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Pyrotechnic Shock Joint Attenuation via Wave Propagation Analysis

Revision B

By Tom Irvine

Email: tom@vibrationdata.com

August 21, 2015

______________________________________________________________________________

Variables

The variables are

B Flexural stiffness

BC Bending wave phase speed

Cg Group speed

E Elastic modulus

f Frequency (Hz)

h Thickness

I Area moment of inertia

L Length

M Mass per length

R Transmission Loss (dB)

r Flexural stiffness ratio (joint/beam)

Coefficient

Coefficient

Wavelength

Frequency (rad/sec)

Note that the flexural stiffness B is

B = E I (1)

2

Introduction

Launch vehicle stages and modules are connected together using bolted or riveted joints, or via clamp

bands as shown in Figure 1. These joints provide mechanical flexibility. They also are potential low pass

filters with respect to propagating shock energy.

Some pyrotechnic shock joint attenuation factors for launch vehicles are given in Reference 1, based on

empirical data. An excerpt is given in Appendix A. This reference also has caveats such as, “This

information is strictly empirical and can be in considerable error for many applications.”

Nevertheless, this 45-year old document remains the industry standard.

The purpose of this paper is to seek a greater understanding of joint attenuation by considering bending

wave propagation in a semi-infinite beam with a thin elastic interlayer as an analogy. The interlayer

represents the joint, and the attenuation is modeled using Reference 2.

Equations of Motion

Note that bending waves are governed by a fourth-order partial differential equation.

Ideally, this paper would consider cylindrical shell structures which are commonly used for launch

vehicle design. These shells are governed by a set of three coupled equations, one fourth-order and two

Figure 1. Spacecraft/Launch Vehicle Clamp Ring Joint (Image Courtesy of Eurocket)

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second-order, which may be formed into the Donnell-Mushtari operator matrix. Alas, cylindrical

shells will require a future paper.

Joint Stiffness

Empirical joint stiffness data is given in Reference 3 for the purpose of low-frequency bending

analysis. The data is insightful, although it may not be directly applicable to high-frequency

bending wave analysis.

This reference notes that:

The transition sections between the standard rocket motors are the most difficult

regions to define and generally are the spaces that contribute a major part to the

flexibility.

Such contributions are consistently encountered from looseness in screwed joints,

thread deflections, flange flexibility, plate and shell deformations that are not within

the confines of beam, theory, etc.

The reference also duly notes joint nonlinearity.

These challenges notwithstanding, this paper will use a beam with a thin elastic interlayer model.

Transmission Loss

The transmission loss for bending waves propagating through a thin elastic interlayer is

dB,

4

24

log10R232

242

32232

(2)

Equation (2) is taken from Reference 2, page 382, equation (180). It was derived by assuming

that the respective shear forces and bending moments on each side of the interlayer are equal.

The coefficients are

L2

h

E

E

1

2 (3)

4

2

1

122

11

1 B

BL2

IE

IEL2

(4)

The subscript 1 represents the beam, and subscript 2 the elastic interlayer.

The wavelength in the primary material is

f

CB1 (5)

The bending wave phase speed is

4/1

2/1B

M

BC

(6)

Equation (6) is taken from Reference 3, page 101, equation (85). Note that bending waves are

dispersive since the phase speed varies with frequency. The group speed Cg is twice the phase

speed for bending waves.

Modeling Method

The method is similar to that in Reference 4.

Assume a semi-infinite beam where the source shock occurs at the free end. This semi-infinite

assumption bypasses the need to consider modal response.

Also assume that the source shock is characterized by a shock response spectrum (SRS). A

series of wavelets can be synthesized to satisfy the shock response spectrum per References 4

and 5. Note that the wavelet amplitudes can be filtered as a function of frequency.

The sample aluminum beam and its interlayer have a uniform rectangular cross-section, 2 in x

0.25 in. The interlayer has a length of 1 in, representing a joint.

The material and cross section assumptions are needed to establish a bending stiffness, for the

wave speed and attenuation calculations.

The interlayer inertia is set equal to that of the beam.

Now assume an independent variable scale factor less than one. Both the interlayer bending

stiffness and elastic modulus are scaled from the beam respective values by this factor.

In practice, a test would be required to measure the interlayer stiffness.

