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Experimental frequency response analysis of flexible mechanisms
Mech. Mach. Theory Vol. 28. No. I. pp. 73-81, 1993 Printed in Great Britain. All nights reserved
EXPERIMENTAL
0094-114X/93 $5.00 + 0.00 Copyright ~ 1992 Pergamon Press Ltd
FREQUENCY
OF FLEXIBLE
RESPONSE
ANALYSIS
MECHANISMS
F. W. LIOU and K. C. PENG Department of Mechanicaland Aerospace Engineeringand EngineeringMechanics,ISC, University of Missouri--Rolla. Rolla, MO 65401. U.S.A. (Receired 8 Februarr 1991; receivedfor publication 15 January 1992)
Abstract--This paper presentssomeexperimentalresultsof the frequencyanalysisof flexiblemechanisms. The objective is to help understand the general trend of vibration characteristics of the flexible mechanisms, and to clarifysomedebatable phenomenain the kineto-clasto-
I. INTRODUCTION Since dynamic analysis of high-speed mechanisms is a very complex process, to design a high-speed mechanism, a lot of factors need to be considered. Recently, experimental studies of flexible mechanisms have been reviewed by Peng and Liou [I]. There are still several problems pending to be solved in the area of kineto-elasto-dynamics (ICED). For instance, the modeling of damping in a mechanism, the quasi-static response that exists at higher speeds [2], and the natural frequencies which appeared to be variable in static testing but rather stable in dynamic condition [3-7]. This paper presents an experimental analysis of frequency response of flexible mechanisms to provide a new understanding of the system behavior. Doubtlessly, the natural frequency is a crucial factor in designing a mechanism. The operating speed must not be too close to the system's critical speed region. When operating within this region, the vibration amplitude of the system will be much higher than that running at the neighborhood speeds. Liao et al. [2] have conducted an experimental study of four-bar linkages to document the different classes of flexural mid-span elasto-dynamic responses of the links. It was found that the coupler and follower links which are different in geometric shape showed different resonant frequencies and at various crank speed these responses could be a combination of three response regimes as shown in Fig. I. The horizontal axis represents different input speeds. These regimes are quasi-state, dynamic and resonance and are defined as follows: (I) Dynamic response : dynamic response is characterized by two waveforms comprising a low-frequency periodic waveform (at the crank frequency) upon which is superimposed a high-frequency sinusoidal wave form at the same frequency of the corresponding link. The amplitude of the high frequency component is generally of the same order of magnitude as the amplitude of the low-frequency component. (2) Quasi-static response: quasi-static response is characterized only by the low-frequency periodic waveform at the crank frequency when the amplitude of any high-frequency ripple superimposed upon the low-frequency carrier was less than 10% of the amplitude of the carrier response. (3) Resonance: the amplitude of this response at a specific crank frequency is significantly larger than the amplitude of the response at the adjacent crank frequencies. It is very interesting to see that in Fig. !, the quasi-static response appears between 300 and 304 rpm, and between 365 and 378 rpm high speed ranges. This finding may lead to a breakthrough in the design process of KED systems if the development of the analytical tool can be developed. This paper is to present the result of some experimental analyses in the frequency domain to help understand the phenomenon. 73
74
F.W. LzoL and K. C, PENG ~,nN ~ o s ¢ O ~ O e
+
roson@nC@
--k-- Clu@Si-StetlC
I
Oy n~rnic te8Dons@
50
100
150
200
250
300
350
400
Ct@~ FrQQugncy[r~m)
Fig. I. Coupler respon~ regimes [2],
2. E X P E R I M E N T A L SETUP In this experimental work, both the frequencies of the mechanism assembly and the individual link were studied. Basic features of the experimental setup are summarized as follows: (I) Accelerometers were used in measuring the flexible mechanism vibration. Using an accelerometer as a sensor to detect the vibrational motion of flexible mechanisms in acceleration has some advantages, such as higher sensitivity to the high frequency vibration and its ease of installation on many different links. One disadvantage is that the accelerometer adds its own weight, which will be slightly lower the natural frequency of the link. However, when compared to the case without the accelerometer, the difference in the natural frequency is within 5 0 for the general cases in this experiment, when the weight of the accelerometer is carefully chosen. Another point is that the measured acceleration is perpendicular to the instantaneous beam curve, but not to the rigid link. However, for most of the practical mechanisms, the rotational degree-of-freedom is relatively small, and therefore the difference can be ignored. (2) The driving speed was controlled through a speed controller (DMC-600 series, Galil Motion Control Inc, Palo Alto, CA94303, U.S.A.) and a balancing flywheel (i in. thick, 8 in. dia steel), and the speed was monitored in a continuous oscilloscope display with the accuracy of 0.02 Hz. (3) in order to obtain the general trend of the system behavior for most of the practical mechanisms, 26 sets of mechanism assembly composed of various links were tested in the experiment. The specifications of the links are: Material-- 1018 carbon steel Link length--varying from 4 to 17 in. Thickness--l/16-1/4, in 1/32 in. gradient Width---constant I in. (4) The clearances of all joints, parallel or perpendicular to the rotating plane, were kept below 0.002 in. and were oil-lubricated to minimize any dry friction and impact. (5) The support frame has been monitored to show no significant vibration, the experimental data have been shown to have very high reproducibility both in amplitude and frequency, and the alignment wotk was carefully carried out to ensure no out-of-plane motion. (6) An HP-3582A spectrum analyzer is utilized along with the accelerometers to find the dynamic response of mechanisms both in time and frequency domains.
