12.9 Generation of Shock Using Shakers

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In about the mid-1950s, with the development of electrodynamic exciters for the realization

of vibration tests, the need for a realization of shocks on this facility was quickly felt.

This simulation on a shaker, when possible, indeed presents a certain number of advantages (Coty

and Sannier, 1966).

12.9.1 Principle Behind the Generation of a Simple Shape Signal versus Time

The objective is to carry out on the shaker a shock of simple shape (half-sine, triangle,

rectangle, etc.) of given amplitude and duration similar to that made on the classical shock

machines. This technique was mainly developed during the years 1955 to 1965 (Wells and Mauer,

1961).

The transfer function between the electric signal of the control applied to the coil and acceleration to

the input of the test item is not constant. It is thus necessary to calculate the signal of control according to

this transfer function and the signal to be realized.

FIGURE 12.47 Inclined plane impact tester (CONBUR

tester). (Source: Lalanne, Chocs Mecaniques, Hermes

Science Publications. With permission.)

12-44 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

The process is as follows (Favour et al., 1969; Magne, 1971):

* Measurement of the transfer function of the installation (including the fixture and the test item)

using a calibration signal; the measurement of the transfer function of the installation can be made

using a calibration signal of the shock type, random vibration or sometimes by fast swept sine

(Favour, 1974; Lalanne, 2002a).

* Calculation of the Fourier transform of the signal specified at the input of the test item.

* By division of this transform by the transfer function, calculation of the Fourier transform of the

signal of control.

* Calculation of the control signal vs. time by inverse transformation.

In all cases, the procedure consists of measurement and calculation of the signal of control to 2n dB

(212, 29, 26, and/or 23). The specified level is applied only after several adjustments on a lower level.

These adjustments are necessary because of the sensitivity of the transfer function to the amplitude of the

signal (nonlinearities). The development can be carried out using a dummy item representative of the

mass of the specimen. However, particularly if the mass of the test specimen is significant (with respect to

that of the moving element), it is definitely preferable to use the real test item or a model with dynamic

behavior very near to it.

If random vibration is used as the calibration signal, its root-mean-square (rms) value is calculated in

order to be lower than the amplitude of the shock (but not too distant in order to avoid the effects of any

nonlinearities). This type of signal can result in application to the test item of many substantial peaks of

acceleration compared with the shock itself.

12.9.2 Main Advantages

The realization of the shocks on shakers has very interesting advantages:

* Possibility of obtaining very diverse shocks shapes.

* Use of the same means for the tests with vibrations and shocks, without disassembly (saving time)

and with the same fixtures (Wells and Mauer, 1961; Hay and Oliva, 1963).

* Possibility of a better simulation of the real environment, in particular by direct reproduction of a

signal of measured acceleration (or of a given shock spectrum).

* Better reproducibility than on the traditional shock machines.

* Very easy realization of the test on two directions of an axis.

* No need to use a shock machine.

In practice, however, one is rather quickly limited by the possibilities of the exciters, which therefore do

not make it possible to generalize their use for shock simulation.

12.9.3 Pre- and Postshocks

12.9.3.1 Requirements

The velocity change, DV ¼

Ðt

0 x€ðtÞdt (t ¼ shock duration), associated with shocks of simple shape

(half-sine, rectangle, TPS, etc.) is different from zero. At the end of the shock, the velocity of

the table of the shaker must, however, be zero. It is thus necessary to devise a method to satisfy

this need.

12.9.3.2 Solutions

One way of bringing back the variation of velocity associated with the shock to zero can be the addition of

a negative acceleration to the principal signal so that the area under the pulse has the same value on the

Mechanical Shock 12-45

© 2005 by Taylor & Francis Group, LLC

side of positive accelerations and on the side of negative accelerations (Lalanne, 2002b). Various solutions

are possible a priori (Figure 12.48):

* A preshock alone.

* A postshock alone.

* Pre- and postshocks, possibly of equal durations.

Preshock alone 1 ; which requires a less powerful power amplifier, thus seems preferable to postshock

alone 3 : The use of symmetrical pre- and postshocks is however better, because of a certain number of

additional advantages (Magne and Leguay, 1972; Figure 12.49).

* The final displacement is minimal. If the specified shock is symmetrical (with respect to the

vertical line t=2), this residual displacement is zero (Young, 1964).

* For the same duration, t; of the specified

shock and for the same value of maximum

velocity, the possible maximum level of

acceleration is twice as big.

* The maximum force is provided at the

moment when acceleration is maximum,

that is, when the velocity is zero (one will

be able to thus have the maximum

current). The solution with symmetrical

pre- and postshocks requires minimal

electric power.

Another parameter is the shape of these pre- and

postshocks, the most used shapes being

the triangle, the half-sine, and the rectangle

(Figure 12.50).

