- Email: [email protected]
Effect of lubrication on slip damping
65
Wear, 154 (1992) 65-76
Effect of lubrication
on slip damping
K. K. Padmanabhan Department (India)
(Received
of Mechanical
Engineering,
NSS
College of Engineering
March 18, 1991; revised and accepted
Palghat 678 008, Kerala
August 20, 1991)
Abstract Metallic resulting response joints. A
mating surfaces in structural joints offer a good source of energy dissipation in a damped dynamic structural response. This paper reports the application of surface methodology to study the effect of lubrication on slip damping in structural central composite rotatable design was used to develop predictive models. Good
correlation between predicted and slip damping test results was obtained. The methodology presented suggests a useful technique for damping research. 1. Introduction
In the design stage of a machine tool, it is necessary to build in sufficient damping to ensure that under all expected operating conditions stress or vibration amplitude levels never cause failure or impair the required functioning of the machine tool. Amongst the possible and available sources of damping, damping at metallic joints play an important role. Material damping is not always sufficient to limit vibration amplitudes and stresses to within the desired level; in such cases joint damping may represent the solution to the problem. Since about 90% of the total damping in a structure originates in joints [l], it would seem worthwhile to optimize such damping, thereby making inherent damping sufficient and eliminating the need for expensive or complex damping devices. Damping at structural joints is associated with frictional losses caused by slip or small relative inter-facial movements of the component parts against one another. Since the energy dissipation is caused by relative slip between surfaces of the joint, this form of damping is termed slip damping. Depending upon the extent of relative motion, the joints are classified as macroslip (gross sliding) and microslip (partial slip) joints. Slip damping offers excellent potential for large energy dissipation at structural joints even under microslip conditions. Such energy loss at various clamped joints throughout the structure can have a significant effect in limiting the magnitude of vibration. Several investigators in the past have been interested in the damping phenomenon which occurs at preloaded metallic interfaces subjected to cyclic tangential forces. Some of these works have been reviewed and the importance of damping at joints emphasized by Padmanabhan and coworkers [2-6]. Andrew ef al. [7] have shown that no significant energy loss occurs at preloaded flat metallic joints subjected to normal oscillations. This suggests that only the tangential component of vibrations can be predominently damped out #by such joints. The area of interest in. the present work is damping occurring at preloaded, flat metallic interfaces subjected to cyclic tangential forces. Masuko et al. [8] found that
0043-1648/92/$5.00
0 1992 - Elsevier
Sequoia.
All rights reserved
66
the tangential-force-displacement characteristics of a preloaded flat metallic interface arc largely influenced, albeit in a complex manner, by the normal load, surface roughness, material and machined lay orientation. Dekonink [9] obtained the energy loss per cycle at metallic joints under dynamic tangential loads, from the tangentialforce-displacement hysteresis loop. Rogers and Boothroyd [lo] have shown the energy loss per cycle under dynamic conditions to be independent of the exciting frequency. Padmanabhan and Das [2] have found experimentally that the energy loss per cycle is closely related to the preload, machining method and machined lay orientation of the mating surfaces. An optimum surface roughness was shown to exist at which the energy dissipation is maximum. It is observed that joint damping has been largely neglected because of fears of fretting and stiffness losses and non-linearities at the joint [ll]. Although the fretting corrosion which accompanies interfacial slip can cause cracks and structural failure by fatigue, this may be preferable to the high dynamic stresses which would otherwise result [12]. Furthermore, fretting can be considerably reduced by suitable treatment of the interface surfaces [13, 141. It is therefore now relevant to consider increasing the joint damping which exists in structures [15]. Realistic joints are normally dependent on several complex parameters and unknowns. A knowledge of the effects of joint parameters on damping at structural joints is necessary, if joint damping is to be used to reduce the vibration amplitudes of a structure. The development of experimental research on damping is required since the theoretical approach does not appear sufficient to acquire good knowledge of the problem and to make satisfactory provisos on the dynamic behaviour of complex structures [16]. However, the conventional experimental techniques of one-variableat-a-time for studying damping in machined joints is time consuming and expensive. The large variety of experimental conditions and multiplicity of conditional conclusions present difficulties in extending these research findings for general use. One means of obviating these difficulties is to use well planned statistical experimental design techniques. This paper reports the application of response surface methodology [17-191 in the prediction of slip damping in structural joints. In applying response surface methodology, the dependent variable or response is viewed as a surface to which a mathematical model is fitted. Only quantitative variables which are controllable can be included in the predictive model. Factors that generally affect slip damping in structures may be grouped into quantitative factors such as normal preload, cyclic tangential load, apparent area of contact and surface roughness, and qualitative factors such as machining method, machined lay orientation, material, surface treatment and lubricant at the interface. In the present investigation the effect of lubricant on slip damping is examined, keeping all other factors constant except the normal preload and cyclic tangential force.
2. Postulation
of the maZhematica1
model
The normal preload and peak cyclic tangential force have a controlling effect on energy dissipation at the joint for given contact conditions and the following secondorder damping model can be adopted. y=bo+blxl
+bzxz+bllx12+
bz2xz2+blgtlx2+e
(1)
wherexl andx2 represent the logarithmically transformed values of peak cyclic tangential force and normal preload respectively; xl2 and x2” are the quadratic effects of these
67
variables and x1x2 represent the interaction between them; bO, br, b2 etc. are the parameters and e the experimental error. The parameters of eqn. (1) are estimated by the method of least squares.