Frequency analysis of flexible mechanisms
75
The following are descriptions of the major experimental setups. A single beam with two pin-sleeved bearings was used to simulate the coupler or the follower link of a four-bar mechanism. A single beam with one end clamped and the other end free with a lump mass was used to simulate the crank link. The assembled four-bar mechanisms were tested for both static and dynamic responses.
2.1. Pin-sleeted end single beam The individual link was placed in the same condition as the coupler or follower link in a four-bar mechanism with pin-sleeve bearings at both ends for testing as shown in Fig. 2. During the experiment, the accelerometer was placed on the mid-point of the link surface while the link was excited with a test hammer hitting at the mid-section of a link. The tests were conducted for different link length and thickness, but With constant width at I in. 2.Z Clamp-freed end single beam The basic experimental setup for the clamp-freed end beam testing is similar to the pin-sleeved end single beam except for different boundary conditions. This testing is designed to simulate the dynamic behavior of a crank link. The accelerometer was placed near the end-mass when an excitation was imposed on the tip of the end-mass by a testing hammer.
Z3. Assembled four-bar mechanism The experimental linkages were the combination of many different link lengths and thicknesses. The setup is shown in Fig. 3. Both the static and the dynamic responses were studied. For the static responses, the crank link was fixed by clamping the crank link. The mechanism was then tested in various positions. For the dynamic case, 26 sets of mechanisms were studied with various input speeds. The accelerometer was placed on either the link's mid-point or joint to measure the vibration in the transverse or axial directions. The input speeds were varied from 2.5 to 6 Hz.
3. EXPERIMENTAL RESULTS Before describing the experimental results, each frequency terminology used in this paper is defined as follows: Input speed--crank rotating speed in Hz Harmonic frequency--integer multiple of the input speed Link natural frequency--constant link frequency in static condition System natural frequency--natural frequency of the assembled mechanism Resonant frequent--frequency in which the amplitude of the resonant response at a specific crank frequency is significantly larger than the amplitude of the response at the adjacent crank frequencies.
Ibushing & Din
joint "-7
, &oce,ero e, q I //i// Fig. 2. Pin-sleeved end beam setup.
personal ' i computer
i HP-7005B ' X-y re~.o_rder...~
OMC- 600
HP-3582A
speed
spectrum
?TT: OC o
DC source &
motor
amplifier
mechanism
accelerometer
Fig. 3. Block diagram of experimental setup.
76
F, W. Llou and K. C. PE.~G
AMPLITUDE(my) 140r 30.6 120~
I
27.0~
tO0
34.2
80
6O 40 I
20'
0 0
20
I0
FREQUENCY (Hz)
30
40
50
Fig, 4. Typical pin sleeved-beam frequency-response curve.
The results are classified as follows: 3. I. Pin.sleeved end single beam This type of beam testing simulates a coupler or a follower link vibrating at its first flexural mode. A typical frequency-response curve is shown in Fig. 4, which indicates that the natural frequency of this pin-sleeved type link has a wide side band. 3.Z Clamp-freed end single beam The crank link vibrates similar to the clamp-freed type of vibration mode. From the data it is found that the natural frequency can be influenced by link length, thickness and end-mass. With extra mass which stands for equivalent end-mass of joints and other two links, the first natural frequency approximation can be found as; co2. = (3EI)/[L 3 * (M + m)],
(I)
where M is the equivalent mass of the crank link, E is the Young's modulus, I is the moment of inertia of the cross-sectional area, and m is the equivalent end mass. Because no viscous damping is present and the material damping is small, the frequency-response curve for the crank type beam was very sharp as a straight line. Thus, the side band effect does not seem to exist in this type of beam. 3.3. Assembled four-bar mechanism 3.3. I. Static response. It is interesting to see whether the static free-vibration modes can reveal any relation with the full vibration response. The assembled mechanisms have been tested at each crank position to find the static vibration mode by placing accelerometers on both the coupler and the follower links. The four-bar mechanism at different static positions is treated as various structures, when excited with a hammer at many different positions around the mechanism. About six significant frequencies were found during the experiments. When the accelerometer was placed at the coupler or follower's mid-point, and the link was hit at the mid-section, it showed that the link's first flexurai natural frequency was the dominant one and was close to the frequency in the single link testing while the other links only showed insignificant vibration. Moreover, this system, natural frequency varies at different crank positions. 3.3.2. Dynamic response. Figure 5 shows the typical vibration frequency spectra of the coupler and the follower links in the motor-driven operation. The results indicate that the frequency
77
Frequency analysis of flexible mechanisms AMPLITUDE (my) / ~ 1 ~
3oo q/
linkage ienOlh 17 2~-3-13-12
input speed
250 I
~
,........
r~armonics
,-,,,.-,,,o
resonance
200i
1
150
4
20
38
FREQUENCY (Hz)
F ' ~ COUPLER
~
FOLLOWER
Fig, 5. Typical coupler and follower vibration spectra.
