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Impact statistics for a simple random rattling system
Journal of Sound and Vibration (1987) 119(3), 529-543
IMPACT STATISTICS FOR A SIMPLE RANDOM RATTLING SYSTEM C. H. LEE AND
K. P. BYRNE
School of Mechanical and Industrial Engineering The University of New South Wales, Kensington, New South Wales 2033, Australia (Received 6 November 1986, and in revised form 24 March 1987)
A small mass (called the ball) constrained to move along a slot of fixed length in a large mass which is vibrating randomly in the direction of the slot is taken as a model of a randomly driven vibroimpact system. Numerical simulation analysis has been used to determine the statistics of the impacts between the rattling ball and the large randomly driven mass. These statistics include the probability densities of the magnitudes of the impacts and the times between impacts. The results presented provide fundamental information which can be used to estimate noise and wear in systems with clearances.
1. INTRODUCTION There has been sustained interest in vibroimpact systems for the last 20 years. This interest and use of vibroimpact systems. An indication is due in part to the common occurrence
of the occurrence and use of vibroimpact impact systems can be obtained from the references given in reference [l]. Although most of the work relating to vibroimpact systems has been concerned with sinusoidally driven vibroimpact systems, work described in references [2-41, has been concerned with randomly driven vibroimpact systems. Randomly driven vibroimpact systems can occur in transportation environments. A substantial body of literature relating to the response of randomly driven non-linear structures has been developed and many of the significant references in this area have been given by Lin [ 51. The techniques developed for computing the response of randomly driven non-linear structures cannot usually be applied to randomly driven vibroimpact systems, often because of the strong non-linearities in the equations of motion of vibroimpact systems. Hence, the randomly driven vibroimpact systems discussed in the literature to date generally have been very simple systems. For example, the randomly driven system considered by Wood and Byrne [3] consisted of a ball which bounced on a massive randomly vibrating table. Further, for the analysis of these simple vibroimpact systems to be tractable it has been necessary to make several potentially restrictive assumptions. The present paper is concerned with a more realistic randomly driven vibroimpact system. The system of interest here consists of a small mass (subsequently called the ball) which is constrained to move along a slot of fixed length in a large mass which is vibrating randomly in the direction of the slot. The random rattling of the ball in the slot is of interest. This system can be used to model practical systems. An example of such a system is a pin which rattles in a clearance hole. The aim of the analysis is to determine the statistics of the impacts between the rattling ball and the large randomly driven mass. These statistics include the probability densities of the magnitudes of the impacts and the times between impacts. Such statistics provide fundamental information which can be used to estimate noise and wear in systems with clearances. 0022-460X/87/240529+ 15 $03.00/O
@ 1987 Pcademic
Press Limited
530
C.
H. LEE
AND
K. P. BYRNE
It is necessary to resort to numerical simulation to investigate this system if assumptions which simplify the analysis are not used. Thus for example, the assumption used by Wood and Byrne [3] that the ball velocity always changes direction after an impact need not be applied with a numerical simulation. The numerical simulation of this randomly driven vibroimpact system is intended to generate useful information which quantifies the effects of parameters such as clearance and coefficient of restitution on the impact statistics. This information is intended to provide designers with useful practical information to aid design decisions. The numerical simulation is also intended to produce results with which the results derived by assumption restricted analytic techniques, such as used in [3], can be compared. 2. MODEL The physical model of the vibroimpact system of interest here is shown in Figure 1. The displacement of the randomly driven mass is denoted x(t) and the displacement of the rattling ball is denoted y(t). The clearance between the rattling ball and the randomly driven mass is d. The velocity of the randomly driven mass is denoted i(t) and the velocity of the rattling ball is denoted i(t). Consider the ith impact which occurs at time ti. The velocities of the ball immediately before and after the ith impact, jti and $, and the velocity of the randomly driven mass at the ith impact, ii, can be related by equation (1) if it is assumed that the randomly driven mass is large and impact is characterized by the coefficient of restitution, e:
(1)
_$:= -e);i + (I+ e)i,.
If it is further assumed that the ball velocity does not change between impacts, the ball velocity after the (i - 1)th impact, jlLl, is the same as the ball velocity before the ith impact, ii. Thus equation (1) can be rewritten as
This equation is similar to equation (3) in reference [3]. However, it is of interest to note that equation (2) does not have the same implied restrictions as equation (3) in the previously cited paper if successive impacts are allowed to occur at the same end of the slot. It was assumed in developing equation (3) in the previously cited paper that the direction of the ball velocity was reversed after impact. This assumption is better satisfied as e, the coefficient of restitution, approaches unity. In the analysis given in reference [3] to determine the probability density of the ball velocity after impact, in addition to requiring the preceding assumption, it also was assumed that the displacement of the
Dtrection *
Figure
1. Model
of vibration t
of the rattling
system.
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table was small when compared with the displacement of the ball so that there was no correlation between the table velocities at successive impacts. This assumption will obviously be unsatisfactory in some of the cases of interest here when the clearance between the ball and the slot in the randomly driven mass is very small. Thus it is necessary to consider the displacements of the ball and the randomly driven mass. Impacts will occur when the magnitude of the displacement of the ball relative to the randomly driven mass is equal to d/2. Since the ball velocity between impacts is assumed to be constant, the time at which the ith impact occurs, fi, must satisfy the equation (3)
Iyi_,+j,:~,!li_ti_l)-X,l=d/7.
