Experimental Mechanics (2012) 52:771–791 DOI 10.1007/s11340-011-9563-3

Slip Damping Mechanism in Welded Structures Using Response Surface Methodology B. Singh & B.K. Nanda

Received: 7 February 2010 / Accepted: 3 October 2011 / Published online: 12 October 2011 # Society for Experimental Mechanics 2011

Abstract Response surface methodology (RSM) is a technique used to determine and represent the cause and effect of relationship between true mean responses and input control variables influencing the responses as an ndimensional hyper surface. Welded joints are used extensively in many modern industries to fabricate jointed structures that contribute significantly to the inherent slip damping. The main problem faced in the manufacture of such structures is the selection of optimum combination of input variables for achieving the required damping. This problem can be solved by developing the mathematical models through effective and strategic planning and executing experiments by RSM. This investigation highlights the use of RSM by designing a four-factor three-level central composite rotatable design matrix with full replication of planning, conducting, executing and developing the mathematical models. This is useful for predicting the mechanism of interfacial slip damping in layered and welded structures. The design utilizes the initial amplitude of excitation, number of tack welded joints and surface roughness at the interfaces as well as the material property to develop a model for the logarithmic damping decrement of layered and welded structures with different end conditions. Experimental results indicate that the proposed mathematical models adequately predict the logarithmic damping decrement within the limits of the factors that are being investigated. B. Singh (*) : B.K. Nanda Department of Mechanical Engineering, National Institute of Technology, Rourkela, Orissa, India e-mail: [email protected] B.K. Nanda e-mail: [email protected]

Keywords Interface slip damping . Welded joint . Surface roughness . Amplitude . Number of tack joints . Response surface methodology . Materials . End conditions

Introduction The purpose of the present work is to experimentally study the effects of a welded joint on the dynamic response of a jointed structure using response surface methodology. The actual physics taking place at the interfaces of a vibrating structure is complex and not fully understood. The existence of many parameters and uncertainties prevent an accurate understanding and modelling of the joint physics which include the interface area, distribution of normal and shear forces at the interfaces, surface finish, and historydependent non-linear effects on the dynamics. Joints have a great potential for reducing the vibration levels of a structure and have attracted the interest of many researchers in the recent past [1–8]. The earlier works by Goodman and Klumpp [9], Masuko et al. [10], Motosh [11], Nishiwaki et al. [12], Nanda and Behera [13] and Mohanty and Nanda [14–16] present different techniques for improving the damping capacity of layered and jointed bolted as well as riveted structures without considering the effects of surface roughness. Many comprehensive review papers on joints and fasteners have appeared in recent years. The small, localized motions during micro-slip result in energy losses at the joints, which are perceived as localized damping of the structures. Berger [17] has studied the effect of microslip on passive damping of a jointed structure. Indeed, most damping effects encountered in practical structures take place at jointed interfaces [18]. Beards [19] performed a series of experiments to show that the damping in joints is

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much larger than material damping. He focused his work to control the effect of vibration with the use of joints of various types. Mayer and Gaul [20] discussed the segment- to-segment contact elements of a structure having both linear and nonlinear constitutive contact behavior in normal and tangential directions including the nonlinear micro-slip behavior. Goodman [21] analytically studied the interfaces that experienced localized slipping. He found that for a wide variety of joints, a power law could be established between the force input and the energy dissipation per cycle of loading in a dynamic system. The exponent of the power law is found to be 3 with Coulomb interface friction which is identical to the contact of two elastic spheres under cyclic tangential loads [22]. Further, investigations into joints have been undertaken at Sandia National Laboratories [23, 24] aiming at identifying the physics of the joint interfaces. A series of closely controlled experiments has established that there exists a power-law relationship between the input force and energy dissipation per cycle [25]. The experiments have been carried out to identify the regions of micro-slip. Further work at Sandia examined the use of Iwan models [26] to describe the dynamics of joints [27, 28]. The Iwan model consists of a continuum network of springs and sliders, with the break-free forces of the sliders being described by a probability distribution function. Different distribution functions lead to different power laws in the diagrams of input force vs. energy dissipation per cycle. There are several approaches for investigating the joint dynamics. One of the approaches is to identify the actual physics taking place within the joint on a micro scale. Alternatively, response surface methodology is also used to study the effect of joint on the overall dynamics of the system. Recently, Liang et al. [29] used the response surface methodology to analyze the effect of design parameters on sound radiation from a vibrating panel. Li and Liang [30] further utilized response surface methodology to analyze the design parameters and optimize the vibro-acoustic properties of damped structures. Further, the response surface methodology has gained importance in structural dynamics of damped structures using the finite element models. In this context Ren and Chen [31] presented a response surface-based finite element model updating procedure for civil engineering structures by formulating explicitly an optimization technique. As established by Nanda [32] and Damisa et al. [33, 34], the damping in jointed structures is due to slip damping arising from kinematic coefficient of friction and relative slip at the interfaces. They considered layered and jointed bolted structures to establish this and found out the parameters influencing the slip damping. Moreover,

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Mohanty and Nanda [14–16] focused their studies on the layered and jointed riveted structures to ascertain the role of slip damping in improving the damping of the structures. All the previous researchers working on bolted and riveted structures have not considered the effect of surface roughness in their analysis. Moreover, no substantial work has been reported on the slip damping of welded structure and therefore the present investigation has been taken up to analyze the effect of surface roughness on the micro-slip and coefficient of friction at the interfaces thereby the damping capacity of the structures. Moreover, reduced order models using response surface methodology have been developed for accurately incorporating the effects of welded joints in the structures. Extensive experimental studies have been performed to identify the principal effects of welded joints on structural dynamics and validate the developed models.

Theoretical Analysis The logarithmic damping decrement, a measure of damping capacity of layered and jointed structures, is usually determined by the energy principle taking into account the relative dynamic slip and the interfacial pressure distribution at the contacting layers. The logarithmic damping decrement is evaluated theoretically for the beams with different end conditions such as; fixed-free, fixed-hinged and fixed-fixed as shown in Fig. 1 (a), (b) and (c) respectively. Interface Pressure Distribution The contact pressure for flat surfaces with rounded corners has been found out by Ciavarella et al. [35], which shows a non-uniform distribution pattern at the interfaces. Contrary to this, the pressure distribution at the interfaces is assumed to be uniform owing to the contact of the upper layer over the lower one. Therefore, the relation for uniform pressure distribution as given by Johnson [36] and Giannakopoulos et al. [37] due to contact of two flat bodies has been considered and the same is given by; pðxÞ ¼

P b

ð1Þ

where P and b are the normal load per unit length and width of the beam, respectively. Evaluation of Relative Dynamic Slip It is assumed that each beam of the jointed cantilever beam being vibrated has the equal bending stiffness with the same bending condition. Moreover, each layer of the beam shows no extension of the neutral axis and no deformation of the cross-

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relative motion called micro-slip. This relative displacement u (x, t) at any distance x from the fixed end in the absence of friction is equal to the sum of Δu1 and Δu2 as shown in Fig. 2 and at a particular position and time is given by; uðx; tÞ ¼ Δu1 þ Δu2 ¼ 2h tan½@yðx; tÞ=@x;

ð2Þ

However, the actual relative dynamic slip [ur(x, t)] at the interfaces in the presence of friction during the vibration will be less and is found out by subtracting the elastic recovery part of the relative displacement from u(x, t) and is rewritten as; ur ðx; t Þ ¼ auðx; tÞ ¼ 2ah tan½@yðx; tÞ=@x;

