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CHAPTER 23

CONCEPTS IN

SHOCK DATA ANALYSIS

Sheldon Rubin

INTRODUCTION

This chapter discusses the interpretation of shock measurements and the reduction

of data to a form adapted to further engineering use. Methods of data reduction also

are discussed. A shock measurement is a trace giving the value of a shock parameter

versus time over the duration of the shock, referred to hereafter as a time-history.

The shock parameter may define a motion (such as displacement, velocity, or accel-

eration) or a load (such as force, pressure, stress, or torque). It is assumed that any

corrections that should be applied to eliminate distortions resulting from the instru-

mentation have been made. The trace may be a pulse or transient. Concepts in vibra-

tion data analysis are discussed in Chap. 22.

Examples of sources of shock to which this discussion applies are earthquakes

(see Chap. 24), free-fall impacts, collisions, explosions, gunfire, projectile impacts,

high-speed fluid entry, aircraft landing and braking loads, and spacecraft launch and

staging loads.

BASIC CONSIDERATIONS

Often, a shock measurement in the form of a time-history of a motion or loading

parameter is not useful directly for engineering purposes. Reduction to a different

form is then necessary, the type of data reduction employed depending upon the

ultimate use of the data.

Comparison of Measured Results with Theoretical Prediction. The correlation

of experimentally determined and theoretically predicted results by comparison of

records of time-histories is difficult. Generally, it is impractical in theoretical analy-

ses to give consideration to all the effects which may influence the experimentally

obtained results. For example, the measured shock often includes the vibrational

response of the structure to which the shock-measuring device is attached. Such

vibration obscures the determination of the shock input for which an applicable the-

ory is being tested; thus, data reduction is useful in minimizing or eliminating the

23.1

23.2

CHAPTER TWENTY-THREE

irrelevancies of the measured data to permit ready comparison of theory with cor-

responding aspects of the experiment. It often is impossible to make such compar-

isons on the basis of original time-histories.

Calculation of Structural Response. In the design of equipment to withstand

shock, the required strength of the equipment is indicated by its response to the

shock. The response may be measured in terms of the deflection of a member of the

equipment relative to another member or by the magnitude of the dynamic loads

imposed upon the equipment. The structural response can be calculated from the

time-history by known means; however, certain techniques of data reduction result

in descriptions of the shock that are related directly to structural response.

As a design procedure it is convenient to represent the equipment by an appro-

priate model that is better adapted to analysis (see Chap. 41). A typical model is

shown in Fig. 23.1; it consists of a secondary structure supported by a primary

structure. Each structure is represented as a lumped-parameter single degree-of-

freedom system with the secondary mass m much smaller than the primary mass M

so that the response of the primary mass is unaffected by the response of the sec-

ondary mass. The response of the primary mass to an input shock motion is the

input shock motion to the secondary structure. Depending upon the ultimate

objective of the design work, certain characteristics of the response of the model

must be known:

1. If design of the secondary structure is to be effected, it is necessary to know the

time-history of the motion of the primary structure. Such motion constitutes the

excitation for the secondary structure.

2. In the design of the primary structure, it is necessary to know the deflection of

such structure as a result of the shock, either the time-history or the maximum

value.

By selection of suitable data reduction methods, response information useful in

the design of the equipment is obtained from the original time-history.

FIGURE 23.1 Commonly used structural model consisting of a primary and a

secondary structure.

Laboratory Simulation of Measured Shock. Because of the difficulty of using

analytical methods in the design of equipment to withstand shock, it is common

practice to prove the design of equipments by laboratory tests that simulate the

anticipated actual shock conditions. Unless the shock can be defined by one of a

23.3

CONCEPTS IN SHOCK DATA ANALYSIS

few simple functions, it is not feasible to reproduce in the laboratory the complete

time-history of the actual shock experienced in service. Instead, the objective is to

synthesize a shock having the characteristics and severity considered significant in

causing damage to equipment. Then, the data reduction method is selected so that

it extracts from the original time-history the parameters that are useful in specify-

ing an appropriate laboratory shock test. Shock testing machines are discussed in

Chap. 26.

EXAMPLES OF SHOCK MOTIONS

Five examples of shock motions are illustrated in Fig. 23.2 to show typical character-

istics and to aid in the comparison of the various techniques of data reduction. The

acceleration impulse and the acceleration step are the classical limiting cases of

shock motions. The half-sine pulse of acceleration, the decaying sinusoidal accelera-

tion, and the complex oscillatory-type motion typify shock motions encountered fre-

quently in practice.

