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Exact stationary solutions of the random response of a single-degree-of-freedom vibro-impact system
Journal
of Sound
EXACT
and Vibrufion (1990) 141(3), 363-373
STATIONARY
RESPONSE
SOLUTIONS
OF THE
RANDOM
OF A SINGLE-DEGREE-OF-FREEDOM VIBRO-IMPACT
SYSTEM
H.-S. JING AND K.-C. SHEU Institute
of Aeronautics
and Astronautics, National Cheng Kung University, Republic of China
(Received
Tainan, Taiwan,
24 May 1989)
The problem of the response of a single-degree-of-freedom system with amplitude constraint on one side subjected to a random excitation is solved exactly. The Hertz law is used to model the contact phenomena between the mass and constraint during vibration. The excitation is limited to be a stationary white Gaussian process with zero mean. By solving the corresponding Fokker-Planck partial differential equation by separation of variables, the exact stationary solutions of the random response are obtained. The changes due to variations of contact stiffness are discussed. 1. INTRODUCTION There has been sustained interest in the non-linear response of a so-called vibro-impact system for many years [ 1,2]. A vibro-impact system is a vibration system with impact interaction. Most of the previous researchers have considered this impact interaction either as a bilinear spring [3], or by using a coefficient of restitution [4] to cover the velocity change after impact. From the analytical solution for the static contact between two elastic bodies with simple shapes [5], it is well known that the force-displacement relationship is non-linear. The contact force is proportional to the 3/2 power of the displacement. A reasonable extension of research along this line would be to incorporate this non-linear force-displacement relation into the analysis. This case has been studied by Nayak [6]. Unfortunately, only an approximate solution was formulated. In this paper, closed form solutions for the motion of a single-degree-of-freedom spring-mass-dashpot system with an elastic amplitude constraint on one side, which is modelled by the Hertz contact law, and subjected to a random excitation are obtained and the behavior of the response is discussed probabilistically. 2. DESCRIPTION
OF THE SYSTEM
A single-degree-of-freedom vibration system is considered. An elastic amplitude constraint is placed under the mass as shown in Figure 1. When the system is vibrating, the mass can move with limited amplitude as in a simple linear system, but for a finite amplitude, the elastic constraint will make its contribution. For simplicity, the constraint is modelled by a non-linear spring according to the Hertz contact law and the inertial effect is neglected. The resulting force-displacement relation, illustrated in Figure 2, clearly shows the non-linear and unsymmetric characteristics. The equation of motion is x312
mf+c~+kx+~g(x)=f(t),
g(x) = 363
0022-460X/90/180363
+ 11 %03.00/O
I
0 @J 1990
Academic
Press
Limited
364
H.-S. JING
Figure
AND
K.-C.
1. The physical
SHEU
model.
x
Figure
2. Force-displacement
relations
for different
values of contact
stiffness
11
Here 77is the contact stiffness, which is a function of the elastic properties and geometries of the two contacting bodies. Its values for several simple cases can be found in the literature [ 51. When the forcing function f( t) is a random process, the response x(t) can be described only in a statistical manner. In this analysis, the excitation is limited to be a stationary white Gaussian process with zero mean and the spectral density is taken to be S,,.
3. EXACT
SOLUTION
OF THE
FOKKER-PLANCK
EQUATION
To extract probabilistic information about the stationary response, it is necessary to find the joint probability density functionp(x, u), which, if the transition time is sufficiently long, satisfies the time-independent Fokker-Planck partial differential equation [7]
(2) Here w. and (Yare the frequency and damping parameters. The coefficients of this equation can be obtained directly from ensemble averages based on the equation of motion and
EXACT
IMPACT
SYSTEM
RANDOM
365
RESPONSE
the excitation. As can be found in the literature [8], the technique of separation of variables can be used if the displacement and velocity are statistically independent and V’/ v= -(2~/7&)U,
X’/X = -(2a/nS,)[o~x+(rl/m)g(x)].
(3,4)
The resulting joint density is
p(x,
u)
=
C exp
X2 V2 -----+j’g(i)dr}, &Y&J; 2~:
mo,w~
(5)
o
where the abbreviation a; = ?Tso/2ffw;
(6)
+m I=+m v) J p(x,
and C is the integration constant to be determined by the normalization
I --Q) --UC
condition
dx dv.
(7)
The integrations in equations (5) and (7) cannot be done analytically for most non-linear systems. In the problem studied here, the integration constant C can be obtained in closed form and is found to be
c-’ = vczu~ql where the non-dimensional
[J
‘+2 2
(-1)” 1 Z_(4n+2,,5r 4n+2 xQ: -5.=0 n! 2” ( 5 )I ’
contact stiffness, or non-linearity
(8)
parameter, 5, is given by
5 = 2nJa,/Smwi.