ð3Þ

where, α is the dynamic slip ratio. Analysis of Energy Dissipated Energy is dissipated due to the relative dynamic slip at the interfaces. For the above cantilever beam of length, l and height, 4h shown in Fig. 1 (a), the interface pressure at x is expressed as p(x) and the normal load acting on the length of dx is p(x)bdx. Thus, the frictional force at the interfaces is given by μp(x)bdx, where μ is the coefficient of kinematic friction. By considering the condition that the interface pressure is uniformly spread over all the contact area, p(x) yields p. For the above cantilever beam with uniform pressure distribution at the interfaces, p, the energy loss due to the frictional force at the interfaces per halfcycle of vibration is given by; p=wn Z l Z

Eloss ¼

mpb½f@ur ðx; t Þ=@t gdxdt : 0

0

Fig. 1 (a) Two layered tack welded beam model for fixed- free ends. (b) Two layered tack welded beam model for fixed- hinged ends. (c) Two layered tack welded beam model for fixed- fixed ends

section. When the jointed cantilever beam is given an initial excitation at the free end, the contacting surfaces undergo

Fig. 2 Mechanism of dynamic slip at the interfaces

ð4Þ

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However, the energy introduced into the layered and jointed cantilever beam in the form of strain energy per half-cycle of vibration is given by;    Ene ¼ 3EI l 3 y2 ðl; 0Þ: ð5Þ   where E , I ¼ bð4hÞ3 =12 and y(l,0) are the modulus of elasticity, cross-sectional moment of inertia and transverse deflection at the free end of the welded beam, respectively. From the above expressions (4) and (5), the ratio of energy is found to be; Z p=wn Z l    Eloss ½mpbf@ur ðx; tÞ=@t gdxdt = 3EI=l 3 y2 ðl; 0Þ : E ne ¼ 0

0

The slope of the cantilever beam @yðx; t Þ=@x being quite small, is modified as tan @yðx; t Þ=@x ’ @yðx; t Þ[email protected], expression (7) is re-written as; 

  Eloss  ¼ 2mbhpa= 3EI=l 3 y2 ðl; 0Þ Ene Z p=wn Z l  2

  @ yðx; t Þ=@x@t dxdt 0

ð8Þ

0

Using the boundary and initial conditions, i.e., yðl; 0Þ ¼ y0 and @yðl; 0Þ=@t ¼ 0, the bending deflection of beam being vibrated y(x, t) can be written as; yðx; t Þ ¼ Y ðxÞfy0 =Y ðlÞg cos wn t;

ð9Þ

where, Y(x) is the space function given by;

ð6Þ Assuming the dynamic slip ratio, α, to be independent of the distance from the fixed end of the cantilever beam and time, the above expression (6) has been modified using expression (3) as;

Y ðxÞ ¼ ðsinh l1 þ sin l1 Þðcosh l1 x=l  coxl1 x=l Þ

 

 Eloss  ¼ 2mbhpa= 3EI=l 3 y2 ðl; 0Þ Ene Z p=wn Z l  ½@ ftan @yðx; t Þ=@xgdxdt =@t

where, λ1 =λ l=1.875 for first mode of natural vibration. Using the above expressions (9) and (10) in expression (8) and changing the limits of the time interval from 0 and π/ωn to 0 and π/2ωn and multiplying the expression by two for yielding definite solution we get;

0

ð7Þ

0

 ðcosh l1 þ cos l1 Þ  ðsinh l1 x=l  sin l1 x=l Þ:

ð10Þ

p=wn Z l Z

Eloss =Ene ¼

@ 2 ½Y ðxÞfy0 =Y ðlÞg cos wn t dxdt=½@x=@t ;

4mbha ð3EI=l 3 Þy2 ðl;0Þ 0

ð11Þ

0

Moreover, using expression (9), (10) and (11), the energy ratio is given by;     Eloss =Ene ¼ ½4mbhpayðl; 0Þ= 3 EI=l 3 y2 ðl; 0Þ ; ð12Þ Replacing 3EI/l3 = kc, i.e., the static bending stiffness of the layered and welded cantilever beam, the above expression (12) reduces to Eloss =Ene ¼ ½4mbhpa=½kc yðl; 0Þ:

where, Eloss and Ene are the energy loss due to interface friction and the energy introduced during the unloading process, respectively. Putting the values of Eloss/Ene from expression (13) in expression (14) we get; y¼

1 : 1 þ ½kc yðl; 0Þ=½4mbpha

ð15Þ

ð13Þ Calculation of Logarithmic Damping Decrement

Evaluation of Damping Ratio The damping ratio, ψ, is expressed as the ratio of energy dissipated due to the relative dynamic slip at the interfaces and the total energy introduced into the system and is found to be; y ¼ ½Eloss =ðEloss þ Ene Þ ¼ 1=½1 þ Ene =Eloss :

ð14Þ

Logarithmic damping decrement is used as a measure of damping capacity of a structure and is influenced by dynamic slip and interface pressure at the contacting surfaces. The logarithmic damping decrement, δ, is usually expressed as, d ¼ lnðan =anþ1 Þ where an is the amplitude of vibration at certain time and an+1 is the amplitude of vibration after elapse of one cycle. The relationship

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between logarithmic damping decrement and damping ratio is written as;

 1 8mapbh d ¼ ln 1 þ 2 kf yð1=2; 0Þ

ð20aÞ

d ¼ lnðan =anþ1 Þ ¼ ½lnf1=ð1  y Þg=2

   m  a ¼ kf 1  e2d yð1=2; 0Þ 8bphe2d

ð20bÞ

ð16Þ

Putting expression (15) in (16) and simplifying, the logarithmic damping decrement is given by;  1 4mapbh d ¼ ln 1 þ ð17Þ 2 kc yðl; 0Þ Determination of the Product of Kinematics Coefficient of Friction and Dynamic Slip Ratio (μ × α) Since the accurate determination of kinematic coefficient of friction, μ, and dynamic slip ratio, α, are difficult, the product of these two parameters, i.e., (μ × α) has been found out from the experimental results for logarithmic damping decrement of 3 mm thickness cantilever beam by modifying expressions (17) as;   m  a ¼ kc 1  e2d yðl; 0Þ=4bphe2d ð18Þ The variations of (μ × α) with natural frequency of vibration at the first mode of transverse vibration have been determined for mild steel, copper and aluminium materials under different initial amplitudes of excitation and plotted as shown in Fig. 3(a) for mild steel as a sample. These plots have been further used for evaluating the numerical results for logarithmic damping decrement with other dimensions of the beam using expression (17). From the above figure, it is evident that the product of coefficient of friction and slip ratio increases with the increase in the frequency and amplitude of vibration. Further, the analysis has been extended to evaluate the damping models of structures with fixed-hinged and fixedfixed end conditions. The mathematical expressions for the logarithmic decrement (δ) and the product of dynamic slip ratio and kinematics coefficient of friction (μ × α) are found out for fixed-hinged and fixed-fixed end conditions of the beam respectively as;  1 8mapbh d ¼ ln 1 þ ð19aÞ 2 kh yð1=2; 0Þ    m  a ¼ kh 1  e2d yð1=2; 0Þ 8bphe2d