1

Pre- shock alone Pre-and post - shocks Pre- shock alone

2 3

FIGURE 12.48 Possibilities for pre- and postshock positioning. (Source: Lalanne, Chocs Mecaniques, Hermes

Science Publications. With permission.)

Displacement

Displacement

Displacement

Acceleration

Velocity

Velocity Velocity 1

2

3

FIGURE 12.49 Kinematics of the movement with preshock alone 1 ; symmetrical pre and postshocks 2

and postshock alone 3 . (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

FIGURE 12.50 Shapes of pre- and postshock pulses.

(Source: Lalanne, Chocs Mecaniques, Hermes Science

Publications. With permission.)

12-46 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Owing to discontinuities at the ends of the pulse, the rectangular compensation is seldom satisfactory

(Smallwood, 1985). One often prefers a versed-sine applied to the entire signal (Hanning window),

which has the advantages of being zero and smoothed at the ends (first zero derivative) and presenting

symmetrical pre- and postshocks.

In all the cases, the amplitude of pre- and postshocks must remain small with respect to that of the

principal shock (preferably lower than approximately 10%), in order not to deform too much the

temporal signal and consequently, the shock spectrum. For a given shape of pre- and postshock, this

choice thus imposes the duration.

12.9.3.2.1 Optimized Pre- and Postpulses

Another method was developed (Fandrich, 1981) in order to take into account the tolerances on the

shape of the signal allowed by the standards (R.T. Fandrich refers to standard MIL-STD 810 C) and to

best use the possibilities of the shaker.

The solution suggested consists of defining the following:

1. One must define a preshock made up of the first two terms of the development in a Fourier series

of a rectangular pulse (with coefficients modified after a parametric analysis). The table being in

equilibrium in a median position before the test, the objective of this preshock is twofold:

* To give to a velocity, just before the principal shock, having a value close to one of the two limits

of the shaker so that, during the shock, the velocity can use the entire range of variation

permitted by the machine.

* To place, in the same way, the table as close as possible to one of the thrusts so that the moving

element can move during the shock in the entire space between the two thrusts (limitation in

displacement equal, according to the machines, to 2.54 or 5.08 cm).

2. One must also define a postshock composed of one period of a signal of the shape, Kty sinð2pf1tÞ;

where the constants K; y; and f1 are evaluated in order to cancel the acceleration, the velocity, and

the displacement at the end of the movement of the table.

The frequency and the exponent are selected in order to respect the ratio of the velocity to the

displacement at the end of the principal shock. The amplitude of the postshock is adjusted to obtain the

desired velocity change.

Figure 12.51 shows the total signal obtained in the case of a principal shock half-sine 30g, 11 msec.

This methodology has been improved to provide a more general solution (Lax, 2001).

Note: The realization of shocks on free or accelerated fall machines imposes de facto preshocks and/or

postshocks, the existence of which the user is not always aware, but that can modify the shock severity at

Time (ms)

ACCELERATION

PRE-SHOCK

POST-SHOCK

DISPLACEMENT

VELOCITY

Half-sine 30 g, 11ms

Acceleration, Velocity, Displacement

0 50 100 150 200 250

FIGURE 12.51 Overall movement in a half-sine shock. (Source: Lalanne, Chocs Mecaniques, Hermes Science

Publications. With permission.)

Mechanical Shock 12-47

© 2005 by Taylor & Francis Group, LLC

low frequencies. The movement of shock starts

with dropping the table from the necessary height

to produce the specified shock and finishes with

stopping the table after rebound on the shock

simulator (Lalanne, 2002b). The preshock takes

place during the fall of the table, the postshock

during its rebound (Figure 12.52). These pre- and

postshocks modify the SRS low frequency and can

lead to an unexpected behavior of material when

its natural frequency is low.

12.9.3.3 Incidence on the Shock Response Spectra

Figure 12.53 shows the response, v20

zðtÞ; of a one-DoF system ðf0 ¼ 4 Hz; j ¼ 0:05Þ to a TPS shock (the

example of Section 12.5.2.5):

* For z0 ¼ z0 ¼ 0 (conditions of the response spectrum).

* In the case of a shock with impact (free fall).

* In the case of a shock on shaker (half-sine symmetrical pre and postshocks with amplitude equal

to 34 m/sec2).

We observed in this example the differences between the theoretical response at 4 Hz and the responses

actually obtained on the shaker and shock machine. According to the test facility used, the shock applied

can undertest or overtest the material. For the estimate of shock severity, one must take account of the

whole of the signal of acceleration.

In Figure 12.54, for j ¼ 0:05; is the SRS of:

* The nominal shock, calculated under the usual conditions of the spectra ðz0 ¼ z0 ¼ 0Þ:

* The realizable shock on shaker, with its pre- and postshocks.