(9)
It should be noted that [ also contains the excitation information. The exact stationary solution of the Fokker-Planck partial differential equation for this problem is then
u2 _-___g (
exp &t&0, [J
X2
2c&;
2a;
u;‘2 >
(-1)” 1 5_(4n+Z,,Sr 4n+2 _n+2 c= -2 5n-o n! 2” ( 5
)3 ’
x
4.
PROBABILISTIC DESCRIPTIONS OF THE RESPONSE
4.1. PROBABILITY DENSITY FUNCTIONS From the preceding analysis, the separated first order probability density functions for
366
H.-S. JING
the displacement
AND
K.-C.
SHEU
pX(x) and for the velocity p,(u) are
x20
P,(X) =
>
1 x2
exp J
~vo+~cTo
( > --72 oo
xc0
1, 5-(4n+2j,5r 4n+2 ’ 2 ( 5 >
m (-1)”
c -n=O n!
PJv) =
(11)
G~o,o ev -( 2u;o;
>
V2
(12)
It is obvious that the probability density of the velocity is still Gaussian but the displacement is non-Gaussian except when 5 = 0: i.e., when the system is reduced to a linear one. The probability density functions of the displacement for different non-dimensional contact stiffnesses 5 are shown in Figure 3. Apparently, they are unsymmetric except when 5 = 0. As the stiffness of the constraint increases, the probability that the mass will be found in the region x > 0 decreases. When 5 approaches infinity, it is reasonable to expect that it would be impossible to find the mass in the x > 0 region at any time. _0.8 0.7 0.6 2 \
0.5
2
0.4 0.3 0.2 0.1 0.0
Figure 3. Stationary linearity parameter 6
-5
probability
-4
density
-3
-2
functions
-1
0
of response
1
2
amplitude
3
4
for different
5
values
of the non-
4.2. MEAN VALUE As can be seen in Figure 3, the mean value changes as 5 varies. From the definition of the mean, the exact form can be expressed as
Pu,= oo
(13)
EXACT
IMPACT
SYSTEM
RANDOM
367
RESPONSE
When the stiffness of the constraint is reduced to zero, i.e., the system is linear, it can be proved that pu, equals zero. On the other hand, when the contact stiffness is very large, the response mean approaches a finite value: (14)
lim CL,= -v?Z+~. 5-a In Figure 4 it is shown how the mean value changes contact stiffness.
with the non-dimensional
4.3. VARIANCE OF THE DISPLACEMENT By using the same procedures as before, the variance of the displacement be found exactly:
(+* X= a*0
a: can also
(19
--EL:.
It is plotted in Figure 5 as a function of 5. Again, when 5 approaches zero, the variance
Figure
4. Stationary
0.41 0
Figure
5. Stationary
mean value of response
1 12
variance
I
I 3
I 4
of response
amplitude
I 5 c amplitude
I 6
versus non-linearity
1 7
I 0
parameter
5.
1 9
versus non-linearity
parameter
g?
368
H.-S. JING
of the displacement finite value:
AND
K.-C. SHEU
is simply ai as expected. As 6 approaches infinity, a: approaches a lim of=&l-2/r). g+==
(16)
4.4. EXPECTED FREQUENCY Since it would be very difficult to solve for the second order statistics, it is appropriate instead to consider a quantity relevant to a narrow-band process which can supply information about the dominant frequency of the response; this is the expected number of zero-crossings with positive slope per unit time, or simply the expected frequency. For a process with a non-zero mean, one might think that the expected frequency should be considered to be the number of mean-crossings with positive slope per unit time. Although the mean is not zero, the key point here is that the process is non-symmetric. The positive portion of the process is compressed towards zero by the elastic constraint, resulting in the non-zero mean. As the process is unsymmetric, as shown in Figure 3, the mean is not the point with the highest probability density. This has the consequence that if only the number of mean-crossings with positive slope is counted, some information will be lost. Therefore, in what follows next, the expected frequency is taken with respect to zero as usual. Let ~0’be the expected number of zero-crossings with positive slope per unit time, as given, according to Rice [9], by substituting x = 0 into the equation 00 v:= up(x, v) dv. (17) I0 After using equation (9), integrating equation (17) and then normalizing by the natural frequency of the corresponding linear system oo/27r, the normalized expected frequency fz is found to be &G
.tT= ?r+2
J 2
m (-l)” I--
5 n=O n!
1 -(4n+2)/5~ s 2”
(18)
4n+2 ’ ( 5 )
It is plotted in Figure 6 as a function of the non-dimensional
contact stiffness 5.