ð19bÞ

Fig. 3 (a) Variation of slip ratio and product of co-efficient of friction„ (μ × α) with the frequency for mild steel beams with fixed-free end condition. (b) Variation of product of co-efficient of friction and slip ratio (μ × α) with the frequency for mild steel beams with fixedhinged end condition. (c) Variation of product of co-efficient of friction and slip ratio (μ × α) with the frequency for mild steel beams with fixed-fixed end condition

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    and kf ¼ 192EI are the static bending where kh ¼ 109:7EI l3 l3 stiffness of the fixed-hinged and fixed-fixed beams, respectively. The expressions (19) and (20) are used to evaluate logarithmic damping decrement of the fixedhinged and fixed-fixed beams of different dimensions. The variations of (μ × α) with natural frequency of vibration at the first mode of transverse vibration have been determined for fixed-hinged and fixed-fixed end conditions under different initial amplitudes of excitation for mild steel, copper and aluminium materials and are plotted as shown in Fig. 3(b) and (c), respectively for mild steel as a sample. These plots have been further used for evaluation of the numerical results for logarithmic damping decrement of the specimen materials with other dimensions of the beam using expressions (19) and (20).

Response Surface Methodology (RSM) RSM is a collection of mathematical and statistical data that are useful for the modelling and analysis of problems in which a response of interest is influenced by several variables with an objective to optimize the response [38]. RSM also quantifies relationships among one or more measured responses and the input factors. Version 8 of the Design Expert software [39] is used to develop the experimental plan for RSM. Multiple response optimizations are performed either by inspecting each response on the interpretation plots or using the graphical and numerical tools. Moreover, RSM designs also help to quantify the relationships between one or more measured responses and the input factors. The data collected is analyzed statistically using regression analysis to establish a relationship between the input factors and response variables. Regression is performed in order to develop a functional relationship between the estimated variables. The performance of the model depends on a large number of factors which interact in a complex manner. A second order response surface model is usually expressed as: O ¼ b0 þ

m X i¼1

b i xi þ

m X i¼1

bi x2i þ

m X m X

bij xi xj þ "

ð21Þ

The response surface analysis is then carried out in terms of the fitted surface. The least square technique is used to fit a model equation containing the input variables by minimizing the residual errors measured by the sum of square deviations between the actual and the estimated responses. This involves the calculation of estimates for the regression coefficients, i.e., the coefficients of the model variables including the intercept or constant term. The calculated coefficients or the model equation is to be tested for statistical significance. In this respect, the following tests are performed. Test for Significance of the Regression Model This test is performed as an ANOVA procedure by calculating the F-ratio, which is the ratio between the regression mean square and the mean square error. The Fratio, also called the variance ratio, is the ratio of variance due to the effect of a factor (in this case the model) and variance due to the error term. The F-ratio representing the test statistics for multiple independent variables is mathematically expressed by; F  ratio ¼

ðMS ÞModel ðMS ÞResidual

where (MS)Model and (MS)Residual are the mean square of the model and residual, respectively. The mean square of the model is used to estimate the model variance given by the model sum of squares divided by the model degrees of freedom. The mean square of the residual is used to estimate the process variance. Mean square (MS) is defined as the mean sum of squares and is mathematically expressed as; MS ¼ SS=k

ð23Þ

where SS and k are the sum of squares and degrees of freedom, respectively. Sum of squares (SS) is defined as the sum of the differences between the individual experimental values and the mean of all the experimental values in the set of experimental data [40]. Sum of squares (SS) is given by;

i¼1 j¼1

where, β0, βi (i=1, 2 . . . m) and βij (i=1, 2 . . . m, j=1, 2 . . . m) are the unknown regression coefficients to be estimated using the method of least squares. In this expression, ε is experimentally random error which may be due to imperfect conditions and measurement errors while conducting the experiments and x1, x2… xm, are the input variables that influence the response (O) and m is the number of input factors. The method of least squares is used to estimate the coefficients of the second order model.

ð22Þ

SS ¼

z X i¼1

z 1X Yi  Yi z i¼1

!2 ð24Þ

where Yi and z are the ith experimental data and number of experimental runs, respectively. The significance level “β” for a given hypothesis test is a value for which a P-value less than or equal to “β” is considered to be statistically significant. Typical value for “β” considered in the present study is 0.05. This value

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“Prob. > F” value on an F-test indicates the proportion of time expected to get the stated F-value if no factor effects are significant. In general, the lowest order polynomial is considered for adequately describing the system. Test for Lack-Of-Fit

Fig. 4 Experimental set-up for the measurement of logarithmic damping decrement

corresponds to the probability of observing an extreme value by chance. Test for Significance on Individual Model Coefficients This test forms the basis for model optimization by adding or deleting coefficients through backward elimination, forward addition or stepwise elimination/addition/exchange. It involves the determination of the P-value or probability value, usually relating the risk of falsely rejecting a given hypothesis. The P-value is the probability of rejecting the hypothesis. In statistics, a given hypothesis is rejected if the P- value is more than 0.05. It is a convention to adopt this value. Moreover, it is common in statistics to adopt this threshold. This value is based on the confidence level of 95%. However, certain researchers use the threshold P- value of 0.1 which is based on the confidence level of 90%. In the present study, P- value of 0.05 has been adopted as the threshold to test the significance of the model coefficients. The selection of P-value depends on the population size [40]. The present analysis is based on CCD design of experiment in which the number of input experimental data (population size) is less and therefore considering the threshold P-value of 0.1 will result in more chances of falsely accepting the hypothesis. This will result in inaccurate model selection. Hence, the threshold P-value of 0.05 has been adopted in which the chances of wrong interpretation are less.

As replicate measurements are available, a test indicating the significance of the replicate error compared to the model dependent error can be performed. This test splits the residual or error sum of squares into two portions; one is due to pure error based on the replicate measurements and the other due to lack-of-fit because of model performance. The test statistic for lack-of-fit is the ratio between the lack-of-fit mean square and the pure error mean square. As established, this F-test statistic can be used to determine whether the lack-of-fit error is significant or not at the desired significance level, β. Insignificant lack-of-fit is desired as significant lackof-fit indicates that there might be contributions in the input variables–response relationship that are not accounted for in the model. Additional checks are required to determine whether the model actually describes the experimental data or not. The checks performed include determining the variance coefficient of determination, R2. These R2 coefficients have values between 0 and 1. R2 is the variation between the mean of the residuals and the individual parameters [40]. It is mathematically expressed by; 

ðSS ÞResidual R ¼1 ðSS ÞResidual þ ðSS ÞModel 2

ð25Þ

where (SS)Model is the summation of the squares of the individual experimental values that are included in the model. (SS)Residual is the summation of the squares of the individual experimental values which are not included in the model. In addition to the above, the adequacy of the model is also investigated by examining the residuals. The residuals

Table 1 Details of specimens used for layered and jointed beams Thickness of the specimen (mm) 3 3 3

Width of the specimen (mm)

Number of layers

Cantilever length (mm)

40.2 40.2 40.2

2 2 2

520.6 520.6 520.6

Type of Welding

Number of Tack Welds

Material and Modulus of elasticity (GPa)

Tack Weld Tack Weld Tack Weld

10,20,30 10,20,30 10,20,30

Mild Steel (203.41) Copper (110.32) Aluminium (69.45)

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Table 2 Important factors and their levels

Sl.No

Factor

Notation

Unit

Levels

1 2 3 4

Modulus of Elasticity Tack number Amplitude Surface roughness

Y N a Ra

GPa

69.45 10 0.1 0.88

representing the differences between the respective observe responses and the predicted responses are examined using the normal probability plots of the residuals and the plots of the residuals versus the predicted response. If the model is adequate, the points on the normal probability plot should

mm μm

110.32 20 0.2 1.53

203.41 30 0.3 2.18

form a straight line. On the other hand, the plots of the residuals versus the predicted response normally do not follow any definite pattern.