Once fi is known, jl can be found from equation (2). Thus once the displacement and velocity time histories of the randomly driven mass are known it is a simple task to determine the ball velocity after each impact and the time between successive impacts. The statistics of these quantities and associated quantities such as the approach or closing velocity can be determined. The approach velocity is the velocity of the ball relative to the randomly driven mass just before impact. Thus the approach velocity associated with the ith impact is given by $ -l;. The approach velocity determines the magnitude of the impact between the ball and the randomly driven mass. Although this process is not elegant it is very simple to implement and it allows the effect of various parameters of the vibroimpact system such as the clearance to be determined. The vibroimpact system of interest here was driven so that the randomly driven mass containing the slot had a velocity spectrum which was white between 0 and w,. The random time history corresponding to this motion was generated by the method described by Shinozuka and Jan [6]. The random time history i(t) with a desired power spectral density S,(ok) at N equally spaced angular frequencies ok = Ao(k - l/2), where do = o, / N can be generated from the equation
i;.(f) =a
;
[S,(w,)Aw]“’
cos (w;t+
(4)
&).
k=l
The term wl, in this equation is given by w;=wk+&.
(5)
&J is a random variable which is uniformly distributed over the range 10.05 Aw. It is introduced to avoid periodicity in the simulated process. The term & which appears in equation (4) is also a random variable which is uniformly distributed over the range 0 to 27r. It was noted in the previous section that the analysis is performed in terms of the displacement and velocity time histories. The displacement time history can be readily generated from the velocity time history defined by equation (4) by integration of the cos terms in equation (4). It is convenient to non-dimensionalize equations (2), (3) and (4) before proceeding further. The velocities can be non-dimensionalized by dividing by a, the R.M.S. velocity of the randomly driven mass containing the slot. It is evident that here CTis given by rs0(W)~Ul”2. The displacements can be non-dimensionalized in terms of the clearance d and time can be non-dimensionalized in terms of w,, the upper frequency of the spectrum of the motion of the randomly driven mass. Thus equations (2), (3) and (4) become ?;“=-e$_,+(l+e)l*,
i*(r*)=m
I$+_,+ (a/dw,)j:_,(
g k=l
cos
r: - tT_1) -x”l
(orr*+$bk),
= ;,
(697)
532
C. H. LEE AND
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where x*=xld,i*=~-/u,yy”=y/d,jl”=y’lu,w*=w/w, and t*=tw,. The nondimensional displacement x*( t*) corresponding to the non-dimensional velocity x*( t*) of the randomly driven mass can be derived from equation (8). It is given by x*(l*)=-&
1.
1
JzINki,$sin(wrr*+&)
u
(9)
The inverse of the term u/do,
which appears in equations (7) and (9) forms an amplitude scaling parameter in equation (9), defined by R = dw,/a.
(10)
3. ANALYSIS AND RESULTS
Equations (8) and (9) were used to generate the non-dimensional velocity and the non-dimensional displacement of the randomly driven mass at any value of the nondimensional time for a motion whose velocity spectrum was white over the nondimensional frequency range 0 to 1. One hundred terms were used in evaluating equations (8) and (9). The random non-dimensional frequency perturbation and the random phases were generated by a random number subroutine. Figure 2 shows the velocity and displacement spectra derived by analysis of the generated time histories. The displacement spectrum shown in Figure 2(b) is that derived by evaluation of the bracketed term in equation (9): that is, for R = 1. The non-dimensional displacements, velocities and times at impact could then be determined from equations (6) and (7) for particular values of e and R. An example of the time histories of the non-dimensional displacements and velocities of both the randomly driven mass and the rattling ball are shown in Figure 3
I
150
I
Non-dlmensionol
Figure 2. Power spectral density (b) of the randomly driven mass.
of the non-dimensional
I
I
(b)
frequency
velocity
(a) and the non-dimensional
displacement
IMPACT
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c
(b)
t
0
(d)
‘d
I 100
200
300
400
500
200
100
0
Non-dlmenslonol
300
400
500
ltme
Figure 3. Sample time histories of the non-dimensional velocity and the non-dimensional the randomly driven mass, (a) and (b), and the ball (c) and (d). e = 0.9 and R = 10.0.
displacement
of
for e = 0.9 and R = 10.0. It can be seen from Figure 3(c) that there are several occurrences of the situation in which the ball velocity does not reverse after an impact. The values of quantities such as the ball velocity at impact were determined for 10 000 impacts at each end of the randomly driven mass containing the slot so that the impact statistics could be established. The normalized frequency of occurrence data, for the non-dimensional ball velocity after impact at the left-hand end of the slotted randomly driven mass shown in Figure 1, are plotted in Figure 4 for the case e = 0.9 and R = 10.0.This data was derived by numerical simulation. Useful descriptors of the non-dimensional ball velocity after impact include the mean and root mean square values of the normalized velocity and these are also shown on Figure 4. It was found that a modified log-normal probability density
0 -5
0
5 Non-dimensional
Figure 4. Normalized frequency of occurrence for the non-dimensional ball velocity after impact. o, = 0.076 and a = 16.7.
I
I
10
15
20
velocity
data and the fitted log-normal probability density function e = 0.9 and R = 10.0. Mean = 2.97, R.M.S. = 3.32. CL,= 2.98,
534
C. H. LEE AND
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function could be fitted to the simulation derived data over the range of e and R values of interest. The equation which defines the modified log-normal probability density function is defined by f(x) = {l/(x + a)syfi}
e-(‘/2)[~‘og(x+a’-~~}/rr~l’.