response of any individual link includes many harmonic frequencies of the input speed and the resonant frequency which is close to their individual link, natural frequency obtained from a single link's test. This phenomenon has been repeated for more than 20 sets of mechanisms with different linkage combinations. When comparing the spectra of the same mechanism but driven under two different speeds, 4 and 4.8 Hz, it was found that both the input speed harmonic frequencies and resonant frequencies were changed as shown in Figs 6 and 7. Moreover, the link's resonant frequency is within the side band of the link's natural frequency. It is certain that whenever a harmonic frequency of the input speed falls into the side band of a link's natural frequency, the dynamic amplification effect will occur. To investigate whether the response of a mechanism with a flexible crank has the similar response as the rigid ones, Fig. 8 shows a typical example of the follower's vibration spectrum for a mechanism with flexible crank. A 1/16 in. thickness crank link was used in this example. It was noted that the spectrum of the mechanism with a flexible crank has the similar phenomenon as the rigid ones, except that the spectrum contains an additional high amplitude frequency which happens to be the resonant frequency of the crank link. This indicates that the flexible crank will affect the follower or coupler link's vibration. AMPLITUDE (my}
140 SPEED-4 HZ
--t.- SPEEDo4 8 HZ
120 IlnklllQe lenglh 17 25-3-12-13 | rll Gkn419~11-1110-1110
100 80
harmonics 80
resonance 40 20 04' o
12
16
20
24
28
32
[TT 36
4O
FREQUENCY (Hz)
Fig. 6. Coupler vibration spectra at two speeds.
44
T 48
78
F . W . Lnot: and K. C. P ~ 6 ~ , n k n ; e ' e - g r ' l l r 2 ~ - 3 - ! 2 "3
AMPLITUDE (my) 4CC I SPEED.4 ~ :
~
SmEED'4 8 ~."
300 d
resonance 200
harmonics
100
8
~2
16
20
24
TI Tr
28
32
36
40
i
44
i
48
FREQUENCY (Hz)
Fig. 7. Follower vibration spectra at two speeds.
4. A N A L Y S I S
AND DISCUSSION
it has been shown above that to avoid resonance, the input speed must not equal any integer division o f the system natural frequencies, i.e.
(...In, ~ l n
'm, # .,,I,l.
(2)
.....
where o.,. = input speed,
O~l = first system natural frequency, w, = second system natural frequency, w3 = third system natural frequency, n = positive integer, n = I, 2, 3 . . . . Therefore, to design a mechanism without resonance, the systems natural frequencies have to be identified. The experimental result has shown that the systems natural frequencies of a Jt~kSQO lenOt~ 2:3 ~ , ? 1 ~ ' ~ 8 " 1 7
AMPLITUDE (my) /~~
70
m:c.,-ess
1I',6 3 / 3 2 - 3 / 3 2 ;/4 3 , 3 2 - 3 / 3 2
',ere~ r~e) I r
(r,;,,3
cra~
1 I~
iIi
60
i
{Hex~lOj
;'3~-e,2~-1e Ir I ~ , ~ !
resonance
50 40
30 20 10 0 2 ::.
':
FREQUENCY
" •
. . . . . .
t
/ '
?
;
:')
P ''
'-:'~ .'~2 2 3 5
II-lz )
Fig. 8. Comparison of follov, cr ,,ibr:ll~
ira with two lyl~s o f crank.
Frequency analysis of flexible mechanisms
79
mechanism can be estimated from the individual link's individual natural frequency without significant error. This is a very simple and useful rule in design. Another interesting phenomenon which can be observed from the experiment is that even when the input speed is not quite an integer division of any link's natural frequency, it could still cause resonance. This is due to the existence of the side band effect as shown in Fig. 4, especially when the viscous damping is present. The vibration in the coupler or the follower link has many harmonic frequencies of the input speed, and as the harmonic frequency gets higher, its magnitude becomes smaller. If any of the first harmonics fall into the link's natural frequency side band, resonance incurs. In other words, the mechanism is exposed to a forced excitation which consists of a series of harmonics. This excitation, which comes from the link's inertial forces, is not a sinusoidal function. Instead, it is a periodic impulse. Since a periodic impulse can be transformed into the Fourier series, the inertial excitative forces can be decomposed into a series of sinusoidal components. The corresponding magnitude of each harmonic excitation depends on the shape and the amplitude of the inertia force. The vibration behavior then depends on the frequency response characteristics of the mechanisms. The following simplified equation of motion for a one-degree-of-freedom system with a series of sinusoidal components, though not a complete representation of a mechanism system, yet can be used to explain how the link in a mechanism responds:
y+o~,,,y=Ft*sin(l*e~,*t)+F2*sin(2*co,*t)+...+Fs*sin(N*co,*t),
(3)
where (o,,, = natural frequency of link m, y = the displacement at the mid-point of the link, F~, = magnitude of the Nth input harmonics. The right-hand side of equation (3) represents the inertia load of a mechanism. Since the homogeneous solution of the system will decay to zero in the steady state condition, to solve for the steady-state (particular) solution with superposition method, let N
y = y~ y . ,
(4)
I
and it can be found that
y,=A,,sin(n,co,,t), ,4, = F./[~L.
- (n .
