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Impact behaviour of an oscillator with limiting stops, part I: A parametric study
Journal of Sound and V~bration (1986) 109(2), 293=307
IMPACT B E H A V I O U R OF A N OSCILLATOR W I T H LIMITING STOPS, PART I: A PARAMETRIC S T U D Y D. T. NGUYEN, S. T. NOAH AND C. F. KETI'LEBOROUGH
Department of Mechanical Engineering, Texas A&M University, College Station, Texas, 77843, U.S.A. (Received 11 July 1985, and in revised form 10 October 1985) The impact behaviour of a periodically forced oscillator with limiting stops is considered. Pai'ametric studies are made to examine the influence of the various excitation and physical Variables of the system on the frequency and intensity of contacts with the stops. The results of these studies and those of a companion paper can be utilized in related studies of noise and wear in mechanical systems with clearances.
1. INTRODUCTION In numerous mechanical systems, some of the components are connected through a backlash [1-3], are loosely supported [4], operate within designed clearances [5] or, due to the nature of their function, experience intermittent motion of cont~ct and separation with other bodies [6]. The complex dynamic responses within such systems can produce the adverse conditions of loss of performance a n d / o r proper functioning. In many cases, higher frequencies and stresses may also be generated, resulting in severe wear and fretting, fatigue, as well as high noise level. Components susceptible to such damage or loss of performance include tubes in steam generators and their support plates, lightly loaded spur gears, cam/follower systems, linkage joints and robotic components, bearings, electrical relays, and circuit breakers. In defence applications, systems of this type include gun mechanisms, ammunition loading devices, power transmission, helicopter and weapon systems. The present paper is the first of two parts concerning a study of the general characteristics of the response o f mechanical systems with clearances [7]. The emphasis of the study is on the aspects of the response which would have the most influence on the wear rate and noise levels in such systems, namely frequency, intensity and duration of contact [8]. In a series of study on fretting wear of tubes (reviewed in reference [9]) Ko concluded that the frequency and intensity of impact forces, as well as the transverse velocity, are the relevant factors affecting the degree of tube wear. Blevins [10] observed a high correlation between impact wear a n d noise levels. Several experimental and analytical studies were conducted to examine the response behaviour o f simple oscillatory systems with nonlinear restoring forces. These include the periodic response of a mass within stops of single [11] or multiple [12] impacts per cycle of excitation and the non-periodic and chaotic responses [13, 14]. For the analysis of a highly non-linear system, as is the case for systems interacting with gaps, computational procedures are mainly used [15]. Semi-analytical procedures are also being used for special simpler cases [16, 17]. To aid in the selection and interpretation of the analysis procedures, qualitative results obtained by using Poincar6 and other methods have been 293 0022-460x186/170293+ 18 503.00/0 9 1986 Academic Press Inc. (London) Limited
294
D. T. N G U Y E N .
S. T. N O A H A N D
C. F. K E " I ' T L E B O R O U ( 3 H
utilized [18]. Approximate methods may also be used if averaged periodic solutions are sufficient to meet the objectives of the intended study [19]. To date, only purely computational procedures are capable of obtaining the general solution for highly non-linear systems. The results of the analysis would yield the most information about the behaviour of the systems provided that an appropriate format for presenting these results has been established, together with an awareness of the expected global behaviour in a given range of parameters. This first part of the study examines the general characteristics of a viscously damped system with limiting compliant stops. Numerical integrations are utilized in determining the response patterns and impact behaviour with respect to variation in the various system parameters. The second part of the study concerns representation of the system's impact behaviour (namely, frequency and intensity of impact) in terms of dimensionless groups of physical and excitation parameters. The results of this study can prove beneficial in the analysis and design of more complex systems with clearances [20]. 2. THE NON-LINEAR SINGLE DEGREE-OF-FREEDOM SYSTEMS 2.1. T H E P I E C E - W I S E L I N E A R M O D E L Figure 1 shows a forced single degree-of-freedom vibratory system with motion limiting stops. The system consists of a mass, a spring, a viscous damper, and a stop on each side
k
c /:s
F i g u r e 1. Piece-wise l i n e a r m o d e l .
of the mass. The limiting stops are modeled as combinations of linear springs and viscous dampers. This representation facilitates a parametric study of the system as described in Part I of this paper and an analysis based on dimensionless groups, as presented in Part II [21]. A slightly different and more representative behaviour of the stops could be taken as a hysteretic or viscously damped non-linear element [22, 23]. It can be reasonably assumed that in most applications the upper and lower stops are of identical properties, and are at the same gap distance, or (a list of nomenclature is given in the Appendix)
lq,=kt=ks,
c,=c,=cs,
du=d,=d.
(1)
2.2. E Q U A T I O N O F M O T I O N The governing equation of motion for the system as depicted in Figure 1 can be described as follows;
my"+(c+c,((y-d)~176
(2)
O S C I L L A T O R WITH L I M I T I N G
where a
prime denotes
STOPS, I
295
differentiation with respect to time, t, and ( a ) " = It0a ' ,
ifa>ifa<~} '
n = 0 , 1.
(3)
For the purpose of performing numerical integration, equation (2) can be written in the finite central difference form 1 m Y.+I=(I+C[y.]At/2m) ( 2 Yn-Y.-I + (At)2(F.-R[Y.])+(C[~mAt)Y.-, },
(4)
totn. C[y], R[y] are defined here as the piece-wiselinear damping and restoringforce constants, respectively, where C[y]=c+cs((y-d)~176 R[y]=ky+k~((y-d)'-(-y-d>'). (5,6)
where F. = Fo sin
3. GENERAL CHARACTERISTICS OF MOTION AND CONTACT FORCES 3.1. GENERAL TYPES OF RESPONSE The response due to sinusoidal excitation of the systems can be classified into two general types, namely periodic response and erratic response. The characteristics of each of these types of response are discussed with the aid of appropriate numerical results. In all cases presented the units of the system parameters and the system responses may be in any consistent units.
Periodicresponses A periodic response consists
3.1.1.
of motion and contact forces that repeat themselves in regular intervals of time. Often responses of the piece-wise linear systems are approximately periodic. For practical purposes, however, any response appearing to be approximately periodic is categorized as periodic response in this study. Some typical periodic responses and the associated contact forces are shown in Figures 2 and 4. It is found that periodic responses are more common among systems having a low frequency of contacts a n d / o r a larger amount of damping. --" c
I.o
I
i
1
36
38
E oo
:~
I
--I 0
I000
..9o o
u -I000
30
32
34
40
Time, I (s)
Figure 2. Single-impact periodic response. F o = 4 0 , t o = 2 . 0 , m = 0 . 1 0 , e =0.024, k = 6 0 , c s = 0 . 0 , k, =6000, d = 0-72.
296 3.1.2.
D . T . NGUYEN, S. T. NOAH AND C. F. KE'I-FLEBOROUGH
Non-periodic (erratic) responses
A n erratic response (as in Figure 3) exhibits r a n d o m - l i k e d y n a m i c b e h a v i o u r and consists o f m o t i o n a n d contact forces that do not repeat in a n y regular and reasonably l o n g periods o f time [13]. T h e f r e q u e n c y o f contacts b e t w e e n the m a s s and the stops in a n o n - p e r i o d i c r e s p o n s e can vary f r o m cycle to cycle o f excitation. A l s o , the contact forces associated w i t h these responses can vary greatly in m a g n i t u d e during the course o f m o t i o n w i t h no apparent repeating patterns.
I.O
I
I
I
22
25
E ca
5
0-0
"o
-I.0
1500 I000 5OO 0
E tj
-500 -I000 -1500 2O
21
24
Time, ! (s)
Figure 3. Multiple-impact erratic response. Fo= 90, m =4.0, m = 0-10, c = 0.005, k=60, c, =0.0, ks = 10 000, d = 0.40.
0,6 - -
T
I
[
T
T - - T - - I ~ -
0-4 0.2 a
0-0
:6
-0,2;
-0-4 _0.6 !
4O0 2O0 0
i[
-200 tj
-400 -600
I
I0
If I II
I
11 I
12
I
I
I
13
Time, t (s)
Figure 4. Multiple-impact (chatter) periodic response. Fo--- 60, m = 2.0, m = 0.02, c = 0-050, k = 80, c, = 0.0, k, = 8000, d = 0.40.
OSCILLATOR
3.2.