1.6 1.5 -
01234567891
1 c
Figure 6. Expected
frequency
of stationary
response
versus non-linearity
parameter 6.
EXACT
1MPACT
SYSTEM
RANDOM
RESPONSE
369
When 5 approaches zero, it can be proved that the normalized expected frequency fi approaches 1. As 5 increases, fc also increases since the system is now becoming more rigid. Inspection of equations (6) and (9) shows that the non-dimensional contact stiffness 5 also includes the excitation information. An increase in the spectral density S, of the excitation corresponds to an increase of input energy, i.e., amplitude in some sense, and 6 will also increase. This results in an increase of fof. It is well known that the response of a non-linear system is also a function of the input amplitude. When the stiffness of the elastic constraint approaches infinity; then, physically, whenever the mass hits the rigid constraint, it will come back immediately. This limiting normalized expected frequency is found to be 2 simply by taking the limiting value of equation (18). This limiting case has also been studied by other researchers under the name “impact oscillator” [lo]. An impact oscillator can also be viewed as a limiting case of a vibration system with a bilinear spring: i.e., a spring with different stiffnesses for positive and negative displacements. It has been found [ 1l] that when one of the stiffnesses approaches infinity, the system becomes an impact oscillator, and the bilinear circular frequency is exactly twice the natural frequency of a corresponding system with the same stiffness for displacements to both sides which confirms the correctness of the results of this paper. On the other hand, if the expected frequency is taken with respect to the mean, one finds, after normalization,
f,z,=
JG
exp (-~Q2ai)
(19)
a (-1)” 1 5-(4n+2,,5r 4n+2 ’ lr+? I-J 2 5,=0 n! 2” ( 5 1
where CL, is given in equation (13). The normalized frequency is found to be l-4547 instead of 2 as 5 approaches infinity. As illustrated in Figure 7, the response in vibration without damping, if 5 is zero, is simply sinusoidal. As 5 increases, the time that the mass stays in the region x > 0 will be shorter since the constraint is more rigid. Hence fO+will be increased. When 6 approaches infinity, it is reasonable that, according to Figure 7, the expected frequency will be twice the natural frequency o. if the average is taken with respect to zero. This corresponds to a zero time interval for the contact period, and hence simply a velocity jump, as has usually been assumed by some previous researchers. Consequently, the rigid constraint assumption will be true only when the contact period is negligible compared to the natural period of the system. DENSITY DISTRIBUTIONS 4.5. PEAK AND ENVELOPE PROBABILITY As information about the probability of mechanical failure, it is useful to find the peak and envelope probability densities of the response. For a linear system these two distributions are identical, but this is not generally true for non-linear systems. In the literature [12], these statistics have been presented for non-linear stiffness. It is thus necessary to separate the calculation in order to obtain a complete view of the peak and envelope densities. Since the mean of the solution is not zero, the statistics such as peak and envelope probability densities should be taken with respect to the mean. As the purpose of this paper is to investigate the influence of amplitude constraint on one side of a vibrating system, it is more relevant here, however, to regard the response from the same standpoint as in the case without constraint: that is to say, to calculate the statistics with respect to zero, not the mean, is more meaningful in this case.
370
H.-S. JING AND K.-C. SHEU
I
Time
Figure 7. Illustration of the change in the free vibration response of the undamped system as 5 goes from zero to infinity.
According to Powell [13], the relative frequency of peaks per unit time occurring between a and a + da as compared with the total number of peaks per unit time occurring at any positive amplitude is p,(a) da = -L
x da. vl da
The same concept can be used to count p,(a) in the x < 0 region. By substituting equations (10) and (17) into equation (20) and similarly for the x < 0 region one obtains a 5 a”’ _i+--- 5
11 2
-blexp
p,(c)=
UO O”
exp
x20 ,
---1 a2
2 u;I > a,( ail2
3
( -;$$), 0
(21)
xc0
1
is shown as a function of ,$ in Figure 8. When 5‘= 0, the system is linear, and both sides are Rayleigh distributions. As 5 increases, the probability of peaks occurring in the x > 0 region shifts closer to x = 0 and increases. On the other hand, p,(a) remains unchanged in the x < 0 region, although intuitively one might think it would be changed. By assuming that the envelope is a smooth, gradual curve joining the peaks and that the time spent between the levels a and a + da is the number of such peaks multiplied by the average period for cycles of amplitude a, the probability density of envelope can
pP( a)
EXACT
IMPACT
SYSTEM
1.50
RANDOM
371
RESPONSE
I
1.25
1.00 2 2
0.75
.z! 0.50
o-25
0.00 -4
Figure
8. Peak probability
-3
density
-2
-1
distributions
1
0
for different
2
3
values of the non-linearity
parameter
6.