3. Experimental
set-up
The joint studied in the present work consists of a cup and lid assembly of grey cast iron (Fig. 1). A rod of the same material was bolted at the centre of the lid to facilitate normal loading of the lid. Centring pins were used to position the rod at the centre of the bore of the cup. The lid sat on the top of the cup, which was bolted securely to a heavy cast iron block rigidly fixed to a heavy cast iron bed. The lid was preloaded through the rod, by means of dead weights acting through a lever system and subjected to cyclic tangential loads along the plane of the interface. The oscillating tangential force was applied along the plane of the intraface by an electrodynamic vibration generator powered by an excitation amplifier-oscillator unit. The vibration generator was rigidly mounted on the heavy cast iron bed and isolated from any structural vibrations of the bed. A load cell was used in conjunction with a carrier amplifier to measure the tangential force. The tangential displacement of the lid relative to the cup was measured by a non-contacting eddy current probe amounted rigidly in the lid. The output of the probe was fed into a matching amplifier. Amplified outputs from the load cell and displacement probe fed into the storage
Cost Normal
Fig. 1. Cup and lid assembly.
force
iron
bed
68
oscilloscope were used to obtain the load displacement hysteresis which indicates the energy loss per cycle during vibration.
4. Experimental
loop, the area of
design
In order to determine the equation for the response surface, several special experimental techniques have been developed which attempt to approximate this equation using the smallest number of experiments possible. An experimental procedure for fitting a second-order equation must have at least three levels of each factor for estimating the model parameters, but with an increasing number of factors the experiment becomes large. The most widely preferred class of second-order response surface design is the central composite rotatable design [20]. The central composite rotatable design for the second-order response surface is shown in Fig. 2. It consists of four corner points, a centre point repeated five times and four axial points called augments. The test conditions are given in Table 1. The
(_l.LILIO)
i
~/
poi"ts
/ (I.L~L,01
~-l~k-pl~ Fig. 2. Central composite rotatable design for K=2. TABLE 1 Levels of independent variables and coding identification Independent variables
Levels in coded form - 1.414
Unlubricated joint Peak cyclic tangential force (N) Normal preload (N) Lubricated joint Peak cyclic tangential force (N) Normal preload (N)
-1
120
131
3000
3129
120
131
4000
4245
+1
0
162.5 3464 162.5 4899
+ 1.414
201
220
3835
4000
201
220
5654
6000
69
coded values of variables for use in eqn. (1) were obtained equations: Unlubricated joint, In T-ln xi=
In 201-In
q=
In 3835-In
from the following transforming
162.5 162.5
In P- In 3464 lubricated
joint,
In T-In “=
3464
In 201-In
162.5 162.5
In P - In 4899 x2= In 5654- In 4899 where x1 is the coded value of the peak cyclic tangential force corresponding to its natural value T and x2 is the coded value of the normal preload corresponding to its natural value P. The normal preload range was increased in the case of lubricated joints in order to limit the experiments within microslip. The above relations were obtained from the following transforming equation: In X, - In X,, ‘=
lnX,l-lnX,o
where x is the coded value of any factor corresponding to its natural value X,, X,,, is the natural value of the factor at + 1 level, and X,,, is the natural value of the factor corresponding to the base or zero level [21]. 5. Experimentation Ground (Ra =0.98 pm), grey cast iron joints having an apparent contact area of 594 mm2 and machined lays perpendicular were used for the tests. Surface conditions and joint geometry were chosen primarily to model practical joints found in machine tools. After surface preparation, the joint surfaces were cleaned with trichloroethylene to obtain real metal contact conditions and stored in a desiccator. Preliminary tests had revealed that energy dissipation is independent of the frequency of the cyclic tangential force. Hence tangential forces were applied at 5 Hz during the tests reported here. Undesirable dynamic effects associated with high frequencies are also taken care of by this selection of low frequency. It was also observed that energy dissipation is higher when the machined lays of joint surfaces were perpendicular than when they were parallel. For given joint surface conditions and normal preload, an oscillating tangential force applied to the lid produced a load-displacement hysteresis loop whose area is proportional to the energy loss per cycle of vibrations. The hysteresis loops displayed on the oscilloscope were photographed and their areas measured. Before conducting each trial, the surfaces were reset by separating the cup and lid and reassembling them. Experimental data presented in this work relate only to those joints in which the surface damage of the interface during the test was insignificant. The scheme of experimentation and the results obtained are given in Table 2, in which D is the energy loss per cycle (10 ELI).