Experimental Details Table 3 Logarithmic damping decrement (δ) response for CCD design of experiment Runs

Factors

Response δ

Y (GPa)

N

a (mm)

Ra (μm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

110.32 110.32 110.32 110.32 110.32 203.41 110.32 110.32 69.45 110.32 69.45 69.45 110.32 69.45 203.41 69.45 203.41

10 20 30 20 10 20 20 30 20 20 30 30 20 10 30 30 30

0.3 0.2 0.2 0.1 0.2 0.2 0.1 0.2 0.2 0.3 0.1 0.3 0.2 0.1 0.1 0.1 0.1

1.53 1.53 2.18 0.88 1.53 1.53 1.53 1.53 1.53 1.53 0.88 0.88 1.53 2.18 0.88 2.18 2.18

0.00379 0.00537 0.00397 0.00896 0.00624 0.00268 0.00874 0.00378 0.00778 0.00366 0.01152 0.00437 0.00485 0.01711 0.00411 0.01138 0.00408

18 19 20 21 22 23 24 25 26 27 28 29 30

110.32 110.32 203.41 203.41 69.45 69.45 203.41 203.41 69.45 203.41 69.45 110.32 203.41

20 30 10 10 10 10 10 10 10 30 30 20 30

0.3 0.3 0.1 0.3 0.1 0.3 0.1 0.3 0.3 0.3 0.3 0.2 0.3

1.53 1.53 2.18 2.18 0.88 0.88 0.88 0.88 2.18 2.18 2.18 1.53 0.88

0.00333 0.00284 0.00629 0.00212 0.01781 0.00596 0.00634 0.00211 0.00599 0.00153 0.00433 0.00493 0.00151

Experimental Set-Up and Experiments The logarithmic damping decrement of layered and jointed welded structures is affected by variables such as number of tack joints, initial amplitude of excitation, surface roughness and material properties like Young’s Modulus. In order to study the effect of these parameters, an experimental set-up has been fabricated as detailed in the succeeding sub-section. Surface roughness (SR) measurements The specimens are prepared from commercial mild steel, copper and aluminium flats with different surface roughness. The surface roughnesses at the interfaces are

Fig. 5 Surface plot for the variation of standard error in the design space

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Fig. 7 Central composite design space cube

Fig. 6 Contour plot for the variation of standard error in the design space

measured with a portable stylus type profilometer (Talysurf, Taylor Hobson Surtronic 3+). The profilometer is set to a cut-off length of 0.8 mm, filter 2CR with transverse speed of 1 mm/s having evaluation length of 4 mm. The roughness has been measured at least for five times and the average value is considered for analysis. The measured profile is digitized and processed through surface finish analysis software Talyprofile [41] for the evaluation of the roughness parameters.

pressure distribution at the interfaces has been considered for evaluating the logarithmic damping decrement. The details of the specimen used in the experiment are presented in Table 1. The number of tack joints varies along the length. The specimens are excited transversally at amplitudes of 0.1, 0.2, and 0.3 mm at the free ends of the fixed-free beams and at the middle for the fixed-hinged and fixed-fixed specimens with the help of a spring arrangement. The free vibration is measured with a vibration pick-up and the corresponding signal is fed to a digital storage oscilloscope, in order to measure the amplitudes of the first cycle (a1), last cycle (an+1) and the number of cycles (n) of the steady signal. The logarithmic damping decrement is then evaluated using the expression d ¼ lnða1 =anþ1 Þ=n.

Logarithmic damping decrement measurements RSM models and experimental design An experimental set-up as shown in Fig. 4 has been fabricated to conduct the experiments with mild steel, copper and aluminium specimens. The specimens are prepared from the above stock of materials by tack welding. Moreover, the beams with different end conditions such as; fixed-free, fixed-hinged and fixed-fixed as shown in Fig. 1 (a), (b) and (c) respectively and uniform intensity of

As explained the logarithmic damping decrement is influenced by number of tack joints, initial amplitude of excitation, surface roughness and Young’s Modulus. The relationship of logarithmic damping decrement response with respect to the first two variables could be estimated by a first-degree model. For the surface

Table 4 Model fit summary Source Linear 2FI Quadratic Cubic

Sequential p-value

Lack of Fit p-value

Adjusted R-Squared

Predicted R-Squared

<0.0001 0.2409 <0.0001 0.0013

<0.0001 <0.0001 0.0058 0.8792

0.8456 0.8608 0.9841 0.9996

0.7569 0.4358 0.9782 0.9990

Not-suggested Not-suggested Suggested Suggested

780 Table 5 Analysis of variance for full quadratic model

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Source Regression Linear Square Interaction Residual Error Lack of Fit Pure Error Total

Sum of squares

df

Mean square

F value

p-value Prob > F

0.000469 0.000385 0.000036 0.000048 0.000007 0.000007 0.000000 0.000475

14 4 4 6 15 12 3 29

0.000033 0.000023 0.000009 0.000008 0.000000 0.000001 0.000000

73.29 49.86 19.19 17.48

0.0000 0.0000 0.0000 0.0000

7.86

0.0058

roughness and Young’s Modulus of the material, a second-degree or third- degree model is necessary. A suitable second and third order polynomial involving linear, quadratic, cubic and cross terms has been selected considering the statistical parameters, i. e., coefficient of determination (R2), adjusted R2, standard error of regression and analysis of variance (ANOVA). In the present work, the input variables are tack number (N), initial amplitude of vibration (a), surface roughness (Ra) and Young’s Modulus (Y) and the output response is the logarithmic damping decrement (δ). The logarithmic damping decrement is analyzed with a standard central composite design (CCD) technique. The star points are at the face of the cube portion which corresponds to the β-value of 1 and this is commonly referred to as a face-centered CCD and the centre points are the locations with coded value set to 0. The important factors and their levels are shown in Table 2. The response surface analysis is carried out in terms of the fitted surface. The lack of fit and the degree of

Table 6 Estimated regression coefficients for full quadratic model

Factor

significance of the model are tested by the analysis of variance (ANOVA) using the Design Expert-8 software. The CCD design of experimental runs with independent control variables in uncoded forms and responses are shown in Table 3.