CHA'ITER
WITH
LIMITING
297
STOPS, I
RESPONSES
When a mass, driven by an external force, strikes a limiting stop and then rebounces from the impact to strike the same stop again during one excitation cycle, the successive contacts between the mass and the stop are usually known as chatter [12]. A chatter response can be identified as a response in which the average number of contacts between the mass and any one o f the stops per cycle of excitation is greater than or equal to two. An example of chatter response is shown in Figure 4. Most chatter responses with high frequency of contacts are found to be erratic. Chatter can be significant in influencing the wear rate in mechanical systems interacting through a gap. Repetitive impacts might cause refinement of micro-ci'acks within the substrates of the contacting solids. 3.3.
SINGLE-SIDED-IMPACT
AND
UNEVEN-CHATTER
RESPONSES
In systems having symmetrical gaps, the frequency of contacts at the upper stop and that at the lower stop are usually approximately equal or exactly the same. In many cases, however, certain combinations o f parameters and initial conditions will yield responses that will have distinct frequency of contacts at each of the limiting stops (Figure 5). A
1.0 ~"
0.5
o
0-0
Q
-o.5 --I-0
~'~. ~oolo 3001
t
~- '~176 Iol I,I ITI I,I I~1 I~1 Iol III Jl
_,oo,,, Ill I ILILI I J
t~ --3001~ I I u I 20 22
I
I I ! 24 26 Time,/ (s)
f
I 28
50
Figure 5. Unevenchatter response. Fo= 30, to = 2.0, m = 0.30, c = 0.10, k = 62.5, c, = 0.0, k, = 2500, d = 0.40. response of this type generally belongs to one of the three following categories: (i) contacts occur at a single stop, (ii) chatter occurs at only one of the stops, or (iii) chatter occurs at both stops but at different frequencies of contacts. Responses of the first category will be hereafter referred to as response with single-sided impacts, and responses of the second and third categories will be referred to as response with uneven chatter at the stops. For systems of symmetrical gaps, the difference in the frequencies of contacts at the u p p e r and the lower stops is observed to be always less than or equal to one contact per cycle of excitation. In cases where the frequency of contacts at the upper and lower stops are equal, the contact forces are usually distributed evenly between the two stops. In such cases, the average contact force at the upper stop is approximately equal to or exactly the same as that at the lower stop. Each of the stops therefore absorbs approximately the same amount
298
D . T. N G U Y E N ,
S. T. N O A H
AND
C. F. K E T T L E B O R O U G H
of energy from impacts. In responses with single-sided impacts or uneven chatter, the relationship between the average contact force and the frequency of contacts is generally one of an inverse order: i.e., the contact force at the stop with the higher frequency o f contacts is generally smaller than that at the other stop. In less frequently encountered cases, the opposite behaviour was observed: e.g., the contact force at the stop with the higher frequency of contacts is also larger than that at the remaining stop. This is always true for systems of single-sided impacts. The uneven distribution of contact forces and impact energy may cause undesirable conditions such as uneven wear that can lead to enlarging of one gap (unsymmetrical gaps) and excessively high contact force at one of the stops. 4. PARAMETRIC STUDY In order to determine the relative degree of influence that each of the involved physical parameters can have on the system response, extensive parametric studies have been conducted. In these studies, each parameter is varied while the remaining parameters are kept unchanged. The results are used to identify the important parameters as well as those which have little or negligible influence on the system response. 4.1. S T A T I S T I C A L R E P R E S E N T A T I O N O F I N T E N S I T Y A N D F R E Q U E N C Y O F C O N T A C T S Since the intensity and frequency of contacts are among the major concerns in this study, the parametric studies are based on the behaviour of the contact force and the frequency of contacts at the limiting stops. In order to examine this behaviour, it is necessai-y to establish some appropriate statistical representations for the contact force and the frequency of contacts. For purposes such as predicting wear rate and noise levels, for example, the most desired information would be the magnitude of the maximum contact force, the average contact force, and the average frequency of contacts between the mass and the stops. It is observed in numerous cases in this study that the overall average contact force coupled with the overall average frequency of contacts consistently reflect the average energy of impact associated with a system, regardless of the type or nature of the response. It can be reasonably established that the average energy of impact is directly proportional to the magnitude o f the average intensity and frequency of contacts in the system. The maximum contact force in the system does not represent the average impact energy and hence is less appropriate for use in comparing or relating system behaviour. However, it may be useful for comparing the extreme and average impact conditions. One significant information is the difference in magnitude between the average contact force and the maximum contact force in a system. As has already been discussed, the average contact force and the average frequency o f contact at the upper stop and those at the lower stops are generally the same or approximately equal in systems having symmetrical gaps. When this is the case, the same characteristic is also usually observed for the maximum contact force at each of the stops. Hence in general, only three quantities are required to describe completely the impact behaviour of any system: namely, the overall maximum contact force, the overall average contact force, and the overall average frequency of contact. However, for the special cases of single sided impacts and uneven-chatter in which the contact forces and the frequency of contacts at each of the stops are distinctly different, it will be necessary to specify the above quantities for each of the stops. The average contact force at any stop is calculated by simply dividing the sum of all contact forces exerted on this stop over a period of time by the corresponding number of contacts (n). The average frequency o f contact at each stop is determined in the same
OSCILLATOR WITH L I M I T I N G STOPS, I
299
manner. These calculations can be summarized by I (F~)""
= n
n
and
,=,
"
2\wAT/"
(7)
It should be noted here that, similar to those in the case of unrestrained vibratory systems, the stationary responses in this study usually begin with a transitional response period. The contact forces and the frequency o f contacts within this transient period are usually distinctly different from those once a stationary response is established and do not reflect the long term behaviour of the system. For this reason, all impact data in the initial transient duration is omitted from the calculations os the average contact force, the average frequency o f contacts, and the maximum contact force at the stops. It is observed in numerous cases in this study that the transitional response will usually disappear after a few cycles of excitation. In most cases, even with the least amount o f damping, approximately 20 cycles os excitation are sufficient for this to occur. 4.2.
RESULTS OF P A R A M E T R I C STUDY
The impact behaviour studies presented herein are based on numerical data obtained for systems having zero initial displacement and velocity os the mass. The effects o f non-zero initial conditions are discussed in a subsequent paper [21]. The particular results are typical and within the practical ranges os interest. 4.2.1. Amplitude of driving force, Fo An increase in the amplitude of the sinusoidal driving force (Fo) will result in an increase in both the contact force and the frequency of contacts, as in Figure 6. There
6.0 ~"
5.0
o
,.~
4.0
"8
3-o
N
z.o
ff
i.o o.o I100 1000
/
,~ u
, . t -'~'~ / 500
=
f,,.
,/
u
o 0
)
)
f
I
40
80
120
160
Amplitudeof
200
excitation force, Fa
Figure 6. Typlcal case showing the ettects of variation in the a mpl i t ude of excitation force on the contact force and the frequency of contacts. ~o = 2.0, ra =0.10, e = 0"050, k = 4 0 , e, = 0"05, k, = 8000, d =0-20. , (Fc).~g; - - -, (F.),.~.
300
O.T. NGUYEN, S. T. NOAIEI AND C. F. KE"I'q'LEBOROUOH
exists a range of Fo ~< F~ where no contact occurs. Hence, in this range the system is effectively unrestrained. The minimum amplitude of driving force (Ft) required for contact to occur is dependent on its frequency, the gap clearance, and the unrestrained system parameters. 4.2.2. Frequency of excitation, to An increase in the frequency of the excitation will generally result in an increase in the average contact force and a decrease in the frequency of contacts. The typical behaviour is illustrated by Figure 7. The average contact force increases gradually with large I
I
I 8
I I0
6
o "6
3 2
0 800
600 .9o
400
200
0 0
I 2
I 4
I 6
12
F r e q u e n c y of e x c i l o t i o n , w (cycles/s)
Figure 7. Typical case showing the effects of variation in the excitation frequency on the contact force and frequency of contacts. Fo=40, m=0-10, c-0-010, k=40, c3=0-10, ks=8000, d=0.20. , (Fc)~,; .
.
.
.