70 TABLE
2
Scheme of experimentation
and results
Standard order
XI
x0
6. Estimation
-1
1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10 11 12 13
-1 -1
1
1 1 0 0
-1 1 - 1.414 1.414 0 0 0 0 0 0 0
of parameters
y=ln
X2
- 1.414 1.414 0 0 0 0 0
in the postulated
D
Unlubricated joint
Lubricated joint
3.081354 4.054963 3.059535 3.781670 3.052155 4.163037 3.531375 3.339837 3.484418 3.484418 3.4147383 3.424995 3.450185
3.388627 5.109124 3.190305 4.12069 3.066861 5.107228 4.182362 3.549558 3.627519 3.623344 3.631677 3.660309 3.707562
models
The parameters b,-,, bl, b2 etc. in eqn. (1) are estimated squares using the basic formula. b = (X=X) - ‘X=Y
by the method
of least (2)
where b is the matrix of parameter estimates, X is the matrix of independent variables or design matrix, XT is the transpose of matrix X, and Y is the matrix of logarithms of the measured responses (energy loss per cycle). For the central composite rotatable design, the (X=X)-r matrix derived from Table 2 is 1
5
0
0
_I
-1
0
10
10
0
0
0
0
0
1 8
0
0
0
-6
0
0
23 160
3 160
0
-- 1 10
0
0
3 160
23 160
0
0
0
0
0
1 4
0
0
1 s
(x=x)-=
0
71
The second-order energy loss equations for unlubricated and lubricated joints were obtained using the result of two sets of 13 tests:
grey cast iron
Yr = 3.451744 + O.4O834&u,- 0.07074~ +0.07111&r,z - 0.014878~~ -0.062868x1x,
(3)
Yz= 3.650077 + 0.69205&, - 026021C& -t 0.212404~~~ + 0.10186fbr,z - 0.197528x,x, 7. Analysis
(4)
of results
Zl. Adequacy of postulated models The usual method for testing the adequacy of a model is to compute the F-ratio of the lack of fit to the pure error and compare it with the standard value. The adequacy can be checked most conveniently by making an analysis of variance (ANOVA) table which includes sum of squares (SS), degrees of freedom (df) and mean square (MS). In order to perform the analysis of variance, the sum of squares of y values is usually partitioned into contributions from the first-order terms, the second-order terms, the lack of fit and experimental error. The lack of fit measures the deviations of the responses from the fitted surface. Experimental error is obtained from the replicated points at the centre. The sum of squares of the lack of fit and the sum of squares of experimental error (pure error} contribute to the residual sum of squares. The mean square is the sum of squares divided by the degrees of freedom. The detailed formulae for the variance analysis used here are given in Table 3. Tables 4 and 5 give the results of the variance analysis, in which F,, is the ratio of the mean square of the respective term to the mean square of experimental error and Ft;;,,, is the standard F-value corresponding to the degrees of freedom and 5% si~ific~ce level [22]. Both the first-order and second-order terms have a significant effect on the energy loss per cycle in the dry and lubricated joints considered; the fit is also adequate. 7.2. Significance of individual variables The sum of squares for any variable xi adjusted for all the others is available by taking bi2/cii. This amount is the sum of squares as xi is added to a model containing all the variables except this particular variable. To test the significance of Xi adjusted for all other variables, F = (b&&s2 is compared with the standard F value. In this way insignificant variables may be eliminated. Tables 6 and 7 were constructed to test the effect of individual variables, adjusted for all other variables, on the experimental results in the case of unlubricated and lubricated joints respectively. In the case of the unlubricated joint (Table 6) the insignificant variable can be identified as x22. Based on the above analysis, eqn. (3) has been modified to Yr = 3.4413948 +0.4083482&-r - O.O707486875(i + 0.073058~~~’
- 0.06286883~~~
(5)
The parameters in eqn. (5) were re-estimated using eqn. (2) and a corresponding new X matrix. Table 8 gives the results of variance analysis, and the fit is found to be adequate.
72
TABLE
3
Formulae
for ANOVA
Source
Sum of squares
First-order
(SS)
Degree of freedom (dt) K
terms
Second-order
terms
K(K+
Lack of fit
Experimental
I)/2
n,+?l,-K(K+3)!2
error
no-l
y, Logarithm of observed response; y,, logarithm of observed responses at centre point with mean 9s; N, total number of experimental points; n,, number of centre points; n,, number of comer points; n,, number of axial points; K, dimension of design; g, grand total of observed responses; (Oy), (ly), (2~) etc., sums of products of each column in the X matrix with the columns of y values; C~ the inverse of the sum of squares of the ith column in the X matrix. TABLE 4 ANOVA
(unlubricated
joint)
Source
ss
df
MS
F c.4,
F tab
First-order terms Second-order terms Lack of fit Experimental error Total
1.3740147 0.0551224 0.003499559 0.004222542 1.4368592
2 3 3 4 12
0.687007 0.018374 0.00116652 0.001055563
650.8 17.4 1.1
6.94 6.59 6.59
In the case of the lubricated joint (Table 7) all the variables are identified as significant. Hence eqns. (5) and (4) are found to represent adequately the models for unlubricated and lubricated joints respectively. 7.3. PrectXm of the prediction The precision of the predicted models is determined by calculating the appropriate confidence intervals and comparing them with the experimental values. The 95% confidence interval for predicted responses Y are given by (Yrt AY). The AY values for the various models are given in Table 9, along with the multiple correlation
73 TABLE 5 ANOVA
(lubricated
joint)
Source
ss
df
MS
F CPI
F tab
First-order terms Second-order terms Lack of fit Experimental error Total
4.3731329 0.5088266 0.018709711 0.004971295 4.9056405
2 3 3 4 12
2.186566 0.169609 0.00623657 0.00124282
1759.35 136.47 5.02
6.94 6.59 6.59
TABLE 6 Test for significance
of individual variables for unlubricated
Source
ss
df
MS
FEd
Remarks
Xl
1.3339716 0.0400426 0.0351838 0.00153975 0.0158097 0.00772210
1 1 1 1 1 7
1.3339716 0.0400426 0.0351838 0.00153975 0.0158097 0.00110316
1209.33 36.29 31.89 1.4 14.33
Significant Significant Significant
x2
x*2 x22 x1x2
Residual
SS
Standard
F value at 5% level is 5.59.
joint
Significant
TABLE 7 Test for significance
of individual variables
Source
ss
df
MS
FCal
Remarks
Xl
3.83146 0.5416714 0.3138413 0.0721769 0.1560692 0.023681006
1 1 1 1 1 7
3.83146 0.5416714 0.3138413 0.0721769 0.1560692 0.003383
1132.56 160.12 92.77 21.34 46.13
Significant Significant Significant Significant Significant
X2 XI2 x22 XIX2
Residual
SS
for lubricated
joint
TABLE 8 ANOVA
(modified
Source
Lack of fit Experimental
error
model) for unlubricated
joint
ss
df
MS
F Cal
F tab
0.00503923 0.00422254
4 4
0.0012598 0.00105563
1.193
6.39
coefficients R2. The 95% confidence intervals for values of Y were found to be quite satisfactory when compared with the corresponding experimental values. Hence the models chosen were found to represent adequately the relationships between the energy loss per cycle and the factors for each of the joints examined.
