Prediction of damping in machined joints

Int. J. Mach. Tools Manufact. Vol. 32, No. 3, pp.305-314, 1992. Printed in Great Britain

PREDICTION

0891~6955/9255.00 + .00 Pergamon Press pie

OF DAMPING

IN MACHINED

JOINTS

K. K. PADMANABHANt (Received 18 December 1990; in final form 19 April 1991) Abstract--The present work is concerned with the experimental investigation of damping at preloaded metallic interfaces subjected to cyclic tangential forces. The situations of various joint materials are compared. Predictive equations for damping in machined joints are developed based on experiments conducted on a rationale of economic data reduction through response surface methodology (RSM). Orthogonal first-order models are developed for energy loss per cycle of vibration in mild steel, grey cast iron and phosphor bronze joints and a central composite rotatable design in the case of aluminium alloy joints. A good correlation between the predicted and actual joint damping test results is obtained. The methodology presented suggests a useful technique for damping research.

1. INTRODUCTION

IN ACHIEVINOsatisfactory dynamic performance characteristics of structures like machine tools, the role of damping has been well appreciated in the past. Amongst the possible and available sources of damping, the damping at metallic joints arising basically from frictional energy dissipation at their mating surfaces plays a prime role. Also it is well known that material damping is inadequate while extra damping devices are costly [1]. Viscoelastic layers are temperature sensitive [2]. Since about 90% of the total damping of vibrational energy takes place in the structural joints, it would seem worthwhile to optimize the joint for maximum damping [3,4]. Damping at joints offers an excellent potentia! for large energy dissipation even under microslip conditions and can have a significant effect in limiting the magnitudes of vibrations of mechanical structures. 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 in Refs [5-8]. Energy dissipation and the slip propagation mechanism at the contact of spherical and cylindrical surfaces with plane surfaces have been studied theoretically [9-13] and experimentally [10, 14-19] as has the interaction of plane surfaces [20-23]. The application of friction damping to real structures such as built up cantilever beams [24-34], roots of cantilever beams [35-43] and compressor and turbine blades [44-47] has all furthered the understanding of the mechanism of dry surface interaction and indicated the conditions under which this type of damping might be used to some advantage. Andrew et al. [48] 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 vibration can be predominantly damped out by such joints. The area of interest in the present work is the damping at preloaded flat metallic interfaces subjected to cyclic tangential forces. Masuko et al. [49] found that the tangential force-displacement characteristics of a preloaded fiat metallic interface are largely influenced, albeit in a complex manner, by the normal preload, surface roughness, material and machined lay orientation. They did not make any measurement of the energy loss. Dekonink [50] obtained the energy loss per cycle at metallic joints under dynamic tangential forces from the tangential force-displacement hysteresis loop. Boothroyd et al. [51] found experimentally that under quasi-static tangential loading, the friction force increases monotonically with displacement. The rate of increase in

tDepartment of Mechanical Engineering, N.S.S. College of Engineering, Palghat 678 008, India. 305

306

K.K. PADMANABHAN

tangential force decreases with increasing displacement until eventually the tangential force becomes constant, when gross slip occurs. If the force is slowly cycled, an essentially bilinear hysteresis loop results. Rogers and Boothroyd [52] have shown that under dynamic conditions, hysteresis loops which are substantially different from the quasi-static bilinear loops are obtained; the energy loss per cycle is shown to be independent of the exciting frequency. However, they have found that the loops reduced in size during the first few cycles of loading. A preload history effect was also noticed by them. If the preload was temporarily raised to a higher level before a test was run, a much thinner hysteresis loop was obtained. The effect could be eliminated by disturbing the interface just prior to running the test. The energy dissipated per cycle was calculated from the measured area of hysteresis loops. It has been found experimentally that the energy loss per cycle is closely related to the preload, machining method and machined lay orientation of the mating surfaces [53]. An optimum surface roughness was shown to exist at which the energy dissipation is a maximum. The energy dissipation at the joint was shown to increase monotonically with an increase in the tangential force and that use of highly viscous oil increases the energy dissipation at the joint. Realistic joints are normally dependent on several complex parameters and unknowns. A knowledge of the effects of the joint parameters on damping at structural joints is necessary if joint damping is to be used to reduce the vibration amplitudes of a structure. A general demand on the development of experimental research on damping has come out since the theoretical approach does not appear sufficient to acquire a good knowledge of the problem and to make satisfactory provisos on the dynamic behaviour of complex structures. However, the conventional experimental techniques of one-variable-at-a-time to study damping in machined joints are time consuming and expensive. The large variety of experimental conditions and the 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 (RSM) [54, 55] in the prediction of damping in machined joints of mild steel, grey cast iron, phosphor bronze and aluminium alloy. 2. RESPONSE SURFACE METHODOLOGY (RSM) Response surface methodology (RSM) [54-56] is a combination of mathematical and statistical techniques used in the empirical study of relationships and optimization, where several independent variables or factors (xl, x: ..... xk) influence a dependent variable or response (y). It is assumed that the independent variables are continuous and controllable with negligible error. The response is assumed to be a random variable. If all these variables are quantifiable, then the relation between the response and the factors can be expressed as: y