(11)
The conventional log-normal probability density function has been modified by the shift parameter a. The fitted probability density curve corresponding to the simulation derived data shown in Figure 4 is also shown in Figure 4 along with the values of the parameters a, aY and CL_” which appear in equation (11). The normalized frequency of occurrence data, for the non-dimensional velocity of the slotted randomly driven mass when impact occurs at the left-hand end of the system shown in Figure 1, are plotted in Figure 5. The mean and root mean square values of the non-dimensional velocity are also given. It was found that the normal probability density function could be fitted to the simulation derived data over the range of e and R values of interest. The normal probability density function here is defined as ,-(u2Kcx-w1’. f(x) = (l/o&) (12)
The fitted probability density curve and its parameters are also shown in Figure 5. It is not surprising that a normal distribution can be used to fit this data as the motion of the randomly driven mass which contains the slot is also normal. However, as shown in reference [3], the distribution of the velocity of the randomly driven mass when impacts occur is slightly different from the distribution of the randomly driven mass at arbitrary sampling times. In particular, the former distribution has a non-zero mean while the mean for the latter distribution is zero. This non-zero mean is, of course, essential so that energy can be introduced to sustain the process. 0.6
0 -4
-3
-2
-1
0
Non-dimensional
1
2
3
4
velocity
Figure 5. Normalized frequency of occurrence data and the fitted normal probability density function for the non-dimensional velocity of the randomly driven mass at impact. e = 0.9 and R = 10.0. Mean = 0.16, R.M.S. = 0.97, y = 0.16, (I = 0.97.
The closing velocity and the time between impacts are also of interest. These quentities are always greater than or equal to zero and it would be expected that versatile distributions such as the Gamma and Weibull distributions could be fitted to the simulation derived data for these quantities. However, it was found that a single distribution function could not be fitted to the simulation derived data over the full range of e and R values of interest. Thus the simulation derived results are presented as frequency of occurrence data. The results relating to the non-dimensional approach or closing velocity are shown
IMPACT
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535
in Figure 6 and those relating to the non-dimensional times between impacts for the case e = 0.9 and R = 10.0 are shown in Figure 7. The difficulty of fitting simple distributions to the closing velocity and time between impacts data can be seen by considering Figure 8. The problem is particularly severe with regard to the non-dimensional time between impacts. The basic shape of the curves is altered as e, the coefficient of restitution, is decreased. It is to be expected, on physical grounds, that the mean of the ball velocity after impact will be reduced if e is reduced while R is kept constant. When the ball is moving more slowly, there is a greater chance of successive impacts occurring at the same end of the slot in the randomly driven mass. The normalized times between impacts associated with such impacts will be much smaller than those in which impacts occur at the opposite ends of the slot. This accounts for the significant peaks in the data of Figure 8(c) and (d) at small normalized time between impacts for e = 0.7. It should be noted that the data in Figures 4, 5 and 6 refer to impacts at one end of the slotted randomly driven mass where the data given in Figure 7 refers to impacts at both ends of the slot. The fitted log-normal and normal distributions for the primary quantities, the nondimensional velocity of the ball after impact and that of the randomly driven mass at impact, are shown in Figure 9. It can be seen from the distribution curves for the velocity of the randomly driven mass at impact that the curves shift to the left as e is increased. Thus the mean velocity of the randomly driven mass at impact decreases as e increases.
Non-dlmewonol
Figure 6. Normalized
frequency of occurrence R = 10.0. Mean = 3.08, R.M.S. = 3.32.
data
Non-dlmensionol
Figure 7. Normalized frequency of occurrence and R = 10.0. Mean =4.80, R.M.S. = 6.15.
velocity
for the non-dimensional
closing
velocity.
e = 0.9 and
hme
data for the non-dimensional
time between
impacts.
e = 0.9
536
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IMPACT
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FOR
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537
SYSTEM
0.4
,”
0.3
h E 0 :: 0
0.2
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STATISTICS
0.1
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0
5
IO
I5
20-5
Non-dlmensionol
-4
-3
-2
-I
0
I
2
3
4
5
velocity
Figure 9. Fitted log-normal distributions for the non-dimensional ball velocity (b) e = 0.8 and (c) e = 0.9 and fitted normal distributions for the non-dimensional driven mass at impact (d) e = 0,7, (e) e = 0.8 and (f) e = 0.9.
after impact (a) e=0,7, velocity of the randomly
This is, of course, to be expected on the basis of a long-term energy balance. The mean and R.M.S. values of the non-dimensional velocity of the ball after impact can be derived from the data used to fit the distributions such as those shown in Figure 9. These non-dimensional velocities are plotted against R with e as a parameter as in Figure 10. The non-dimensional mean impact velocities of the randomly driven mass are shown in Figure 11. It can be seen from Figure 10 that the ball velocities after impact are high when the coefficient of restitution is high. The reverse applies to the impact velocities of the randomly driven mass. This situation is to be expected as high values of e are associated with small energy loss. The non-dimensional R.M.S. impact velocities of the randomly driven mass are close to 1 for all values of e and R and so are not plotted. The mean and R.M.S. non-dimensional closing velocities are plotted in Figure 12 and the mean and R.M.S. values of the non-dimensional time between impacts are plotted on Figure 13. It can be seen from Figures lo-12 that the mean and RMS non-dimensional impact velocities of both the ball and the randomly driven mass stabilize as R increases. The value of R at which the velocities stabilize depends on the coefficient of restitution, e. The stabilization occurs at smaller values of R as e decreases. Generally, the mean and R.M.S. values of these velocities do not vary greatly once R > 100.R can be interpreted
538
C. H. LEE
AND
K. P. BYRNE
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---
5.0 4.0
~-J+=WWC-&A..%-._...
3.0
_~~~ooo-___o_d_____o-___
2.0 I.0 0 IO.0 9.0 El.0 7.0 6.0 5.0 4.0
A_&-.---. _~____~~____~____
3.0 2.0 I.0 0
I
2
5
IO
50
20
100
200
500
1000
R
Figure x
10. Mean (a) and RMS (b) non-dimensional
velocity
of the ball after impact
0 e = 0.7; & e = 0.8;
) e = 0.9, 0, e = 0.95.