~.,)2],
n - - l , 2 . . . . . N,
(5)
(6)
where A, is the peak amplitude of the nth response. As the nth harmonic frequency near the link's natural frequency, the link's vibration amplitude will be significantly increased. From equations (5) and (6), one can see how the similar super-harmonic resonance happens due to the inertia force nonlinearity in a mechanism. Equation (3) does not include the damping effect. If damping is to be considered, then the side band effect should be taken into account. In Fig. 4, the side band of the link's natural frequency is about 7.2 Hz. If the input speed of the mechanism is below 7.2 Hz (i.e. co, < 7.2 Hz), then at least one of the harmonic frequencies (I • co~,. 2 , co¢,. . . . . n • o~,) will fall into the side band and cause resonance. This first explains why in this case. a high frequency vibration is always superimposed with the carrying vibration, if the input speed is higher than 7.2 Hz, then there is a chance that the serial harmonics would skip the side band of a link's natural frequency and not incur any resonance. This can be used to explain how the quasi-static phenomenon occurs in the higher input speeds as shown in Fig. I. In order to vertify the above statements, an experimental testing case of four-bar linkage has been conducted, with link lengths and thicknesses (in.) as follows: ground 23.5, crank 7.125 x 0.0625, coupler 18.0 x 0.09375. and follower 17.0 x 0.09375. The natural frequency of the follower link is at about 28.6 Hz from the single link test. and its side band is about 3.0 Hz. The
F. W. Llou and K. C. PENG
80
crank s p e e d • 3.8 Hz
.o
0 0
I
c r a n k e p e e d • 3 . 8 HZ
,~
i i
i
0
. . . . . . . . .
Time
Fig. 9. Quasi-static response at higher speed (time-domainsignal).
time response of the follower link mid-point vibration is shown in Fig. 9. The follower link's vibration response at the input speed 3.6 Hz has serious high frequency vibration amplitude (upper curve), but the one below 3.8 Hz obviously has no significant high frequency vibration which is similar to a quasi-static response, although in a higher speed. This can be well explained by examining the follower link's natural frequency side band 28.6 + 1.5 Hz. One of the super-harmonic frequencies of 3.6 Hz is within this side band, but none of 3.8 Hz. Although the accelerometer may possess some error (about 5%), the repeated results from different cases can pretty well support the final conclusion. 5. C O N C L U S I O N This paper presents the frequency aspect of the dynamic response of flexible mechanisms. The conclusion can be summarized as follows: (I) In the steady state condition, the dynamic response of each link behaves like an individual beam so that the resonant frequency of each link can be different and is very close to the corresponding link's natural frequency. However, in the static case, the system's natural frequencies of a mechanism are varied with respect to different crank positions. (2) The super-harmonic frequencies (! . ( o , , 2*oJ~. . . . . . n . ( a , ) , which are induced by the inertia force nonlinearity, are integer multiples of the input speed w~,. Whenever a harmonic frequency falls into a link's resonance region, a dynamic amplification will occur, and if the harmonic frequency equals the natural frequency, a resonance will occur. (3) The system's natural frequencies of a flexible mechanism can be estimated based on the individual link's natural frequency. For example, a coupler link may be estimated to be at resonance when the input speed or one of its harmonics is close to its link's natural frequency. (4) If the input speed is carefully chosen so that all of its harmonic frequencies can avoid falling into the resonance region of any link, then a quasi-static response will occur. There will be a good chance for a designer to synthesize a flexible four-bar mechanism to operate at higher input speed and reduce the vibration problem. Since the above observations are quite different from the result of the current analytical models, it is very challenging to develop more analytical tools to stimulate the above phenomena.
Frequency analysis of flexible mechanisms
81
Acknowledgements--The authors would like to express their appreciation for the financial support of the Weldon Spring Inter-campus Endowment Fund, Grant No. R-3--42024, and the support from Department of Mechanical and Aerospace Engineering. and Engineering Mechanics, University of Missouri--Rolla.
REFERENCES 1. K. C. Peng and F. W. Liou, Proc. 1990 A S M E Mechanisms Conf., DE-VOL-28. pp. 37-44 (1990). 2. D. X. Liao, C. K. Sung. B. S. Thompson and K. Soong, ASME Paper No. 86-DET-146 (1986). 3. R. M. Alexander and K. L. Lawrence. ASME Paper No. 73-DET-27 (1973). 4. R. M. Alexander and K. L. Lawrence. ASME Paper No. 74-DET-33 (1974). 5. K. L. Lawrence and R. M. Alexander, !. Mech, E, pp. 133-137 (1975). 6. J. R. Sanders and D. Tesar, Trans. ASME, 100, 762-768 (1978). 7. D. A. Turcic, A. Midha and J. R. Bosnik, ASME Paper No. 83-WA, DSC-38 (1983). 8. C. Bagel and M. H. Dado, ASME Paper No. 84-DET-141 (1984). 9. C. Bagci and F. Khosravi. 9th Conf. Vol. 2 Session VII.A-1 (1985). 10. F. W. Liou and A. G. Erdman, A S M E A
ANALYSE
SPECTRALE
DE MI~CANISMES
FLEXIBLES
R~sum6---Cet article pr~sente des r~sultats exp~rimentaux concernant I'analys¢ spectrale de m~anismes flexibles. L'objectif est de mieux comprcndre les vibrations de rnL'canisme~ flexibles et de clarifier certain ph~nom~ncs dynamiques complexes par une analyse spectrale. Des tests ont ~te conduits :i la lois pour un seul ~l~ment et pour un m~canisme complet de faqon :i pouvoir comparer les fr~quences proprcs de I'el,~ment par lui-m~me avcc les fr~quences propres du syst~me. Certains r~sultats exp~rimentaux font apparaitrc des differences sensibles par rapport au predictions obtenues avec les modeles existants.