* The realizable shock by impact, taking of account of the fall and rebound phases.

One notes in this example that for:

* f0 # 6 Hz; the spectrum of the shock by impact is lower than the nominal spectrum, but higher

than the spectrum of the shock on the shaker.

* 6 Hz , f0 , 30 Hz; the spectrum of the shock on the shaker is much overestimated.

* f0 , 30 Hz; all the spectra are superimposed.

ti tR

g

xm

··

τ

FIGURE 12.52 Shock performed. (Source: Lalanne,

Chocs Mecaniques, Hermes Science Publications. With

permission.)

Response to a half-sine shock 500 m/s2 10 ms

Theoretical

Shaker

Impact

f0 = 5 Hz

x = 0.05

0.00

−100

−80

−60

−40

−20

−0

20

40

60

80

100

0.05 0.10 0.15 0.20

Time (s)

0.25 0.30 0.35 0.40

ω20

z(t) (m/s2)

FIGURE 12.53 Influence of the realization mode of a TPS shock on the response of a one-DoF system. (Source:

Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

12-48 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

This result appears logical when we remember that the slope of the shock spectrum at the origin is, for

zero damping, proportional to the velocity change associated with the shock. The compensation signal,

added to bring the velocity change back to zero, thus makes the slope of the spectrum at the origin zero.

In addition, the response spectrum of the compensated signal can be larger than the spectrum of the

theoretical signal close to the frequency corresponding to the inverse of the duration of the compensation

signal. It is thus advisable to make sure that the variations observed are not in a range that includes the

resonant frequencies of the test item.

12.9.4 Limitations of Electrodynamic Shakers

12.9.4.1 Mechanical Limitations

Performances of electrodynamic shakers (see Chapter 15) are limited in the following fields (Miller, 1964;

Magne and Leguay, 1972).

* They are limited in terms of the maximum stroke of the coil-table unit (according to the machines

being used, 25.4 to 75 mm peak-to-peak). At the time of the realization of a usual simple shock on

shaker, the displacement starts from the equilibrium position (rest) of the coil, passes through a

maximum, then returns to the initial position. In fact only half of the available stroke is used.

For better use of the capacities of the machine, it is possible to shift the rest position from the

central value towards one of the extreme values (Figure 12.55; Miller, 1964; McClanahan and

Fagan, 1966; Smallwood and Witte, 1972).

Theoretical

TPS shock 340 m/s2 9 ms

Impact

Shaker, with

pre-and post-shocks

x = 0.05

Frequency f 0 (Hz)

10−1 100 101 102 104

450

400

350

300

250

200

150

100

50

0

w20

zsup (m/s2)

FIGURE 12.54 Influence of the realization mode of a TPS shock on the SRS. (Source: Lalanne, Chocs Mecaniques,

Hermes Science Publications. With permission.)

It is necessary to add pre- and/or postshock to the specified shock in order to bring back the

velocity of the shaker table to zero at the end of the shock pulse. The use of pre- and postshocks is

best. Their amplitude must remain small with respect to that of the principal shock (lower than

approximately 10%).

The realization of shocks on free or accelerated fall machines also imposes pre- and postshocks.

These pre- and postshocks lead to differences at low frequency between the spectrum of the

specified shock and the spectrum of the shock actually carried out on the test facility.

Mechanical Shock 12-49

© 2005 by Taylor & Francis Group, LLC

* The maximum velocity is also limited (Young, 1964): 1.5 to 3 m/sec in sine mode (in shock,

one can admit a larger velocity with nontransistorized amplifiers (electronic tubes), because

these amplifiers can generally accept a very short overvoltage). During the movement of the

moving element in the air-gap of the magnetic coils, there is an electromotive force (emf)

produced which is opposed to the voltage supply. The velocity must thus have a value such

that this emf is lower than the acceptable maximum output voltage of the amplifier. The

velocity must in addition be zero at the end of the shock movement (Smallwood and Witte,

1972; Galef, 1973).

* There is a limit to maximum acceleration, related to the maximum force. McClanahan and

Fagan (1965) consider that the realizable maxima shock levels are approximately 20% below

the vibratory limit levels in velocity and in displacement. The majority of authors agree that

the limits in force are, for the shocks, larger than those indicated by the manufacturer (in

sine mode). The determination of the maximum force and the maximum velocity is based,

in vibration, on considerations of the fatigue of the shaker mechanical assembly. Since the

number of shocks that the shaker will carry out is very much lower than the number of

cycles of vibrations that it will undergo during its life, the parameter maximum force can be,

for the shock applications, increased considerably.