0.5
.-mx-&--A-n--_h-
0.4
0.3
?G-,.-,,_,___x
.._..
0.2
0.1
0
I
2
5
IO
20
50
100
200
500
1000
R
Figure x
11. Mean
non-dimensional
velocity
of the randomly
driven
mass at impact.
0,
e = 0.7; A, e =0.8;
( e = 0.9; cl, e = 0.95.
as the ratio of the clearance (d) to a measure of the displacement of the randomly driven mass (a/w,). Thus when the clearance, d, is about 100 times the measure of the displacement of the randomly driven mass, u/w,, the motion of the ball becomes weakly correlated with the motion of the randomly driven mass. When this same interpretation is placed on R it is evident that as R approaches zero the clearance between the ball and the randomly driven mass approaches zero. Thus it would be expected that the ball velocity after impact would approach the velocity of the randomly driven mass. It can be seen from Figure IO(b) that the non-dimensional velocity approaches 1 as R approaches zero as expected. It can be seen also from Figures 9(a), (b) and (c) that for small values of R
IMPACT
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539
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540
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0.3
H.
LEE
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K. P. BYRNE
I
I
I
I I
(0)
0.3
I
I (b)
I
I
Non-damenslonal
Figure 14. Comparison frequency of occurrence (b) e = 0.8, (c) e = 0.9.
velocity
of the theoretical probability density functions with the simulation derived normalized data for the non-dimensional ball velocity after impact. R = 100.0 (a) e = 0.7,
IMPACT
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541
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0.6
0.5 A r
0.4
: < 1 n
0.3
2 e a
0.2
0.1
0 0.6
,
,
,
,
,
,
,
I,
I
,
I
,
1 (cl
0.5-
Non-dtmenslonol
veloctty
of the theoretical probability density functions with the sikdation derived normalized Figure 15. Comparison frequency of occurrence data for the non-dimensional velocity of the randomly driven mass at impact. R = 100.0 (a) e = 0.7, (b) e = 0.8, (c) e = 0.9.
542
C. Ii. LEE AND
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the distribution is almost Gaussian. This is expected as the ball motion has a strong correlation with the motion of the randomly driven mass. When R is large the ball motion is weakly correlated with the motion of the randomly driven mass and the distribution would be expected to be close to the distribution developed from the theory in [3]. It is of interest to compare the simulation derived results for large values of R with the theoretical results derived by solution of the integral equation developed in reference [3]. Figure 14 shows the theoretically derived and simulation derived results for the nondimensional ball velocity after impact for R = 100. Since the theoretical results were based on a model for which it was assumed that the ball velocity was always reversed after an impact, the obvious discrepancies at negative values of the non-dimensional ball velocity after impact are expected. However, at positive velocities the agreement is good. The
Figure 16. Comparison frequency of occurrence (c) e ~0.9.
of the theoretical probability density functions with the simulation derived normalized data for the non-dimensional closing velocity. R = 100.0 (a) e = 0.7, (b) e = 0.8,
IMPACT STATISTICS
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543
comparison of the simulation derived data for the table velocities with the theoretically derived velocities for e = 0*7,0*8, and 0.9 is shown in Figure 15. The comparison of closing velocities is shown in Figure 16. It can be seen, that these two sets of curves show a good agreement. The theoretically derived and the simulation derived non-dimensional mean ball velocities after impact are shown as a function of e in Figure 17. It can be seen that the mean ball velocities derived by simulation converge to the theoretically derived data as R increases and the agreement is also good when R becomes large (R > 100).
I
0.6
I
I
I
0.7
0.8
0.9
,
I.0
e
Figure 17. Non-dimensional
mean velocities ofthe ball after impact along with the theoretically derived results.
4. CONCLUSIONS The equations which govern a simple but practically important random rattling system have been formulated and non-dimensionalized so that they are in convenient form. The system, as a result, can be described by two parameters, the coefficient of restitution, c, and a parameter, R, which incorporates the clearance, d, the R.M.S. velocity of the randomly driven mass, C, and the upper frequency, o,, of the spectrum of the velocity of the randomly driven mass. These equations have been solved numerically to yield the impact statistics. The impact statistics are given as functions of the parameters e and R. These impact statistics enable practically useful quantities such as the mean time between impacts and the mean magnitude of the impacts to be found. These useful quantities can be obtained from Figures 10-13. A comparison has been made between simulation derived results and results derived from the theoretical model of Wood and Byrne [3]. It can be concluded from this comparison between the impact statistics derived by simulation and those derived by the theoretical method that if R > 100 there is good agreement.
REFERENCES 1. C. N. BAPAT and N. POPPLEWELL 1983 Journal of Sound and Vibration 87, 19-40. Stable periodic motions of an impact-pair. 2. S. F. MASRI and A. M. ~BRAHIM 1973 Journal of the Acoustical Society of America 53,200-211. Response of the impact damper to stationary random excitation. _. : L. A. WOOD and K. P. BYRNE 1981 Journal of Sound and Vibration 78, 329-345.
Analysis of a random repeated impact process. 4. L. A. WOOD and K. P. BYRNE 1982 Journal of Sound and Vibration 86, 53-59. Experimental investigation of a random repeated impact process. 5. Y. K. LIN 1967 Probabilitic Theory of Structural Dynamics. New York: McGraw-Hill. 6. M. SHINOZUKA and C. M. JAN 1972 Journal of Sound and Vibration 25, 111-128. Digital simulation of random process and its applications.