EXPERIMENTAL
0094-114X/93 $5.00 + 0.00 Copyright ~ 1992 Pergamon Press Ltd
FREQUENCY
OF FLEXIBLE
RESPONSE
ANALYSIS
MECHANISMS
F. W. LIOU and K. C. PENG Department of Mechanicaland Aerospace Engineeringand EngineeringMechanics,ISC, University of Missouri--Rolla. Rolla, MO 65401. U.S.A. (Receired 8 Februarr 1991; receivedfor publication 15 January 1992)
Abstract--This paper presentssomeexperimentalresultsof the frequencyanalysisof flexiblemechanisms. The objective is to help understand the general trend of vibration characteristics of the flexible mechanisms, and to clarifysomedebatable phenomenain the kineto-clasto-
I. INTRODUCTION Since dynamic analysis of high-speed mechanisms is a very complex process, to design a high-speed mechanism, a lot of factors need to be considered. Recently, experimental studies of flexible mechanisms have been reviewed by Peng and Liou [I]. There are still several problems pending to be solved in the area of kineto-elasto-dynamics (ICED). For instance, the modeling of damping in a mechanism, the quasi-static response that exists at higher speeds [2], and the natural frequencies which appeared to be variable in static testing but rather stable in dynamic condition [3-7]. This paper presents an experimental analysis of frequency response of flexible mechanisms to provide a new understanding of the system behavior. Doubtlessly, the natural frequency is a crucial factor in designing a mechanism. The operating speed must not be too close to the system's critical speed region. When operating within this region, the vibration amplitude of the system will be much higher than that running at the neighborhood speeds. Liao et al. [2] have conducted an experimental study of four-bar linkages to document the different classes of flexural mid-span elasto-dynamic responses of the links. It was found that the coupler and follower links which are different in geometric shape showed different resonant frequencies and at various crank speed these responses could be a combination of three response regimes as shown in Fig. I. The horizontal axis represents different input speeds. These regimes are quasi-state, dynamic and resonance and are defined as follows: (I) Dynamic response : dynamic response is characterized by two waveforms comprising a low-frequency periodic waveform (at the crank frequency) upon which is superimposed a high-frequency sinusoidal wave form at the same frequency of the corresponding link. The amplitude of the high frequency component is generally of the same order of magnitude as the amplitude of the low-frequency component. (2) Quasi-static response: quasi-static response is characterized only by the low-frequency periodic waveform at the crank frequency when the amplitude of any high-frequency ripple superimposed upon the low-frequency carrier was less than 10% of the amplitude of the carrier response. (3) Resonance: the amplitude of this response at a specific crank frequency is significantly larger than the amplitude of the response at the adjacent crank frequencies. It is very interesting to see that in Fig. !, the quasi-static response appears between 300 and 304 rpm, and between 365 and 378 rpm high speed ranges. This finding may lead to a breakthrough in the design process of KED systems if the development of the analytical tool can be developed. This paper is to present the result of some experimental analyses in the frequency domain to help understand the phenomenon. 73
74
F.W. LzoL and K. C, PENG ~,nN ~ o s ¢ O ~ O e
+
roson@nC@
--k-- Clu@Si-StetlC
I
Oy n~rnic te8Dons@
50
100
150
200
250
300
350
400
Ct@~ FrQQugncy[r~m)
Fig. I. Coupler respon~ regimes [2],
2. E X P E R I M E N T A L SETUP In this experimental work, both the frequencies of the mechanism assembly and the individual link were studied. Basic features of the experimental setup are summarized as follows: (I) Accelerometers were used in measuring the flexible mechanism vibration. Using an accelerometer as a sensor to detect the vibrational motion of flexible mechanisms in acceleration has some advantages, such as higher sensitivity to the high frequency vibration and its ease of installation on many different links. One disadvantage is that the accelerometer adds its own weight, which will be slightly lower the natural frequency of the link. However, when compared to the case without the accelerometer, the difference in the natural frequency is within 5 0 for the general cases in this experiment, when the weight of the accelerometer is carefully chosen. Another point is that the measured acceleration is perpendicular to the instantaneous beam curve, but not to the rigid link. However, for most of the practical mechanisms, the rotational degree-of-freedom is relatively small, and therefore the difference can be ignored. (2) The driving speed was controlled through a speed controller (DMC-600 series, Galil Motion Control Inc, Palo Alto, CA94303, U.S.A.) and a balancing flywheel (i in. thick, 8 in. dia steel), and the speed was monitored in a continuous oscilloscope display with the accuracy of 0.02 Hz. (3) in order to obtain the general trend of the system behavior for most of the practical mechanisms, 26 sets of mechanism assembly composed of various links were tested in the experiment. The specifications of the links are: Material-- 1018 carbon steel Link length--varying from 4 to 17 in. Thickness--l/16-1/4, in 1/32 in. gradient Width---constant I in. (4) The clearances of all joints, parallel or perpendicular to the rotating plane, were kept below 0.002 in. and were oil-lubricated to minimize any dry friction and impact. (5) The support frame has been monitored to show no significant vibration, the experimental data have been shown to have very high reproducibility both in amplitude and frequency, and the alignment wotk was carefully carried out to ensure no out-of-plane motion. (6) An HP-3582A spectrum analyzer is utilized along with the accelerometers to find the dynamic response of mechanisms both in time and frequency domains.