Another reasoning consists of considering the acceptable maximum force, given by the manufacturer

in random vibration mode, expressed by its rms value. Knowing that one can observe random peaks

being able to reach 4.5 times this value (limitation of control system), one can admit the same limitation

in shock mode. One finds other values in the literature, such as:

* # 4 times the maximum force in sine mode, with the proviso of not exceeding 300g on the

armature assembly (Hug, 1972).

* . 8 times the maximum force in sine mode in certain cases (very short shocks; 0.4 msec, for

example; Gallagher and Adkins, 1966). Dinicola (1964) and Keegan (1973) give a factor of about

ten for the shocks of duration lower than 5 msec.

The limits of velocity, displacement, and force are not affected by the mass of the specimen.

12.9.4.1.1 Abacuses

For a given shock and for given pre- and postshocks shapes, the velocity and the displacement can be

calculated as a function of time by integration of the expressions of the acceleration, as well as the

maximum values of these parameters, in order to compare them with the characteristics of the

facilities.

From these data, abacuses can be established allowing quick evaluation of the possibility of

realization of a specified shock on a given test facility (characterized by its limits of velocity and

Rest

Rest

Maximum

displacement during

the shock movement

FIGURE 12.55 Displacement of the coil of the shaker. (Source: Lalanne, Chocs Mecaniques, Hermes Science

Publications. With permission.)

12-50 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

of displacement). These abacuses are made up of straight-line segments on logarithmic scales

(Figure 12.56).

* AA0 corresponding to the limitation of velocity: the condition vm # vL (vL ¼ acceptable

maximumvelocity on the facility considered) results in a relationship of the form x€mt # constant

(independent of p; the ratio of the absolute values of the pre- and postshocks amplitude and of the

principal shock amplitude).

* CC, DD, and so on, the greater slope corresponding to the limitation in displacement for various

values of p ( p ¼ 0.05, 0.10, 0.25, 0.50, and 1.00).

A particular shock will be thus realizable on the shaker only if the point of coordinates t; x€m (duration

and amplitude of the shock considered) is located under these lines, this useful domain increasing when

p increases.

Example

TPS shock pulse, 340 m/sec2, 9 msec

(example of Section 12.5.2.5)

Unit table mass: 192 kg

Shaker: 135 kN (maximum velocity:

1.78 m/sec, maximum stroke: ^12.7 mm; see

Figure 12.57)

Test item þ fixture mass: 150 kg

Maximum acceleration without load:

(135,000/192) < 703 m=sec2

Maximum acceleration with test item and

fixture: (135,000/(192 þ 150)) < 395 m=sec2

The TPS shock pulse (340 m/sec2, 9 msec)

is realizable on this shaker with p ¼ 0:05:

VELOCITY

LIMITATION

FORCE

LIMITATION

DISPLACEMENT

LIMITATION

Shock duration (s)

p = 0.05

0.10

0.50

A

C

E

D

F

G

1.00

A′

C′ D′ F′

E′

G′

0.25

Shock amplitude (m/s2)

10−4 10−3 10−2 10−1 101

102

103

104

105

FIGURE 12.56 Abacus of the realization domain of a shock. (Source: Lalanne, Chocs Mecaniques, Hermes Science

Publications. With permission.)

Shock Duration (s)

No load

Test item + fixture mass: 150 kg

0.009

340

0.25

0.5

1.0

0.1

p = 0.05

10−4 101

102

103

10−3 10−2 10−1

m/s2

FIGURE 12.57 Shaker 135 kN, ^12.7 mm — TPS

pulse with half-sine symmetrical pre- and postshocks.

Mechanical Shock 12-51

© 2005 by Taylor & Francis Group, LLC

The limitation can also be due to:

* The resonance of the moving element (a few thousands Hertz; although it is kept to the maximum

by design, the resonance of this element can be excited in the presence of signals with very short

rise time).

* The strength of the material (very great accelerations can involve a separation of the coil of the

moving component).

12.9.4.2 Electronic Limitations

1. Limitation of the output voltage of the amplifier (Smallwood, 1974), which limits coil velocity.

2. Limitation of the acceptable maximum current in the amplifier, related to the acceptable

maximum force (i.e., with acceleration).

3. Limitation of the bandwidth of the amplifier.

4. Limitation in power, which relates to the shock duration (and the maximum displacement) for a

given mass.

Current transistor amplifiers make it possible to increase the low frequency bandwidth but do not

handle even short overtensions well, and thus are limited in mode shock (Miller, 1964).

12.9.5 The Use of Electrohydraulic Shakers

Shocks are realizable on the electrohydraulic exciters, but with additional stresses.

* Contrary to the case of the electrodynamic shakers, one cannot obtain via these means shocks of

amplitude larger than realizable accelerations in the steady mode.

* The hydraulic vibration machines are in addition strongly nonlinear (Favour, 1974).

However, their long stroke, required for long duration shocks, is an advantage.

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