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Impulse response of a dynamic system with statistical properties
Journal of Sound and Vibration (1973) 31(3), 309-314
IMPULSE RESPONSE OF A DYNAMIC SYSTEM WITH STATISTICAL PROPERTIES P.-C. CHEN Scientific Development, Bechtel Corporation, San Francisco, Calif 94105, U.S.A. AND
W. W.
SOROKA
Department ofMechanical Engineering, University of California, Berkeley, Calif. 94720, U.S.A. (Received 17 May 1973)
The impulse response function of a simple dynamic system with statistical properties is obtained by solving the random differential equation of motion by a perturbation method. Statistical moments of the impulse response function are determined and presented together with numerical results. The influence of system properties on the statistical moments is discussed. 1. INTRODUCTION
In determining the response of a dynamic system, it is customary to assume that the given system has precisely defined properties. Often, however, system properties can at best be determined within a certain range due to inability to calculate them precisely. The uncertainty inherent in system response induced by uncertainty in properties raises concern for reliabiity in engineering design since design relies upon predicted system behavior. To insure reliable design, one needs to know system response variations due to variations in system properties. This consideration leads to the treatment of system properties as random variables in order to predict system behavior. The investigation of the dynamic response of a simple physical system with random properties seems to have been originated by Chenea and Bogdanoff [1,2). They studied the impulse and frequency response functions of a one-degree-of-freedom dynamic system with random parameters. The statistical moments of the impulse and frequency response functions were subsequently determined. Later Soong and Bogdanoff [3] investigated the impulse and frequency response functions of a disordered (random) linear chain of N degrees offreedom. This paper takes a different approach for the impulse response function of a harmonic oscillator with random parameters. The formulation solves the second-order random differential equation by a perturbation method. Statistical moments of the impulse response function are determined and presented, together with numerical results. 2. ANALYSIS
The equation of motion of a simple harmonic oscillator is of the form
x + 2(wn x+ w~x=f(t), where WII is the natural frequency and ( is the viscous damping factor. 309
(I)
310
P.-C. CHEN AND W. W. SOROKA
Let the natural frequency
OJ n
be a random variable which has the representation (2)
in which wn is a constant and the perturbation wn is a random variable with zero mean. The mean value of the natural frequency W n is W". The relative magnitude of perturbation is indicated by the parameter e. When B = 0, the equation of motion has constant coefficients. A solution of equation (1) is sought in the form x(t) = xo(t)
+ aXl(t),
(3)
where Xl(t) is the perturbation on xo(t). Substitution of equation (2) and equation (3) into equation (1) and equating the coefficients of equal powers of a to zero yields (4)
and
(5) Although equation (1) has random coefficients, equation (4) does not. The solution of xo(t) in equation (4) is obtained readily and becomes the forcing function to determine xl(t). The statistical moments of x(t) can be determined subsequently by applying the expectation operator on x(t). This approach of solving an ordinary differential equation with random coefficients was given by Samuels and Eringen [4]. The solutions xo(t) of equation (4) and xl(t) of equation (5) can be represented by the integrals t
xo(t)
= JhCt-'t)!('t)d't
(6)
o
and t
xl(t) = -2(
t
f h(t - Ll)w.xo('tl)d'tl - 2w" Jh(t- 't2)W"XO('t2) d'2
o
(7)
0
where e-,tiint
h(t) = =
sinw..VT=12 t, wn v'T="[2 1 - (2
t;;. 0,
0,
t < 0.
(8)
In particular, if!(t) = 10 oCt), where 0 is the Dirac delta function, then xo(t)
=
IohU)
(9)
and t
XlU)
t
J
J o
= -2(10 h(t - 'tl) W,Ji('I) d't"l - 2w.lo h(r - 't"2) W"h('t2) dL2'
(10)
0
It is clear from equations (3), (9) and (10) that the impulse response functions of a harmonic oscillator with and without statistical variations in frequency are not the same. However, their mean values are identical from the fact that the random variable w" has zero mean, (II)
·in which
Jh(t -
t
t
E {Xl} = -2(10
o
't"l) E{w.}/i(LI) d't l - 2w. 10
f h(t - 't2) E 0
{W,,}h('t2) d't2 = 0.