74
TABLE 9 Assessment of precision of prediction Joint
s
Unlubricated Lubricated Material
0.0332138 0.0581636
: Grey Cost
AY values at Corner points
Central points
Axial points
Multiple correlation coefficient R2
0.0620997 0.1087481
0.0351289 0.0615172
0.0620997 0.1087481
0.994626 0.995172
Iron
machining : Grinding R,= 0.98 Urn
P (N)-
Fig. 3. Response surface of energy loss at unlubricated grey cast iron joint. Figures 3 and 4 show the response surfaces of the energy loss per cycle in unlubricated and lubricated joints. The effect of lubricant on energy loss is also indicated in Fig. 5. It is found that the energy loss is drastically improved by lubricating the joint surfaces.
8. Conclusion An experimental investigation of slip damping can be carried out much more economically by response surface methodology. It provides a large amount of information through a small amount of experimentation; the methodology is practical and easy to use. For both dry and lubricated joints, the high precision mathematical equations developed can be used to estimate the energy loss per cycle of vibration and to obtain response surfaces. The accuracy of the prediction can be calculated. The response surface models make it possible to visualize slip damping and to study optimum selection.
75 Oiled
Joint
Material :Grey Cast Iron
: Grinding
Machining R,=0.98
JJrn
f = 5 HZ
Fig. 4. Response
surface of energy loss at lubricated
Material
: Grey
Normal
Pre-load
R.
cost
Iran
qCoOO N
: O.SBum
Lubricated
Peak
Fig. 5.
Effect
grey cast iron joint.
cyclic
tangential
of lubrication
joint
force
(N)
on energy
loss vs. tangential
force.
The presence of oil in the interface of the joint improves the slip damping capacity considerably; damping decreases with increasing normal preload and increases with increasing tangential exciting force.
The author wishes to express his gratitude to the department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India for the facilities rendered in carrying out this work.
References 1 C. F. Beards, Damping in structural joints, Shock yibr. Dig., 14, (6) (1982) P-11. 2 K. K. Pa~anabhan and S. C. Das, Slip damping in machine tool joints, &QC. 7th AZ2India Mach. Tool Des. Res. Co& 1976, PSG Tech, India, 1976, pp. 421-424. 3 A. S. R. Murty and K. K. Padmanabh~, Effect of surface topography on damping in machine joints, Precis. Eng., 4 (1982) 185-190. 4 K. IL Padmanabhan, Damping in machine tool joints, Proc. 14th All India Mach. TOOIDes. Res. Conf., 1990, McGraw-Hill, India, 1990, pp. 71-76. 5 K. K. Padmanabhan and A. S. R. Murty, Damping in structural joints subjected to tangential loads, Proc. Inst. Mech. Eng. C: J. Mech. Eng. Sci., 205 (C2) (1991) 121-129. 6 K. K. Padmanabhan, Prediction of damping in machined joints, Znt. J. Mach. Took Manufuct., 32 (1) (1992). 7 C. Andrew, J. A. Cockbum and A. E. Waring, Metal surfaces in contact under normal forces: some dynamic stiffness and damping characteristics, Pzoc. Inst. Me&. Eng., 182 (3K) (1967-1968) 92-100. 8 M. Masuko, Y. Ito and C. Fujimoto, Behaviour of the horizontal stiffness and the microsliding on the bolted joint under the normal preload, Proc. 12th Int. Conf: Mach. Tool Des. Rex, 1971, pp. 81-88. 9 C. Dekonink, Deformation properties of metallic contact surfaces of joints under the influence of dynamic tangential loads, Int. J. Me&. Toof Des. Res., 12 (1972) 193-199. 10 P. F. Rogers and G. Boothroyd, Damping at metallic interfaces subjected to oscillating tangential loads, ASME Water Annu. Meet., 1974, ASME, New York, 1974, paper 74WA/ Prod. 9. 11 C. F. Beards, The control of structural vibration by frictional damping in joints, J. Sot. Environ. Eng., 19 (2) (1980) 23-27. 12 S. W. E. Earles and C. F. Beards, Some aspects of frictional damping as applied to vibrating beams, ht. X Mach. Tool Des. Res., 10 (1970) 123-131. 13 C. F. Beards, Some effects of interface preparation on frictional damping in joints, Znt. 1. Mach. Tbol Des. Rex, 15 (1975) 77-83. 14 C. F. Beards and A. Neroutsopoulos, The control of structural vibration by frictional damping in electrodischarge machined joints, J. Mech. Des., Trans. ASME, 102 (1980) 54-57. 15 C. F. Beards and I. M. A. Imam, The damping of plate vibration by interfacial slip between layers, Int. 1. Mach. Tool Des. Res., 18 (1978) 131-137. 16 K. K. Padmanabhan and A. S. R. Murty, Evaluation of frictional damping .by response surface methodology, Int. I. Much. To& Munufact., 31 (1991) 95-105. 17 K. C. Peng, The Design and Analysis of Scientific Eperimenfs, Addison Wesley, 1967, London. 18 G. E. P. Box and K. B. Wilson, On the experimental attainment of optimum conditions, J. R. Stat. Sot., BI3 (1951) l-45. 19 S. M. Wu, Tool life testing by response surface methodology, parts I and II, J. Eng. Ind., Trans. ASME, 86 (1964) 105-116. 20 D. C. Montgomery, Design and Analysis of Experiments, Wiley, New York, 1976. 21 K. C. Lo and N. N. S. Chen, Prediction of tool lie in hot machining of alloy steels, Int. J. Prod. Res., X5 (1977) 47-63. 22 K. K. Padmanabhan, Analysis of measured energy dissipation on an annular surface with harmonic loading, Tribof. Znt., in the press.