IMPACT STATISTICS FOR A SIMPLE RANDOM RATTLING SYSTEM C. H. LEE AND
K. P. BYRNE
School of Mechanical and Industrial Engineering The University of New South Wales, Kensington, New South Wales 2033, Australia (Received 6 November 1986, and in revised form 24 March 1987)
A small mass (called the ball) constrained to move along a slot of fixed length in a large mass which is vibrating randomly in the direction of the slot is taken as a model of a randomly driven vibroimpact system. Numerical simulation analysis has been used to determine the statistics of the impacts between the rattling ball and the large randomly driven mass. These statistics include the probability densities of the magnitudes of the impacts and the times between impacts. The results presented provide fundamental information which can be used to estimate noise and wear in systems with clearances.
1. INTRODUCTION There has been sustained interest in vibroimpact systems for the last 20 years. This interest and use of vibroimpact systems. An indication is due in part to the common occurrence
of the occurrence and use of vibroimpact impact systems can be obtained from the references given in reference [l]. Although most of the work relating to vibroimpact systems has been concerned with sinusoidally driven vibroimpact systems, work described in references [2-41, has been concerned with randomly driven vibroimpact systems. Randomly driven vibroimpact systems can occur in transportation environments. A substantial body of literature relating to the response of randomly driven non-linear structures has been developed and many of the significant references in this area have been given by Lin [ 51. The techniques developed for computing the response of randomly driven non-linear structures cannot usually be applied to randomly driven vibroimpact systems, often because of the strong non-linearities in the equations of motion of vibroimpact systems. Hence, the randomly driven vibroimpact systems discussed in the literature to date generally have been very simple systems. For example, the randomly driven system considered by Wood and Byrne [3] consisted of a ball which bounced on a massive randomly vibrating table. Further, for the analysis of these simple vibroimpact systems to be tractable it has been necessary to make several potentially restrictive assumptions. The present paper is concerned with a more realistic randomly driven vibroimpact system. The system of interest here consists of a small mass (subsequently called the ball) which is constrained to move along a slot of fixed length in a large mass which is vibrating randomly in the direction of the slot. The random rattling of the ball in the slot is of interest. This system can be used to model practical systems. An example of such a system is a pin which rattles in a clearance hole. The aim of the analysis is to determine the statistics of the impacts between the rattling ball and the large randomly driven mass. These statistics include the probability densities of the magnitudes of the impacts and the times between impacts. Such statistics provide fundamental information which can be used to estimate noise and wear in systems with clearances. 0022-460X/87/240529+ 15 $03.00/O
@ 1987 Pcademic
Press Limited
530
C.
H. LEE
AND
K. P. BYRNE
It is necessary to resort to numerical simulation to investigate this system if assumptions which simplify the analysis are not used. Thus for example, the assumption used by Wood and Byrne [3] that the ball velocity always changes direction after an impact need not be applied with a numerical simulation. The numerical simulation of this randomly driven vibroimpact system is intended to generate useful information which quantifies the effects of parameters such as clearance and coefficient of restitution on the impact statistics. This information is intended to provide designers with useful practical information to aid design decisions. The numerical simulation is also intended to produce results with which the results derived by assumption restricted analytic techniques, such as used in [3], can be compared. 2. MODEL The physical model of the vibroimpact system of interest here is shown in Figure 1. The displacement of the randomly driven mass is denoted x(t) and the displacement of the rattling ball is denoted y(t). The clearance between the rattling ball and the randomly driven mass is d. The velocity of the randomly driven mass is denoted i(t) and the velocity of the rattling ball is denoted i(t). Consider the ith impact which occurs at time ti. The velocities of the ball immediately before and after the ith impact, jti and $, and the velocity of the randomly driven mass at the ith impact, ii, can be related by equation (1) if it is assumed that the randomly driven mass is large and impact is characterized by the coefficient of restitution, e:
(1)
_$:= -e);i + (I+ e)i,.
If it is further assumed that the ball velocity does not change between impacts, the ball velocity after the (i - 1)th impact, jlLl, is the same as the ball velocity before the ith impact, ii. Thus equation (1) can be rewritten as
This equation is similar to equation (3) in reference [3]. However, it is of interest to note that equation (2) does not have the same implied restrictions as equation (3) in the previously cited paper if successive impacts are allowed to occur at the same end of the slot. It was assumed in developing equation (3) in the previously cited paper that the direction of the ball velocity was reversed after impact. This assumption is better satisfied as e, the coefficient of restitution, approaches unity. In the analysis given in reference [3] to determine the probability density of the ball velocity after impact, in addition to requiring the preceding assumption, it also was assumed that the displacement of the
Dtrection *
Figure
1. Model
of vibration t
of the rattling
system.
IMPACT
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FOR
RATTLING
531
SYSTE.M
table was small when compared with the displacement of the ball so that there was no correlation between the table velocities at successive impacts. This assumption will obviously be unsatisfactory in some of the cases of interest here when the clearance between the ball and the slot in the randomly driven mass is very small. Thus it is necessary to consider the displacements of the ball and the randomly driven mass. Impacts will occur when the magnitude of the displacement of the ball relative to the randomly driven mass is equal to d/2. Since the ball velocity between impacts is assumed to be constant, the time at which the ith impact occurs, fi, must satisfy the equation (3)
Iyi_,+j,:~,!li_ti_l)-X,l=d/7.
Once fi is known, jl can be found from equation (2). Thus once the displacement and velocity time histories of the randomly driven mass are known it is a simple task to determine the ball velocity after each impact and the time between successive impacts. The statistics of these quantities and associated quantities such as the approach or closing velocity can be determined. The approach velocity is the velocity of the ball relative to the randomly driven mass just before impact. Thus the approach velocity associated with the ith impact is given by $ -l;. The approach velocity determines the magnitude of the impact between the ball and the randomly driven mass. Although this process is not elegant it is very simple to implement and it allows the effect of various parameters of the vibroimpact system such as the clearance to be determined. The vibroimpact system of interest here was driven so that the randomly driven mass containing the slot had a velocity spectrum which was white between 0 and w,. The random time history corresponding to this motion was generated by the method described by Shinozuka and Jan [6]. The random time history i(t) with a desired power spectral density S,(ok) at N equally spaced angular frequencies ok = Ao(k - l/2), where do = o, / N can be generated from the equation
i;.(f) =a
;
[S,(w,)Aw]“’
cos (w;t+
(4)
&).
k=l
The term wl, in this equation is given by w;=wk+&.