, (F~)o,~.
increments in the frequency of driving force, as opposed to a rapid reduction in the frequency of contacts. The contact force displays significant fluctuations with variation in the frequency of excitation force. The degree of fluctuation in the contact force is found to be significantly high in systems having relatively large gap clearance (characterized by low frequency of contacts) and low in systems having relatively small gap clearance (characterized by high frequency of contacts). Large and irregular fluctuations occur in the maximum contact force ((Fc)m=,) as opposed to the relatively much smaller fluctuations in the average contact force ((Fc)at, g ). (This is a demonstration of why the maximum contact force does not reflect the average impact energy associated with the system.) The large fluctuations in the maximum contact forces are the results of transitions from erratic to periodic responses. The magnitudes of the maximum contact forces tend to decrease significantly when a response is making a transition from erratic to periodic. There exists a range of forcing frequencY only within which contact can occur. The values of the upper and the lower limit of this frequency range are dependent on the amplitude of the driving force, the gap size, and the unrestrained system parameters.
OSCILLATOR WITH LIMITING STOPS. I
301
4.2.3. Stiffness at the stops, ks An increase in the stiffness of the upper and lower stops will result in an increase in both the contact force and the frequency of contacts in the system, as shown in Figure 8. The frequency of contacts tends to increase rapidly with increasing stop stiffness in .4
~o u
2
o
g
u~
0 1500
u
I000
u o c ~.
500
0
I I IO,OOO 20,OO0 Sliffness of slops, ks
50,00{
F i g u r e 8. T y p i c a l c a s e s h o w i n g t h e effects o f v a r i a t i o n in t h e s t o p stiffness o n t h e c o n t a c t f o r c e a n d f r e q u e n c y o f c o n t a c t s . F o = 100, to = 2 . 0 , m = 0 . 2 0 , c = 0 . 0 1 0 , k = 4 0 , c, = 0 . 0 1 , d = 0 . 2 0 . K e y as F i g u r e 7.
the low stiffness range, as compared to the more gradual but significant increase in the contact force. As the stop stiffness is increased in the high stiffness range, the frequency of contacts and the contact force eventually reach their respective extreme magnitudes, at which points they essentially remain constant for any further increases in the stop stiffness. The frequency of contacts reaches its extreme values at a much lower stop stiffness than that required for the contact force. The values of stop stiffness at which these extremes occur are dictated by the magnitudes of the remaining parameters in the system. 4.2.4. Gap size, d An increase in the gap clearance between the mass and the stops will generally result in a decrease in the frequency of contacts between the mass and the stops, although irregular fluctuations in the contact forces occur with variation in gap size. The general behaviour of the contact force and the frequency Of contacts are shown in Figure 9. The frequency of contacts generally decreases, but not smoothly, with increasing gap clearance. There are "jumps" or discontinuities in the frequency of contacts as the gap size is varied, which identify the large and sudden changes in the dynamic response at the critical gap sizes (point a on Figure 9). These abrupt changes in the frequency of contacts are responsible for the highly irregular fluctuation in the contact force. An abrupt change in the frequency of contacts is generally accompanied by an abrupt change in the
302
D, T.
NGUYEN,
6
S. T .
NOAH
I
AND
C.
F.
KE'VFLEBOROUGH
I
l
I
I 0.8
I 1.2
I 1.6
5
w o
4 3
21 I 0
1200
~, 8OO
~_~ 4 0 0
//._.;-L,' i/
0 0.0
I 0.4
2-0
Gopsize, d
Figure 9. Typical case showing the effects of variation in gap size on the contact force and frequency of contacts. Fo=40, to =2.0, m =0-10, c=0.010, k=40, q =0.10, k, = 8000. 9 ( F , ) ~ ; ---, (F~)o~. contact force, and the two are generally inversely related: i.e., a decrease in the frequency of contacts is generally accompanied by an increase in the contact force, and vice versa. In those ranges of gap clearance in which the frequency o f contacts remains constant, the contact force generally decreases rapidly with increasing gap clearance between the mass and the stops. An upper limit of the gap size exists beyond which contact cannot occur. The maximum gap clearance in a restrained system is dependent on the excitation parameters (including the initial conditions) and the unrestrained system parameters o f t h e system. Those systems having a "relatively small" gap are characterized by high frequency of contacts, and those said to have a "relatively large" gap are characterized by low frequency of contacts. 4.2.5. Damping at the stop, cs This study revealed that, in general, damping at the stops significantly affects the impact behaviour only o f those systems having relatively small gap size a n d / o r relatively low stop stiffness (characterized by large stop deflection). In systems in which the above conditions exist, the contact force and the frequency o f contacts are generally unaffected by small variations in the amount of damping at the stops. However, a significant increase in the damping at the stops can generally result in a decrease in the contact force and either an increase or a decrease in the frequency of contacts, depending on the amount o f damping at the stops. For a typical system having relatively small gap clearance, the average contact force decreases gradually as the coefficient o f damping at the stops is first increased, then remains nearly unchanged as the coefficient of damping is further increased. The frequency o f contacts, on the other hand, first increases then decreases with increase of cs. A different behaviour is displayed in Figure 10 for a system having a relatively large gap. In this system the frequency of contacts remains unchanged as the amount of damping
OSCILLATOR
3"0
o
WITH
I
LIMITING
STOPS,
I
303
I
I
2.0
o c
g g u~
f-O
0"0 450 400
3oo l - . . O- . . . . . . . . . . . .
8
200 t.)
I0
0-0
I 1.0
I 2.0
I 3.0
4.0
D a m p i n g at t h e stops, cs
Figure 10. Typical case showing the effectsof variation in the damping at the stops on the contact force and frequency of contacts in systemswith relativelylarge gaps. Fo = 30, to = 2.0, m = 0.10, c = 0.050, k = 40, ks = 8000, d = 0.40. Key as Figure 9. at the stops is incremented from zero (sr, = 0) to a relatively large damping coefficient o f 4.0 (~:, =0.07). Based on the known relationship between the coefficient of restitution (e,) and the spring-damper model at the stops, namely, es = e - ~ ' , the coefficient of restitution (e,) for the system of Figure 10 ranges from that for perfect elastic (e, = 1) to e, = 0.80. Typical values in various applications for metals are between 1.0 and 0.6. The average contact force also remains unchanged for the most part, except for a sudden decrease between cs = 0.3 and 0.4 (points A and B on Figure 10). It is observed that this abrupt reduction in the contact force is caused by a change from erratic to periodic response under the influence of damping at the stops. The motion of the mass remains relatively unchanged with large increments in the damping at the stops once periodic motion is attained. 4.2.6. System damping, c The amount of damping in the system may have different effects on the contact force and the frequency of contacts in different systems. In some systems, small variations in the coefficient of damping will result in irregular fluctuations in the contact force and the frequency of contacts, while in others they will be essentially unaffected by large variations in the amount of damping. However, a very large increase in damping, in which the system is brought near the critical damping, will result in a decrease in both the frequency of contacts and the contact force. In general, if the frequency of contacts is unaffected by a change in the amount o f damping in the system, the contact force will also remain unchanged. An exception to this general behaviour is observed when the frequency of contacts of the system is unity--i.e., one contact at each stop per cycle of excitation--in which case the contact force generally decreases with increasing damping in the system. However, if the frequency of contacts changes or fluctuates due to variations in the amount of system damping, an
304
D.T.
NGI.JYEN,
S. T . N O A H
AND
C . F. K E T T L E B O R O U G H
increase in the frequency of contacts is generally accompanied by a decrease in the contact force, and vice versa. It is also observed in this study that damping generally has a larger effect on the systems whose responses are erratic than in those systems whose responses are periodic. With a sufficiently large increase in the amount of damping, most erratic responses would become periodic. The contact force and frequency of contacts for a typical system of relatively small gap size and damping ratio in the range from zero to 0.2 (~ = 0 ~ 0-2) typically fluctuate highly in a large range of smaller damping ratios. However, they remain relatively unchanged in the heavier damping range. Figure 11 shows the contact force and frequency of contacts for a system having a relatively large gap. The contact force in this case decreases with increase in damping in the system in the range of damping where the frequency of contacts is unity.
Q !
2-0
c
8
"8 t.o
I
I
I 0-4
I 0.6
i
g
0.0 35O 3O0
200 2 I00
O 0-0
I 0-2
I 0.8
1.0
Coefficient of system damping, r
Figure 11. Typical case showing the ettects of variation in the system damping on the contact force and frequency of contacts in systems with relatively large gaps. Fo = 25, (a = 2.0, m = 0.10, c = 0.0, k = 62-5, ks = 5500, d = 0.40. Key as Figure 9.