5

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

10 100 1000 10000 20K

r = 0.1r = 0.02r = 0.01

FREQUENCY (Hz)

RE

MA

ININ

G R

AT

IO

REMAINING RATIO BEAM BENDING WITH INTERLAYER

Figure 2.

The remaining ratio for the transmitted shock is show in Figure 2. The transmitted acceleration

amplitude is equal to the incident amplitude multiplied by the remaining ratio, as a function of

frequency.

These curves apply to both of the following examples. The remaining ratio is to be applied to

the spectral wavelet components of the incident shock, in order to construct the transmitted

shock time history.

The r value is a separate ratio: (joint bending stiffness/beam bending stiffness).

Each curve has a 10 dB/octave roll-off, which would be a straight line if the plot was in log-log

format instead of semi-log format.

The frequency at which the peak occurs for each curve is called the total transmission frequency.

It is the only frequency at which no reflection occurs.

Example 1

The joint is assumed to be very near the source, so that the source shock is the incident wave to

the interlayer.

6

1

10

100

1000

10000

20000

10 100 1000 10000

Incidentr = 0.1r = 0.02r = 0.01

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

SRS Q=10 JOINT NEAR SOURCE

Figure 3.

An SRS comparison is shown in Figure 3.

Each curve with an “r” represents the transmitted shock.

The plateau attenuation is very sensitive to the bending stiffness ratio r.

Now idealize each transmitted SRS as a ramp & plateau, where the plateau is set to the peak in

each curve. Neglect ramp attenuation.

The results under these assumptions are:

r Attenuation

0.1 Negligible

0.02 Negligible

0.01 6 dB

7

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

0 0.02 0.04 0.06 0.08 0.10-2000

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

r = 0.01, Right Scale

Incident, Left Scale

TIME (SEC)

AC

CE

L (

G)

AC

CE

L (

G)

BENDING WAVE TIME HISTORIES, JOINT NEAR SOURCE

Figure 4.

A sample pair of time histories is shown in Figure 4. The peak attenuation for this case is about

6 dB.

Note that a typically assumption for joint attenuation is about 3 dB.

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Example 2

1

10

100

1000

10000

10 100 1000 10000

Incidentr = 0.1r = 0.02r = 0.01

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

SRS Q=10 JOINT AT 100 INCH FROM SOURCE

Figure 5.

An SRS comparison is shown in Figure 5.

The joint is placed at a distance 100 inches from the source, following the example in Reference

4. The source shock remains the same. The incident shock at the joint is the source shock

subjected to distance attenuation for the 3.25% damping case, again from Reference 4.

Now idealize each SRS as a ramp & plateau, where the plateau is set to the peak in each curve.

Neglect ramp attenuation.

The results under these assumptions are:

r Attenuation

0.1 Negligible

0.02 Negligible

0.01 3 dB

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-1200

-1000

-800

-600

-400

-200

0

200

400

0 0.02 0.04 0.06 0.08 0.10-400

-200

0

200

400

600

800

1000

1200

r = 0.01, Right Scale

Incident, Left Scale

TIME (SEC)

AC

CE

L (

G)

AC

CE

L (

G)

BENDING WAVE TIME HISTORIES, JOINT AT 100 INCH FROM SOURCE

Figure 6.

A sample pair of time histories is shown in Figure 6 for the 100 inch case. The red curve is the

transmitted curve for the r=0.01 stiffness ratio case.

Conclusions

Shock propagation in a launch vehicle with cylindrical and frustum shells is significantly more

complicated than the beam/interlayer approach used in this paper. Nevertheless, the beam

analogy yields the following insights.

Joint attenuation is highly dependent on the ratio of the joint bending stiffness to that of the beam

or primary structure.

The frequency content of the incident energy is another significant factor. Joints closer to the

source are more effective attenuators. The reason is that both material and joints act as low pass

filters. So the material has already filtered out much of the high frequency source energy from

by the time the waveform reaches a distant joint.

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Again, the attenuation curves in Figure 2 were highly dependent on the stiffness and geometry

parameters. Furthermore, the 10 dB/octave roll-off may be too steep. The method is thus not

ready for use as a predictive tool.

A better approach would be the development of conservative attenuation curves referenced to

joint type and with unity gain up to the assumed total transmission frequency. The roll-off slope

could be changed to, say, 3 dB/octave. The resulting attenuation factors could then be applied to

the wavelet components for the transmission time history calculation, from which the SRS can

then be calculated. This development remains as future work along with the other points in the

next section. An example of this approach is given in Appendix C.