be derived as [ 121
p,(a) =P,&bGw),
(22)
where T(U) is the period of undamped free vibration at amplitude a, which can be formulated by integrating the energy curve in the phase plane [14] as a r(a) = 2
i\i0
dx ~211~+~1~5/~_o~x~_-4-z!x5/~ 0 5m
+T 0
(23)
wO 5m
same concept can be used in the x < 0 region. By substituting equations (17); (20) and (23) into equation (22), pe( a) can be formulated separately for the x 2 0 and x < 0
The
1.50
1.25
1.00 ^o b 2 0.75 7
:
0.25
-4
Figure 9. Envelope
probability
-3
density
-2
-1
distributions
0
1
for different
2
3
values of the non-linearity
parameter
5.
372
H.-S. JING
AND
K.-C.
SHEU
regions as \
x20
pe(4 =4
b .
p
9
1
(24)
x-co J
From equation (24), it can be seen that when 6 = 0, p,(a) is a Rayleigh distribution. When 5 increases, p,(a) can be evaluated only numerically. In Figure 9 is shown p,(a) for both the x 3 0 and x < 0 regions as a function of 5. 5. CONCLUSIONS The stationary response of a single-degree-of-freedom vibro-impact system with a Hertz contact law and subjected to a random excitation has been found in closed form. Being unsymmetric, the response has to be formulated separately for the x s 0 and x < 0 regions. Several conclusions can be drawn from the results. (1) When the contact stiffness 6 increases, the mean value of the response will shift from zero, since the system is unsymmetric. The variance will decrease because the system is more rigid. (2) It is more relevant for those statistics such as the expected frequency, peak and envelope distributions to be taken with respect to zero, not the mean, in order to extract more information from the response. (3) The expected frequency of the response will increase up to a limiting value of two times the natural frequency of the corresponding linear system as 5 increases to infinity. (4) The assumption of velocity discontinuity in the phase plane made in the usual analysis of vibro-impact phenomenon is valid only when 5 approaches infinity: i.e., the contact stiffness is much larger than the stiffness of the linear system, or the contact period is much shorter than the natural period of the linear system. (5) Both the peak and envelope distributions of the response in the x > 0 region are compressed closer to x = 0 when e increases; actually, they are identical. On the left side, i.e., XCO, the peak distribution remains unchanged no matter how large 6 is, while the envelope distribution changes a little. It is interesting to note that the envelope distributions for 5 = 0 and 5 + cc are identical and equal to the peak distribution. ACKNOWLEDGMENTS The support of the National Science Council through Grant NSC77-0401-E006-19 is gratefully acknowledged.
EXACT
IMPACT
SYSTEM
RANDOM
RESPONSE
373
REFERENCES 1956 American Society of Mechanical Engineers, Journal of Applied Mechanics 23, 373-378. On the theory of the acceleration damper. 2. M. I. FEIGIN 1966 fikladnaia Matematika i Makhanika 30, 942-946. Resonance behavior of
1. C. GRUBIN
a dynamical system with collisions. 3. F. C. MOON and S. W. SHAW 1983 International Journal of Non-linear Mechanics l&465-477. Chaotic vibrations of a beam with non-linear boundary conditions. 4. S. W. SHAW 1985 American Society of Mechanical Engineers, Journal of Applied Mechanics 52, 453-464. The dynamics of a harmonically excited system having rigid amplitude constraints. 5. W. GOLDSMITH 1960 Impact. London: Edward Arnold. 6. P. R. NAYAK 1972 Journal of Sound and Vibration 22, 297-322. Contact vibrations. 1963 Journal of the Acoustical Society of America 35, 1683-1692. Derivation 7. T. K. CAUGHEY and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white random excitation. 8. S. H. CRANDALL 1962 American Society of Mechanical Engineers, Journal of Applied Mechanics 29, 477-482. Random vibration of a nonlinear system with a set-up spring 9. S. 0. RICE 1944 Bell System Technical Journal 23, 282-332. Mathematical analysis of random noise. 10. J. M. T. THOMPSON and R. GHAFFARI 1982 Physics letters 91A, 5-8. Chaos after perioddoubling bifurcations in the response of an impact oscillator. 11. J. M. T. THOMPSON and H. B. STEWART 1986 Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists. Chichester, England: John Wiley. 12. S. H. CRANDALL 1963Journalof the AcousticalSociety ofAmerica 35,1693-1699. Zero crossings, peaks, and other statistical measures of random responses. 13. A. POWELL 1958 Journal of the Acoustical Society of America 30, 1130-1135. On the fatigue failure of structures due to vibrations excited by random pressure fields. 14. J.J.STOKER 1950 Nonlinear Vibration. New York: Interscience.
