CALCULATION OF SHOCK RESPONSE SPECTRUM

VSB – TU of OstravaFaculty of Mechanical Engineering Jiří Tůma & Petr Kočí

MotivationLaunch Vehicle Stage Separation Test

http://www.vibrationdata.com/SRS.htm

OutlineMotivation

Research work for the Visteon CompanyCalculation Shock Response Spectrum

Single-Degree-of-Freedom ModelTransfer Functions for Relative Velocity and Displacement ResponseApproximation of Continuous Transfer Function by Discrete Transfer FunctionApproximation of the transfer function G(s) = 1/(s + a) by G(z) = (β0 + β1z-1)/(1 + α1z-1)Assumptions about the function of x1(t) in the time interval [ nTS , (n+1)TS ]Ramp invariant methodCalculation SRSSRS of A Half-Sine ImpulseExamplesConclusion

Shock Response Spectrum

fn1 fn2 fn3 fn4 fnK< < <

Resonance gain Q = 10 Input impulse

Response

Natural frequencies for SDOF system

A set of SDOF systems - substructures

A base structure exposed by vibration transient

Amplitude of vibration

Single-Degree-of-Freedom Model

m

c k

x2

x1

( ) ( )12122 xxkxxcxm −−−−= &&&&

( ) ( )( ) kcsms

kcssXsXsG

+++

== 21

2

( ) ( )( ) ( ) ( )

( ) kcsmskcs

sAsAsG

kcsmskcs

sVsVsG

+++

==++

+== 2

1

22

1

2

kmc

QckmQ

mkfn 22

1,,21

==== ξπ

( )( ) 22

2

1

2

nn

nn

Qss

Qs

sAsA

ωω

ωω

++

+=

( ){ } ( ) ( ){ } ( ) ( ){ } ( )( ){ } ( ) ( ){ } ( ) ( ){ } ( )sAtaLsVtvLsXtxL

sAtaLsVtvLsXtxL

222222

111111

,,,,

======

2222

1111

,,

avvxavvx

====

&&

&&

Equation of motion

Laplace transform

Transfer functions with parameters: mass, damping and stiffness

Transfer functions with parameters: natural frequency and resonance gain

( ) ( )( ) 2

1

2

ωωω

ωωω

mkcjkcj

jXjXjG

−++

==

101 102 103 104

10-2

100

102

Frequency [Hz]

abs(

G(jw

))

Frequency response

Q

fn

Transfer Functions for Relative Velocity and Displacement Response

kcsmsms

AVV

++−

=−

21

12

kcsmsm

AXX

++−

=−

21

12

kcsmsm

AXX n

n ++−

=−

21

12 ωωkcsms

mA

XX nn ++

−=

−2

22

1

12 ωω

Relative displacement response spectrum expressed as equivalent static acceleration

Pseudo velocity response spectrum

Relative displacement response spectrumRelative velocity response

Approximation of the Continuous Transfer Function by the Discrete Transfer Function

( ) ( )12122 xxkxxcxm −−−−= &&&&

( )( ) 2

21

1

22

110

1

2

1)( −−

−−

++++

==zzzz

zAzAzH

ααβββ

21210 ,,,, ααβββ

[ ] [ ] [ ] [ ] [ ] [ ]2121 22211211102 −−−−−+−+= kxkxkxkxkxkx ααβββ

( )( ) 22

2

1

2

nn

nn

Qss

Qs

sAsA

ωω

ωω

++

+=

Differential equation of motion Difference equation

... parameters

( )( ) *

!