In selecting data reduction methods to be used in a particular circumstance, the

applicable physical conditions must be considered. The original record, usually a

time-history, may indicate any of several physical parameters; e.g., acceleration,

force, velocity, or pressure. Data reduction methods discussed in subsequent sections

of this chapter are applicable to a time-history of any parameter. For purposes of

illustration in the following examples, the primary time-history is that of accelera-

tion; time-histories of velocity and displacement are derived therefrom by integra-

tion. These examples are included to show characteristic features of typical shock

motions and to demonstrate data reduction methods.

ACCELERATION IMPULSE OR STEP VELOCITY

The delta function d ( t ) is defined mathematically as a function consisting of an infi-

nite ordinate (acceleration) occurring in a vanishingly small interval of abscissa

(time) at time t

0 such that the area under the curve is unity. An acceleration time-

history of this form is shown diagrammatically in Fig. 23.2 A. If the velocity and dis-

placement are zero at time t

0, the corresponding velocity time-history is the

velocity step and the corresponding displacement time-history is a line of constant

slope, as shown in the figure. The mathematical expressions describing these time

histories are

ü ( t ) = u 0 d ( t )

(23.1)

∞

where d ( t ) = 0 when t ≠ 0, d ( t ) =∞when t = 0, and

d ( t ) dt = 1. The acceleration can

−∞

be expressed alternatively as

ü ( t )

lim

→

u 0 /

]

(23.2)

where ü ( t ) = 0 when t < 0 and t > . The corresponding expressions for velocity and

displacement for the initial conditions u = u = 0 when t < 0 are

u ( t )

u 0

[ t

(23.3)

u ( t ) = u 0 t

[ t > 0]

(23.4)

23.4

CHAPTER TWENTY-THREE

FIGURE 23.2

Five examples of shock motions.

ACCELERATION STEP

The unit step function 1 ( t ) is defined mathematically as a function which has a value

of zero at time less than zero ( t < 0) and a value of unity at time greater than zero

( t > 0). The mathematical expression describing the acceleration step is

ü ( t )

ü 0 1 ( t )

(23.5)

23.5

CONCEPTS IN SHOCK DATA ANALYSIS

where 1 ( t )

0. An acceleration time-history of the unit

step function is shown in Fig. 23.2 B; the corresponding velocity and displacement

time-histories are also shown for the initial conditions u

1 for t

0 and 1 ( t )

0 for t

0 when t

u ( t ) = ü 0 t

[ t > 0]

(23.6)

u ( t )

1 ⁄ 2 ü 0 t 2

[ t

(23.7)

The unit step function is the time integral of the delta function:

1 ( t )

d ( t ) dt

[ t

(23.8)

−∞

HALF-SINE ACCELERATION

A half-sine pulse of acceleration of duration τ is shown in Fig. 23.2 C; the correspon-

ding velocity and displacement time-histories also are shown, for the initial condi-

tions u = u = 0 when t = 0. The applicable mathematical expressions are

π t

ü ( t )

ü 0 sin

<τ

]

(23.9)

ü ( t ) = 0

when t < 0

and t >τ

ü 0 τ

u ( t ) =

1 − cos

[0 < t <τ]

(23.10)

2 ü 0 τ

u ( t ) =

[ t >τ]

ü 0 τ

π t

u ( t )

−

sin

<τ

]

(23.11)

ü 0 τ

2 t

u ( t )

−

[ t

>τ

]

This example is typical of a class of shock motions in the form of acceleration pulses

not having infinite slopes.

DECAYING SINUSOIDAL ACCELERATION

A decaying sinusoidal trace of acceleration is shown in Fig. 23.2 D; the corresponding

time-histories of velocity and displacement also are shown for the initial conditions

˙ =− u 0 and u = 0 when t = 0. The applicable mathematical expression is

u 0 ω 1

ü ( t )

e −ζ 1 ω 1 t sin (

− ζ 1 2

ω 1 t

sin −1 (2

ζ 1

− ζ 1 2

))

[ t

(23.12)

− ζ 1 2

where ω 1 is the frequency of the vibration and ζ 1 is the fraction of critical damping

corresponding to the decrement of the decay. Corresponding expressions for veloc-

ity and displacement are

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