of Sound
EXACT
and Vibrufion (1990) 141(3), 363-373
STATIONARY
RESPONSE
SOLUTIONS
OF THE
RANDOM
OF A SINGLE-DEGREE-OF-FREEDOM VIBRO-IMPACT
SYSTEM
H.-S. JING AND K.-C. SHEU Institute
of Aeronautics
and Astronautics, National Cheng Kung University, Republic of China
(Received
Tainan, Taiwan,
24 May 1989)
The problem of the response of a single-degree-of-freedom system with amplitude constraint on one side subjected to a random excitation is solved exactly. The Hertz law is used to model the contact phenomena between the mass and constraint during vibration. The excitation is limited to be a stationary white Gaussian process with zero mean. By solving the corresponding Fokker-Planck partial differential equation by separation of variables, the exact stationary solutions of the random response are obtained. The changes due to variations of contact stiffness are discussed. 1. INTRODUCTION There has been sustained interest in the non-linear response of a so-called vibro-impact system for many years [ 1,2]. A vibro-impact system is a vibration system with impact interaction. Most of the previous researchers have considered this impact interaction either as a bilinear spring [3], or by using a coefficient of restitution [4] to cover the velocity change after impact. From the analytical solution for the static contact between two elastic bodies with simple shapes [5], it is well known that the force-displacement relationship is non-linear. The contact force is proportional to the 3/2 power of the displacement. A reasonable extension of research along this line would be to incorporate this non-linear force-displacement relation into the analysis. This case has been studied by Nayak [6]. Unfortunately, only an approximate solution was formulated. In this paper, closed form solutions for the motion of a single-degree-of-freedom spring-mass-dashpot system with an elastic amplitude constraint on one side, which is modelled by the Hertz contact law, and subjected to a random excitation are obtained and the behavior of the response is discussed probabilistically. 2. DESCRIPTION
OF THE SYSTEM
A single-degree-of-freedom vibration system is considered. An elastic amplitude constraint is placed under the mass as shown in Figure 1. When the system is vibrating, the mass can move with limited amplitude as in a simple linear system, but for a finite amplitude, the elastic constraint will make its contribution. For simplicity, the constraint is modelled by a non-linear spring according to the Hertz contact law and the inertial effect is neglected. The resulting force-displacement relation, illustrated in Figure 2, clearly shows the non-linear and unsymmetric characteristics. The equation of motion is x312
mf+c~+kx+~g(x)=f(t),
g(x) = 363
0022-460X/90/180363
+ 11 %03.00/O
I
0 @J 1990
Academic
Press
Limited
364
H.-S. JING
Figure
AND
K.-C.
1. The physical
SHEU
model.
x
Figure
2. Force-displacement
relations
for different
values of contact
stiffness
11
Here 77is the contact stiffness, which is a function of the elastic properties and geometries of the two contacting bodies. Its values for several simple cases can be found in the literature [ 51. When the forcing function f( t) is a random process, the response x(t) can be described only in a statistical manner. In this analysis, the excitation is limited to be a stationary white Gaussian process with zero mean and the spectral density is taken to be S,,.
3. EXACT
SOLUTION
OF THE
FOKKER-PLANCK
EQUATION
To extract probabilistic information about the stationary response, it is necessary to find the joint probability density functionp(x, u), which, if the transition time is sufficiently long, satisfies the time-independent Fokker-Planck partial differential equation [7]
(2) Here w. and (Yare the frequency and damping parameters. The coefficients of this equation can be obtained directly from ensemble averages based on the equation of motion and
EXACT
IMPACT
SYSTEM
RANDOM
365
RESPONSE
the excitation. As can be found in the literature [8], the technique of separation of variables can be used if the displacement and velocity are statistically independent and V’/ v= -(2~/7&)U,
X’/X = -(2a/nS,)[o~x+(rl/m)g(x)].
(3,4)
The resulting joint density is
p(x,
u)
=
C exp
X2 V2 -----+j’g(i)dr}, &Y&J; 2~:
mo,w~
(5)
o
where the abbreviation a; = ?Tso/2ffw;
(6)
+m I=+m v) J p(x,
and C is the integration constant to be determined by the normalization
I --Q) --UC
condition
dx dv.
(7)
The integrations in equations (5) and (7) cannot be done analytically for most non-linear systems. In the problem studied here, the integration constant C can be obtained in closed form and is found to be
c-’ = vczu~ql where the non-dimensional
[J
‘+2 2
(-1)” 1 Z_(4n+2,,5r 4n+2 xQ: -5.=0 n! 2” ( 5 )I ’
contact stiffness, or non-linearity
(8)
parameter, 5, is given by
5 = 2nJa,/Smwi.