Results and Discussion Development of the Quadratic and Cubic Response Surface Models for the Logarithmic Damping Decrement (δ) of Welded Beams with Fixed-Free Condition The results as shown in Table 3 are used as the input data to the Design Expert-8 software for further analysis. Figures 5 and 6 show the standard error of the design which is found to be uniform and thus favorable. Initially, the sequential or extra sums of squares for the linear, quadratic and cubic terms in the model are computed and a fit summary based on this has been generated as presented in the Table 4. The Fit

Coefficient estimate

Standard error

Probability (P)

Intercept A-Y B-N C-a D-Ra

0.042202 −0.000216 −0.000573 −0.130057 0.002671

0.002295 0.000029 0.000162 0.016008 0.003474

0.000 0.000 0.003 0.000 0.454

A2 B2 C2 D2 AB AC AD BC BD CD

0.000000 0.000004 0.139127 −0.000989 0.000001 0.000209 0.000001 0.000777 0.000009 0.000426

0.000000 0.000004 0.035910 0.001136 0.000000 0.000025 0.000004 0.000160 0.000026 0.002564

0.001 0.284 0.001 0.398 0.004 0.000 0.865 0.000 0.723 0.870

(Not-Significant) (Not-Significant) (Not-Significant)

(Not-Significant) (Not-Significant) (Not-Significant)

Exp Mech (2012) 52:771–791

781

damping decrement of fixed—free beam:

Summary output as shown in Table 4 has been examined without performing any transformation of the response. The summary revealed that the quadratic and cubic models are statistically significant and therefore used for fitting the data. The Central composite design space cube representing the input variables at the star points and the face of the cube is shown in Fig. 7. The following expression of quadratic model in terms of the uncoded factors has been found out for the logarithmic Table 7 Analysis of variance for full cubic model

d ¼ 0:042  2:16E  4  Y  5:73E  4  N  1:3E  1  a þ 2:67E  3  Ra þ 4E  6  N  N þ 1:39E  1  a  a  9:89E  4  Ra  Ra þ 1E  6  Y  N þ 2:09E  4  Y  a þ 7:77E  4  N  a þ 9E  6  N  Ra þ 4:26E  4  a  Ra ð26Þ

Source

Sum of squares

df

Mean square

F value

Model A-Y B-N C-a D-Ra AB AC AD BC BD CD

1.378E-003 4.625E-005 9.140E-006 7.224E-005 7.392E-009 1.771E-005 9.172E-005 2.369E-011 2.688E-005 1.096E-009 1.688E-008

30 1 1 1 1 1 1 1 1 1 1

4.593E-005 4.625E-005 9.140E-006 7.224E-005 7.392E-009 1.771E-005 9.172E-005 2.369E-011 2.688E-005 1.096E-009 1.688E-008

466.64 469.95 92.86 733.97 0.075 179.97 931.98 2.407E-004 273.16 0.011 0.17

0.0001 0.0001 0.0001 0.0001 0.7851 0.0001 0.0001 0.9877 0.0001 0.9164 0.6803

A2 B2 C2 D2 ABC ABD ACD BCD A2B A2C A2D AB2 AC2 AD2 B2C B2D BC2 BD2

6.180E-007 1.461E-008 4.108E-005 5.331E-010 4.161E-006 3.686E-011 5.490E-011 8.844E-011 4.552E-007 4.370E-007 3.545E-009 7.637E-007 1.348E-005 9.996E-010 1.480E-007 1.109E-010 2.587E-006 3.823E-010

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

6.180E-007 1.461E-008 4.108E-005 5.331E-010 4.161E-006 3.686E-011 5.490E-011 8.844E-011 4.552E-007 4.370E-007 3.545E-009 7.637E-007 1.348E-005 9.996E-010 1.480E-007 1.109E-010 2.587E-006 3.823E-010

6.28 0.15 417.42 5.416E-003 42.28 3.746E-004 5.578E-004 8.986E-004 4.62 4.44 0.036 7.76 136.95 0.010 1.50 1.127E-003 26.29 3.884E-003

0.0151 0.7015 0.0001 0.9416 0.0001 0.9846 0.9812 0.9762 0.0358 0.0396 0.8502 0.0073 0.0001 0.9201 0.2253 0.9201 0.0001 0.9505

C2D CD2 A3 B3 C3 D3 Residual Lack of Fit Pure Error Cor Total

9.687E-009 4.611E-014 0.000 0.000 0.000 0.000 5.511E-006 5.511E-006 0.000 1.383E-003

1 1 0 0 0 0 56 50 6 86

9.687E-009 4.611E-014

0.098 4.685E-007

0.7549 0.995

9.842E-008 1.102E-007 0.000

21.54

p-value Prob > F

0.0073

Significant

Significant

782

Exp Mech (2012) 52:771–791

The analysis has been further carried out for the cubic model and the following expression of cubic model in terms of the uncoded factors has been found out for the logarithmic damping decrement of fixed—free beam: d ¼ þ5:11E  3  2:39E  3  Y  1:06E  3  N  3:03E  3  a þ 3:06E  5  Ra þ 6:69E  4  Y  N þ 1:56E  3  Y  a þ 7:83E  7  Y  Ra þ 8:44E  4  N  a  5:33E  6  N  Ra þ 2:12E  5  a  Ra þ 1:86E  4  Y 2  2:82E  5  N 2 þ1:46E  3  a2 þ 5:29E  6  Ra2  4:02E  4  Y  N  a þ 1:18E  6  Y  N

Table 9 Estimated regression coefficients for reduced quadratic model Factor Intercept A-Y B-N C-a A2 C2 AB AC BC

Coefficient estimate

df

Standard error

Probability (P)

0.041678 −0.000134 −0.000395 −0.126231 0.000000 0.133252 0.000001 0.000206 0.000777

1 1 1 1 1 1 1 1 1

0.001545 0.000020 0.000044 0.012020 0.000000 0.027624 0.000000 0.000022 0.000142

0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000

 Ra  1:47E  6  Y  a  Ra þ 1:86E  6  N  a  Ra þ 1:92E  4  Y 2  N  1:89E  4  Y 2  a  1:69E  5  Y 2  Ra þ 2:49E  4  Y  N 2  1:01E  3  Y  a  8:74E  6  Y  Ra 2

2

 1:10E  4  N 2  a þ 3:01E  6  N 2  Ra  4:43E  4  N  a2  5:40E  6  N  Ra2  2:73E  5  a2  Ra  6:05E  8  a  Ra2

The Model F-value of 73.29 as given in Table 5 implies that the model is significant. There is only 0.01% chance that a high “Model F-Value” could occur due to noise. Model F-value is calculated to test the adequacy of the model [40] and is mathematically expressed as;

ð27Þ

R2 =k ð1  R2 Þ=½z  ðk þ 1Þ

Analysis of Variance (ANOVA) for Full Quadratic and Cubic Models

ModelF  value ¼

The tests for significance of the regression model, individual model coefficients and lack-of-fit are performed to ensure adequacy of the model. Usually an ANOVA table is used to summarize the statistical data obtained from the tests.

where k and z are degrees of freedom and number of experimental runs for RSM analysis, respectively. The value of “Prob > F” less than 0.050 indicates that the model terms are significant as shown in Table 6. In this case Y, N, a, Y2, a2, Y × N, Y × a and N × a are significant model terms.Values greater than 0.05 indicate that the model terms are insignificant. The reduction of terms may improve the model further if there are many insignificant terms.The “Lack of Fit F-value” of 7.86 implies that this is significant. There is only 0.5% chance that a “Lack of Fit F-value” is insignificant and could occur due to noise. The model can be improved further by eliminating the insignificant interaction terms from it. Insignificant factors are

Quadratic model The ANOVA result for the quadratic response full model of logarithmic damping decrement is presented in Table 5. The value of “P” in Table 5 is less than 0.05 indicating that the model and its terms have a significant effect on the response. Further, the significance of each coefficient in the full model has been examined by the P-values and the results are listed in Table 6. Table 8 Analysis of Variance for reduced quadratic model