(12)
311
SYSTEM WITH STATISTICAL PROPERTIES
The interchangeability of the expectation operation and integration is assumed to exist in achieving equation (11). The second moment, or mean square, and the variance of the impulse response function x(t) are, respectively, of the forms E{x 2} = E{X5} (j~ = E{(x -
+ 82E{xOX1} + 82 E{xn =
xi+
8
x')2} = E{x 2} - E{X}2 = e2E{x1}
=
2
E{x1},
(13)
82 0';,.
(14)
They are related to the variance ofx1(t) through the parameter e. Equations (11), (13) and (14) indicate that the second-order statistics of the impulse response function of a system with statistical variation in natural frequency are determined completely by the second-order statistics of the perturbation X1(t). The second statistical moment of the perturbation xJ(t) is evaluated by applying the expectation operator to the square of equation (7), r t
E{xf(t)}
=
E
{
JJfl(t -1:1)h(t -1:3)XO(TJ)Xo(T3)(2()l [wllw
o
r t
n]
0
d1: 1 dT 3 +
f I (40h(t - T2)
x
0 0 r I
X
h(t-T3)XO(1:2)Xo(T3)WII[W"wII]d1:2dT3+
II (4011(t-T1) x ci Ii I
X h(t-T4)WIIXO(T1)XO(t'4)[WnwIIJdT1d1:4
I
+ II ci
(4w;)h(t-T:z) x
0 (15)
x h(t - T4)Xo(T2) XO(T4)[WIIWII]dT2dT4}.
Upon interchanging the expectation operation and integration and noting that the mean square of W" is equal to the variance of wII for this case, i.e., E {w~}
=
(j~n'
( 16)
equation (15) becomes E{xf(t)}
=
(]'~n [4e
(i
r
+ 8(w II }
h(t - T1) Xo(T1) d1: 1
x ( h(t - T2) XO(1:2) d1: 2 + Ii
4w~ ( (h(t \0
h(t - 't't)xO(1:1)dT1 x
1:2) XO(1:2) d1: 2)' 2] .
(17)
Substituting equations (8) and (9) into equation (17) (after some algebra) yields (18)
in which A(t)=
0
w"
~e-'&nt{sinY1="'fiw"t(!..+ ~ II sin2~w"t)+ 1- ( 2 4 1- (
+ X
0
W
.hCOS~W"f(~1 - (2 2 4
vb
1 - (2 Wn
I
sin2~Wnt) - 4 ~ x 1 - (2 W/I
(~sin~Wllf+cOS~W/lf) O-COS2Vl-(2W/lt)},
(19)
P.~C.
312
CHEN AND W. W. SOROKA
(20)
3. NUMERICAL RESULTS
For convenience, the mean and the standard deviations of x(t), without the parameter 1l 2 , may be written, respectively, as (21)
and (22)
in which (23)
C(O)
=
~{Sin~O(-tO+ ~sin2~e) + vbCOSv'I(2e x I - (2 4 I- e I _ (2
x(teX (
k
4
~sin2~e)1_(2 4
b
1 _ (2
x
sin v 1 - (20 + cos VI=(2 e) (1 - cos 2 -J 1 -
(2
e)} ,
(24)
(25)
Plots of equations (21) and (22) as functions of roll t are shown in Figure 1 for ( = 0·05. It is seen that both the mean and the standard deviation have oscillatory forms, with the latter being approximately 90 degrees out of phase from the mean. The amplitude of the standard deviation increases as roll t increases, and gradually dampens out after it has reached a certain level. The amplitude of standard deviation in the figure has a factor of w,,/uwn , as cOJJ:?pared to the mean. For a given value of U W , the value of standard deviation diminishes as the mean natural frequency of the system inc;eases. This observation indicates that for a system with very high frequency, the uncertainty in natural frequency may be neglected in predicting the system response. For a system with low natural frequency, the influence of uncertainty in frequency to system response is significant. The effect of the damping factor on the standard deviation is shown in Figure 2. It is clear that the dispersion from the mean becomes substantial as the damping factor decreases. This
313
SYSTEM WITH STATISTICAL PROPERTIES
60
.5';
,
4'0
t{ "-
ti
C:3
2·0
-2,0 '--------'-2"----4'-".----"-6".----8='-".----10"--".---"'12-".---.,..,14"--,,.---'
Figure 1. Mean and standard deviation of impulse function, ,
=
0·05.
