplitude of oscillation Fo; period

T ¼ 2p; and frequency f ¼ 1=T: The second is a periodic loading with a period T: Using Fourier analysis,

any periodic function with a period, T; may be decomposed into a series of harmonic sines and cosines

with frequencies f1 ¼ 1=T; 2f1; 3f1; …; nf1: Figure 9.21c shows a transient load with a duration T:

Some forcing functions do not lend themselves to simple frequency or time specifications. Figure 9.21d

shows a typical time history of a forcing function of such a category that may be considered as random

excitation. Transient and cyclic forcing functions are normally specified as force vs. time or frequency,

respectively. On the other hand, random forcing functions are commonly specified as the magnitude of

the input acceleration squared vs. frequency. This input data is normally called a power spectral density

(PSD) input curve. Time history input may be used in the analysis of random excitation and the random

input may be treated as transient. This would normally require extensive computer resources and CPU

time due to the very large number of time steps that would be required to capture the peak response.

Random excitation may occur from random sources such as road undulation on vehicles, noise,

earthquakes, and seismic events on buildings, and wind and turbulent loading on airplanes. For practical

purposes, PSD curves have been compiled for various random events and are normally available in most

FE commercial, programs.

In addition to the load specification, the second factor affecting the decision on the type of analysis is

the natural frequency of the structure. Structures with mass and stiffness characteristics are capable of free

vibrations after removing the initial excitation on the structure. Depending on the initial conditions of

excitation, the structure may vibrate in one natural frequency or in a combination of more than one

Time

Force

Time

Fo

(a) Steady

Force

Fo

T = 2π

Harmonic with period = 2π Force

Time

T

Periodic with period = T

Force

Time

T

(c) Transient

Force

Time

(d) Random

(b) Cyclic

FIGURE 9.21 Various loading types.

9-20 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

natural frequency. In the latter case, the vibration will result in a complex periodic motion that may be

decomposed into a series of harmonic motions vibrating at the first natural frequency (fundamental one)

and its multiples. As indicated, these frequencies are called the natural or resonant frequencies of the

system and they depend on the mass and elastic or stiffness characteristics of the system. In other words,

the natural frequencies of a system are those frequencies that the system tends to vibrate under conditions

of free vibration. Theoretically, a continuous system or structure has infinite DoF and infinite natural

frequencies. From a practical modeling point of view, the structure will have a number of natural

frequencies equal to the number of DoF used to model the structure. The mode shapes give the relative

displacements of each point in the structure when it vibrates in one of the natural frequencies. Each

natural frequency has a corresponding mode shape. The first mode shape corresponding to the first

natural frequency of a structure represents the most flexible way in which the structure may deform or

vibrate and corresponds to the least strain energy level in all modes. It is important to note that, a

symmetric structure will have symmetric and antisymmetric mode shapes. This may be realized by

considering the mode shapes of a simple beam as shown in Figure 9.22 where all odd number modes are

symmetric and all even number modes are antisymmetric.

The information provided by the natural frequencies and mode shapes of a structure is vital in

understanding the behavior of the structure under general excitation. If a structure is excited in one of its

natural frequencies, then theoretical analysis of the structure response as an undamped system shows that

the amplitude of the resulting vibration response reaches infinity. In practice, all structures possess a

certain amount of damping that will limit the amplitude of vibration.

Determining the type of analysis required for a structure depends on the nature of the applied load and

the magnitude of the first natural frequency or the fundamental frequency of the structure. The two main

categories of analysis types are static and dynamic analyses. A steady load, i.e., a load that does not change

with time (Figure 9.21a), would require simple static analysis. If the load varies with time, it does not

necessarily mean that dynamic analysis needs to be performed. For example, if the loading is harmonic

with a frequency less than approximately one third of the first natural frequency of the structure then

static analysis will provide an accurate solution and we only need to solve the static problem for the peak

load values. (For this frequency range, static analysis may provide a maximum difference of 12.5% in the

response for undamped structures and for a forcing frequency less than one fourth of the natural

frequency, the difference is only 6.7%.) In this case, the load is normally called “quasi-static.” We may

apply the same rule to periodic loading after decomposing it into its harmonic components. In transient

loading, we should consider the time of application of the load or the “rise time.” If the longest natural

period corresponding to the first natural frequency of the structure is more than about twice the rise

time, the loading should be classified as shock or impact loading and transient dynamic analysis would be

required. If the longest natural period of a system is less than about one third of the rise time, it would be

sufficient to perform static analysis and consider the loading to be quasi-static. If the longest natural

period falls between the quasi-static and shock conditions then a transient dynamic analysis would be

required and the load would be classified as transient loading. If the loading cannot be categorized as

frequency dependent or time dependent, as for example the one shown schematically in Figure 9.21d,

then it should be considered random.