= F(xI,x2

.....

Xk) + e.

The function F is called the response surface. The residual e measures the experimental error. A knowledge of the function gives a complete summary of the results of the experiment and also enables the response for the values of factors that were not tested experimentally to be predicted [57-59]. Response surface methodology was initially developed and described by Box and Wilson [60] for obtaining optimum conditions in chemical investigations. Hill and Hunter [61] and Mead and Pike [62] reviewed the earlier work on RSM. Response surface methodology has been successfully applied in a wide variety of situations for developing predictive models [63--67] and it has been used in determining the optimum values of the independent variables as far as the response is concerned [68]. It also assists in the understanding of the basic mechanism of the system under investigation. In applying the RSM, the response or dependent variable is viewed as a surface [69,

Prediction of Damping in Machined Joints

307

70] to which a mathematical model is fitted. Only quantitative variables which are controllable can be included in the predictive model; however, by building a number of models for each of the qualitative variables, their effects also could be studied. The present investigation illustrates a few damping predictive models for joints of various materials. The general procedure adopted for the development of damping predictive models based on RSM covers the following: (1) Postulation of the mathematical model, experimental design and determination of test regions for independent variables; (2) Experiment; (3) Estimation of parameters in the postulated model; and (4) Analysis of r e s u l t s (a) checking the adequacy of the postulated model and the test for significance of individual variables; and (b) precision of prediction, i.e. the estimation of confidence intervals. 3. MATHEMATICAL MODEL

Since the form of the relationship between the response and the independent variables is unknown, the first step in RSM is to find a suitable approximation for the response surface (mathematical surface of a multi-variable plot) F. Usually a low-order polynomial within the experimental region of independent variables is employed. Polynomial response surfaces have the great advantage that they are easy to fit. Normal preload and peak cyclic tangential force have a controlling effect on the energy dissipation at the joint under given contact conditions and the following first-order or second-order polynomial can be adopted to represent the cyclic energy dissipation [55]. The first-order polynomial y = bo + blxl + b2x2 + e.

(1)

The second-order polynomial y = bo + blXl + b2x2 + bllxZt + bz2x~ + b~2xlx2 + e

(2)

where y is the logarithm of the observed cyclic energy dissipation; xl and x2 represent the coded values of the peak cyclic tangential force and the normal preload, respectively; bo, bl, bE, bll, b22, bl2 are the parameters which are estimated by the method of least squares and e the experimental error. Depending upon the adequacy based on tests, either a first- or second-order model can be chosen. In order to determine the equation of the response surface, several special experimental techniques have been developed which attempt to approximate this equation using the smallest number of experiments possible. The most widely preferred classes of response surface design are the orthogonal first-order design and the central composite rotatable second-order design [57]. The test conditions are given in Table 1. The coded values of the variables for use in equations (1) and (2) were obtained from the following transforming equations: In T - In 141 x~ = In 181 - In 141 In P - In 3464 x2 = In 5108 - In 3464 where xl is the coded value of the peak cyclic tangential force corresponding to its natural value T and x2 the coded value of the normal preload corresponding to its HTH 32:3-D

308

K . K . PADMANABHAN TABLE 1. LEVELS OF INDEPENDENT VARIABLES

Independent variables

Peak cyclic tangential force (N) Normal preload (N)

Levels in coded form -1.414

-1

0

1

1.414

lifo 2000

111 2349

141 3464

181 5108

200 6000

natural value P. The above relations were obtained from the following transforming equation: X=

in X . - In X . o In X . l - in )(no

where x is the coded value of any factor corresponding to its natural value Xn, X . I the natural value of the factor at the +1 level and X.II the natural value of the factor corresponding to the base or zero level [66]. 4. EXPERIMENT