20·0,------r----,----,----.,-----.-------r----,---,
16·0
.
~
12·0
t!,'
"b ' NO
°3
8'0
4·0
Figure 2. Standard deviation of impulse response function, <: = 0,02, 0·05, 0'10,0'20.
observation suggests that for lightly damped system, little confidence can be placed ill the response if the natural frequency cannot be calculated precisely. 4. DISCUSSION
The observations from the two figures in general agree with those given by Soong and Bogdanoff [3]. However, no quantitative comparison was made as the results in the previous case [3] are for a system of multiple degrees offreedom. The method presented provides an alternative for the impulse response of a system with random natural frequency. The method also accommodates in principle the situation in which
314
P.-C. CHEN AND W. W. SOROKA
both the natural frequency and damping factor are random variables. However, its algebraic manipulation becomes tedious when the order of perturbation is greater than one. The accuracy in solution gained by determining higher order terms does not seem to warrant the effort in view of the fact that the parameter e is also present. REFERENCES
1. P. F. CHENEA and J. L. BOGDANOFF 1958 ASME Colloquium on Mechanical Impedance Methodsfor Mechanical Vibration, 125-128. Impedance of some disordered systems. 2. J. L. BOGDANOFF and P. F. CHENEA 1961 International Journal of Mechanical Science 3, 157-169.
Dynamics of some disordered linear systems. 3. T. T. SOONG and J. L. BOGDANOFF 1964 International Journal of Mechanical Science 6, 225-23.7. On the impulsive admittance and frequency response of a disordered linear chain of N degrees of freedom. 4. J. C. SAMUELS and A. C. ERINGEN 1959 Journal of Mathematics and Physics 38,2, 83-104. On stochastic linear systems.
IMPULSE RESPONSE OF A DYNAMIC SYSTEM WITH STATISTICAL PROPERTIES P.-C. CHEN Scientific Development, Bechtel Corporation, San Francisco, Calif 94105, U.S.A. AND
W. W.
SOROKA
Department ofMechanical Engineering, University of California, Berkeley, Calif. 94720, U.S.A. (Received 17 May 1973)
The impulse response function of a simple dynamic system with statistical properties is obtained by solving the random differential equation of motion by a perturbation method. Statistical moments of the impulse response function are determined and presented together with numerical results. The influence of system properties on the statistical moments is discussed. 1. INTRODUCTION
In determining the response of a dynamic system, it is customary to assume that the given system has precisely defined properties. Often, however, system properties can at best be determined within a certain range due to inability to calculate them precisely. The uncertainty inherent in system response induced by uncertainty in properties raises concern for reliabiity in engineering design since design relies upon predicted system behavior. To insure reliable design, one needs to know system response variations due to variations in system properties. This consideration leads to the treatment of system properties as random variables in order to predict system behavior. The investigation of the dynamic response of a simple physical system with random properties seems to have been originated by Chenea and Bogdanoff [1,2). They studied the impulse and frequency response functions of a one-degree-of-freedom dynamic system with random parameters. The statistical moments of the impulse and frequency response functions were subsequently determined. Later Soong and Bogdanoff [3] investigated the impulse and frequency response functions of a disordered (random) linear chain of N degrees offreedom. This paper takes a different approach for the impulse response function of a harmonic oscillator with random parameters. The formulation solves the second-order random differential equation by a perturbation method. Statistical moments of the impulse response function are determined and presented, together with numerical results. 2. ANALYSIS
