Vietnam Journal of Mechanics, VAST, Vol 41, No (2019), pp 349 – 361 DOI: https://doi.org/10.15625/0866-7136/14566 EXACT RECEPTANCE FUNCTION AND RECEPTANCE CURVATURE OF A CLAMPED-CLAMPED CONTINUOUS CRACKED BEAM Nguyen Viet Khoa∗ , Cao Van Mai, Dao Thi Bich Thao Institute of Mechanics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam ∗ E-mail: nvkhoa@imech.vast.vn Received: 21 October 2019 / Published online: 24 December 2019 Abstract The receptance function has been studied and applied widely since it interrelates the harmonic excitation and the response of a structure in the frequency domain This paper presents the derivation of the exact receptance function of continuous cracked beams and its application for crack detection The receptance curvature is defined as the second derivative of the receptance The influence of the crack on the receptance and receptance curvature is investigated It is concluded that when there are cracks the receptance curvature has sharp changes at the crack positions This can be applied for the crack detection purpose In this paper, the numerical simulations are provided Keywords: receptance, curvature of receptance, frequency response function, crack, crack detection INTRODUCTION The receptance method which was first introduced by Bishop and Johnson [1] has been applied wildly in mechanical system and structural dynamics Yang [2] presented the exact receptances of non-proportionally damped dynamic systems Based on a decomposition of the damping matrix, an iteration procedure is developed which does not require matrix inversion Mottershead [3] investigated the measured zeros form frequency response functions and its application to model assessment and updating Gurgoze [4] was concerned with receptance matrices of viscously damped systems subject to several constraint equations The frequency response matrix of the constrained system was established in terms of the frequency response matrix of the unconstrained system ă oze ¨ [5] preand the coefficient vectors of the constraint equations Karakas and Gurg sented a formulation of the receptance matrix of non-proportionally damped dynamic systems The receptance matrix was obtained directly without using the iterations as presented in [1] Albertelli et al [6] proposed a method using receptance coupling substructure method to improve chatter free cutting conditions prediction Recently, Muscolino and Santoro [7] presented the explicit frequency response function of beams with c 2019 Vietnam Academy of Science and Technology 350 Nguyen Viet Khoa, Cao Van Mai, Dao Thi Bich Thao cracks of uncertain depths in order to evaluate the main statistics as well as the upper and lower bounds of the response The cracks can influence significantly the dynamic characteristics of structures such as natural frequencies, mode shapes, etc These dynamic characteristics have been investigated and applied wildly for crack detection of structures Lee and Chung [8] studied the change in natural frequencies of beams caused by a crack Zheng and Kessissoglou [9] investigated the relationship between natural frequency of a cracked beam to the depth and location of the crack The results in these papers showed that the natural frequency of the cracked beam decreases as the crack depth increases Gudmundson [10] investigated the influence of cracks on the natural frequencies of slender structures using a flexibility matrix approach Thalapil and Maiti [11] revealed the change in natural frequencies caused by longitudinal cracks for crack detection Khoa [12] proposed a method for monitoring a sudden crack of a beam-like bridge appeared during earthquake excitation based on the instantaneous frequency extracted from wavelet power spectrum Some authors presented methods to calculate and apply the mode shape of cracked