4.2.7. System stiffness, k A significant increase in the spring stiffness (k) will generally result in a decrease in both the contact force and the frequency of contacts, although they usually display significant fluctuations with changes in the spring stiffness. As the stiffness is increased, the system will eventually become effectively unrestrained. 4.2.8. Mass, m The effect of variation in the magnitude of the mass on the force and frequency of contacts is similar to the effect of variation in the gap size in the system. An increase in the magnitude of the mass will generally result in a decrease in the frequency of contacts, while the contact force is generally subjected to large and irregular fluctuations. The
OSCILLATOR WITH
LIMITING
305
STOPS, I
frequency of contacts is usually subjected to abrupt changes with variation in the magnitude of the mass, which in effect causes the irregular fluctuations in the contact force, similar to the behaviour shown in Figure 9. 5. CONCLUSIONS The following summarizes the results and observations in this study concerning the dynamic and impact behaviour of the restrained vibratory system considered. (a) Periodic responses occur more frequently with systems having a lower frequency of contacts and/or a larger amount of damping. The opposite is true for erratic responses. (b) The intensity and frequency of contacts at the individual stops in systems having symmetrical gaps are generally the same; however, certain combinations of parameters and initial conditions can result in single-sided impacts or uneven-chatter response in which the frequency of contacts and the contact forces at the individual stops are distinctly different. (c) An increase in the amplitude of the driving force or in the stiffness of the stops will generally result in an increase in both the frequency of contacts and the contact force. (d) An increase in the frequency of the driving force, gap size, or mass will generally result in a decrease in the frequency of contacts. The contact force tends to display Significant fluctuations but generally increases with increase inthe driving frequency, gap size, or mass. (e) A substantial increase in the spring stiffness will result in a decrease in both the frequency of contacts and the contact force. (f) In general, damping at the stops, cs, only affects the contact force and the frequency of contacts in systems having relatively small gap size and/or low stop stiffness. In these systems, a substantial increase in the amount of damping at the stops will generally result in a decrease in the contact force. The frequency of contacts generally increases with increasing damping at the stops for 0 < c~<~csa and decreases for larger values of c, where c~t is some value of damping coefficient greater than zero. (g) A variation in the system damping generally brings about different effects on the frequency of contacts and the contact force in different systems. An increase in the damping near the critical damping range will generally result in a decrease in both the contact force and frequency of contacts. (h) With exception of the variation in the amplitude of the driving force and stop stiffness, an increase (decrease) in the frequency of contacts due to variations in the magnitude of any one of the remaining parameters is generally accompanied by a decrease (increase) in the contact force. -
.
,
-
ACKNOWLEDGMENT This study was carried out as part of a research project supported by the National Science Foundation under Grant MEA-8211288. This support and that provided by the Mechanical Engineering Department of Texas A&M University are gratefully acknowledged. REFERENCES 1. R. C. AZAR 1974 Ph.D. Thesis, University of Massachusetts, Amherst. An investigationof impact phenomenon in geared torsional system. 2. G. HA1DL1976 Messerschmitt-Bolkow-Blohm GMBH, Report No. LIFE-1273. Non-linear effects in aircraft ground and flight vibration tests. (Also presented at the 43rd Meeting of the Structures and Materials Panel of AGARD at London, U.K., 26 September-1 October 1976.)
306
D . T . NGUYEN, S. T. NOAH AND (2. F. KETrLEBOROUGH
3. R. C. WINFREY,'R~ V. ANDERSON and C. W. GHILKA 1973 Journal of Engineeringfor Industry 95, 695-703. Analysis of elastic machinery with clearances. 4. P. O. STEVENS-GUILLE 1973 AECL-4449, Chalk River Nuclear Laboratories, Chalk Ricer, Ontario. Steam generator tube failures: a world survey of water-cooled nuclear power reactors to the end of 1971. 5. R. C. HUANG, E. J. HAUG, JR. and J. G. ANDREWS 1978 American Society of Mechanical Engineers, Journal of Mechanical Design 100, 492-499. Sensitivity analysis and optimal design of a mechanical system with intermittent motion. 6. M. P. KOSTER, 1975 American Society of Mechanical Engineers, Journal of Engineering for Industry 97, 555-602. The effects of the flexibility of the driving shaft on the dynamic behaviour of a cam mechanism. 7. D. T. NGUYEN 1984 M.S. Thesis, Texas A&M Unicersity. Study of single degree of freedom restrained vibratory systems. 8. S. T. NOAH, C. F. KETTLEBOROUGH and R. B. GRIFFIN 1984 Proceedings of the Society of
Manufacturing Engineers, Eleventh Conferenceon Production Research and Technology, Pittsburg, Pennsylvania. Dynamics and wear of mechanical systems with clearances. 9. P. L. K o 1985 American Society of Mechanical Engineers Journal of Tribology 107, 149-156. Heat exchanger tube fretting wear: Review and application to design. 10. R. D. BLEVINS 1978 ASME Paper No. 78-JPGC-NE-8. Fretting wear of heat exchanger tubes, Part 1: Experiments. 11. D. F. MASRI 1965 Ph.D. Thesis, California Institute of Technology. Analytical and experimental studies of impact dampers. 12. M. A. VELUSWAMI, F. R. E. CROSSLEY and G. HORVAY 1975 Journal of Engineering for Industry 97, 828-835. Multiple impacts of a ball between two plates, Part 2: Mathematical modelling. 13. :J. P. WOLF and P. E. SKRIKERUND 1980 Nuclear Engineering and Design 57, 253-275. M u t u a l pounding of adjacent structures during earthquakes. 14. J. M. T. THOMPSON and R. GHAFFARI 1982 Physics Letters 91A, 5-8. Chaos after period doubling bifurcations in the resonance of an impact oscillator. 15. S. DUaOWSKV and F. FREUDENSTEtN 1971 Journal of Engineering for Industry 93, 310-316. Dynamic analysis of mechanical systems with clearances, Part II: Dynamic response. 16. S. MAEZAWA 1963 Zagadnienia Drgan Nieliniowych, 156-163. Perfect Fourier series solution for subharmonic vibration of piecewise-linear system. 17. N. POPPLEWELL, C. N. BAPAT and K. MCLACHLAN 1983 Journal of Sound and Vibration 87, 41-59. Stable periodic vibroimpacts of an oscillator. 18. S. W. SHAW and P. J. HOLMES 1983 Journal of Sound and Vibration 90, 129-155. A periodically forced piecewise linear oscillator. 19. W. D. IWAN 1973 International Journal of Non-Linear Mechanics 8, 279-287. A generalization of the concept of equivalent lineafization. 20. S. DUaOWSKY 1974 ASME Journal of Engineeringfor Industry 93, 324-329. On predicting the dynamic effects of clearance in one dimensional closed loop systems. 21. D.T. NGUYEN, S. T. NOAH and C. F. KETTLEBOROUGH 1986 Journal of Sound and Vibration 109, 309-325. Impact behaviour of an oscillator with limiting stops, Part II: Dimensionless design parameters. 22. K. H. HUNT and F. R. E. CROSSLEY 1975 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 42, 440-445. Coefficient of restitution interpreted as damping in vibroimpact. 23. T. W. LEE and A. C. WANG 1982 ASME Paper Number 82-DET-64. On the dynamics of intermittent-motion mechanisms, Part 1: Dynamic model and response. 24. S. LEVY and J. P. D. WILKINSON 1976 The Component Element Method in Dynamics. New York: McGraw-Hill.