Wear, 154 (1992) 65-76
Effect of lubrication
on slip damping
K. K. Padmanabhan Department (India)
(Received
of Mechanical
Engineering,
NSS
College of Engineering
March 18, 1991; revised and accepted
Palghat 678 008, Kerala
August 20, 1991)
Abstract Metallic resulting response joints. A
mating surfaces in structural joints offer a good source of energy dissipation in a damped dynamic structural response. This paper reports the application of surface methodology to study the effect of lubrication on slip damping in structural central composite rotatable design was used to develop predictive models. Good
correlation between predicted and slip damping test results was obtained. The methodology presented suggests a useful technique for damping research. 1. Introduction
In the design stage of a machine tool, it is necessary to build in sufficient damping to ensure that under all expected operating conditions stress or vibration amplitude levels never cause failure or impair the required functioning of the machine tool. Amongst the possible and available sources of damping, damping at metallic joints play an important role. Material damping is not always sufficient to limit vibration amplitudes and stresses to within the desired level; in such cases joint damping may represent the solution to the problem. Since about 90% of the total damping in a structure originates in joints [l], it would seem worthwhile to optimize such damping, thereby making inherent damping sufficient and eliminating the need for expensive or complex damping devices. Damping at structural joints is associated with frictional losses caused by slip or small relative inter-facial movements of the component parts against one another. Since the energy dissipation is caused by relative slip between surfaces of the joint, this form of damping is termed slip damping. Depending upon the extent of relative motion, the joints are classified as macroslip (gross sliding) and microslip (partial slip) joints. Slip damping offers excellent potential for large energy dissipation at structural joints even under microslip conditions. Such energy loss at various clamped joints throughout the structure can have a significant effect in limiting the magnitude of vibration. Several investigators in the past have been interested in the damping phenomenon which occurs at preloaded metallic interfaces subjected to cyclic tangential forces. Some of these works have been reviewed and the importance of damping at joints emphasized by Padmanabhan and coworkers [2-6]. Andrew ef al. [7] have shown that no significant energy loss occurs at preloaded flat metallic joints subjected to normal oscillations. This suggests that only the tangential component of vibrations can be predominently damped out #by such joints. The area of interest in. the present work is damping occurring at preloaded, flat metallic interfaces subjected to cyclic tangential forces. Masuko et al. [8] found that
0043-1648/92/$5.00
0 1992 - Elsevier
Sequoia.
All rights reserved
66
the tangential-force-displacement characteristics of a preloaded flat metallic interface arc largely influenced, albeit in a complex manner, by the normal load, surface roughness, material and machined lay orientation. Dekonink [9] obtained the energy loss per cycle at metallic joints under dynamic tangential loads, from the tangentialforce-displacement hysteresis loop. Rogers and Boothroyd [lo] have shown the energy loss per cycle under dynamic conditions to be independent of the exciting frequency. Padmanabhan and Das [2] have found experimentally that the energy loss per cycle is closely related to the preload, machining method and machined lay orientation of the mating surfaces. An optimum surface roughness was shown to exist at which the energy dissipation is maximum. It is observed that joint damping has been largely neglected because of fears of fretting and stiffness losses and non-linearities at the joint [ll]. Although the fretting corrosion which accompanies interfacial slip can cause cracks and structural failure by fatigue, this may be preferable to the high dynamic stresses which would otherwise result [12]. Furthermore, fretting can be considerably reduced by suitable treatment of the interface surfaces [13, 141. It is therefore now relevant to consider increasing the joint damping which exists in structures [15]. Realistic joints are normally dependent on several complex parameters and unknowns. A knowledge of the effects of joint parameters on damping at structural joints is necessary, if joint damping is to be used to reduce the vibration amplitudes of a structure. The development of experimental research on damping is required since the theoretical approach does not appear sufficient to acquire good knowledge of the problem and to make satisfactory provisos on the dynamic behaviour of complex structures [16]. However, the conventional experimental techniques of one-variableat-a-time for studying damping in machined joints is time consuming and expensive. The large variety of experimental conditions and multiplicity of conditional conclusions present difficulties in extending these research findings for general use. One means of obviating these difficulties is to use well planned statistical experimental design techniques. This paper reports the application of response surface methodology [17-191 in the prediction of slip damping in structural joints. In applying response surface methodology, the dependent variable or response is viewed as a surface to which a mathematical model is fitted. Only quantitative variables which are controllable can be included in the predictive model. Factors that generally affect slip damping in structures may be grouped into quantitative factors such as normal preload, cyclic tangential load, apparent area of contact and surface roughness, and qualitative factors such as machining method, machined lay orientation, material, surface treatment and lubricant at the interface. In the present investigation the effect of lubricant on slip damping is examined, keeping all other factors constant except the normal preload and cyclic tangential force.
2. Postulation
of the maZhematica1
model
The normal preload and peak cyclic tangential force have a controlling effect on energy dissipation at the joint for given contact conditions and the following secondorder damping model can be adopted. y=bo+blxl
+bzxz+bllx12+
bz2xz2+blgtlx2+e
(1)
wherexl andx2 represent the logarithmically transformed values of peak cyclic tangential force and normal preload respectively; xl2 and x2” are the quadratic effects of these
67
variables and x1x2 represent the interaction between them; bO, br, b2 etc. are the parameters and e the experimental error. The parameters of eqn. (1) are estimated by the method of least squares.