Source Regression Linear Square Interaction Residual Error Lack of Fit Pure Error Total

ð28Þ

Sum of squares

df

Mean square

F value

p-value Prob>F

0.000468 0.000384 0.000036 0.000047 0.000008 0.000007 0.000001 0.000475

8 3 2 3 21 8 13 29

0.000058 0.000042 0.000017 0.000016 0.000000 0.000001 0.000000

162.60 117.66 47.92 43.85

0.000 0.000 0.000 0.000

22.37

0.000

Exp Mech (2012) 52:771–791 Table 10 Analysis of variance for reduced cubic model

783

Source

Sum of squares

df

Mean square

F value

p-value Prob>F

Model A-Y B-N C-a AB AC BC A2 C2 ABC A2B A2C AC2 BC2

1.377E-003 9.920E-005 1.261E-005 1.737E-004 1.810E-005 9.207E-005 2.703E-005 6.426E-007 4.088E-005 4.127E-006 4.813E-007 4.304E-007 1.365E-005 2.661E-006

13 1 1 1 1 1 1 1 1 1 1 1 1 1

1.059E-004 9.920E-005 1.261E-005 1.737E-004 1.810E-005 9.207E-005 2.703E-005 6.426E-007 4.088E-005 4.127E-006 4.813E-007 4.304E-007 1.365E-005 2.661E-006

1197.31 1121.43 142.56 1963.25 204.56 1040.88 305.57 7.26 462.17 46.65 5.44 4.87 154.35 30.08

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Residual Lack of Fit Pure Error Cor Total

6.457E-006 6.457E-006 0.000 1.383E-003

73 67 6 86

8.846E-008 9.638E-008 0.000

27.64

0.000

Significant

ð29Þ

further. The “Lack of Fit F-value” of 21.54 implies that this is significant. There is only 0.3% chance that a “Lack of Fit F-value” is insignificant and could occur due to noise. The model can be improved further by eliminating the insignificant interaction terms from it. Insignificant factors are removed from the full model by implementing the backward elimination technique for its improvement. Thus,

The ANOVA result for the cubic response full model of logarithmic damping decrement is presented in Table 7. The value of “P” in Table 7 is less than 0.05 indicating that the model and its terms have a significant effect on the response. Further, the significance of each coefficient in the full cubic model has been examined by the P-values and the results are listed in Table 7. The Model F-value of 466.64 as given in Table 7 implies that the model is significant. There is only 0.01% chance that a high “Model F-Value” could occur due to noise. Values of “P” for model terms, greater than 0.05 indicate that the model terms are insignificant. In this case, Ra, Y×Ra, N×Ra, a×Ra, Y2, N2, Ra2, Y×N×Ra, Y×a×Ra, N×a×Ra, Y 2×N, Y 2×a, Y 2×Ra, Y×N 2 , Y×Ra2 , N 2 ×a, N2×Ra, N×Ra2, a2×Ra, a×Ra2 are the insignificant terms. The elimination of these terms may improve the model

Fig. 8 Response surface plot: effect of tack number and surface roughness on the logarithmic damping decrement (δ)

removed from the full model by implementing the backward elimination technique for its improvement. Thus, the full quadratic model for the logarithmic decrement (δ) with fixedfree configuration of the beam has been reduced to: d ¼ 0:042  1:33E  4  Y  3:95E  4  N  1:26E  1  a þ 1:33E  1  a  a þ 1E  6  Y  N þ 2:06E  4  Y  a þ 7:77E  4  N  a

Cubic model

784

Exp Mech (2012) 52:771–791

Fig. 9 Contour plot: effect of tack number and surface roughness on the logarithmic damping decrement (δ)

the full cubic model for the logarithmic decrement (δ) with fixed-free configuration of the beam has been reduced to: d ¼ 5:10E  3  2:22E  3  Y  1:06E  3  N  3:11E

Fig. 11 Contour plot: Effect of amplitude and surface roughness on the logarithmic damping decrement (δ)

Analysis of Variance (ANOVA) for Reduced Quadratic and Cubic Models

 3  a þ 6:75E  4  Y  N þ 1:56E  3  Y  a þ 8:45E  4  N  a þ 1:87E  4  Y 2 þ 1:46E  3  a2  3:99E  4  Y  N  a þ 1:97E  4  Y 2  N  1:88E  4  Y 2  a  1:02E  3  Y  a2  4:49E  4  N  a2

The tests for significance of the reduced quadratic and cubic regression models, individual model coefficients and lack-of-fit have been performed to check adequacy of the models. The resulting ANOVA tables have been used to summarize the statistical data obtained from these tests.

ð30Þ

Fig. 10 Response surface plot: Effect of amplitude and surface roughness on the logarithmic damping decrement (δ)

Fig. 12 Response surface plot: Effect of amplitude and tack number on the logarithmic damping decrement

Exp Mech (2012) 52:771–791

785

Fig. 13 Contour plot: Effect of amplitude and tack number on the logarithmic damping decrement

Fig. 14 Main effects plot for logarithmic decrement (δ)

Quadratic model

Quadratic model

The resulting ANOVA table for the reduced quadratic model for logarithmic damping decrement is shown in Table 8. The Estimated Regression Coefficients in the reduced quadratic model for logarithmic damping decrement is shown in Table 9. The Model F-value of 162.60 in Table 8 implies that the model is significant. There is only 0.01% chance that this high “Model F-Value” might be due to noise. The value of “Prob > F” less than 0.050 indicate that the model terms are significant. The “Lack of Fit F-value” of 22.37 implies that this is significant. There is only 0.01% chance that a “Lack of Fit F-value” is large due to noise

The quadratic damping models for the logarithmic decrement (δ) of structures with fixed-hinged and fixed-fixed end conditions are given by the expressions (31) and (32), respectively as; d ¼ 0:022  1:1E  4  Y  2:08E  4  N  6:64E  2  a þ 7:01E  2  a  a þ 1:08E  4  Y  a þ 4:09E  4  N  a ð31Þ

Cubic model The resulting ANOVA table for the reduced cubic model for logarithmic damping decrement is shown in Table 10. The Model F-value of 1197.31 in Table 10 implies that the model is significant. There is only 0.01% chance that this high “Model F-Value” might be due to noise. Development of the Quadratic and Cubic Response Surface Models for the Logarithmic Damping Decrement (δ) of Welded Beams with Fixed-Hinged and Fixed-Fixed Conditions The same procedure is repeated and the analysis has been extended to evaluate the quadratic and cubic RSM damping models of structures with fixed-hinged and fixed-fixed end conditions.