APPENDIX: N O M E N C L A T U R E r
% cj c~ c~
viscous damping coefficient of the system critical damping at the stops, =2mco~ damping coefficient at the lower stop damping coefficient at the upper stop damping coefficient at the stops
OSCILLATOR
WITH
LIMITING
STOPS,
I
k, ks k~
gap size lower gap size upper gap size coefficient of restitution at the stops system stiffness stiffness of the lower stop stiffness of the stops stiffness of the upper stop
m
mass
Fo (F~)~ (Fo)~ F,,
amplitude of excitation force average contact force maximum contact force peak contact force at the ith contact average number of contacts per side per cycle of excitation frequency time displacement of mass velocity of mass, = d y / d t frequency of excitation (cycles/second) natural frequency based on stiffness of the stops, =dkJm damping ratio at the stops, =cs/cc, time interval of response
d
,t, d. e+ k
t
y y' to to s
4T
307
IMPACT B E H A V I O U R OF A N OSCILLATOR W I T H LIMITING STOPS, PART I: A PARAMETRIC S T U D Y D. T. NGUYEN, S. T. NOAH AND C. F. KETI'LEBOROUGH
Department of Mechanical Engineering, Texas A&M University, College Station, Texas, 77843, U.S.A. (Received 11 July 1985, and in revised form 10 October 1985) The impact behaviour of a periodically forced oscillator with limiting stops is considered. Pai'ametric studies are made to examine the influence of the various excitation and physical Variables of the system on the frequency and intensity of contacts with the stops. The results of these studies and those of a companion paper can be utilized in related studies of noise and wear in mechanical systems with clearances.
1. INTRODUCTION In numerous mechanical systems, some of the components are connected through a backlash [1-3], are loosely supported [4], operate within designed clearances [5] or, due to the nature of their function, experience intermittent motion of cont~ct and separation with other bodies [6]. The complex dynamic responses within such systems can produce the adverse conditions of loss of performance a n d / o r proper functioning. In many cases, higher frequencies and stresses may also be generated, resulting in severe wear and fretting, fatigue, as well as high noise level. Components susceptible to such damage or loss of performance include tubes in steam generators and their support plates, lightly loaded spur gears, cam/follower systems, linkage joints and robotic components, bearings, electrical relays, and circuit breakers. In defence applications, systems of this type include gun mechanisms, ammunition loading devices, power transmission, helicopter and weapon systems. The present paper is the first of two parts concerning a study of the general characteristics of the response o f mechanical systems with clearances [7]. The emphasis of the study is on the aspects of the response which would have the most influence on the wear rate and noise levels in such systems, namely frequency, intensity and duration of contact [8]. In a series of study on fretting wear of tubes (reviewed in reference [9]) Ko concluded that the frequency and intensity of impact forces, as well as the transverse velocity, are the relevant factors affecting the degree of tube wear. Blevins [10] observed a high correlation between impact wear a n d noise levels. Several experimental and analytical studies were conducted to examine the response behaviour o f simple oscillatory systems with nonlinear restoring forces. These include the periodic response of a mass within stops of single [11] or multiple [12] impacts per cycle of excitation and the non-periodic and chaotic responses [13, 14]. For the analysis of a highly non-linear system, as is the case for systems interacting with gaps, computational procedures are mainly used [15]. Semi-analytical procedures are also being used for special simpler cases [16, 17]. To aid in the selection and interpretation of the analysis procedures, qualitative results obtained by using Poincar6 and other methods have been 293 0022-460x186/170293+ 18 503.00/0 9 1986 Academic Press Inc. (London) Limited
294
D. T. N G U Y E N .
S. T. N O A H A N D
C. F. K E " I ' T L E B O R O U ( 3 H
utilized [18]. Approximate methods may also be used if averaged periodic solutions are sufficient to meet the objectives of the intended study [19]. To date, only purely computational procedures are capable of obtaining the general solution for highly non-linear systems. The results of the analysis would yield the most information about the behaviour of the systems provided that an appropriate format for presenting these results has been established, together with an awareness of the expected global behaviour in a given range of parameters. This first part of the study examines the general characteristics of a viscously damped system with limiting compliant stops. Numerical integrations are utilized in determining the response patterns and impact behaviour with respect to variation in the various system parameters. The second part of the study concerns representation of the system's impact behaviour (namely, frequency and intensity of impact) in terms of dimensionless groups of physical and excitation parameters. The results of this study can prove beneficial in the analysis and design of more complex systems with clearances [20]. 2. THE NON-LINEAR SINGLE DEGREE-OF-FREEDOM SYSTEMS 2.1. T H E P I E C E - W I S E L I N E A R M O D E L Figure 1 shows a forced single degree-of-freedom vibratory system with motion limiting stops. The system consists of a mass, a spring, a viscous damper, and a stop on each side
k
c /:s
F i g u r e 1. Piece-wise l i n e a r m o d e l .
of the mass. The limiting stops are modeled as combinations of linear springs and viscous dampers. This representation facilitates a parametric study of the system as described in Part I of this paper and an analysis based on dimensionless groups, as presented in Part II [21]. A slightly different and more representative behaviour of the stops could be taken as a hysteretic or viscously damped non-linear element [22, 23]. It can be reasonably assumed that in most applications the upper and lower stops are of identical properties, and are at the same gap distance, or (a list of nomenclature is given in the Appendix)
lq,=kt=ks,
c,=c,=cs,
du=d,=d.
(1)
2.2. E Q U A T I O N O F M O T I O N The governing equation of motion for the system as depicted in Figure 1 can be described as follows;
my"+(c+c,((y-d)~176
(2)
O S C I L L A T O R WITH L I M I T I N G
where a
prime denotes
STOPS, I
295
differentiation with respect to time, t, and ( a ) " = It0a ' ,
ifa>ifa<~} '
n = 0 , 1.
(3)
For the purpose of performing numerical integration, equation (2) can be written in the finite central difference form 1 m Y.+I=(I+C[y.]At/2m) ( 2 Yn-Y.-I + (At)2(F.-R[Y.])+(C[~mAt)Y.-, },
(4)
totn. C[y], R[y] are defined here as the piece-wiselinear damping and restoringforce constants, respectively, where C[y]=c+cs((y-d)~176 R[y]=ky+k~((y-d)'-(-y-d>'). (5,6)
where F. = Fo sin
3. GENERAL CHARACTERISTICS OF MOTION AND CONTACT FORCES 3.1. GENERAL TYPES OF RESPONSE The response due to sinusoidal excitation of the systems can be classified into two general types, namely periodic response and erratic response. The characteristics of each of these types of response are discussed with the aid of appropriate numerical results. In all cases presented the units of the system parameters and the system responses may be in any consistent units.
Periodicresponses A periodic response consists
3.1.1.
of motion and contact forces that repeat themselves in regular intervals of time. Often responses of the piece-wise linear systems are approximately periodic. For practical purposes, however, any response appearing to be approximately periodic is categorized as periodic response in this study. Some typical periodic responses and the associated contact forces are shown in Figures 2 and 4. It is found that periodic responses are more common among systems having a low frequency of contacts a n d / o r a larger amount of damping. --" c
I.o
I
i
1
36
38
E oo
:~
I
--I 0
I000
..9o o
u -I000
30
32
34
40
Time, I (s)
Figure 2. Single-impact periodic response. F o = 4 0 , t o = 2 . 0 , m = 0 . 1 0 , e =0.024, k = 6 0 , c s = 0 . 0 , k, =6000, d = 0-72.
296 3.1.2.
D . T . NGUYEN, S. T. NOAH AND C. F. KE'I-FLEBOROUGH
Non-periodic (erratic) responses
A n erratic response (as in Figure 3) exhibits r a n d o m - l i k e d y n a m i c b e h a v i o u r and consists o f m o t i o n a n d contact forces that do not repeat in a n y regular and reasonably l o n g periods o f time [13]. T h e f r e q u e n c y o f contacts b e t w e e n the m a s s and the stops in a n o n - p e r i o d i c r e s p o n s e can vary f r o m cycle to cycle o f excitation. A l s o , the contact forces associated w i t h these responses can vary greatly in m a g n i t u d e during the course o f m o t i o n w i t h no apparent repeating patterns.
I.O
I
I
I
22
25
E ca
5
0-0
"o
-I.0
1500 I000 5OO 0
E tj
-500 -I000 -1500 2O
21
24
Time, ! (s)
Figure 3. Multiple-impact erratic response. Fo= 90, m =4.0, m = 0-10, c = 0.005, k=60, c, =0.0, ks = 10 000, d = 0.40.
0,6 - -
T
I
[
T
T - - T - - I ~ -
0-4 0.2 a
0-0
:6
-0,2;
-0-4 _0.6 !
4O0 2O0 0
i[
-200 tj
-400 -600
I
I0
If I II
I
11 I
12
I
I
I
13
Time, t (s)
Figure 4. Multiple-impact (chatter) periodic response. Fo--- 60, m = 2.0, m = 0.02, c = 0-050, k = 80, c, = 0.0, k, = 8000, d = 0.40.