Frequency analysis of flexible mechanisms
75
The following are descriptions of the major experimental setups. A single beam with two pin-sleeved bearings was used to simulate the coupler or the follower link of a four-bar mechanism. A single beam with one end clamped and the other end free with a lump mass was used to simulate the crank link. The assembled four-bar mechanisms were tested for both static and dynamic responses.
2.1. Pin-sleeted end single beam The individual link was placed in the same condition as the coupler or follower link in a four-bar mechanism with pin-sleeve bearings at both ends for testing as shown in Fig. 2. During the experiment, the accelerometer was placed on the mid-point of the link surface while the link was excited with a test hammer hitting at the mid-section of a link. The tests were conducted for different link length and thickness, but With constant width at I in. 2.Z Clamp-freed end single beam The basic experimental setup for the clamp-freed end beam testing is similar to the pin-sleeved end single beam except for different boundary conditions. This testing is designed to simulate the dynamic behavior of a crank link. The accelerometer was placed near the end-mass when an excitation was imposed on the tip of the end-mass by a testing hammer.
Z3. Assembled four-bar mechanism The experimental linkages were the combination of many different link lengths and thicknesses. The setup is shown in Fig. 3. Both the static and the dynamic responses were studied. For the static responses, the crank link was fixed by clamping the crank link. The mechanism was then tested in various positions. For the dynamic case, 26 sets of mechanisms were studied with various input speeds. The accelerometer was placed on either the link's mid-point or joint to measure the vibration in the transverse or axial directions. The input speeds were varied from 2.5 to 6 Hz.
3. EXPERIMENTAL RESULTS Before describing the experimental results, each frequency terminology used in this paper is defined as follows: Input speed--crank rotating speed in Hz Harmonic frequency--integer multiple of the input speed Link natural frequency--constant link frequency in static condition System natural frequency--natural frequency of the assembled mechanism Resonant frequent--frequency in which the amplitude of the resonant response at a specific crank frequency is significantly larger than the amplitude of the response at the adjacent crank frequencies.
Ibushing & Din
joint "-7
, &oce,ero e, q I //i// Fig. 2. Pin-sleeved end beam setup.
personal ' i computer
i HP-7005B ' X-y re~.o_rder...~
OMC- 600
HP-3582A
speed
spectrum
?TT: OC o
DC source &
motor
amplifier
mechanism
accelerometer
Fig. 3. Block diagram of experimental setup.
76
F, W. Llou and K. C. PE.~G
AMPLITUDE(my) 140r 30.6 120~
I
27.0~
tO0
34.2
80
6O 40 I
20'
0 0
20
I0
FREQUENCY (Hz)
30
40
50
Fig, 4. Typical pin sleeved-beam frequency-response curve.
The results are classified as follows: 3. I. Pin.sleeved end single beam This type of beam testing simulates a coupler or a follower link vibrating at its first flexural mode. A typical frequency-response curve is shown in Fig. 4, which indicates that the natural frequency of this pin-sleeved type link has a wide side band. 3.Z Clamp-freed end single beam The crank link vibrates similar to the clamp-freed type of vibration mode. From the data it is found that the natural frequency can be influenced by link length, thickness and end-mass. With extra mass which stands for equivalent end-mass of joints and other two links, the first natural frequency approximation can be found as; co2. = (3EI)/[L 3 * (M + m)],
(I)
where M is the equivalent mass of the crank link, E is the Young's modulus, I is the moment of inertia of the cross-sectional area, and m is the equivalent end mass. Because no viscous damping is present and the material damping is small, the frequency-response curve for the crank type beam was very sharp as a straight line. Thus, the side band effect does not seem to exist in this type of beam. 3.3. Assembled four-bar mechanism 3.3. I. Static response. It is interesting to see whether the static free-vibration modes can reveal any relation with the full vibration response. The assembled mechanisms have been tested at each crank position to find the static vibration mode by placing accelerometers on both the coupler and the follower links. The four-bar mechanism at different static positions is treated as various structures, when excited with a hammer at many different positions around the mechanism. About six significant frequencies were found during the experiments. When the accelerometer was placed at the coupler or follower's mid-point, and the link was hit at the mid-section, it showed that the link's first flexurai natural frequency was the dominant one and was close to the frequency in the single link testing while the other links only showed insignificant vibration. Moreover, this system, natural frequency varies at different crank positions. 3.3.2. Dynamic response. Figure 5 shows the typical vibration frequency spectra of the coupler and the follower links in the motor-driven operation. The results indicate that the frequency
77
Frequency analysis of flexible mechanisms AMPLITUDE (my) / ~ 1 ~
3oo q/
linkage ienOlh 17 2~-3-13-12
input speed
250 I
~
,........
r~armonics
,-,,,.-,,,o
resonance
200i
1
150
4
20
38
FREQUENCY (Hz)
F ' ~ COUPLER
~
FOLLOWER
Fig, 5. Typical coupler and follower vibration spectra.