The damping source studied in this work is that of a preloaded metallic interface, such as that found in machine tools. Due to the mechanical complexity of a machine, results obtained from vibration experiments performed directly on it would be very difficult to analyse. Thus an experimental model of the interface was designed and built in an attempt to isolate the effects of interest. The set-up developed provides a plane annular interface and uniform pressure distribution, variability of normal load and driving force and allows the influence of the material and surface conditions to be investigated. In order to evaluate the energy loss in the interface, tangential forcedisplacement hysteresis loops could be obtained. The experimental set-up and the procedure used for the investigation have been described earlier [55]. Ground (Ra = 0.65 p.m) and unlubricated mild steel, grey cast iron, phosphor bronze and aluminium alloy joints, having an apparent contact area of 594 mm 2 at the interface were used with the machined lays perpendicular during the tests. Surface conditions and joint geometry were chosen primarily to model practical joints found in metal cutting machines. Several sets of cup-lid combinations were manufactured from each of the above materials in order to facilitate testing. After surface preparation, the joint surfaces were cleaned with trichloroethylene to obtain real metal contact condition and then stored in a dessicator. A preliminary investigation to study the phenomenon varying from microslip to gross slip was initially undertaken [8] to obtain the critical microslip and the corresponding tangential force (gross slip force) for various normal preloads. This enabled the microslip region for the present investigation to be obtained. The energy loss measurements reported in this paper were only within microslip. For a given surface condition and normal preload, an oscillating tangential force applied to the lid at 5 Hz produced a force-displacement hysteresis loop whose area is proportional to the energy loss per cycle of vibration. The hysteresis loops displayed on the oscilloscope were photographed and their areas measured using a planimeter. The scheme of experimentation and the results obtained are given in Table 2, in which D is the energy loss per cycle (10 p3). 5. ESTIMATION OF PARAMETERS

The regression parameters of the selected model were estimated by the method of

Prediction of Damping in Machined Joints

309

TABLE 2. SCHEME OF EXPERIMENTA'nOS Arid RESULTS

Standard order

xo

xl

x2

y = lnD Mild steel Yl

1 2 3 4 5 6 7 8 9 10 11 12 13

1 1 1 1 1 1 1 1 1 1 1 1 1

-1 1 - 1 1 -1.414 1.414 0 0 0 0 0 0 0

-1 - 1 1 1 0 0 - 1.414 1.414 0 0 0 0 0

3.1698224 3.8346212 2.9086996 3.6218075 2.8158059 3.7232077 3.5177637 3.3141647 3.1510131 3.2813749 3.2924247 3.1882845 3.1446639

Grey cast iron Y2

3.0241687 3.9288217 2.9819684 3.2522587 2.8133456 3.7498391 3.6444786 3.3343236 3.205119 3.2188108 3.4101617 3.2255904 3.3097825

Phosphor bronze Y3

Aluminium alloy Y4

2.9587113 3.4927938 2.6591947 3.3118506 2.4863519 3.9321604 3.1361188 2.916663 2.9587105 2.9288576 2.8065187 2.9876981 3.0102939

3.0798850 4.4920541 2.8394585 3.6976856 2.6904224 3.8317604 3.7360683 3.1648799 3.2702417 3.1386111 3.0553964 3.1868608 3.1627775

least squares, using the basic formula:

b = ( X r X ) - ~x'ry

(3)

where b is the matrix o f p a r a m e t e r estimates, X the matrix of i n d e p e n d e n t variables or design matrix, X "r the transpose of matrix X and Y the matrix of logarithms of m e a s u r e d responses (cyclic e n e r g y loss). T h e first-order e n e r g y loss equations for the joints of mild steel, grey cast iron, p h o s p h o r b r o n z e and aluminium alloy were o b t a i n e d using the results of four sets of nine tests: Ys -- 3.288078 + 0.344478xt - 0.118485x2

(4)

Yc -- 3.284069 + 0.293733xl - 0.179684x2

(5)

Yp = 3.012732 + 0.296685xl - 0.120111x2

(6)

Ya = 3.324774 + 0.5676xt - 0.2587x2.