OSCILLATOR
3.2.
CHA'ITER
WITH
LIMITING
297
STOPS, I
RESPONSES
When a mass, driven by an external force, strikes a limiting stop and then rebounces from the impact to strike the same stop again during one excitation cycle, the successive contacts between the mass and the stop are usually known as chatter [12]. A chatter response can be identified as a response in which the average number of contacts between the mass and any one o f the stops per cycle of excitation is greater than or equal to two. An example of chatter response is shown in Figure 4. Most chatter responses with high frequency of contacts are found to be erratic. Chatter can be significant in influencing the wear rate in mechanical systems interacting through a gap. Repetitive impacts might cause refinement of micro-ci'acks within the substrates of the contacting solids. 3.3.
SINGLE-SIDED-IMPACT
AND
UNEVEN-CHATTER
RESPONSES
In systems having symmetrical gaps, the frequency of contacts at the upper stop and that at the lower stop are usually approximately equal or exactly the same. In many cases, however, certain combinations o f parameters and initial conditions will yield responses that will have distinct frequency of contacts at each of the limiting stops (Figure 5). A
1.0 ~"
0.5
o
0-0
Q
-o.5 --I-0
~'~. ~oolo 3001
t
~- '~176 Iol I,I ITI I,I I~1 I~1 Iol III Jl
_,oo,,, Ill I ILILI I J
t~ --3001~ I I u I 20 22
I
I I ! 24 26 Time,/ (s)
f
I 28
50
Figure 5. Unevenchatter response. Fo= 30, to = 2.0, m = 0.30, c = 0.10, k = 62.5, c, = 0.0, k, = 2500, d = 0.40. response of this type generally belongs to one of the three following categories: (i) contacts occur at a single stop, (ii) chatter occurs at only one of the stops, or (iii) chatter occurs at both stops but at different frequencies of contacts. Responses of the first category will be hereafter referred to as response with single-sided impacts, and responses of the second and third categories will be referred to as response with uneven chatter at the stops. For systems of symmetrical gaps, the difference in the frequencies of contacts at the u p p e r and the lower stops is observed to be always less than or equal to one contact per cycle of excitation. In cases where the frequency of contacts at the upper and lower stops are equal, the contact forces are usually distributed evenly between the two stops. In such cases, the average contact force at the upper stop is approximately equal to or exactly the same as that at the lower stop. Each of the stops therefore absorbs approximately the same amount
298
D . T. N G U Y E N ,
S. T. N O A H
AND
C. F. K E T T L E B O R O U G H
of energy from impacts. In responses with single-sided impacts or uneven chatter, the relationship between the average contact force and the frequency of contacts is generally one of an inverse order: i.e., the contact force at the stop with the higher frequency o f contacts is generally smaller than that at the other stop. In less frequently encountered cases, the opposite behaviour was observed: e.g., the contact force at the stop with the higher frequency of contacts is also larger than that at the remaining stop. This is always true for systems of single-sided impacts. The uneven distribution of contact forces and impact energy may cause undesirable conditions such as uneven wear that can lead to enlarging of one gap (unsymmetrical gaps) and excessively high contact force at one of the stops. 4. PARAMETRIC STUDY In order to determine the relative degree of influence that each of the involved physical parameters can have on the system response, extensive parametric studies have been conducted. In these studies, each parameter is varied while the remaining parameters are kept unchanged. The results are used to identify the important parameters as well as those which have little or negligible influence on the system response. 4.1. S T A T I S T I C A L R E P R E S E N T A T I O N O F I N T E N S I T Y A N D F R E Q U E N C Y O F C O N T A C T S Since the intensity and frequency of contacts are among the major concerns in this study, the parametric studies are based on the behaviour of the contact force and the frequency of contacts at the limiting stops. In order to examine this behaviour, it is necessai-y to establish some appropriate statistical representations for the contact force and the frequency of contacts. For purposes such as predicting wear rate and noise levels, for example, the most desired information would be the magnitude of the maximum contact force, the average contact force, and the average frequency of contacts between the mass and the stops. It is observed in numerous cases in this study that the overall average contact force coupled with the overall average frequency of contacts consistently reflect the average energy of impact associated with a system, regardless of the type or nature of the response. It can be reasonably established that the average energy of impact is directly proportional to the magnitude o f the average intensity and frequency of contacts in the system. The maximum contact force in the system does not represent the average impact energy and hence is less appropriate for use in comparing or relating system behaviour. However, it may be useful for comparing the extreme and average impact conditions. One significant information is the difference in magnitude between the average contact force and the maximum contact force in a system. As has already been discussed, the average contact force and the average frequency o f contact at the upper stop and those at the lower stops are generally the same or approximately equal in systems having symmetrical gaps. When this is the case, the same characteristic is also usually observed for the maximum contact force at each of the stops. Hence in general, only three quantities are required to describe completely the impact behaviour of any system: namely, the overall maximum contact force, the overall average contact force, and the overall average frequency of contact. However, for the special cases of single sided impacts and uneven-chatter in which the contact forces and the frequency of contacts at each of the stops are distinctly different, it will be necessary to specify the above quantities for each of the stops. The average contact force at any stop is calculated by simply dividing the sum of all contact forces exerted on this stop over a period of time by the corresponding number of contacts (n). The average frequency o f contact at each stop is determined in the same
OSCILLATOR WITH L I M I T I N G STOPS, I
299
manner. These calculations can be summarized by I (F~)""
= n
n
and
,=,
"
2\wAT/"
(7)
It should be noted here that, similar to those in the case of unrestrained vibratory systems, the stationary responses in this study usually begin with a transitional response period. The contact forces and the frequency o f contacts within this transient period are usually distinctly different from those once a stationary response is established and do not reflect the long term behaviour of the system. For this reason, all impact data in the initial transient duration is omitted from the calculations os the average contact force, the average frequency o f contacts, and the maximum contact force at the stops. It is observed in numerous cases in this study that the transitional response will usually disappear after a few cycles of excitation. In most cases, even with the least amount o f damping, approximately 20 cycles os excitation are sufficient for this to occur. 4.2.
RESULTS OF P A R A M E T R I C STUDY
The impact behaviour studies presented herein are based on numerical data obtained for systems having zero initial displacement and velocity os the mass. The effects o f non-zero initial conditions are discussed in a subsequent paper [21]. The particular results are typical and within the practical ranges os interest. 4.2.1. Amplitude of driving force, Fo An increase in the amplitude of the sinusoidal driving force (Fo) will result in an increase in both the contact force and the frequency of contacts, as in Figure 6. There
6.0 ~"
5.0
o
,.~
4.0
"8
3-o
N
z.o
ff
i.o o.o I100 1000
/
,~ u
, . t -'~'~ / 500
=
f,,.
,/
u
o 0
)
)
f
I
40
80
120
160
Amplitudeof
200
excitation force, Fa
Figure 6. Typlcal case showing the ettects of variation in the a mpl i t ude of excitation force on the contact force and the frequency of contacts. ~o = 2.0, ra =0.10, e = 0"050, k = 4 0 , e, = 0"05, k, = 8000, d =0-20. , (Fc).~g; - - -, (F.),.~.
300
O.T. NGUYEN, S. T. NOAIEI AND C. F. KE"I'q'LEBOROUOH
exists a range of Fo ~< F~ where no contact occurs. Hence, in this range the system is effectively unrestrained. The minimum amplitude of driving force (Ft) required for contact to occur is dependent on its frequency, the gap clearance, and the unrestrained system parameters. 4.2.2. Frequency of excitation, to An increase in the frequency of the excitation will generally result in an increase in the average contact force and a decrease in the frequency of contacts. The typical behaviour is illustrated by Figure 7. The average contact force increases gradually with large I
I
I 8
I I0
6
o "6
3 2
0 800
600 .9o
400
200
0 0
I 2
I 4
I 6
12
F r e q u e n c y of e x c i l o t i o n , w (cycles/s)
Figure 7. Typical case showing the effects of variation in the excitation frequency on the contact force and frequency of contacts. Fo=40, m=0-10, c-0-010, k=40, c3=0-10, ks=8000, d=0.20. , (Fc)~,; .
.
.
.