Fig. 15 Normal probability plot of the residuals

786

Exp Mech (2012) 52:771–791

Fig. 18 Typical time history plot for amplitude (a): 0.1 mm, number of tack welds (N): 10 and surface roughness (Ra): 1.53 μm for mild steel beams with fixed-free end conditions

d ¼ 2:13E  3  9:25E  4  Y  4:42E  4  N  1:29E  3  a þ 2:81E  4  Y  N þ 6:50E  4

Fig. 16 Residuals versus order of the data

d ¼ 0:017  8:7E  5  Y  1:65E  4  N  5:26E  2  a þ 5:55E  2  a  a þ 8:6E  5  Y  a þ 3:24E  4  N  a

ð32Þ

 Y  a þ 3:52E  4  N  a þ 7:76E  5  Y 2 þ 6:08E  4  a2  1:66E  4  Y  N  a þ 8:29E  5  Y 2  N  7:87E  5  Y 2  a  4:24E  4  Y  a2  1:87E  4  N  a2

Cubic model

ð34Þ The cubic damping models for the logarithmic decrement (δ) of welded beams with fixed-hinged and fixed-fixed end conditions are given by the expressions (33) and (34), respectively as; d ¼ 2:69E  3  1:67E  3  Y  5:58E  4  N  1:64E  3  a þ 3:54E  4  Y  N þ 8:21E  4  Y  a þ 4:46E  4  N  a þ 9:53E  5  Y 2 þ 7:69E  4  a2  2:09E  4  Y  N  a þ 1:047E  4  Y 2  N  9:29E  5  Y 2  a  5:37E  4  Y  a2  2:36E  4  N  a2

Fig. 17 Residuals versus the fitted values

ð33Þ

Surface and Contour Plots for Logarithmic Damping Decrement (δ) The effects of the interactions of the parameter such as; tack number/amplitude, tack number/surface roughness, and amplitude/surface roughness on the logarithmic damping decrement are shown in Figs. 8, 9, 10, 11, 12, and 13. The initial amplitude of excitation of free vibration is an important parameter influencing the logarithmic damping decrement of layered and jointed welded structures. The logarithmic damping decrement of such structures decreases with an increase in initial amplitude of excitation. This decrease is due to the introduction of higher strain energy into the system compared to that of the dissipated energy due to interface friction. The product of the kinematic coefficient of friction and dynamic slip ratio “(μ×α)” is the key factor in the determination of damping capacity of layered and jointed welded structures as found by Singh and Nanda [42]. The kinematic coefficient of friction and the relative dynamic slip at the interfaces result in energy dissipation and therefore the damping of the built-up structure. These two vital parameters are interdependent with each other depending on the interface surface roughness. The friction at the interfaces increases and the relative dynamic slip decreases with an increase in surface roughness and vice – versa, hence the product of these two parameters is

Exp Mech (2012) 52:771–791

787

the dynamic slip at the interfaces increases causing an increase in the logarithmic damping decrement of the layered and jointed tack welded structure. Main Effects Plot or Perturbation Plot

Fig. 19 Typical time history plot for amplitude (a): 0.1 mm, number of tack welds (N): 20 and surface roughness (Ra): 2.83 μm for mild steel beams with fixed-free ends end conditions

assumed to remain constant. The product “(μ × α)” also depends on the initial amplitude of excitation and frequency of vibration. This product increases with the increase in the natural frequency of vibration and the initial amplitude of excitation. The product “(μ × α)” remains almost constant with respect to the surface roughness and thereby the logarithmic damping decrement remains constant for a given jointed interface of same material irrespective of the surface roughness as shown in Figs. 8, 9, 10, 11, 12, and 13. The logarithmic damping decrement increases with a decrease in the number of tack joints. The frequency of vibration depends on stiffness and mass. With a decrease in the number of the tack welds, the static bending stiffness remains the same, but the overall mass decreases as there is less weld material. The frequency of vibration increases due to decrease in mass deposition in case of tack welded joints. Hence the product “(μ × α)” is enhanced resulting in an increase in the logarithmic damping decrement. Further, the relative spacing between the consecutive tacks is increased with the decrease in the number of tack weld joints. Thus,

The main effects plot for logarithmic damping decrement is shown in Fig. 14. These plots are used to compare the changes in the mean levels of response and to ascertain the factors influencing the response the most. The surface roughness effect line is almost parallel to the X-axis implying that the effect of surface roughness on logarithmic decrement is almost negligible. Further, the slope of the amplitude is greater in comparison to the number of tack welds indicating that the effect of the amplitude is more predominant. Residual Plots for Logarithmic Damping Decrement (δ) The regression model is used for determining the residuals of each individual experimental run. The differences between the measured and predicted values are termed as the residuals. The residuals are calculated and ranked in ascending order. The normal probability plot indicates whether the residuals follow a normal or random distribution. The points follow a straight and zigzag line in case of normal and random distributions respectively. If a pattern like “S-shape” is obtained, the transformation of response may provide a better fitting analysis. The normal probability of residuals for the response is shown in Fig. 15. The normal probability plot is used to verify the adequacy of normality [39]. The data is spread roughly along the straight line for the response as shown in Fig. 15 establishing that the residuals are normally distributed. Further, the plot for the residuals versus the experimental run order is shown in Fig. 16 to check the influence of lurking variables on the response. The plot shows a random

Table 11 Comparison of theoretical and RSM results for logarithmic damping decrement of mild steel specimens with fixed-free end conditions Parameters

Logarithmic damping decrement

N

a(mm)

Ra(μm)

Theoretical

RSM (Quadrilateral)

RSM (Cubic)

%Deviation between Theoretical and RSM (Quadrilateral)

%Deviation between Theoretical and RSM (cubic)

15 20 18 12 10

0.1 0.3 0.1 0.4 0.5

1.86 0.94 2.64 2.88 1.77

0.016973 0.001945 0.005271 0.003441 0.007539

0.018648 0.002123 0.005789 0.003729 0.008346

0.018335 0.002103 0.005712 0.003711 0.008086

8.90 9.15 9.82 8.37 8.07

8.02 8.12 8.37 7.85 7.26

Avg. deviation %Deviation between Theoretical and RSM (Quadrilateral): 8.862% Avg. deviation %Deviation between Theoretical and RSM (Cubic): 7.924%

788

Exp Mech (2012) 52:771–791

Table 12 Comparison of theoretical and RSM results for logarithmic damping decrement of mild steel specimens with fixed –hinged conditions Parameters N

15 20 18 12 10

Logarithmic damping decrement

a(mm) Ra(μm) Theoretical RSM (Quadrilateral) RSM (Cubic) %Deviation between Theoretical %Deviation between Theoretical and RSM (Quadrilateral) and RSM (cubic) 0.1 0.3 0.1 0.4 0.5

1.83 0.99 2.63 2.85 1.75

0.008901 0.001144 0.002774 0.001864 0.003968

0.009634 0.001252 0.002998 0.002068 0.004399

0.009510 0.001244 0.002982 0.002031 0.004324

7.60 9.44 8.07 10.94 10.86

6.84 8.74 7.49 8.96 8.97

Avg. deviation %Deviation between Theoretical and RSM (Quadrilateral): 9.382% Avg. deviation %Deviation between Theoretical and RSM (Cubic): 8.200%

scatter indicating the accuracy of analysis. The correlation between the residuals is shown in Fig. 16 to check the independency of the variables. The plot shows that the residuals are distributed evenly in both positive and negative directions along the run signifying the independency of the variables [39]. The residual versus predicted response is shown in Fig. 17 to check the accuracy of the model. Since the plot shows a random scatter without any pattern, the fitted model is considered to be correct [39]. Checking the Adequacy of Mathematical Models The accuracy of the fit for the mathematical models has also been tested by coefficient of determination (R2) and adjusted coefficient of determination (Radj2). The R2 is the proportion of the variation in the dependent variable indicated by the regression model. On the other hand, Radj2 is the coefficient of determination adjusted for the number of independent variables in the regression model. The R2 and Radj2 values of reduced quadratic model are found to be 98.4 and 97.8%, respectively. The R2 and Radj2 values of reduced cubic model are found to be 99.53 and 99.45%,

respectively establishing the excellent correlation between the predicted and experimental values of the cubic response. Thus, from the R2 and Radj2, it is inferred that the cubic models provides better precision in evaluating the logarithmic damping decrement of layered and welded structures. Validity of the Model The performance of the developed models is tested using five experimental values that are not used during the experimentation in the modeling process. The results for the RSM quadratic and cubic response as predicted by the models in expressions (29)–(34) are compared with the theoretical values. The time history plots as recorded in the digital storage oscilloscope for two experimental samples are presented in Figs. 18 and 19. Quadratic model The average percentage of deviation between the RSM quadratic results and theoretical values of logarithmic damping decrement for various configurations and loading