(7)

6. ANALYSIS OF RESULTS 6.1. Adequacy o f postulated models T h e analysis of variance ( A N O V A ) technique [55] is used to check the a d e q u a c y of the d e v e l o p e d models. As per this technique, the F-ratio of the m o d e l is calculated and c o m p a r e d with the s t a n d a r d tabulated value o f the F-ratio for a specific level of confidence. If the calculated value o f the F-ratio does not exceed the tabulated value, then with the c o r r e s p o n d i n g confidence probability, the m o d e l m a y be considered adequate. Tables 3 - 6 give the results of the variance analysis. First-order terms have a significant effect on cyclic e n e r g y loss in all the cases. Fit is a d e q u a t e in the case of joints of mild steel (Table 3), grey cast iron (Table 4) and p h o s p h o r b r o n z e (Table 5). H o w e v e r , in the case o f an aluminium alloy joint (Table 6) fit is i n a d e q u a t e and h e n c e a s e c o n d - o r d e r m o d e l is p r o p o s e d based on the results of 13 tests: Ya = 3.162771 + 0.485564xl - 0.230323x2 + 0.092030x~ + 0.186723x 2 - 0.138485xlx2.

(8)

310

K . K . PADMANABHAN

TABLE 3. ANOVA Source First-order terms Lack of fit Experimental error Total

FOR MILD STEEL JOINT

SS

df

MS

F~ I

Eta,

0.53081 0.06646 0.02010 0.61737

2 2 4 8

0.2654 0.03323 0.00502

52.8 6.6

6.94 6.94

TABLE 4. ANOVA FOR GREY CAST IRON JOINT

Source First-order terms Lack of fit Experimental error Total

TABLE 5.

Source First-order terms Lack of fit Experimental error Total

TABLE 6.

Source First-order terms Lack of fit Experimental error Total

SS

df

MS

F~I

F.,~,

0.474278 0.101773 0.029952 0.606003

2 2 4 8

0.237139 0.050886 0.007488

31.67 6.7

6.94 6.94

A N O V A FOR PHOSPHOR BRONZE JOINT

SS

df

MS

F~,I

Ft~b

0.4097974 0.0656546 0.0254958 0.5009478

2 2 4 8

0.2048987 0.0328273 0.00637395

32.15 5.1

6.94 6.94

A N O V A FOR ALUMINIUM ALLOY JOINT

SS

df

MS

F~,,~

Ft~b

1.55638 0.3715968 0.0242432 1.95222

2 2 4 8

0.77819 0.1857984 0.0060608

128.4 30.6

6.94 6.94

T a b l e 7 gives the results of the variance analysis. Both first- a n d s e c o n d - o r d e r t e r m s have a significant effect o n the cyclic e n e r g y loss at the a l u m i n i u m alloy j o i n t a n d the fit is also f o u n d to be a d e q u a t e . T a b l e 8 was c o n s t r u c t e d to test the effect of i n d i v i d u a l variables, a d j u s t e d for all o t h e r variables, o n the e x p e r i m e n t a l results in the case of a l u m i n i u m alloy j o i n t . F o r this p u r p o s e , the variance ratio:

F

= (bi2/cii)/s 2

for a p a r t i c u l a r v a r i a b l e xi, a d j u s t e d for all o t h e r variables, is c o m p a r e d with a s t a n d a r d F - v a l u e having o n e a n d seven degrees of f r e e d o m for a 95% confidence interval [71].

Prediction of Damping in Machined Joints

311

TABLE 7. A N O V A FOR ALUMINIUMALLOY JOINT (SECOND-ORDER MODEL) Source First-order terms Second-order terms Lack of fit Experimental error Total

SS

df

MS

FcaI

Etab

2.3105400 0.351649 0.1190264 0.0242432 2.8054515

2 3 3 4 12

1.15527 0.1172139 0.0396754 0.0060608

190.61 19.34 6.5

6.94 6.59 6.59

TABLE 8. TEST FOR SIGNIFICANCE OF INDEPENDENT VARIABLES(ALUM1NIUMALLOY JOINT) Source x~ xa x~ x~ xlxz

Res. SS

SS

df

MS

Fr,~,,

Remarks

1.8861481 0.4243857 0.0589145 0.2425294 0.0767129 0.1432696

1 1 1 1 1 7

1.8861481 0.4243857 0.0589145 0.2425294 0.0767129 0.02467

92.16 20.74 2.9 11.85 3.75

Significant Significant Non-significant Significant Non-significant

The significant variables can be identified as xi, x2 and x~. Based on the above analysis, equation (8) has been modified to: Ya = 3.2267904 + 0.485564429Xl - 0.23032337x2 + 0.17472047x~.