*1

1

1

22

2

1

2

ssR

ssR

Qss

Qs

sAsA

nn

nn

−+

−=

++

+=

ωω

ωω

( ) ( ) ( ) ( ) ( ) ( )tatvtxtatvtx 111222 ,,,,, [ ] [ ] [ ] [ ] [ ] [ ]kakvkxkakvkx 111222 ,,,,,

Continuous time signals Sampled time signals

k ... Index of a sample t ... Continuous time

Decomposition into partial fractions

[ ] ( )[ ] ( )S

S

TkxkxTkxkx

22

11

==

Approximation of the transfer function G(s) = 1/(s + a) by G(z) = (β0 + β1z-1)/(1 + α1z-1)

Convolution integral

( ) ( ){ } ( )tasGLtg −== − exp1

( ) ( ) ( )∫ −=t

dxtgtx0

12 τττ

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )∫

∫∫∫

+−+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+−−−=−+−=+

++

S

SS

S

SSS

T

SSSS

TnT

nT

nT

SS

TnT

SSSS

dunTuxuaaTnTxaT

ddxnTaaTdxTnTaTnTx

012

01

012

expexpexp

...expexpexp τττττττ

( ) ( ) ( ) ( ) ( )SSSSSSS nTxTnTxnTxaTTnTx 111022 exp ββ +++−=+

( ) ( )( ) ( ) 1

110

1

2

exp1 −

−

−−+

==zaT

zzXzXzG

S

ββ

The output signal for t = nTS

The input signal is an unknown function except for a sequence of samples [ ] [ ] [ ] ...,2,1,0 111 xxx

[ ]01x[ ]11x [ ]21x

[ ]31x[ ]41x [ ]51x

[ ]61x[ ]71x

( )tx1

0 TS 2TS 3TS 4TS 5TS 6TS 7TS t

Impulse response

The result depends on the assumption dealing with x1(t) in between nTS and (n+1)TS

http://www.bth.se/tek/amt/nonlinear.nsf/(WebFiles)/F75E7709B9C6F47DC12571D50033078C/$FILE/ISMA_2006_444.pdf

[ ] ( )STnxnx 11 =

Assumptions about the function of x1(t) in the time interval [ nTS , (n+1)TS ]

Impulse invariant

( ) ( )( ) ( )( )SS TntTnxtx 1111 +−+= δ

( ) ( ) ( ) SSS TntTnTnxtx 1,11 +<≤=

( ) ( ) ( )( )( ) ( ) ( )⎩

⎨⎧

+<≤+++<≤

=SSS

SSS

TntTnTnxTntTnnTx

tx15.0,1

5.0,

1

11

( ) ( ) ( )( ) ( )( )( ) ( ) SSSSSSS TntTnTTntnTxTnxnTxtx 1,1 1111 +<≤−−++=

nTS (n+1)TS

x1(t)

x1(n) x1(n+1)

nTS (n+1)TS

x1(t)

x1(n) x1(n+1)

nTS (n+1)TS

x1(t)

x1(n) x1(n+1)

nTS (n+1)TS

x1(t)

x1(n) x1(n+1)

Step invariant

Ramp invariant

Centered step invariant

( ) ( )

( )( ) ( )SS

T

S

TaTnx

duTnuxuaS

exp1

exp

1

01

+=

=+∫

( ) ( )

( ) ( )( ) aTaTnx

duTnuxua

SS

T

S

S

1exp

exp

1

01

−=

=+∫

Ramp Invariant Method

( )( ) 2

21

1

22

110

1

2

1)( −−

−−

++++

==zzzz

zAzAzH

ααβββ

BBA /)sin()exp(10 −−=β

[ ])cos(/)sin()exp(21 BBBA −−=β

BBAA /)sin()exp()2exp(2 −−−=β

)cos()exp(21 BA−−=α

)2exp(2 A−=α

2411,

2 QTB

QTA Sn

Sn −== ωω

Digital filter

Coefficients

where

Accuracy

( )

22

2

ωωωωωωω

ω

−++

=

=

nn

nn

QjQj

jG

101 102 103 10410-4

10-3

10-2

10-1

100

101

102Frequency response

Frequency [Hz]

abs(

G(jw

))

fs /2

Continuous system

Ramp invariant method

Bilinear transform with taking on account prewarpingfrequency

fs / 2

Calculation SRS

Input impulse

-10000

1000

200030004000

0,0 0,1 0,2 0,3 0,4 0,5

Time [s]