, (F~)o,~.
increments in the frequency of driving force, as opposed to a rapid reduction in the frequency of contacts. The contact force displays significant fluctuations with variation in the frequency of excitation force. The degree of fluctuation in the contact force is found to be significantly high in systems having relatively large gap clearance (characterized by low frequency of contacts) and low in systems having relatively small gap clearance (characterized by high frequency of contacts). Large and irregular fluctuations occur in the maximum contact force ((Fc)m=,) as opposed to the relatively much smaller fluctuations in the average contact force ((Fc)at, g ). (This is a demonstration of why the maximum contact force does not reflect the average impact energy associated with the system.) The large fluctuations in the maximum contact forces are the results of transitions from erratic to periodic responses. The magnitudes of the maximum contact forces tend to decrease significantly when a response is making a transition from erratic to periodic. There exists a range of forcing frequencY only within which contact can occur. The values of the upper and the lower limit of this frequency range are dependent on the amplitude of the driving force, the gap size, and the unrestrained system parameters.
OSCILLATOR WITH LIMITING STOPS. I
301
4.2.3. Stiffness at the stops, ks An increase in the stiffness of the upper and lower stops will result in an increase in both the contact force and the frequency of contacts in the system, as shown in Figure 8. The frequency of contacts tends to increase rapidly with increasing stop stiffness in .4
~o u
2
o
g
u~
0 1500
u
I000
u o c ~.
500
0
I I IO,OOO 20,OO0 Sliffness of slops, ks
50,00{
F i g u r e 8. T y p i c a l c a s e s h o w i n g t h e effects o f v a r i a t i o n in t h e s t o p stiffness o n t h e c o n t a c t f o r c e a n d f r e q u e n c y o f c o n t a c t s . F o = 100, to = 2 . 0 , m = 0 . 2 0 , c = 0 . 0 1 0 , k = 4 0 , c, = 0 . 0 1 , d = 0 . 2 0 . K e y as F i g u r e 7.
the low stiffness range, as compared to the more gradual but significant increase in the contact force. As the stop stiffness is increased in the high stiffness range, the frequency of contacts and the contact force eventually reach their respective extreme magnitudes, at which points they essentially remain constant for any further increases in the stop stiffness. The frequency of contacts reaches its extreme values at a much lower stop stiffness than that required for the contact force. The values of stop stiffness at which these extremes occur are dictated by the magnitudes of the remaining parameters in the system. 4.2.4. Gap size, d An increase in the gap clearance between the mass and the stops will generally result in a decrease in the frequency of contacts between the mass and the stops, although irregular fluctuations in the contact forces occur with variation in gap size. The general behaviour of the contact force and the frequency Of contacts are shown in Figure 9. The frequency of contacts generally decreases, but not smoothly, with increasing gap clearance. There are "jumps" or discontinuities in the frequency of contacts as the gap size is varied, which identify the large and sudden changes in the dynamic response at the critical gap sizes (point a on Figure 9). These abrupt changes in the frequency of contacts are responsible for the highly irregular fluctuation in the contact force. An abrupt change in the frequency of contacts is generally accompanied by an abrupt change in the
302
D, T.
NGUYEN,
6
S. T .
NOAH
I
AND
C.
F.
KE'VFLEBOROUGH
I
l
I
I 0.8
I 1.2
I 1.6
5
w o
4 3
21 I 0
1200
~, 8OO
~_~ 4 0 0
//._.;-L,' i/
0 0.0
I 0.4
2-0
Gopsize, d
Figure 9. Typical case showing the effects of variation in gap size on the contact force and frequency of contacts. Fo=40, to =2.0, m =0-10, c=0.010, k=40, q =0.10, k, = 8000. 9 ( F , ) ~ ; ---, (F~)o~. contact force, and the two are generally inversely related: i.e., a decrease in the frequency of contacts is generally accompanied by an increase in the contact force, and vice versa. In those ranges of gap clearance in which the frequency o f contacts remains constant, the contact force generally decreases rapidly with increasing gap clearance between the mass and the stops. An upper limit of the gap size exists beyond which contact cannot occur. The maximum gap clearance in a restrained system is dependent on the excitation parameters (including the initial conditions) and the unrestrained system parameters o f t h e system. Those systems having a "relatively small" gap are characterized by high frequency of contacts, and those said to have a "relatively large" gap are characterized by low frequency of contacts. 4.2.5. Damping at the stop, cs This study revealed that, in general, damping at the stops significantly affects the impact behaviour only o f those systems having relatively small gap size a n d / o r relatively low stop stiffness (characterized by large stop deflection). In systems in which the above conditions exist, the contact force and the frequency o f contacts are generally unaffected by small variations in the amount of damping at the stops. However, a significant increase in the damping at the stops can generally result in a decrease in the contact force and either an increase or a decrease in the frequency of contacts, depending on the amount o f damping at the stops. For a typical system having relatively small gap clearance, the average contact force decreases gradually as the coefficient o f damping at the stops is first increased, then remains nearly unchanged as the coefficient of damping is further increased. The frequency o f contacts, on the other hand, first increases then decreases with increase of cs. A different behaviour is displayed in Figure 10 for a system having a relatively large gap. In this system the frequency of contacts remains unchanged as the amount of damping
OSCILLATOR
3"0
o
WITH
I
LIMITING
STOPS,
I
303
I
I
2.0
o c
g g u~
f-O
0"0 450 400
3oo l - . . O- . . . . . . . . . . . .
8
200 t.)
I0
0-0
I 1.0
I 2.0
I 3.0
4.0
D a m p i n g at t h e stops, cs
Figure 10. Typical case showing the effectsof variation in the damping at the stops on the contact force and frequency of contacts in systemswith relativelylarge gaps. Fo = 30, to = 2.0, m = 0.10, c = 0.050, k = 40, ks = 8000, d = 0.40. Key as Figure 9. at the stops is incremented from zero (sr, = 0) to a relatively large damping coefficient o f 4.0 (~:, =0.07). Based on the known relationship between the coefficient of restitution (e,) and the spring-damper model at the stops, namely, es = e - ~ ' , the coefficient of restitution (e,) for the system of Figure 10 ranges from that for perfect elastic (e, = 1) to e, = 0.80. Typical values in various applications for metals are between 1.0 and 0.6. The average contact force also remains unchanged for the most part, except for a sudden decrease between cs = 0.3 and 0.4 (points A and B on Figure 10). It is observed that this abrupt reduction in the contact force is caused by a change from erratic to periodic response under the influence of damping at the stops. The motion of the mass remains relatively unchanged with large increments in the damping at the stops once periodic motion is attained. 4.2.6. System damping, c The amount of damping in the system may have different effects on the contact force and the frequency of contacts in different systems. In some systems, small variations in the coefficient of damping will result in irregular fluctuations in the contact force and the frequency of contacts, while in others they will be essentially unaffected by large variations in the amount of damping. However, a very large increase in damping, in which the system is brought near the critical damping, will result in a decrease in both the frequency of contacts and the contact force. In general, if the frequency of contacts is unaffected by a change in the amount o f damping in the system, the contact force will also remain unchanged. An exception to this general behaviour is observed when the frequency of contacts of the system is unity--i.e., one contact at each stop per cycle of excitation--in which case the contact force generally decreases with increasing damping in the system. However, if the frequency of contacts changes or fluctuates due to variations in the amount of system damping, an
304
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C . F. K E T T L E B O R O U G H
increase in the frequency of contacts is generally accompanied by a decrease in the contact force, and vice versa. It is also observed in this study that damping generally has a larger effect on the systems whose responses are erratic than in those systems whose responses are periodic. With a sufficiently large increase in the amount of damping, most erratic responses would become periodic. The contact force and frequency of contacts for a typical system of relatively small gap size and damping ratio in the range from zero to 0.2 (~ = 0 ~ 0-2) typically fluctuate highly in a large range of smaller damping ratios. However, they remain relatively unchanged in the heavier damping range. Figure 11 shows the contact force and frequency of contacts for a system having a relatively large gap. The contact force in this case decreases with increase in damping in the system in the range of damping where the frequency of contacts is unity.
Q !
2-0
c
8
"8 t.o
I
I
I 0-4
I 0.6
i
g
0.0 35O 3O0
200 2 I00
O 0-0
I 0-2
I 0.8
1.0
Coefficient of system damping, r
Figure 11. Typical case showing the ettects of variation in the system damping on the contact force and frequency of contacts in systems with relatively large gaps. Fo = 25, (a = 2.0, m = 0.10, c = 0.0, k = 62-5, ks = 5500, d = 0.40. Key as Figure 9.