Mode-1

Mode-2 Mode-4

Mode-3

Simple Beam

FIGURE 9.22 First four mode shapes of a simple beam.

Finite Element Applications in Dynamics 9-21

© 2005 by Taylor & Francis Group, LLC

From the above discussion, the following categories of dynamic analysis may be classified (Mirovitch,

1980):

Modal and natural frequency analysis: This is normally performed before any other type of dynamic

analysis and will produce the natural frequencies and mode shapes of the structure. This information

is vital for understanding the dynamic behavior of the structure and provides data for decoupling the

dynamic equations in other analyses. One objective of this analysis is to make sure that the structure

is not operating at a frequency close to one of its natural frequencies. A comfortable range for an

operating frequency is three times higher than the nearest natural frequency from the lower side and

three times smaller than the nearest natural frequency from the higher side. If the operating

frequency is higher than the first natural frequency, then large vibration amplitudes or “shudder” will

be evident upon passing the natural frequency and certain startup procedures may have to be

devised.

Frequency response analysis: This type of analysis is performed if the loading on the structure is

harmonic or periodic. In periodic loading, the loading function is first decomposed into its sine and

cosine components using Fourier analysis. In this analysis, the response will be harmonic with the same

loading frequency and with a possible phase shift. The output of such analysis would be displacements,

velocities, and accelerations that may be used to calculate forces and stresses in the structure. The

displacement output of this analysis defines the deformed shape of the structure, which, in general, is

different from the structural mode shapes and may be used to calculate stresses and strains in the

structure. Assuming linear conditions, the response to multiple frequency inputs may be simply summed

up using the superposition technique.

Transient response analysis: This type of analysis is performed if the loading on the structure is classified

as transient or shock. There are two general approaches to solving the equation of motion in transient

analysis. One is the “direct integration approach” and the other is the “modal superposition approach.”

In the direct integration approach, the system equations of motion are integrated directly in the time

domain. The required number of time steps depends on the period and the assumptions specified for

displacement, velocities, and accelerations within the time step. This number may be quite large and the

solution to large size problems may become a computationally difficult task.

In the second method, the modal superposition approach, the equations of motion are first

transformed to modal generalized displacements. In order to perform this transformation, the mode

shapes of the structure should first be determined through a modal analysis. The transformation yields a

set of decoupled second-order differential equations for the system that are easier to solve. The basic

assumption in this approach is that the superposition of the structural response due to the first few lower

mode shapes adequately represents the dynamic response of the structure under general transient

loading. In practice, this means that only a fraction of the mode shapes of the structure are needed to

accurately represent the dynamic behavior. This approach will be generally less accurate than the direct

integration approach but will provide substantial savings in computation time for large size problems.

The output of transient analysis is time histories of displacements, velocities, and accelerations of the

system that may be used to calculate forces and stresses.

Random response analysis: If the loading on the structure cannot be classified as frequency or time

dependent, it is considered random. Problems with random loading may be solved using a “time history

approach” or a “power spectral density (PSD) approach.” The time history approach treats the random

input as a transient one and performs a step-by-step integration over the excitation period. This is

normally very costly and requires intensive use of computational resources. If the structure has multiple

random inputs in more than one direction, then the transient time history approach is more accurate and

may be the preferred approach to solve the problem. Structures with random inputs in more than one

direction may be analyzed by a modal superposition approach with special procedures to combine the

modal responses (see Section 9.5.4 for details). In the PSD approach, the input will be acceleration as a

function of frequency. This is normally specified as a discrete or continuous spectrum by providing values

of the ðG2=f Þ vs. f ; where G is the input acceleration and f is the frequency. This approach is most

appropriate for a single input in one direction.

9-22 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

As mentioned above, most programs will have standard time history inputs or PSD inputs to simulate

commonly used random excitations such as vibrations due to road irregularities, earthquakes and seismic

inputs, vibration due to wind loads for aerospace parts, and wind loads for wind tunnel tests.

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