(5)
&J is a random variable which is uniformly distributed over the range 10.05 Aw. It is introduced to avoid periodicity in the simulated process. The term & which appears in equation (4) is also a random variable which is uniformly distributed over the range 0 to 27r. It was noted in the previous section that the analysis is performed in terms of the displacement and velocity time histories. The displacement time history can be readily generated from the velocity time history defined by equation (4) by integration of the cos terms in equation (4). It is convenient to non-dimensionalize equations (2), (3) and (4) before proceeding further. The velocities can be non-dimensionalized by dividing by a, the R.M.S. velocity of the randomly driven mass containing the slot. It is evident that here CTis given by rs0(W)~Ul”2. The displacements can be non-dimensionalized in terms of the clearance d and time can be non-dimensionalized in terms of w,, the upper frequency of the spectrum of the motion of the randomly driven mass. Thus equations (2), (3) and (4) become ?;“=-e$_,+(l+e)l*,
i*(r*)=m
I$+_,+ (a/dw,)j:_,(
g k=l
cos
r: - tT_1) -x”l
(orr*+$bk),
= ;,
(697)
532
C. H. LEE AND
K. P. BYRNE
where x*=xld,i*=~-/u,yy”=y/d,jl”=y’lu,w*=w/w, and t*=tw,. The nondimensional displacement x*( t*) corresponding to the non-dimensional velocity x*( t*) of the randomly driven mass can be derived from equation (8). It is given by x*(l*)=-&
1.
1
JzINki,$sin(wrr*+&)
u
(9)
The inverse of the term u/do,
which appears in equations (7) and (9) forms an amplitude scaling parameter in equation (9), defined by R = dw,/a.
(10)
3. ANALYSIS AND RESULTS
Equations (8) and (9) were used to generate the non-dimensional velocity and the non-dimensional displacement of the randomly driven mass at any value of the nondimensional time for a motion whose velocity spectrum was white over the nondimensional frequency range 0 to 1. One hundred terms were used in evaluating equations (8) and (9). The random non-dimensional frequency perturbation and the random phases were generated by a random number subroutine. Figure 2 shows the velocity and displacement spectra derived by analysis of the generated time histories. The displacement spectrum shown in Figure 2(b) is that derived by evaluation of the bracketed term in equation (9): that is, for R = 1. The non-dimensional displacements, velocities and times at impact could then be determined from equations (6) and (7) for particular values of e and R. An example of the time histories of the non-dimensional displacements and velocities of both the randomly driven mass and the rattling ball are shown in Figure 3
I
150
I
Non-dlmensionol
Figure 2. Power spectral density (b) of the randomly driven mass.
of the non-dimensional
I
I
(b)
frequency
velocity
(a) and the non-dimensional
displacement
IMPACT
STATISTICS
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RATTLING
533
SYSTEM
c
(b)
t
0
(d)
‘d
I 100
200
300
400
500
200
100
0
Non-dlmenslonol
300
400
500
ltme
Figure 3. Sample time histories of the non-dimensional velocity and the non-dimensional the randomly driven mass, (a) and (b), and the ball (c) and (d). e = 0.9 and R = 10.0.
displacement
of
for e = 0.9 and R = 10.0. It can be seen from Figure 3(c) that there are several occurrences of the situation in which the ball velocity does not reverse after an impact. The values of quantities such as the ball velocity at impact were determined for 10 000 impacts at each end of the randomly driven mass containing the slot so that the impact statistics could be established. The normalized frequency of occurrence data, for the non-dimensional ball velocity after impact at the left-hand end of the slotted randomly driven mass shown in Figure 1, are plotted in Figure 4 for the case e = 0.9 and R = 10.0.This data was derived by numerical simulation. Useful descriptors of the non-dimensional ball velocity after impact include the mean and root mean square values of the normalized velocity and these are also shown on Figure 4. It was found that a modified log-normal probability density
0 -5
0
5 Non-dimensional
Figure 4. Normalized frequency of occurrence for the non-dimensional ball velocity after impact. o, = 0.076 and a = 16.7.
I
I
10
15
20
velocity
data and the fitted log-normal probability density function e = 0.9 and R = 10.0. Mean = 2.97, R.M.S. = 3.32. CL,= 2.98,
534
C. H. LEE AND
K. P. BYRNE
function could be fitted to the simulation derived data over the range of e and R values of interest. The equation which defines the modified log-normal probability density function is defined by f(x) = {l/(x + a)syfi}
e-(‘/2)[~‘og(x+a’-~~}/rr~l’.