3. Experimental
set-up
The joint studied in the present work consists of a cup and lid assembly of grey cast iron (Fig. 1). A rod of the same material was bolted at the centre of the lid to facilitate normal loading of the lid. Centring pins were used to position the rod at the centre of the bore of the cup. The lid sat on the top of the cup, which was bolted securely to a heavy cast iron block rigidly fixed to a heavy cast iron bed. The lid was preloaded through the rod, by means of dead weights acting through a lever system and subjected to cyclic tangential loads along the plane of the interface. The oscillating tangential force was applied along the plane of the intraface by an electrodynamic vibration generator powered by an excitation amplifier-oscillator unit. The vibration generator was rigidly mounted on the heavy cast iron bed and isolated from any structural vibrations of the bed. A load cell was used in conjunction with a carrier amplifier to measure the tangential force. The tangential displacement of the lid relative to the cup was measured by a non-contacting eddy current probe amounted rigidly in the lid. The output of the probe was fed into a matching amplifier. Amplified outputs from the load cell and displacement probe fed into the storage
Cost Normal
Fig. 1. Cup and lid assembly.
force
iron
bed
68
oscilloscope were used to obtain the load displacement hysteresis which indicates the energy loss per cycle during vibration.
4. Experimental
loop, the area of
design
In order to determine the equation for the response surface, several special experimental techniques have been developed which attempt to approximate this equation using the smallest number of experiments possible. An experimental procedure for fitting a second-order equation must have at least three levels of each factor for estimating the model parameters, but with an increasing number of factors the experiment becomes large. The most widely preferred class of second-order response surface design is the central composite rotatable design [20]. The central composite rotatable design for the second-order response surface is shown in Fig. 2. It consists of four corner points, a centre point repeated five times and four axial points called augments. The test conditions are given in Table 1. The
(_l.LILIO)
i
~/
poi"ts
/ (I.L~L,01
~-l~k-pl~ Fig. 2. Central composite rotatable design for K=2. TABLE 1 Levels of independent variables and coding identification Independent variables
Levels in coded form - 1.414
Unlubricated joint Peak cyclic tangential force (N) Normal preload (N) Lubricated joint Peak cyclic tangential force (N) Normal preload (N)
-1
120
131
3000
3129
120
131
4000
4245
+1
0
162.5 3464 162.5 4899
+ 1.414
201
220
3835
4000
201
220
5654
6000
69
coded values of variables for use in eqn. (1) were obtained equations: Unlubricated joint, In T-ln xi=
In 201-In
q=
In 3835-In
from the following transforming
162.5 162.5
In P- In 3464 lubricated
joint,
In T-In “=
3464
In 201-In
162.5 162.5
In P - In 4899 x2= In 5654- In 4899 where x1 is the coded value of the peak cyclic tangential force corresponding to its natural value T and x2 is the coded value of the normal preload corresponding to its natural value P. The normal preload range was increased in the case of lubricated joints in order to limit the experiments within microslip. The above relations were obtained from the following transforming equation: In X, - In X,, ‘=
lnX,l-lnX,o
where x is the coded value of any factor corresponding to its natural value X,, X,,, is the natural value of the factor at + 1 level, and X,,, is the natural value of the factor corresponding to the base or zero level [21]. 5. Experimentation Ground (Ra =0.98 pm), grey cast iron joints having an apparent contact area of 594 mm2 and machined lays perpendicular were used for the tests. Surface conditions and joint geometry were chosen primarily to model practical joints found in machine tools. After surface preparation, the joint surfaces were cleaned with trichloroethylene to obtain real metal contact conditions and stored in a desiccator. Preliminary tests had revealed that energy dissipation is independent of the frequency of the cyclic tangential force. Hence tangential forces were applied at 5 Hz during the tests reported here. Undesirable dynamic effects associated with high frequencies are also taken care of by this selection of low frequency. It was also observed that energy dissipation is higher when the machined lays of joint surfaces were perpendicular than when they were parallel. For given joint surface conditions and normal preload, an oscillating tangential force applied to the lid produced a load-displacement hysteresis loop whose area is proportional to the energy loss per cycle of vibrations. The hysteresis loops displayed on the oscilloscope were photographed and their areas measured. Before conducting each trial, the surfaces were reset by separating the cup and lid and reassembling them. Experimental data presented in this work relate only to those joints in which the surface damage of the interface during the test was insignificant. The scheme of experimentation and the results obtained are given in Table 2, in which D is the energy loss per cycle (10 ELI).