response of any individual link includes many harmonic frequencies of the input speed and the resonant frequency which is close to their individual link, natural frequency obtained from a single link's test. This phenomenon has been repeated for more than 20 sets of mechanisms with different linkage combinations. When comparing the spectra of the same mechanism but driven under two different speeds, 4 and 4.8 Hz, it was found that both the input speed harmonic frequencies and resonant frequencies were changed as shown in Figs 6 and 7. Moreover, the link's resonant frequency is within the side band of the link's natural frequency. It is certain that whenever a harmonic frequency of the input speed falls into the side band of a link's natural frequency, the dynamic amplification effect will occur. To investigate whether the response of a mechanism with a flexible crank has the similar response as the rigid ones, Fig. 8 shows a typical example of the follower's vibration spectrum for a mechanism with flexible crank. A 1/16 in. thickness crank link was used in this example. It was noted that the spectrum of the mechanism with a flexible crank has the similar phenomenon as the rigid ones, except that the spectrum contains an additional high amplitude frequency which happens to be the resonant frequency of the crank link. This indicates that the flexible crank will affect the follower or coupler link's vibration. AMPLITUDE (my}
140 SPEED-4 HZ
--t.- SPEEDo4 8 HZ
120 IlnklllQe lenglh 17 25-3-12-13 | rll Gkn419~11-1110-1110
100 80
harmonics 80
resonance 40 20 04' o
12
16
20
24
28
32
[TT 36
4O
FREQUENCY (Hz)
Fig. 6. Coupler vibration spectra at two speeds.
44
T 48
78
F . W . Lnot: and K. C. P ~ 6 ~ , n k n ; e ' e - g r ' l l r 2 ~ - 3 - ! 2 "3
AMPLITUDE (my) 4CC I SPEED.4 ~ :
~
SmEED'4 8 ~."
300 d
resonance 200
harmonics
100
8
~2
16
20
24
TI Tr
28
32
36
40
i
44
i
48
FREQUENCY (Hz)
Fig. 7. Follower vibration spectra at two speeds.
4. A N A L Y S I S
AND DISCUSSION
it has been shown above that to avoid resonance, the input speed must not equal any integer division o f the system natural frequencies, i.e.
(...In, ~ l n
'm, # .,,I,l.
(2)
.....
where o.,. = input speed,
O~l = first system natural frequency, w, = second system natural frequency, w3 = third system natural frequency, n = positive integer, n = I, 2, 3 . . . . Therefore, to design a mechanism without resonance, the systems natural frequencies have to be identified. The experimental result has shown that the systems natural frequencies of a Jt~kSQO lenOt~ 2:3 ~ , ? 1 ~ ' ~ 8 " 1 7
AMPLITUDE (my) /~~
70
m:c.,-ess
1I',6 3 / 3 2 - 3 / 3 2 ;/4 3 , 3 2 - 3 / 3 2
',ere~ r~e) I r
(r,;,,3
cra~
1 I~
iIi
60
i
{Hex~lOj
;'3~-e,2~-1e Ir I ~ , ~ !
resonance
50 40
30 20 10 0 2 ::.
':
FREQUENCY
" •
. . . . . .
t
/ '
?
;
:')
P ''
'-:'~ .'~2 2 3 5
II-lz )
Fig. 8. Comparison of follov, cr ,,ibr:ll~
ira with two lyl~s o f crank.
Frequency analysis of flexible mechanisms
79
mechanism can be estimated from the individual link's individual natural frequency without significant error. This is a very simple and useful rule in design. Another interesting phenomenon which can be observed from the experiment is that even when the input speed is not quite an integer division of any link's natural frequency, it could still cause resonance. This is due to the existence of the side band effect as shown in Fig. 4, especially when the viscous damping is present. The vibration in the coupler or the follower link has many harmonic frequencies of the input speed, and as the harmonic frequency gets higher, its magnitude becomes smaller. If any of the first harmonics fall into the link's natural frequency side band, resonance incurs. In other words, the mechanism is exposed to a forced excitation which consists of a series of harmonics. This excitation, which comes from the link's inertial forces, is not a sinusoidal function. Instead, it is a periodic impulse. Since a periodic impulse can be transformed into the Fourier series, the inertial excitative forces can be decomposed into a series of sinusoidal components. The corresponding magnitude of each harmonic excitation depends on the shape and the amplitude of the inertia force. The vibration behavior then depends on the frequency response characteristics of the mechanisms. The following simplified equation of motion for a one-degree-of-freedom system with a series of sinusoidal components, though not a complete representation of a mechanism system, yet can be used to explain how the link in a mechanism responds:
y+o~,,,y=Ft*sin(l*e~,*t)+F2*sin(2*co,*t)+...+Fs*sin(N*co,*t),
(3)
where (o,,, = natural frequency of link m, y = the displacement at the mid-point of the link, F~, = magnitude of the Nth input harmonics. The right-hand side of equation (3) represents the inertia load of a mechanism. Since the homogeneous solution of the system will decay to zero in the steady state condition, to solve for the steady-state (particular) solution with superposition method, let N
y = y~ y . ,
(4)
I
and it can be found that
y,=A,,sin(n,co,,t), ,4, = F./[~L.
- (n .