Joint attenuation can be neglected for conservatism if there are any doubts about its effect. This

assumes that the resulting transmitted shock level can be tolerated.

Future Work

The analysis can be repeated for other sample source waveforms.

A method for predicting joint stiffness is needed for the purpose of pyrotechnic shock

propagation in the early stages of a launch vehicle program. The joint stiffness should be

measured when hardware becomes available. Note that the data in Appendix B shows that the

compliance may vary by a few orders of magnitude between an “Excellent” joint and a “Loose”

joint.

The structure was considered to be uniform on either side of the joint. A more generalized

approach is needed for non-uniformity.

The approach in this paper was essentially an “impedance mismatch” one. Energy dissipation

within the joints due to frictional and other effects was not considered, but should be in future

work.

Nonlinearity must also be considered. Joints may slip under higher shock loads, thus increasing

the energy dissipation.

A joint attenuation equation needs to be derived for cylindrical and frustum shells, as well as for

realistic launch vehicle geometry and constraints.

References

1. W. Kacena, M. McGrath, A. Rader; Aerospace Systems Pyrotechnic Shock Data, Vol. VI, NASA CR

116406, Goddard Space Flight Center, 1970.

2. L. Cremer, M. Heckl, E. Ungar, Structure-Borne Sound, Springer-Verlag, New York, 1988.

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3. V. Alley and S. Leadbetter, Prediction and Measurement of Natural Frequencies of

Multistage Launch Vehicles, AIAA Journal, Vol. 1, No. 2, 1963.

4. T. Irvine, Pyrotechnic Shock Distance Attenuation via Wave Propagation Analysis,

Vibrationdata, 2015.

5. T. Irvine, Webinar 27, SRS Synthesis, Vibrationdata, 2014.

https://vibrationdata.wordpress.com/2014/08/11/webinar-27-srs-synthesis/

APPENDIX A

Excerpt from Reference 1

3.2 Effects of Structural Interfaces

Certain structural interfaces will attenuate a shock pulse.

From general information and Reference 54, the following table has been compiled.

Interface Percent Reduction

1. Solid Joint

2. Riveted butt joint

3. Matched angle joint

4. Solid Joint with layer of

different material in Joint

0

0

30 – 60

0 - 30

Some reduction in shock levels can be expected from intervening structure in a shell type

structure.

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APPENDIX B

Excerpt from Reference 2

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APPENDIX C

Example with Prescribed Attenuation Function

0.1

0.2

0.5

1

2

10 100 1000 10000

FREQUENCY (Hz)

RE

MA

ININ

G R

AT

IO

REMAINING RATIO FREQ = 2000 Hz, SLOPE = -3 dB/octave

Figure C-1.

Assume a vehicle with a 36 inch diameter, aluminum cylindrical module. Analyze the

transmitted shock for a joint near the source, where the source shock is the same as that from

Reference 4 and also the main text of this paper.

The ring frequency for this configuration is 1737 Hz. The author’s experience from numerous

pyrotechnic shock separation tests is that there is often a high modal density near the ring

frequency, about which the highest transmission occurs. So assume that the ring frequency is the

total transmission frequency, but raise it to 2000 Hz for conservatism, with unity gain up to this

frequency. Also assume a 3 dB/octave roll-off. Coincidently, this slope is that same as that

which would transform white noise into pink noise.

The attenuation function for this case is shown in Figure C-1. It is used to filter the wavelet

table.

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1

10

100

1000

10000

20000

10 100 1000 10000

IncidentTransmitted

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

SRS Q=10 JOINT NEAR SOURCE

Figure C-2.

The shock spectra for the Incident and Transmitted pulses are shown in Figure C-2. The plateau

attenuation is about 4 dB, which seems reasonable for typical launch vehicle joints.

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-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

0 0.02 0.04 0.06 0.08 0.10-2000

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

Incident, Left Scale

Transmitted, Right Scale

TIME (SEC)

AC

CE

L (

G)

AC

CE

L (

G)

BENDING WAVE TIME HISTORIES, JOINT NEAR SOURCE

Figure C-3.

The time history pair is shown in Figure C-3.