The equation of motion of a simple harmonic oscillator is of the form
x + 2(wn x+ w~x=f(t), where WII is the natural frequency and ( is the viscous damping factor. 309
(I)
310
P.-C. CHEN AND W. W. SOROKA
Let the natural frequency
OJ n
be a random variable which has the representation (2)
in which wn is a constant and the perturbation wn is a random variable with zero mean. The mean value of the natural frequency W n is W". The relative magnitude of perturbation is indicated by the parameter e. When B = 0, the equation of motion has constant coefficients. A solution of equation (1) is sought in the form x(t) = xo(t)
+ aXl(t),
(3)
where Xl(t) is the perturbation on xo(t). Substitution of equation (2) and equation (3) into equation (1) and equating the coefficients of equal powers of a to zero yields (4)
and
(5) Although equation (1) has random coefficients, equation (4) does not. The solution of xo(t) in equation (4) is obtained readily and becomes the forcing function to determine xl(t). The statistical moments of x(t) can be determined subsequently by applying the expectation operator on x(t). This approach of solving an ordinary differential equation with random coefficients was given by Samuels and Eringen [4]. The solutions xo(t) of equation (4) and xl(t) of equation (5) can be represented by the integrals t
xo(t)
= JhCt-'t)!('t)d't
(6)
o
and t
xl(t) = -2(
t
f h(t - Ll)w.xo('tl)d'tl - 2w" Jh(t- 't2)W"XO('t2) d'2
o
(7)
0
where e-,tiint
h(t) = =
sinw..VT=12 t, wn v'T="[2 1 - (2
t;;. 0,
0,
t < 0.
(8)
In particular, if!(t) = 10 oCt), where 0 is the Dirac delta function, then xo(t)
=
IohU)
(9)
and t
XlU)
t
J
J o
= -2(10 h(t - 'tl) W,Ji('I) d't"l - 2w.lo h(r - 't"2) W"h('t2) dL2'
(10)
0
It is clear from equations (3), (9) and (10) that the impulse response functions of a harmonic oscillator with and without statistical variations in frequency are not the same. However, their mean values are identical from the fact that the random variable w" has zero mean, (II)
·in which
Jh(t -
t
t
E {Xl} = -2(10
o
't"l) E{w.}/i(LI) d't l - 2w. 10
f h(t - 't2) E 0
{W,,}h('t2) d't2 = 0.
(12)
311
SYSTEM WITH STATISTICAL PROPERTIES
The interchangeability of the expectation operation and integration is assumed to exist in achieving equation (11). The second moment, or mean square, and the variance of the impulse response function x(t) are, respectively, of the forms E{x 2} = E{X5} (j~ = E{(x -
+ 82E{xOX1} + 82 E{xn =
xi+
8
x')2} = E{x 2} - E{X}2 = e2E{x1}
=
2
E{x1},
(13)
82 0';,.
(14)
They are related to the variance ofx1(t) through the parameter e. Equations (11), (13) and (14) indicate that the second-order statistics of the impulse response function of a system with statistical variation in natural frequency are determined completely by the second-order statistics of the perturbation X1(t). The second statistical moment of the perturbation xJ(t) is evaluated by applying the expectation operator to the square of equation (7), r t
E{xf(t)}
=
E
{
JJfl(t -1:1)h(t -1:3)XO(TJ)Xo(T3)(2()l [wllw
o
r t
n]
0
d1: 1 dT 3 +
f I (40h(t - T2)
x
0 0 r I
X
h(t-T3)XO(1:2)Xo(T3)WII[W"wII]d1:2dT3+
II (4011(t-T1) x ci Ii I
X h(t-T4)WIIXO(T1)XO(t'4)[WnwIIJdT1d1:4
I
+ II ci
(4w;)h(t-T:z) x
0 (15)
x h(t - T4)Xo(T2) XO(T4)[WIIWII]dT2dT4}.
Upon interchanging the expectation operation and integration and noting that the mean square of W" is equal to the variance of wII for this case, i.e., E {w~}
=
(j~n'
( 16)
equation (15) becomes E{xf(t)}
=
(]'~n [4e
(i
r
+ 8(w II }
h(t - T1) Xo(T1) d1: 1
x ( h(t - T2) XO(1:2) d1: 2 + Ii
4w~ ( (h(t \0
h(t - 't't)xO(1:1)dT1 x
1:2) XO(1:2) d1: 2)' 2] .