(11)
The conventional log-normal probability density function has been modified by the shift parameter a. The fitted probability density curve corresponding to the simulation derived data shown in Figure 4 is also shown in Figure 4 along with the values of the parameters a, aY and CL_” which appear in equation (11). The normalized frequency of occurrence data, for the non-dimensional velocity of the slotted randomly driven mass when impact occurs at the left-hand end of the system shown in Figure 1, are plotted in Figure 5. The mean and root mean square values of the non-dimensional velocity are also given. It was found that the normal probability density function could be fitted to the simulation derived data over the range of e and R values of interest. The normal probability density function here is defined as ,-(u2Kcx-w1’. f(x) = (l/o&) (12)
The fitted probability density curve and its parameters are also shown in Figure 5. It is not surprising that a normal distribution can be used to fit this data as the motion of the randomly driven mass which contains the slot is also normal. However, as shown in reference [3], the distribution of the velocity of the randomly driven mass when impacts occur is slightly different from the distribution of the randomly driven mass at arbitrary sampling times. In particular, the former distribution has a non-zero mean while the mean for the latter distribution is zero. This non-zero mean is, of course, essential so that energy can be introduced to sustain the process. 0.6
0 -4
-3
-2
-1
0
Non-dimensional
1
2
3
4
velocity
Figure 5. Normalized frequency of occurrence data and the fitted normal probability density function for the non-dimensional velocity of the randomly driven mass at impact. e = 0.9 and R = 10.0. Mean = 0.16, R.M.S. = 0.97, y = 0.16, (I = 0.97.
The closing velocity and the time between impacts are also of interest. These quentities are always greater than or equal to zero and it would be expected that versatile distributions such as the Gamma and Weibull distributions could be fitted to the simulation derived data for these quantities. However, it was found that a single distribution function could not be fitted to the simulation derived data over the full range of e and R values of interest. Thus the simulation derived results are presented as frequency of occurrence data. The results relating to the non-dimensional approach or closing velocity are shown
IMPACT
STATISTICS
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RATTLING
SYSTEM
535
in Figure 6 and those relating to the non-dimensional times between impacts for the case e = 0.9 and R = 10.0 are shown in Figure 7. The difficulty of fitting simple distributions to the closing velocity and time between impacts data can be seen by considering Figure 8. The problem is particularly severe with regard to the non-dimensional time between impacts. The basic shape of the curves is altered as e, the coefficient of restitution, is decreased. It is to be expected, on physical grounds, that the mean of the ball velocity after impact will be reduced if e is reduced while R is kept constant. When the ball is moving more slowly, there is a greater chance of successive impacts occurring at the same end of the slot in the randomly driven mass. The normalized times between impacts associated with such impacts will be much smaller than those in which impacts occur at the opposite ends of the slot. This accounts for the significant peaks in the data of Figure 8(c) and (d) at small normalized time between impacts for e = 0.7. It should be noted that the data in Figures 4, 5 and 6 refer to impacts at one end of the slotted randomly driven mass where the data given in Figure 7 refers to impacts at both ends of the slot. The fitted log-normal and normal distributions for the primary quantities, the nondimensional velocity of the ball after impact and that of the randomly driven mass at impact, are shown in Figure 9. It can be seen from the distribution curves for the velocity of the randomly driven mass at impact that the curves shift to the left as e is increased. Thus the mean velocity of the randomly driven mass at impact decreases as e increases.
Non-dlmewonol
Figure 6. Normalized
frequency of occurrence R = 10.0. Mean = 3.08, R.M.S. = 3.32.
data
Non-dlmensionol
Figure 7. Normalized frequency of occurrence and R = 10.0. Mean =4.80, R.M.S. = 6.15.
velocity
for the non-dimensional
closing
velocity.
e = 0.9 and
hme
data for the non-dimensional
time between
impacts.
e = 0.9
536
C.
H.
LEE
AND
K. P. BYRNE
IMPACT
z. =
z
FOR
RATTLING
537
SYSTEM
0.4
,”
0.3
h E 0 :: 0
0.2
a”
STATISTICS
0.1
0.0 0.5
0.0 -5
0
5
IO
I5
20-5
Non-dlmensionol
-4
-3
-2
-I
0
I
2
3
4
5
velocity
Figure 9. Fitted log-normal distributions for the non-dimensional ball velocity (b) e = 0.8 and (c) e = 0.9 and fitted normal distributions for the non-dimensional driven mass at impact (d) e = 0,7, (e) e = 0.8 and (f) e = 0.9.
after impact (a) e=0,7, velocity of the randomly
This is, of course, to be expected on the basis of a long-term energy balance. The mean and R.M.S. values of the non-dimensional velocity of the ball after impact can be derived from the data used to fit the distributions such as those shown in Figure 9. These non-dimensional velocities are plotted against R with e as a parameter as in Figure 10. The non-dimensional mean impact velocities of the randomly driven mass are shown in Figure 11. It can be seen from Figure 10 that the ball velocities after impact are high when the coefficient of restitution is high. The reverse applies to the impact velocities of the randomly driven mass. This situation is to be expected as high values of e are associated with small energy loss. The non-dimensional R.M.S. impact velocities of the randomly driven mass are close to 1 for all values of e and R and so are not plotted. The mean and R.M.S. non-dimensional closing velocities are plotted in Figure 12 and the mean and R.M.S. values of the non-dimensional time between impacts are plotted on Figure 13. It can be seen from Figures lo-12 that the mean and RMS non-dimensional impact velocities of both the ball and the randomly driven mass stabilize as R increases. The value of R at which the velocities stabilize depends on the coefficient of restitution, e. The stabilization occurs at smaller values of R as e decreases. Generally, the mean and R.M.S. values of these velocities do not vary greatly once R > 100.R can be interpreted
538
C. H. LEE
AND
K. P. BYRNE
10.0 9.0 8.0 7.0 6.0
-))
.‘~‘*-_’
---
5.0 4.0
~-J+=WWC-&A..%-._...
3.0
_~~~ooo-___o_d_____o-___
2.0 I.0 0 IO.0 9.0 El.0 7.0 6.0 5.0 4.0
A_&-.---. _~____~~____~____
3.0 2.0 I.0 0
I
2
5
IO
50
20
100
200
500
1000
R
Figure x
10. Mean (a) and RMS (b) non-dimensional
velocity
of the ball after impact
0 e = 0.7; & e = 0.8;
) e = 0.9, 0, e = 0.95.