(9)
It should be noted that [ also contains the excitation information. The exact stationary solution of the Fokker-Planck partial differential equation for this problem is then
u2 _-___g (
exp &t&0, [J
X2
2c&;
2a;
u;‘2 >
(-1)” 1 5_(4n+Z,,Sr 4n+2 _n+2 c= -2 5n-o n! 2” ( 5
)3 ’
x
4.
PROBABILISTIC DESCRIPTIONS OF THE RESPONSE
4.1. PROBABILITY DENSITY FUNCTIONS From the preceding analysis, the separated first order probability density functions for
366
H.-S. JING
the displacement
AND
K.-C.
SHEU
pX(x) and for the velocity p,(u) are
x20
P,(X) =
>
1 x2
exp J
~vo+~cTo
( > --72 oo
xc0
1, 5-(4n+2j,5r 4n+2 ’ 2 ( 5 >
m (-1)”
c -n=O n!
PJv) =
(11)
G~o,o ev -( 2u;o;
>
V2
(12)
It is obvious that the probability density of the velocity is still Gaussian but the displacement is non-Gaussian except when 5 = 0: i.e., when the system is reduced to a linear one. The probability density functions of the displacement for different non-dimensional contact stiffnesses 5 are shown in Figure 3. Apparently, they are unsymmetric except when 5 = 0. As the stiffness of the constraint increases, the probability that the mass will be found in the region x > 0 decreases. When 5 approaches infinity, it is reasonable to expect that it would be impossible to find the mass in the x > 0 region at any time. _0.8 0.7 0.6 2 \
0.5
2
0.4 0.3 0.2 0.1 0.0
Figure 3. Stationary linearity parameter 6
-5
probability
-4
density
-3
-2
functions
-1
0
of response
1
2
amplitude
3
4
for different
5
values
of the non-
4.2. MEAN VALUE As can be seen in Figure 3, the mean value changes as 5 varies. From the definition of the mean, the exact form can be expressed as
Pu,= oo
(13)
EXACT
IMPACT
SYSTEM
RANDOM
367
RESPONSE
When the stiffness of the constraint is reduced to zero, i.e., the system is linear, it can be proved that pu, equals zero. On the other hand, when the contact stiffness is very large, the response mean approaches a finite value: (14)
lim CL,= -v?Z+~. 5-a In Figure 4 it is shown how the mean value changes contact stiffness.
with the non-dimensional
4.3. VARIANCE OF THE DISPLACEMENT By using the same procedures as before, the variance of the displacement be found exactly:
(+* X= a*0
a: can also
(19
--EL:.
It is plotted in Figure 5 as a function of 5. Again, when 5 approaches zero, the variance
Figure
4. Stationary
0.41 0
Figure
5. Stationary
mean value of response
1 12
variance
I
I 3
I 4
of response
amplitude
I 5 c amplitude
I 6
versus non-linearity
1 7
I 0
parameter
5.
1 9
versus non-linearity
parameter
g?
368
H.-S. JING
of the displacement finite value:
AND
K.-C. SHEU
is simply ai as expected. As 6 approaches infinity, a: approaches a lim of=&l-2/r). g+==
(16)
4.4. EXPECTED FREQUENCY Since it would be very difficult to solve for the second order statistics, it is appropriate instead to consider a quantity relevant to a narrow-band process which can supply information about the dominant frequency of the response; this is the expected number of zero-crossings with positive slope per unit time, or simply the expected frequency. For a process with a non-zero mean, one might think that the expected frequency should be considered to be the number of mean-crossings with positive slope per unit time. Although the mean is not zero, the key point here is that the process is non-symmetric. The positive portion of the process is compressed towards zero by the elastic constraint, resulting in the non-zero mean. As the process is unsymmetric, as shown in Figure 3, the mean is not the point with the highest probability density. This has the consequence that if only the number of mean-crossings with positive slope is counted, some information will be lost. Therefore, in what follows next, the expected frequency is taken with respect to zero as usual. Let ~0’be the expected number of zero-crossings with positive slope per unit time, as given, according to Rice [9], by substituting x = 0 into the equation 00 v:= up(x, v) dv. (17) I0 After using equation (9), integrating equation (17) and then normalizing by the natural frequency of the corresponding linear system oo/27r, the normalized expected frequency fz is found to be &G
.tT= ?r+2
J 2
m (-l)” I--
5 n=O n!
1 -(4n+2)/5~ s 2”
(18)
4n+2 ’ ( 5 )
It is plotted in Figure 6 as a function of the non-dimensional
contact stiffness 5.
1.6 1.5 -
01234567891
1 c
Figure 6. Expected
frequency
of stationary
response
versus non-linearity
parameter 6.