4.2.7. System stiffness, k A significant increase in the spring stiffness (k) will generally result in a decrease in both the contact force and the frequency of contacts, although they usually display significant fluctuations with changes in the spring stiffness. As the stiffness is increased, the system will eventually become effectively unrestrained. 4.2.8. Mass, m The effect of variation in the magnitude of the mass on the force and frequency of contacts is similar to the effect of variation in the gap size in the system. An increase in the magnitude of the mass will generally result in a decrease in the frequency of contacts, while the contact force is generally subjected to large and irregular fluctuations. The
OSCILLATOR WITH
LIMITING
305
STOPS, I
frequency of contacts is usually subjected to abrupt changes with variation in the magnitude of the mass, which in effect causes the irregular fluctuations in the contact force, similar to the behaviour shown in Figure 9. 5. CONCLUSIONS The following summarizes the results and observations in this study concerning the dynamic and impact behaviour of the restrained vibratory system considered. (a) Periodic responses occur more frequently with systems having a lower frequency of contacts and/or a larger amount of damping. The opposite is true for erratic responses. (b) The intensity and frequency of contacts at the individual stops in systems having symmetrical gaps are generally the same; however, certain combinations of parameters and initial conditions can result in single-sided impacts or uneven-chatter response in which the frequency of contacts and the contact forces at the individual stops are distinctly different. (c) An increase in the amplitude of the driving force or in the stiffness of the stops will generally result in an increase in both the frequency of contacts and the contact force. (d) An increase in the frequency of the driving force, gap size, or mass will generally result in a decrease in the frequency of contacts. The contact force tends to display Significant fluctuations but generally increases with increase inthe driving frequency, gap size, or mass. (e) A substantial increase in the spring stiffness will result in a decrease in both the frequency of contacts and the contact force. (f) In general, damping at the stops, cs, only affects the contact force and the frequency of contacts in systems having relatively small gap size and/or low stop stiffness. In these systems, a substantial increase in the amount of damping at the stops will generally result in a decrease in the contact force. The frequency of contacts generally increases with increasing damping at the stops for 0 < c~<~csa and decreases for larger values of c, where c~t is some value of damping coefficient greater than zero. (g) A variation in the system damping generally brings about different effects on the frequency of contacts and the contact force in different systems. An increase in the damping near the critical damping range will generally result in a decrease in both the contact force and frequency of contacts. (h) With exception of the variation in the amplitude of the driving force and stop stiffness, an increase (decrease) in the frequency of contacts due to variations in the magnitude of any one of the remaining parameters is generally accompanied by a decrease (increase) in the contact force. -
.
,
-
ACKNOWLEDGMENT This study was carried out as part of a research project supported by the National Science Foundation under Grant MEA-8211288. This support and that provided by the Mechanical Engineering Department of Texas A&M University are gratefully acknowledged. REFERENCES 1. R. C. AZAR 1974 Ph.D. Thesis, University of Massachusetts, Amherst. An investigationof impact phenomenon in geared torsional system. 2. G. HA1DL1976 Messerschmitt-Bolkow-Blohm GMBH, Report No. LIFE-1273. Non-linear effects in aircraft ground and flight vibration tests. (Also presented at the 43rd Meeting of the Structures and Materials Panel of AGARD at London, U.K., 26 September-1 October 1976.)
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3. R. C. WINFREY,'R~ V. ANDERSON and C. W. GHILKA 1973 Journal of Engineeringfor Industry 95, 695-703. Analysis of elastic machinery with clearances. 4. P. O. STEVENS-GUILLE 1973 AECL-4449, Chalk River Nuclear Laboratories, Chalk Ricer, Ontario. Steam generator tube failures: a world survey of water-cooled nuclear power reactors to the end of 1971. 5. R. C. HUANG, E. J. HAUG, JR. and J. G. ANDREWS 1978 American Society of Mechanical Engineers, Journal of Mechanical Design 100, 492-499. Sensitivity analysis and optimal design of a mechanical system with intermittent motion. 6. M. P. KOSTER, 1975 American Society of Mechanical Engineers, Journal of Engineering for Industry 97, 555-602. The effects of the flexibility of the driving shaft on the dynamic behaviour of a cam mechanism. 7. D. T. NGUYEN 1984 M.S. Thesis, Texas A&M Unicersity. Study of single degree of freedom restrained vibratory systems. 8. S. T. NOAH, C. F. KETTLEBOROUGH and R. B. GRIFFIN 1984 Proceedings of the Society of
Manufacturing Engineers, Eleventh Conferenceon Production Research and Technology, Pittsburg, Pennsylvania. Dynamics and wear of mechanical systems with clearances. 9. P. L. K o 1985 American Society of Mechanical Engineers Journal of Tribology 107, 149-156. Heat exchanger tube fretting wear: Review and application to design. 10. R. D. BLEVINS 1978 ASME Paper No. 78-JPGC-NE-8. Fretting wear of heat exchanger tubes, Part 1: Experiments. 11. D. F. MASRI 1965 Ph.D. Thesis, California Institute of Technology. Analytical and experimental studies of impact dampers. 12. M. A. VELUSWAMI, F. R. E. CROSSLEY and G. HORVAY 1975 Journal of Engineering for Industry 97, 828-835. Multiple impacts of a ball between two plates, Part 2: Mathematical modelling. 13. :J. P. WOLF and P. E. SKRIKERUND 1980 Nuclear Engineering and Design 57, 253-275. M u t u a l pounding of adjacent structures during earthquakes. 14. J. M. T. THOMPSON and R. GHAFFARI 1982 Physics Letters 91A, 5-8. Chaos after period doubling bifurcations in the resonance of an impact oscillator. 15. S. DUaOWSKV and F. FREUDENSTEtN 1971 Journal of Engineering for Industry 93, 310-316. Dynamic analysis of mechanical systems with clearances, Part II: Dynamic response. 16. S. MAEZAWA 1963 Zagadnienia Drgan Nieliniowych, 156-163. Perfect Fourier series solution for subharmonic vibration of piecewise-linear system. 17. N. POPPLEWELL, C. N. BAPAT and K. MCLACHLAN 1983 Journal of Sound and Vibration 87, 41-59. Stable periodic vibroimpacts of an oscillator. 18. S. W. SHAW and P. J. HOLMES 1983 Journal of Sound and Vibration 90, 129-155. A periodically forced piecewise linear oscillator. 19. W. D. IWAN 1973 International Journal of Non-Linear Mechanics 8, 279-287. A generalization of the concept of equivalent lineafization. 20. S. DUaOWSKY 1974 ASME Journal of Engineeringfor Industry 93, 324-329. On predicting the dynamic effects of clearance in one dimensional closed loop systems. 21. D.T. NGUYEN, S. T. NOAH and C. F. KETTLEBOROUGH 1986 Journal of Sound and Vibration 109, 309-325. Impact behaviour of an oscillator with limiting stops, Part II: Dimensionless design parameters. 22. K. H. HUNT and F. R. E. CROSSLEY 1975 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 42, 440-445. Coefficient of restitution interpreted as damping in vibroimpact. 23. T. W. LEE and A. C. WANG 1982 ASME Paper Number 82-DET-64. On the dynamics of intermittent-motion mechanisms, Part 1: Dynamic model and response. 24. S. LEVY and J. P. D. WILKINSON 1976 The Component Element Method in Dynamics. New York: McGraw-Hill.
APPENDIX: N O M E N C L A T U R E r
% cj c~ c~
viscous damping coefficient of the system critical damping at the stops, =2mco~ damping coefficient at the lower stop damping coefficient at the upper stop damping coefficient at the stops
OSCILLATOR
WITH
LIMITING
STOPS,
I
k, ks k~
gap size lower gap size upper gap size coefficient of restitution at the stops system stiffness stiffness of the lower stop stiffness of the stops stiffness of the upper stop
m
mass
Fo (F~)~ (Fo)~ F,,
amplitude of excitation force average contact force maximum contact force peak contact force at the ith contact average number of contacts per side per cycle of excitation frequency time displacement of mass velocity of mass, = d y / d t frequency of excitation (cycles/second) natural frequency based on stiffness of the stops, =dkJm damping ratio at the stops, =cs/cc, time interval of response
d
,t, d. e+ k
t
y y' to to s
4T
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