(9)

The parameters in equation (9) were re-estimated by using equation (3) and a new Xmatrix. Table 9 gives the results of variance analysis. Since the model is found to be inadequate the original model, given by equation (8) itself, is selected. 6.2. Precision of prediction The precision of the predicted model can be determined by calculating the appropriate confidence intervals and comparing them with the experimental values. The 95% confidence interval for the predicted responses Y are given by (Y -+ AY), where:

A y = tdf,,,os,2 x/IV(V)) t is the value of the horizontal coordinate of the t-distribution corresponding to the specified degrees of freedom and level of confidence and V(Y) the variance of predicted response Y. The A y values for various models are given in Table 10 along with multiple correlation coefficients (R2). The 95% confidence interval for Y-values were found to be quite satisfactory when compared with the corresponding experimental values. Hence the models were found to be fully adequate to represent the relationships between the response (cyclic energy loss) and the factors for each of the materials examined. Decoding the equations, the first-order models given by equations (4)-(6) can be represented as: Ds = 3.022297 T 1'44)77969p-(I.3O5o4626 Dc = 3 0 . 3 5 9 9 6 5 T l 2 { ~ 4 3 ° 9 p -°'46262759 O p = 3 0 . 1 3 4 2 5 1 T l 2 1 2 4 8 1 5 p -°'49013352

in which D is the energy loss per cycle in ixJ.

312

K . K . PADMANARHAN TABLE 9. A N O V A FOR ALUMINIUMALLOYJOINT (MODIFIED MODEL)

Source Lack of fit Experimental error

SS

df

MS

F¢~

Ft~h

0.2547220 0.0242432

5 4

0.0509444 0.0060608

8.406

6.26

TABLE I0. ASSESSMENTOF PRECISION OF PREDICTION Model

Equation Equation Equation Equation

S

(6) (7) (8) (10)

A Y values at

0.1201137 0.1481698 0.1232546 0.1430629

R2

Corner points

Central points

Axial points

0.2297664 0.2834351 0.2357747 0.2674842

0.0979727 0.1208571 0.1005346 0.1513119

---0.2674842

0.859796 0.782619 0.81809 0.94891

Figure 1 shows the typical response surface of cyclic energy loss in an aluminium alloy joint. Similar surfaces may be obtained for other materials utilizing the predictive equations. Amongst the various materials considered, aluminium alloy shows the maximum energy loss whereas phosphor bronze shows the minimum; the mild steel joint gives higher energy dissipation than the grey cast iron joint in the whole range of the experiment. The effect of joint material on the energy loss is further demonstrated in Fig. 2 under a constant normal preload of 3950 N. The mild steel joint yields the highest damping up to a certain value of tangential force, after which the aluminium alloy joint shows higher values; phosphor bronze gives only the minimum amongst all materials within the whole range of experimentation.

Moterlol = Alumin~um olloy Mochlnlng : Grinding Ro = 0.65 ;urn

800

r00}_ 200t-- I 0U

"-

~

!

/ / / o 1 " / / / /

I I ~z0 ", , , e

P (NI

FIG. 1. Response surface of energy loss at aluminium alloy joint.

Prediction of Damping in Machined Joints

313

Mochinln 9 ; Gr~ndin9 Ra = 0.65/am f =SHz

P = 3950 N 400

Alumlnium olloy _, S

350

,~/ Mild steel

, 300

//~,.jcost,ron

250

or Bronze

Q

2O0

150 110

I 130

I 150 Tm (N)

I 170

I 190

F=G. 2. Energy loss vs tangential force for various materials predicted by the models.

7. CONCLUSIONS

Experimental investigation of damping at structural joints can be much more economically conducted by a statistical technique called response surface methodology (RSM). The high precision mathematical models developed can be utilized to estimate the energy loss during vibration and to obtain response surfaces. The reliability of predictive models is shown by confirmatory tests. It is possible to visualize damping through response surfaces, which also aids in choosing optimum combinations of factor levels for a particular situation. Energy loss is maximum in the case of the aluminium alloy joint whereas phosphor bronze yields the minimum; the mild steel joint provides higher energy dissipation than grey cast iron. The effects of normal and tangential forces on the cyclic energy loss can be seen through Fig. 1. In general, an increase in tangential force causes an increase in cyclic energy loss, whereas an increase in normal preload causes a decrease in cyclic energy loss. Acknowledgements--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.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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