EXACT
1MPACT
SYSTEM
RANDOM
RESPONSE
369
When 5 approaches zero, it can be proved that the normalized expected frequency fi approaches 1. As 5 increases, fc also increases since the system is now becoming more rigid. Inspection of equations (6) and (9) shows that the non-dimensional contact stiffness 5 also includes the excitation information. An increase in the spectral density S, of the excitation corresponds to an increase of input energy, i.e., amplitude in some sense, and 6 will also increase. This results in an increase of fof. It is well known that the response of a non-linear system is also a function of the input amplitude. When the stiffness of the elastic constraint approaches infinity; then, physically, whenever the mass hits the rigid constraint, it will come back immediately. This limiting normalized expected frequency is found to be 2 simply by taking the limiting value of equation (18). This limiting case has also been studied by other researchers under the name “impact oscillator” [lo]. An impact oscillator can also be viewed as a limiting case of a vibration system with a bilinear spring: i.e., a spring with different stiffnesses for positive and negative displacements. It has been found [ 1l] that when one of the stiffnesses approaches infinity, the system becomes an impact oscillator, and the bilinear circular frequency is exactly twice the natural frequency of a corresponding system with the same stiffness for displacements to both sides which confirms the correctness of the results of this paper. On the other hand, if the expected frequency is taken with respect to the mean, one finds, after normalization,
f,z,=
JG
exp (-~Q2ai)
(19)
a (-1)” 1 5-(4n+2,,5r 4n+2 ’ lr+? I-J 2 5,=0 n! 2” ( 5 1
where CL, is given in equation (13). The normalized frequency is found to be l-4547 instead of 2 as 5 approaches infinity. As illustrated in Figure 7, the response in vibration without damping, if 5 is zero, is simply sinusoidal. As 5 increases, the time that the mass stays in the region x > 0 will be shorter since the constraint is more rigid. Hence fO+will be increased. When 6 approaches infinity, it is reasonable that, according to Figure 7, the expected frequency will be twice the natural frequency o. if the average is taken with respect to zero. This corresponds to a zero time interval for the contact period, and hence simply a velocity jump, as has usually been assumed by some previous researchers. Consequently, the rigid constraint assumption will be true only when the contact period is negligible compared to the natural period of the system. DENSITY DISTRIBUTIONS 4.5. PEAK AND ENVELOPE PROBABILITY As information about the probability of mechanical failure, it is useful to find the peak and envelope probability densities of the response. For a linear system these two distributions are identical, but this is not generally true for non-linear systems. In the literature [12], these statistics have been presented for non-linear stiffness. It is thus necessary to separate the calculation in order to obtain a complete view of the peak and envelope densities. Since the mean of the solution is not zero, the statistics such as peak and envelope probability densities should be taken with respect to the mean. As the purpose of this paper is to investigate the influence of amplitude constraint on one side of a vibrating system, it is more relevant here, however, to regard the response from the same standpoint as in the case without constraint: that is to say, to calculate the statistics with respect to zero, not the mean, is more meaningful in this case.
370
H.-S. JING AND K.-C. SHEU
I
Time
Figure 7. Illustration of the change in the free vibration response of the undamped system as 5 goes from zero to infinity.
According to Powell [13], the relative frequency of peaks per unit time occurring between a and a + da as compared with the total number of peaks per unit time occurring at any positive amplitude is p,(a) da = -L
x da. vl da
The same concept can be used to count p,(a) in the x < 0 region. By substituting equations (10) and (17) into equation (20) and similarly for the x < 0 region one obtains a 5 a”’ _i+--- 5
11 2
-blexp
p,(c)=
UO O”
exp
x20 ,
---1 a2
2 u;I > a,( ail2
3
( -;$$), 0
(21)
xc0
1
is shown as a function of ,$ in Figure 8. When 5‘= 0, the system is linear, and both sides are Rayleigh distributions. As 5 increases, the probability of peaks occurring in the x > 0 region shifts closer to x = 0 and increases. On the other hand, p,(a) remains unchanged in the x < 0 region, although intuitively one might think it would be changed. By assuming that the envelope is a smooth, gradual curve joining the peaks and that the time spent between the levels a and a + da is the number of such peaks multiplied by the average period for cycles of amplitude a, the probability density of envelope can
pP( a)
EXACT
IMPACT
SYSTEM
1.50
RANDOM
371
RESPONSE
I
1.25
1.00 2 2
0.75
.z! 0.50
o-25
0.00 -4
Figure
8. Peak probability
-3
density
-2
-1
distributions
1
0
for different
2
3
values of the non-linearity
parameter
6.