(17)
Substituting equations (8) and (9) into equation (17) (after some algebra) yields (18)
in which A(t)=
0
w"
~e-'&nt{sinY1="'fiw"t(!..+ ~ II sin2~w"t)+ 1- ( 2 4 1- (
+ X
0
W
.hCOS~W"f(~1 - (2 2 4
vb
1 - (2 Wn
I
sin2~Wnt) - 4 ~ x 1 - (2 W/I
(~sin~Wllf+cOS~W/lf) O-COS2Vl-(2W/lt)},
(19)
P.~C.
312
CHEN AND W. W. SOROKA
(20)
3. NUMERICAL RESULTS
For convenience, the mean and the standard deviations of x(t), without the parameter 1l 2 , may be written, respectively, as (21)
and (22)
in which (23)
C(O)
=
~{Sin~O(-tO+ ~sin2~e) + vbCOSv'I(2e x I - (2 4 I- e I _ (2
x(teX (
k
4
~sin2~e)1_(2 4
b
1 _ (2
x
sin v 1 - (20 + cos VI=(2 e) (1 - cos 2 -J 1 -
(2
e)} ,
(24)
(25)
Plots of equations (21) and (22) as functions of roll t are shown in Figure 1 for ( = 0·05. It is seen that both the mean and the standard deviation have oscillatory forms, with the latter being approximately 90 degrees out of phase from the mean. The amplitude of the standard deviation increases as roll t increases, and gradually dampens out after it has reached a certain level. The amplitude of standard deviation in the figure has a factor of w,,/uwn , as cOJJ:?pared to the mean. For a given value of U W , the value of standard deviation diminishes as the mean natural frequency of the system inc;eases. This observation indicates that for a system with very high frequency, the uncertainty in natural frequency may be neglected in predicting the system response. For a system with low natural frequency, the influence of uncertainty in frequency to system response is significant. The effect of the damping factor on the standard deviation is shown in Figure 2. It is clear that the dispersion from the mean becomes substantial as the damping factor decreases. This
313
SYSTEM WITH STATISTICAL PROPERTIES
60
.5';
,
4'0
t{ "-
ti
C:3
2·0
-2,0 '--------'-2"----4'-".----"-6".----8='-".----10"--".---"'12-".---.,..,14"--,,.---'
Figure 1. Mean and standard deviation of impulse function, ,
=
0·05.
20·0,------r----,----,----.,-----.-------r----,---,
16·0
.
~
12·0
t!,'
"b ' NO
°3
8'0
4·0
Figure 2. Standard deviation of impulse response function, <: = 0,02, 0·05, 0'10,0'20.
observation suggests that for lightly damped system, little confidence can be placed ill the response if the natural frequency cannot be calculated precisely. 4. DISCUSSION
The observations from the two figures in general agree with those given by Soong and Bogdanoff [3]. However, no quantitative comparison was made as the results in the previous case [3] are for a system of multiple degrees offreedom. The method presented provides an alternative for the impulse response of a system with random natural frequency. The method also accommodates in principle the situation in which
314
P.-C. CHEN AND W. W. SOROKA
both the natural frequency and damping factor are random variables. However, its algebraic manipulation becomes tedious when the order of perturbation is greater than one. The accuracy in solution gained by determining higher order terms does not seem to warrant the effort in view of the fact that the parameter e is also present. REFERENCES
1. P. F. CHENEA and J. L. BOGDANOFF 1958 ASME Colloquium on Mechanical Impedance Methodsfor Mechanical Vibration, 125-128. Impedance of some disordered systems. 2. J. L. BOGDANOFF and P. F. CHENEA 1961 International Journal of Mechanical Science 3, 157-169.
Dynamics of some disordered linear systems. 3. T. T. SOONG and J. L. BOGDANOFF 1964 International Journal of Mechanical Science 6, 225-23.7. On the impulsive admittance and frequency response of a disordered linear chain of N degrees of freedom. 4. J. C. SAMUELS and A. C. ERINGEN 1959 Journal of Mathematics and Physics 38,2, 83-104. On stochastic linear systems.