~.,)2],
n - - l , 2 . . . . . N,
(5)
(6)
where A, is the peak amplitude of the nth response. As the nth harmonic frequency near the link's natural frequency, the link's vibration amplitude will be significantly increased. From equations (5) and (6), one can see how the similar super-harmonic resonance happens due to the inertia force nonlinearity in a mechanism. Equation (3) does not include the damping effect. If damping is to be considered, then the side band effect should be taken into account. In Fig. 4, the side band of the link's natural frequency is about 7.2 Hz. If the input speed of the mechanism is below 7.2 Hz (i.e. co, < 7.2 Hz), then at least one of the harmonic frequencies (I • co~,. 2 , co¢,. . . . . n • o~,) will fall into the side band and cause resonance. This first explains why in this case. a high frequency vibration is always superimposed with the carrying vibration, if the input speed is higher than 7.2 Hz, then there is a chance that the serial harmonics would skip the side band of a link's natural frequency and not incur any resonance. This can be used to explain how the quasi-static phenomenon occurs in the higher input speeds as shown in Fig. I. In order to vertify the above statements, an experimental testing case of four-bar linkage has been conducted, with link lengths and thicknesses (in.) as follows: ground 23.5, crank 7.125 x 0.0625, coupler 18.0 x 0.09375. and follower 17.0 x 0.09375. The natural frequency of the follower link is at about 28.6 Hz from the single link test. and its side band is about 3.0 Hz. The
F. W. Llou and K. C. PENG
80
crank s p e e d • 3.8 Hz
.o
0 0
I
c r a n k e p e e d • 3 . 8 HZ
,~
i i
i
0
. . . . . . . . .
Time
Fig. 9. Quasi-static response at higher speed (time-domainsignal).
time response of the follower link mid-point vibration is shown in Fig. 9. The follower link's vibration response at the input speed 3.6 Hz has serious high frequency vibration amplitude (upper curve), but the one below 3.8 Hz obviously has no significant high frequency vibration which is similar to a quasi-static response, although in a higher speed. This can be well explained by examining the follower link's natural frequency side band 28.6 + 1.5 Hz. One of the super-harmonic frequencies of 3.6 Hz is within this side band, but none of 3.8 Hz. Although the accelerometer may possess some error (about 5%), the repeated results from different cases can pretty well support the final conclusion. 5. C O N C L U S I O N This paper presents the frequency aspect of the dynamic response of flexible mechanisms. The conclusion can be summarized as follows: (I) In the steady state condition, the dynamic response of each link behaves like an individual beam so that the resonant frequency of each link can be different and is very close to the corresponding link's natural frequency. However, in the static case, the system's natural frequencies of a mechanism are varied with respect to different crank positions. (2) The super-harmonic frequencies (! . ( o , , 2*oJ~. . . . . . n . ( a , ) , which are induced by the inertia force nonlinearity, are integer multiples of the input speed w~,. Whenever a harmonic frequency falls into a link's resonance region, a dynamic amplification will occur, and if the harmonic frequency equals the natural frequency, a resonance will occur. (3) The system's natural frequencies of a flexible mechanism can be estimated based on the individual link's natural frequency. For example, a coupler link may be estimated to be at resonance when the input speed or one of its harmonics is close to its link's natural frequency. (4) If the input speed is carefully chosen so that all of its harmonic frequencies can avoid falling into the resonance region of any link, then a quasi-static response will occur. There will be a good chance for a designer to synthesize a flexible four-bar mechanism to operate at higher input speed and reduce the vibration problem. Since the above observations are quite different from the result of the current analytical models, it is very challenging to develop more analytical tools to stimulate the above phenomena.
Frequency analysis of flexible mechanisms
81
Acknowledgements--The authors would like to express their appreciation for the financial support of the Weldon Spring Inter-campus Endowment Fund, Grant No. R-3--42024, and the support from Department of Mechanical and Aerospace Engineering. and Engineering Mechanics, University of Missouri--Rolla.
REFERENCES 1. K. C. Peng and F. W. Liou, Proc. 1990 A S M E Mechanisms Conf., DE-VOL-28. pp. 37-44 (1990). 2. D. X. Liao, C. K. Sung. B. S. Thompson and K. Soong, ASME Paper No. 86-DET-146 (1986). 3. R. M. Alexander and K. L. Lawrence. ASME Paper No. 73-DET-27 (1973). 4. R. M. Alexander and K. L. Lawrence. ASME Paper No. 74-DET-33 (1974). 5. K. L. Lawrence and R. M. Alexander, !. Mech, E, pp. 133-137 (1975). 6. J. R. Sanders and D. Tesar, Trans. ASME, 100, 762-768 (1978). 7. D. A. Turcic, A. Midha and J. R. Bosnik, ASME Paper No. 83-WA, DSC-38 (1983). 8. C. Bagel and M. H. Dado, ASME Paper No. 84-DET-141 (1984). 9. C. Bagci and F. Khosravi. 9th Conf. Vol. 2 Session VII.A-1 (1985). 10. F. W. Liou and A. G. Erdman, A S M E A
ANALYSE
SPECTRALE
DE MI~CANISMES
FLEXIBLES
R~sum6---Cet article pr~sente des r~sultats exp~rimentaux concernant I'analys¢ spectrale de m~anismes flexibles. L'objectif est de mieux comprcndre les vibrations de rnL'canisme~ flexibles et de clarifier certain ph~nom~ncs dynamiques complexes par une analyse spectrale. Des tests ont ~te conduits :i la lois pour un seul ~l~ment et pour un m~canisme complet de faqon :i pouvoir comparer les fr~quences proprcs de I'el,~ment par lui-m~me avcc les fr~quences propres du syst~me. Certains r~sultats exp~rimentaux font apparaitrc des differences sensibles par rapport au predictions obtenues avec les modeles existants.