70 TABLE
2
Scheme of experimentation
and results
Standard order
XI
x0
6. Estimation
-1
1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10 11 12 13
-1 -1
1
1 1 0 0
-1 1 - 1.414 1.414 0 0 0 0 0 0 0
of parameters
y=ln
X2
- 1.414 1.414 0 0 0 0 0
in the postulated
D
Unlubricated joint
Lubricated joint
3.081354 4.054963 3.059535 3.781670 3.052155 4.163037 3.531375 3.339837 3.484418 3.484418 3.4147383 3.424995 3.450185
3.388627 5.109124 3.190305 4.12069 3.066861 5.107228 4.182362 3.549558 3.627519 3.623344 3.631677 3.660309 3.707562
models
The parameters b,-,, bl, b2 etc. in eqn. (1) are estimated squares using the basic formula. b = (X=X) - ‘X=Y
by the method
of least (2)
where b is the matrix of parameter estimates, X is the matrix of independent variables or design matrix, XT is the transpose of matrix X, and Y is the matrix of logarithms of the measured responses (energy loss per cycle). For the central composite rotatable design, the (X=X)-r matrix derived from Table 2 is 1
5
0
0
_I
-1
0
10
10
0
0
0
0
0
1 8
0
0
0
-6
0
0
23 160
3 160
0
-- 1 10
0
0
3 160
23 160
0
0
0
0
0
1 4
0
0
1 s
(x=x)-=
0
71
The second-order energy loss equations for unlubricated and lubricated joints were obtained using the result of two sets of 13 tests:
grey cast iron
Yr = 3.451744 + O.4O834&u,- 0.07074~ +0.07111&r,z - 0.014878~~ -0.062868x1x,
(3)
Yz= 3.650077 + 0.69205&, - 026021C& -t 0.212404~~~ + 0.10186fbr,z - 0.197528x,x, 7. Analysis
(4)
of results
Zl. Adequacy of postulated models The usual method for testing the adequacy of a model is to compute the F-ratio of the lack of fit to the pure error and compare it with the standard value. The adequacy can be checked most conveniently by making an analysis of variance (ANOVA) table which includes sum of squares (SS), degrees of freedom (df) and mean square (MS). In order to perform the analysis of variance, the sum of squares of y values is usually partitioned into contributions from the first-order terms, the second-order terms, the lack of fit and experimental error. The lack of fit measures the deviations of the responses from the fitted surface. Experimental error is obtained from the replicated points at the centre. The sum of squares of the lack of fit and the sum of squares of experimental error (pure error} contribute to the residual sum of squares. The mean square is the sum of squares divided by the degrees of freedom. The detailed formulae for the variance analysis used here are given in Table 3. Tables 4 and 5 give the results of the variance analysis, in which F,, is the ratio of the mean square of the respective term to the mean square of experimental error and Ft;;,,, is the standard F-value corresponding to the degrees of freedom and 5% si~ific~ce level [22]. Both the first-order and second-order terms have a significant effect on the energy loss per cycle in the dry and lubricated joints considered; the fit is also adequate. 7.2. Significance of individual variables The sum of squares for any variable xi adjusted for all the others is available by taking bi2/cii. This amount is the sum of squares as xi is added to a model containing all the variables except this particular variable. To test the significance of Xi adjusted for all other variables, F = (b&&s2 is compared with the standard F value. In this way insignificant variables may be eliminated. Tables 6 and 7 were constructed to test the effect of individual variables, adjusted for all other variables, on the experimental results in the case of unlubricated and lubricated joints respectively. In the case of the unlubricated joint (Table 6) the insignificant variable can be identified as x22. Based on the above analysis, eqn. (3) has been modified to Yr = 3.4413948 +0.4083482&-r - O.O707486875(i + 0.073058~~~’
- 0.06286883~~~
(5)
The parameters in eqn. (5) were re-estimated using eqn. (2) and a corresponding new X matrix. Table 8 gives the results of variance analysis, and the fit is found to be adequate.
72
TABLE
3
Formulae
for ANOVA
Source
Sum of squares
First-order
(SS)
Degree of freedom (dt) K
terms
Second-order
terms
K(K+
Lack of fit
Experimental
I)/2
n,+?l,-K(K+3)!2
error
no-l
y, Logarithm of observed response; y,, logarithm of observed responses at centre point with mean 9s; N, total number of experimental points; n,, number of centre points; n,, number of comer points; n,, number of axial points; K, dimension of design; g, grand total of observed responses; (Oy), (ly), (2~) etc., sums of products of each column in the X matrix with the columns of y values; C~ the inverse of the sum of squares of the ith column in the X matrix. TABLE 4 ANOVA
(unlubricated
joint)
Source
ss
df
MS
F c.4,
F tab
First-order terms Second-order terms Lack of fit Experimental error Total
1.3740147 0.0551224 0.003499559 0.004222542 1.4368592
2 3 3 4 12
0.687007 0.018374 0.00116652 0.001055563
650.8 17.4 1.1
6.94 6.59 6.59
In the case of the lubricated joint (Table 7) all the variables are identified as significant. Hence eqns. (5) and (4) are found to represent adequately the models for unlubricated and lubricated joints respectively. 7.3. PrectXm of the prediction The precision of the predicted models is determined by calculating the appropriate confidence intervals and comparing them with the experimental values. The 95% confidence interval for predicted responses Y are given by (Yrt AY). The AY values for the various models are given in Table 9, along with the multiple correlation
73 TABLE 5 ANOVA
(lubricated
joint)
Source
ss
df
MS
F CPI
F tab
First-order terms Second-order terms Lack of fit Experimental error Total
4.3731329 0.5088266 0.018709711 0.004971295 4.9056405
2 3 3 4 12
2.186566 0.169609 0.00623657 0.00124282
1759.35 136.47 5.02
6.94 6.59 6.59
TABLE 6 Test for significance
of individual variables for unlubricated
Source
ss
df
MS
FEd
Remarks
Xl
1.3339716 0.0400426 0.0351838 0.00153975 0.0158097 0.00772210
1 1 1 1 1 7
1.3339716 0.0400426 0.0351838 0.00153975 0.0158097 0.00110316
1209.33 36.29 31.89 1.4 14.33
Significant Significant Significant
x2
x*2 x22 x1x2
Residual
SS
Standard
F value at 5% level is 5.59.
joint
Significant
TABLE 7 Test for significance
of individual variables
Source
ss
df
MS
FCal
Remarks
Xl
3.83146 0.5416714 0.3138413 0.0721769 0.1560692 0.023681006
1 1 1 1 1 7
3.83146 0.5416714 0.3138413 0.0721769 0.1560692 0.003383
1132.56 160.12 92.77 21.34 46.13
Significant Significant Significant Significant Significant
X2 XI2 x22 XIX2
Residual
SS
for lubricated
joint
TABLE 8 ANOVA
(modified
Source
Lack of fit Experimental
error
model) for unlubricated
joint
ss
df
MS
F Cal
F tab
0.00503923 0.00422254
4 4
0.0012598 0.00105563
1.193
6.39
coefficients R2. The 95% confidence intervals for values of Y were found to be quite satisfactory when compared with the corresponding experimental values. Hence the models chosen were found to represent adequately the relationships between the energy loss per cycle and the factors for each of the joints examined.