be derived as [ 121
p,(a) =P,&bGw),
(22)
where T(U) is the period of undamped free vibration at amplitude a, which can be formulated by integrating the energy curve in the phase plane [14] as a r(a) = 2
i\i0
dx ~211~+~1~5/~_o~x~_-4-z!x5/~ 0 5m
+T 0
(23)
wO 5m
same concept can be used in the x < 0 region. By substituting equations (17); (20) and (23) into equation (22), pe( a) can be formulated separately for the x 2 0 and x < 0
The
1.50
1.25
1.00 ^o b 2 0.75 7
:
0.25
-4
Figure 9. Envelope
probability
-3
density
-2
-1
distributions
0
1
for different
2
3
values of the non-linearity
parameter
5.
372
H.-S. JING
AND
K.-C.
SHEU
regions as \
x20
pe(4 =4
b .
p
9
1
(24)
x-co J
From equation (24), it can be seen that when 6 = 0, p,(a) is a Rayleigh distribution. When 5 increases, p,(a) can be evaluated only numerically. In Figure 9 is shown p,(a) for both the x 3 0 and x < 0 regions as a function of 5. 5. CONCLUSIONS The stationary response of a single-degree-of-freedom vibro-impact system with a Hertz contact law and subjected to a random excitation has been found in closed form. Being unsymmetric, the response has to be formulated separately for the x s 0 and x < 0 regions. Several conclusions can be drawn from the results. (1) When the contact stiffness 6 increases, the mean value of the response will shift from zero, since the system is unsymmetric. The variance will decrease because the system is more rigid. (2) It is more relevant for those statistics such as the expected frequency, peak and envelope distributions to be taken with respect to zero, not the mean, in order to extract more information from the response. (3) The expected frequency of the response will increase up to a limiting value of two times the natural frequency of the corresponding linear system as 5 increases to infinity. (4) The assumption of velocity discontinuity in the phase plane made in the usual analysis of vibro-impact phenomenon is valid only when 5 approaches infinity: i.e., the contact stiffness is much larger than the stiffness of the linear system, or the contact period is much shorter than the natural period of the linear system. (5) Both the peak and envelope distributions of the response in the x > 0 region are compressed closer to x = 0 when e increases; actually, they are identical. On the left side, i.e., XCO, the peak distribution remains unchanged no matter how large 6 is, while the envelope distribution changes a little. It is interesting to note that the envelope distributions for 5 = 0 and 5 + cc are identical and equal to the peak distribution. ACKNOWLEDGMENTS The support of the National Science Council through Grant NSC77-0401-E006-19 is gratefully acknowledged.
EXACT
IMPACT
SYSTEM
RANDOM
RESPONSE
373
REFERENCES 1956 American Society of Mechanical Engineers, Journal of Applied Mechanics 23, 373-378. On the theory of the acceleration damper. 2. M. I. FEIGIN 1966 fikladnaia Matematika i Makhanika 30, 942-946. Resonance behavior of
1. C. GRUBIN
a dynamical system with collisions. 3. F. C. MOON and S. W. SHAW 1983 International Journal of Non-linear Mechanics l&465-477. Chaotic vibrations of a beam with non-linear boundary conditions. 4. S. W. SHAW 1985 American Society of Mechanical Engineers, Journal of Applied Mechanics 52, 453-464. The dynamics of a harmonically excited system having rigid amplitude constraints. 5. W. GOLDSMITH 1960 Impact. London: Edward Arnold. 6. P. R. NAYAK 1972 Journal of Sound and Vibration 22, 297-322. Contact vibrations. 1963 Journal of the Acoustical Society of America 35, 1683-1692. Derivation 7. T. K. CAUGHEY and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white random excitation. 8. S. H. CRANDALL 1962 American Society of Mechanical Engineers, Journal of Applied Mechanics 29, 477-482. Random vibration of a nonlinear system with a set-up spring 9. S. 0. RICE 1944 Bell System Technical Journal 23, 282-332. Mathematical analysis of random noise. 10. J. M. T. THOMPSON and R. GHAFFARI 1982 Physics letters 91A, 5-8. Chaos after perioddoubling bifurcations in the response of an impact oscillator. 11. J. M. T. THOMPSON and H. B. STEWART 1986 Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists. Chichester, England: John Wiley. 12. S. H. CRANDALL 1963Journalof the AcousticalSociety ofAmerica 35,1693-1699. Zero crossings, peaks, and other statistical measures of random responses. 13. A. POWELL 1958 Journal of the Acoustical Society of America 30, 1130-1135. On the fatigue failure of structures due to vibrations excited by random pressure fields. 14. J.J.STOKER 1950 Nonlinear Vibration. New York: Interscience.