Table 13 Comparison of theoretical and RSM results for logarithmic damping decrement of mild steel specimens with fixed –fixed conditions Parameters N

15 20 18 12 10

Logarithmic damping decrement

a(mm) Ra(μm) Theoretical RSM (Quadrilateral) RSM (Cubic) %Deviation between Theoretical %Deviation between Theoretical and RSM (Quadrilateral) and RSM (cubic) 0.1 0.3 0.1 0.4 0.5

1.79 0.96 2.64 2.87 1.76

0.000708 0.000846 0.002312 0.001483 0.003236

0.000786 0.000946 0.002525 0.001642 0.003555

0.000765 0.000921 0.002485 0.001620 0.0003506

Avg. deviation %Deviation between Theoretical and RSM (Quadrilateral): 10.526% Avg. deviation %Deviation between Theoretical and RSM (Cubic): 8.394%

11.02 11.82 9.21 10.72 9.86

8.05 8.86 7.48 9.24 8.34

Exp Mech (2012) 52:771–791

789

Table 14 Comparison of the theoretical and RSM results for logarithmic damping decrement of copper specimens with fixed-free end condition Parameters N

15 20 18 12 10

Logarithmic damping decrement

a(mm) Ra(μm) Theoretical RSM (Quadrilateral) RSM (Cubic) %Deviation between Theoretical %Deviation between Theoretical and RSM (Quadrilateral) and RSM (cubic) 0.1 0.3 0.1 0.4 0.5

1.85 0.99 2.78 2.86 1.77

0.028891 0.036182 0.009752 0.006939 0.013797

0.031342 0.038968 0.010498 0.007487 0.014936

0.031152 0.038731 0.010422 0.007432 0.014840

8.48 7.69 7.65 7.89 8.26

7.83 7.04 6.87 7.10 7.56

Avg. deviation %Deviation between Theoretical and RSM (Quadrilateral): 7.994% Avg. deviation %Deviation between Theoretical and RSM (Cubic): 7.280%

conditions are evaluated and presented in Tables 11, 12, and 13. The results indicate that the RSM quadratic model predicting the values of logarithmic damping decrement has good validity with acceptable percentage deviation of 8.86, 9.38 and 10.53% for fixed-free, fixed-hinged and fixedfixed beams respectively. Moreover, the average percentage of deviation between the RSM results and theoretical values of logarithmic damping decrement for mild steel, copper and aluminium specimens have been evaluated and presented in Tables 11, 12, 13, 14 and 15. The results indicate that both the values are close to each other with 8.86, 7.99, and 10.04% deviation for mild steel, copper and aluminium beams respectively, thereby authenticating the accuracy of the analysis. Cubic model The average percentage of deviation between the RSM cubic results and theoretical values of logarithmic damping decrement for various configurations and loading conditions are evaluated and presented in Table 11, 12, and 13. The results indicate that the RSM cubic model predicting

the values of logarithmic damping decrement has good validity with acceptable percentage deviation of 7.92, 8.2 and 8.39% for fixed-free, fixed-hinged and fixed-fixed beams respectively. Moreover, the average percentage of deviation between the RSM results and theoretical values of logarithmic damping decrement for mild steel, copper and aluminium specimens have been evaluated and presented in Tables 11, 12, 13, 14 and 15. The results indicate that both the values are close to each other with 7.92, 7.28, and 8.92% deviation for mild steel, copper and aluminium beams respectively, thereby authenticating the accuracy of the analysis. The average percentage of deviation between the RSM cubic results and theoretical values of logarithmic damping decrement for various configurations and loading conditions is less than that of the quadratic models thus establishing that the cubic models are more appropriate for the estimation of logarithmic damping decrement in layered and welded structures. It is observed that the logarithmic damping decrement is maximum in case of aluminium and minimum in mild steel beams. The present model establishes the relationship

Table 15 Comparison of the theoretical and RSM results for logarithmic damping decrement of aluminium specimens with fixed-free end condition Parameters N

15 20 18 12 10

Logarithmic damping decrement

a(mm) Ra(μm) Theoretical RSM (Quadrilateral) RSM (Cubic) %Deviation between Theoretical %Deviation between Theoretical and RSM (Quadrilateral) and RSM (cubic) 0.1 0.3 0.1 0.4 0.5

1.87 0.92 2.66 2.87 1.78

0.049562 0.005738 0.015444 0.010148 0.021712

0.054293 0.006266 0.017074 0.011254 0.023884

0.053972 0.006211 0.016846 0.011083 0.023702

Avg. deviation %Deviation between Theoretical and RSM (Quadrilateral): 10.038% Avg. deviation %Deviation between Theoretical and RSM (Cubic): 8.918%

9.55 9.20 10.55 10.89 10.00

8.89 8.24 9.08 9.21 9.17

790

between the logarithmic damping decrement and number of tack welded joints in structures of various materials with different end conditions vibrating at various amplitudes of excitation. It is evident from this analysis that an increase in number of joints and initial amplitude of transverse excitation reduces the damping capacity. On comparing the results of the previous works on bolted and riveted structures with welded ones, it is established that the bolted and welded joints contribute maximum and minimum damping to the system respectively.

Conclusion This work presents the findings of both theoretical and experimental investigation considering the effects of number of tack joints, initial amplitude of excitation and surface roughness on the logarithmic damping decrement of layered and jointed welded structures of various materials with different configurations of the beam. In this study, a theoretical model has been developed to estimate the logarithmic damping decrement of layered welded beams. Further, the design of experiments approach has been employed to develop second and third order polynomial expressions for predicting the values of logarithmic damping decrement of layered and tack welded structures. It is concluded from the analysis of variance (ANOVA) and various statistical tests that the cubic model is more accurate and statistically significant as compared to the quadratic one. The statistically designed experiments have been performed to estimate the coefficients of the mathematical models for predicting the response and check the adequacy of the models. The RSM approach provides an effective tool for a wide range of information on the interrelationships of control and response variables with a relatively small number of test runs. The relationship between the logarithmic damping decrement with interaction parameters has been successfully obtained using RSM at 95% confidence level. Further, the response regression and variance analysis of the second and third order model for the response shows that the surface roughness is statistically insignificant and the logarithmic damping decrement is constant for a jointed interface of same material irrespective of surface condition. It is concluded from the analysis of variance (ANOVA) that the logarithmic damping decrement decreases with an increase in amplitude of excitation and number of tack joints. The results of theoretical model are compared with that of the RSM models and are found to be in good agreement. This analysis can be utilized effectively to evaluate the logarithmic damping decrement of layered and welded structures

Exp Mech (2012) 52:771–791

with various materials and configurations in which the slip damping is predominant.

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