G

-2000

0

2000

4000

6000

0,0 0,1 0,2 0,3 0,4 0,5

Time [s]

G

fn1 = 10 Hz

Resonance gain Q = 10

fn2 = 100 Hz fn3 = 1000 Hz fn4 = 10000 Hz

-2000

02000

40006000

0,0 0,1 0,2 0,3 0,4 0,5

Time [s]

G

-2000

02000

40006000

0,0 0,1 0,2 0,3 0,4 0,5

Time [s]

G

-2000

02000

40006000

0,0 0,1 0,2 0,3 0,4 0,5

Time [s]

G

Shock Response Spectrum

100

1000

10000

10 100 1000 10000

G

Natural frequency in Hz

Input signal

SRS

A set of responsesto the input signal

Peak values of responses as a function of the natural frequency.

Sub

stru

ctur

es

A base structure

SRS of A Half-Sine ImpulseHalf-sine impulse (11 ms)

0

2

4

6

8

10

12

0,00 0,01 0,02 0,03 0,04 0,05Time [s]

G

Shock Response Spectrum

0,1

1,0

10,0

100,0

1 10 100 1000Frequency [Hz]

G

05

101520

abs(G)

0,00 0,02 0,04 0,06 0,08 0,10 0,120

612

1824

Time [s]

Index ofthe SDOF natural

frequency

Input impulse

Absolute value of SDOF-system responses

Shock Response Spectrum

Peak values

Peak values as a function of fn

G

Signal Analyzer Script‘Shock Response Spectrum’; ‘CrLf’;ymax = [ ]; ff = [ ]; fmin = 1; fmax = 1000; n = 90;qv = (fmax/fmin)^(1/n); T = 1/get(input1,'freq'); Q = 10;for(i=0;i<n;i=i+1){

fn = fmin*qv^i; ff = [ff,fn]; wn = 2*pi*fn;A = wn*T/2/Q;B = wn*T*sqr(1-1/4/Q/Q);b0 = 1-exp(-A)*sin(B)/B;b1 = 2*exp(-A)*(sin(B)/B-cos(B));b2 = exp((-2)*A)-exp(-A)*sin(B)/B;a1 = (-2)*exp((-1)*A)*cos(B);a2 = exp((-2)*A);BB = [b0,b1,b2];AA = [-a1,-a2];y = filter(input1,BB,AA);yy = max(y);ymax = [ymax,yy];

};save(ff);save(ymax);

Metal hammer and plastic shield

Examples: impact of metal to plasticTime History : Acc

-400-300-200-100

0100200300400

0,0 0,1 0,2 0,3 0,4

Time [s]

G

Shock Response Spectrum

1

10

100

1000

10000

1 10 100 1000 10000

Natural frequency [Hz]

G

Time History : Acc - Hamer

-4000

-3000

-2000

-1000

0

1000

0,00 0,05 0,10 0,15 0,20Time [s]

m/s

2

Shock Response Spectrum

10

100

1000

10000

1 10 100 1000 10000 100000Natural frequency [Hz]

m/s

2

Acceleration during impact of a steel hammer to the plastic shield and corresponding SRS

Acceleration during impact of a car light system and corresponding SRS

Conclusion

The paper presents the method of calculation the shock response spectrum, which is corresponding to an acceleration signal exciting the resonance vibration of substructures. SRS determines the maximum or minimum of the substructure acceleration response as a function of the natural frequencies of a set of the single-degree-of-freedom systems modeling the mentioned substructures.

The vibration (or shock) is recorded in digital form, commonly as acceleration signal. The single-degree-of-freedom systems are approximated by an IIR digital filter and the filter response to the sampled acceleration signal may be easily calculated. This shock response spectrum shows how the individual component of the impulse signal excites the mechanical structure to resonate.