74
TABLE 9 Assessment of precision of prediction Joint
s
Unlubricated Lubricated Material
0.0332138 0.0581636
: Grey Cost
AY values at Corner points
Central points
Axial points
Multiple correlation coefficient R2
0.0620997 0.1087481
0.0351289 0.0615172
0.0620997 0.1087481
0.994626 0.995172
Iron
machining : Grinding R,= 0.98 Urn
P (N)-
Fig. 3. Response surface of energy loss at unlubricated grey cast iron joint. Figures 3 and 4 show the response surfaces of the energy loss per cycle in unlubricated and lubricated joints. The effect of lubricant on energy loss is also indicated in Fig. 5. It is found that the energy loss is drastically improved by lubricating the joint surfaces.
8. Conclusion An experimental investigation of slip damping can be carried out much more economically by response surface methodology. It provides a large amount of information through a small amount of experimentation; the methodology is practical and easy to use. For both dry and lubricated joints, the high precision mathematical equations developed can be used to estimate the energy loss per cycle of vibration and to obtain response surfaces. The accuracy of the prediction can be calculated. The response surface models make it possible to visualize slip damping and to study optimum selection.
75 Oiled
Joint
Material :Grey Cast Iron
: Grinding
Machining R,=0.98
JJrn
f = 5 HZ
Fig. 4. Response
surface of energy loss at lubricated
Material
: Grey
Normal
Pre-load
R.
cost
Iran
qCoOO N
: O.SBum
Lubricated
Peak
Fig. 5.
Effect
grey cast iron joint.
cyclic
tangential
of lubrication
joint
force
(N)
on energy
loss vs. tangential
force.
The presence of oil in the interface of the joint improves the slip damping capacity considerably; damping decreases with increasing normal preload and increases with increasing tangential exciting force.
The author wishes to express his gratitude to the department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India for the facilities rendered in carrying out this work.
References 1 C. F. Beards, Damping in structural joints, Shock yibr. Dig., 14, (6) (1982) P-11. 2 K. K. Pa~anabhan and S. C. Das, Slip damping in machine tool joints, &QC. 7th AZ2India Mach. Tool Des. Res. Co& 1976, PSG Tech, India, 1976, pp. 421-424. 3 A. S. R. Murty and K. K. Padmanabh~, Effect of surface topography on damping in machine joints, Precis. Eng., 4 (1982) 185-190. 4 K. IL Padmanabhan, Damping in machine tool joints, Proc. 14th All India Mach. TOOIDes. Res. Conf., 1990, McGraw-Hill, India, 1990, pp. 71-76. 5 K. K. Padmanabhan and A. S. R. Murty, Damping in structural joints subjected to tangential loads, Proc. Inst. Mech. Eng. C: J. Mech. Eng. Sci., 205 (C2) (1991) 121-129. 6 K. K. Padmanabhan, Prediction of damping in machined joints, Znt. J. Mach. Took Manufuct., 32 (1) (1992). 7 C. Andrew, J. A. Cockbum and A. E. Waring, Metal surfaces in contact under normal forces: some dynamic stiffness and damping characteristics, Pzoc. Inst. Me&. Eng., 182 (3K) (1967-1968) 92-100. 8 M. Masuko, Y. Ito and C. Fujimoto, Behaviour of the horizontal stiffness and the microsliding on the bolted joint under the normal preload, Proc. 12th Int. Conf: Mach. Tool Des. Rex, 1971, pp. 81-88. 9 C. Dekonink, Deformation properties of metallic contact surfaces of joints under the influence of dynamic tangential loads, Int. J. Me&. Toof Des. Res., 12 (1972) 193-199. 10 P. F. Rogers and G. Boothroyd, Damping at metallic interfaces subjected to oscillating tangential loads, ASME Water Annu. Meet., 1974, ASME, New York, 1974, paper 74WA/ Prod. 9. 11 C. F. Beards, The control of structural vibration by frictional damping in joints, J. Sot. Environ. Eng., 19 (2) (1980) 23-27. 12 S. W. E. Earles and C. F. Beards, Some aspects of frictional damping as applied to vibrating beams, ht. X Mach. Tool Des. Res., 10 (1970) 123-131. 13 C. F. Beards, Some effects of interface preparation on frictional damping in joints, Znt. 1. Mach. Tbol Des. Rex, 15 (1975) 77-83. 14 C. F. Beards and A. Neroutsopoulos, The control of structural vibration by frictional damping in electrodischarge machined joints, J. Mech. Des., Trans. ASME, 102 (1980) 54-57. 15 C. F. Beards and I. M. A. Imam, The damping of plate vibration by interfacial slip between layers, Int. 1. Mach. Tool Des. Res., 18 (1978) 131-137. 16 K. K. Padmanabhan and A. S. R. Murty, Evaluation of frictional damping .by response surface methodology, Int. I. Much. To& Munufact., 31 (1991) 95-105. 17 K. C. Peng, The Design and Analysis of Scientific Eperimenfs, Addison Wesley, 1967, London. 18 G. E. P. Box and K. B. Wilson, On the experimental attainment of optimum conditions, J. R. Stat. Sot., BI3 (1951) l-45. 19 S. M. Wu, Tool life testing by response surface methodology, parts I and II, J. Eng. Ind., Trans. ASME, 86 (1964) 105-116. 20 D. C. Montgomery, Design and Analysis of Experiments, Wiley, New York, 1976. 21 K. C. Lo and N. N. S. Chen, Prediction of tool lie in hot machining of alloy steels, Int. J. Prod. Res., X5 (1977) 47-63. 22 K. K. Padmanabhan, Analysis of measured energy dissipation on an annular surface with harmonic loading, Tribof. Znt., in the press.