0.5
.-mx-&--A-n--_h-
0.4
0.3
?G-,.-,,_,___x
.._..
0.2
0.1
0
I
2
5
IO
20
50
100
200
500
1000
R
Figure x
11. Mean
non-dimensional
velocity
of the randomly
driven
mass at impact.
0,
e = 0.7; A, e =0.8;
( e = 0.9; cl, e = 0.95.
as the ratio of the clearance (d) to a measure of the displacement of the randomly driven mass (a/w,). Thus when the clearance, d, is about 100 times the measure of the displacement of the randomly driven mass, u/w,, the motion of the ball becomes weakly correlated with the motion of the randomly driven mass. When this same interpretation is placed on R it is evident that as R approaches zero the clearance between the ball and the randomly driven mass approaches zero. Thus it would be expected that the ball velocity after impact would approach the velocity of the randomly driven mass. It can be seen from Figure IO(b) that the non-dimensional velocity approaches 1 as R approaches zero as expected. It can be seen also from Figures 9(a), (b) and (c) that for small values of R
IMPACT
STATISTICS
FOR
RATTLING;
SYSTEM
539
E I
I
I
I
I
I
I
P
1
fir;
E
.e
h+pOlaA
IOUOISUauIp-UON
540
C.
0.3
H.
LEE
AND
K. P. BYRNE
I
I
I
I I
(0)
0.3
I
I (b)
I
I
Non-damenslonal
Figure 14. Comparison frequency of occurrence (b) e = 0.8, (c) e = 0.9.
velocity
of the theoretical probability density functions with the simulation derived normalized data for the non-dimensional ball velocity after impact. R = 100.0 (a) e = 0.7,
IMPACT
STATISTICS
FOR
RATTLING
541
SYSTEM
0.6
0.5 A r
0.4
: < 1 n
0.3
2 e a
0.2
0.1
0 0.6
,
,
,
,
,
,
,
I,
I
,
I
,
1 (cl
0.5-
Non-dtmenslonol
veloctty
of the theoretical probability density functions with the sikdation derived normalized Figure 15. Comparison frequency of occurrence data for the non-dimensional velocity of the randomly driven mass at impact. R = 100.0 (a) e = 0.7, (b) e = 0.8, (c) e = 0.9.
542
C. Ii. LEE AND
K. P. BYRNE
the distribution is almost Gaussian. This is expected as the ball motion has a strong correlation with the motion of the randomly driven mass. When R is large the ball motion is weakly correlated with the motion of the randomly driven mass and the distribution would be expected to be close to the distribution developed from the theory in [3]. It is of interest to compare the simulation derived results for large values of R with the theoretical results derived by solution of the integral equation developed in reference [3]. Figure 14 shows the theoretically derived and simulation derived results for the nondimensional ball velocity after impact for R = 100. Since the theoretical results were based on a model for which it was assumed that the ball velocity was always reversed after an impact, the obvious discrepancies at negative values of the non-dimensional ball velocity after impact are expected. However, at positive velocities the agreement is good. The
Figure 16. Comparison frequency of occurrence (c) e ~0.9.
of the theoretical probability density functions with the simulation derived normalized data for the non-dimensional closing velocity. R = 100.0 (a) e = 0.7, (b) e = 0.8,
IMPACT STATISTICS
FOR
RATTLING
SYSTEM
543
comparison of the simulation derived data for the table velocities with the theoretically derived velocities for e = 0*7,0*8, and 0.9 is shown in Figure 15. The comparison of closing velocities is shown in Figure 16. It can be seen, that these two sets of curves show a good agreement. The theoretically derived and the simulation derived non-dimensional mean ball velocities after impact are shown as a function of e in Figure 17. It can be seen that the mean ball velocities derived by simulation converge to the theoretically derived data as R increases and the agreement is also good when R becomes large (R > 100).
I
0.6
I
I
I
0.7
0.8
0.9
,
I.0
e
Figure 17. Non-dimensional
mean velocities ofthe ball after impact along with the theoretically derived results.
4. CONCLUSIONS The equations which govern a simple but practically important random rattling system have been formulated and non-dimensionalized so that they are in convenient form. The system, as a result, can be described by two parameters, the coefficient of restitution, c, and a parameter, R, which incorporates the clearance, d, the R.M.S. velocity of the randomly driven mass, C, and the upper frequency, o,, of the spectrum of the velocity of the randomly driven mass. These equations have been solved numerically to yield the impact statistics. The impact statistics are given as functions of the parameters e and R. These impact statistics enable practically useful quantities such as the mean time between impacts and the mean magnitude of the impacts to be found. These useful quantities can be obtained from Figures 10-13. A comparison has been made between simulation derived results and results derived from the theoretical model of Wood and Byrne [3]. It can be concluded from this comparison between the impact statistics derived by simulation and those derived by the theoretical method that if R > 100 there is good agreement.
REFERENCES 1. C. N. BAPAT and N. POPPLEWELL 1983 Journal of Sound and Vibration 87, 19-40. Stable periodic motions of an impact-pair. 2. S. F. MASRI and A. M. ~BRAHIM 1973 Journal of the Acoustical Society of America 53,200-211. Response of the impact damper to stationary random excitation. _. : L. A. WOOD and K. P. BYRNE 1981 Journal of Sound and Vibration 78, 329-345.
Analysis of a random repeated impact process. 4. L. A. WOOD and K. P. BYRNE 1982 Journal of Sound and Vibration 86, 53-59. Experimental investigation of a random repeated impact process. 5. Y. K. LIN 1967 Probabilitic Theory of Structural Dynamics. New York: McGraw-Hill. 6. M. SHINOZUKA and C. M. JAN 1972 Journal of Sound and Vibration 25, 111-128. Digital simulation of random process and its applications.