structures for crack detection purposes Caddemi and Calio [13, 14] presented the exact closed-form solution for the mode shapes of the Euler-Bernoulli beam with multiple open cracks Lien et al [15] presented a mode shape analysis of multiple cracked functionally graded beamlike structures by using dynamic stiffness method for crack detection purpose One of authors of this paper applied 3D finite elements to investigate the change in mode shapes at the crack positions [16] The results showed that the sharp changes in mode shapes at the crack positions can be applied for detecting small cracks In most of the previous works the receptance of beams was derived discretely In this paper the exact formulas of receptance function and receptance curvature of a cracked beam will be established The receptance curvature is defined as the second derivative of the receptance with respect to the coordinate of beam The effect of the crack on the receptance curvature of cracked beams is investigated The result showed that the receptance curvature has significant changes at crack positions This result can be used for crack detection The numerical simulations are provided in this paper DERIVATION OF THE RECEPTANCE FUNCTION OF A CRACKED BEAM 2.1 Intact beam In this work, the undamped Euler-Bernoulli beam is considered The forced vibration equation of undamped beam can be written as follows ∂2 v (ξ, t) ∂2 v (ξ, t) ∂2 EI ξ + L m ξ = L4 P (ξ, t) , ( ) ( ) ∂ξ ∂ξ ∂t2 where ξ = the form (1) x is the non-dimensional coordinate The solution of Eq (1) can be found in L ∞ v (ξ, t) = ∑ φk (ξ ) Yk (t), k =1 (2) Exact receptance function and receptance curvature of a clamped-clamped continuous cracked beam 351 where φk ( x ) is the kth mode shape of the beam, Yk (t) is the time-dependent amplitude which is referred to as generalized coordinate Substituting Eq (2) into Eq (1), multiplying both sides of the equation with φn ( x ) integrating and applying orthogonality conditions of beam gives Yăn (t) m ( ) n2 (ξ ) dξ + Yn (t) ωn2 m (ξ ) φn2 (ξ ) dξ = φn (ξ ) P (ξ, t) dξ (3) ˆ Eq (3) becomes If the force P excites at a point ξ = , 1 Yăn (t) m ( ) n2 (ξ ) dξ ˆ t m (ξ ) φn2 (ξ ) dξ = φn ξˆ P ξ, + Yn (t) ωn2 (4) ˆ t) = Pˆ sin ωt exciting at the point ξ, ˆ then we have Yn (t) = If the force is harmonic P(ξ, Y¯n sin ωt, where Y¯n is the amplitude Multiply both sides of Eq (4) by φn (ξ ), yields ωn2 −ω ˆ m (ξ ) φn2 (ξ ) dξ = φn (ξ ) φn ξˆ P Y¯n φn (ξ ) (5) From Eq (5) the following formula is derived ∞ ∞ Y¯n (ω ) φn (ξ ) = ∑ P n =1 n =1 φn (ξ ) φn ξˆ ∑ (6) m (ξ ) φn2 (ξ ) dξ (ωn2 − ω ) Therefore, the receptance at ξ due to the force at ξˆ is ∞ αξ ξˆ (ω ) = φn (ξ ) φn ξˆ ∑ ω2 − ω2 n =1 n (7) m (ξ ) φn2 (ξ ) dξ 2.2 Cracked beam Although in general the change in mode shapes caused by the crack at the crack position is small when the crack depth is small, the curvature of the mode shape at the crack position can be significant since the mode shape is changed sharply at the crack position In this paper define the “receptance curvature” as the second derivative of the receptance function with respect to ξ variable as follows ∂2 αξ ξˆ (ω ) ∂ξ ∞ = − ω2 ω n =1 n φn ξˆ ∑ φn2 (ξ ) dξ m d2 φn (ξ ) dξ (8) 352 Nguyen Viet Khoa, Cao Van Mai, Dao Thi Bich Thao Here we consider the elementary case with m( x ) set equal to constant m The exact closed form of the mode shape of a clamped-clamped beam with n cracks is adopted from [13] as follows 2αk φk (ξ ) = C1 2αk + ∑ λi µi [sin αk (ξ − ξ 0i ) + sinh αk (ξ − ξ 0i )] U (ξ − ξ 0i ) + sin αk ξ i =1 n ∑ λi vi [sin αk (ξ − ξ 0i ) + sinh αk (ξ − ξ 0i )] U (ξ − ξ 0i ) + cos αk ξ i =1 2αk − C1 2αk − n n ∑ λ j ζ i [sin αk (ξ − ξ 0i ) + sinh αk (ξ − ξ si )] U (ξ − ξ 0i ) + sinh αk ξ i =1 n ∑ λi ηi [sin αk (ξ − ξ 0i ) + sinh αk (ξ − ξ 0i )] U (ξ − ξ 0i ) + cosh αk ξ , i =1 (9) where C1 = − 2αk 2αk n ∑ λi (νi − ηi ) [sin αk (1 − ξ oi ) + sinh αk (1 − ξ oi )] + cos αk − cosh αk i =1 n , ∑ λi (µi − ζ i ) [sin αk (1 − ξ oi ) + sinh αk (1 − ξ oi )] + sin αk − sinh αk (10) i =1 C2 = −C4 = 1, C3 = −C1 , ωk2 mL4 ; ξ 0i is the position of the ith EI crack, where < ξ 01 < ξ 02 < < ξ 0n < 1; U is Heaviside function The terms µi , νi , ζ i , η i are calculated recurrently by the following equations αk is the dimensionless frequency parameter α4k = µj = vj = α α α ζj = α ηj = j −1 ∑ λi µi − sin α ξ 0j − ξ 0i + sinh αk ξ 0j − ξ 0i − α2 sin αξ 0j , − sin α ξ 0j − ξ 0i + sinh αk ξ 0j − ξ 0i − α2 cos αξ 0j , i =1 j −1 ∑ λi vi i =1 (11) j −1 ∑ λi si − sin α ξ 0j − ξ 0i + sinh αk ξ 0j − ξ 0i + α sinh αξ 0j , − sin α ξ 0j − ξ 0i + sinh αk ξ 0j − ξ 0i + α2 cosh αξ 0j i =1 j −1 ∑ λ i ηi i =1 In order to derive the exact formulas of receptance and curvature receptance as presented in Eqs (9) and (10), the second derivative of the mode shape and the integral of the square of the mode shape need to be calculated For simplicity, the operator S (αk , ξ ) = sin αk (ξ − ξ 0i ) + sinh αk (ξ − ξ 0i ) is presented The second derivative of the mode shape with respected to ξ can be obtained as follows (SU ) = S U + SU + 2S U (12) Exact receptance function and receptance curvature of a clamped-clamped continuous cracked beam 353 Applying the following properties of Heaviside function and Dirac delta function [17] U (ξ ) = δ (ξ ) , f (ξ ) δ (ξ ) = − f (ξ ) δ (ξ ) , (13) (SU ) = α2k [sinh αk (ξ − ξ 0i ) − sin αk (ξ − ξ 0i )] U (ξ − ξ 0i ) + αk [cos αk (ξ − ξ 0i ) + cosh αk (ξ − ξ 0i )] δ (ξ − ξ 0i ) (14) yields From Eq (9) and Eq (14), the second derivative of the mode shape can be derived as follows φk (ξ ) =C1 n λi µi [αk (sinh αk (ξ − ξ 0i ) − sin αk (ξ − ξ 0i )) U (ξ − ξ 0i ) i∑ =1 + (cos αk (ξ − ξ 0i ) + cosh αk (ξ − ξ 0i )) δ (ξ − ξ 0i )] − α2k sin αk ξ + n λi vi [αk (sinh αk (ξ − ξ 0i ) − sin αk (ξ − ξ 0i )) U (ξ − ξ 0i ) i∑ =1 + (cos αk (ξ − ξ 0i ) + cosh αk (ξ − ξ 0i )) δ (ξ − ξ 0i )] − α2k cos αk ξ n λi ζ i [αk (sinh αk (ξ − ξ 0i ) − sin αk (ξ − ξ 0i )) U (ξ − ξ 0i ) i∑ =1 − C1 (15) + (cos αk (ξ − ξ 0i ) + cosh αk (ξ − ξ 0i )) δ (ξ − ξ 0i )] + α2k sinh αk ξ − n λi ηi [αk (sinh αk (ξ − ξ 0i ) − sin αk (ξ − ξ 0i )) U (ξ − ξ 0i ) i∑ =1 + (cos αk (ξ − ξ 0i ) + cosh αk (ξ − ξ 0i )) δ (ξ − ξ 0i )] + α2k cosh αk ξ Applying the property of Heaviside function, we have 1 f (ξ ) U (ξ − ξ 0i ) dξ = f (ξ ) dξ = F (1) − F (ξ 0i ) , (16) ξ 0i where F is the antiderivative function of f It is noted that U (ξ − ξ 0i ) U ξ − ξ 0j = U (ξ − ξ 0i ) , i ≥ j U ξ − ξ 0j , i < j (17) From Eqs (16) and (17) we have f (ξ ) U (ξ − ξ 0i ) U ξ − ξ 0j dξ = F (1) − F (ξ 0i ) U ξ 0i − ξ 0j − F ξ 0j U ξ oj − ξ 0i + F (ξ 0i ) δij where δij is the Kronecker delta (18) 354 Nguyen Viet Khoa, Cao Van Mai, Dao Thi Bich Thao Analytical calculations show that the term F (ξ 0i ) δij in Eq (18) vanishes From Eqs (9) to (18), the following equation is obtained φk2 (ξ )dξ = n n sin αk − ξ 0i − ξ 0j λi λ j A1 × cos αk ξ 0i − ξ 0j − ∑ ∑ 2α 8αk i=1 j=1 k sinh αk (1 − ξ 0i ) cosh αk − ξ 0j αk 1 − cos αk (1 − ξ 0i ) sinh αk − ξ 0j + sin αk (1 − ξ 0i ) cosh αk − ξ 0j αk αk 1 − cos αk − ξ 0j sinh αk (1 − ξ 0i ) + sin αk − ξ 0j cosh αk (1 − ξ 0i ) αk αk − ξ 0i cos αk ξ 0i − ξ 0j + sin αk ξ 0i − ξ 0j − ξ 0i cosh αk ξ 0i − ξ 0j 2αk − cosh αk ξ 0i − ξ 0j + − sinh αk ξ 0i − ξ 0j αk U ξ 0i − ξ 0j − ξ 0j cos αk ξ 0j − ξ 0i + sin αk (ξ 0i − ξ 0i ) − ξ 0j cosh αk ξ 0j − ξ 0i 2αk + 1 sin αk ξ 0i − sin αk (2 − ξ 0i ) (1 − ξ 0i ) cos αk ξ 0i − 2αk ∑in=1 λi A2 2αk 2αk + 1 sin αk cosh αk (1 − ξ 0i ) − cos αk sinh αk (1 − ξ 0i ) αk αk U ξ 0j − ξ 0i n 1 λi A3 (ξ 0i − 1) sin αk ξ 0i − cos αk ξ 0i − cos αk (2 − ξ 0i ) 2αk i∑ 2α 2α k k =1 1 cos αk cosh αk (1 − ξ 0i ) + sin αk sinh αk (1 − ξ 0i ) + αk αk + n 1 λi A4 (ξ 0i − 1) cosh αk ξ 0i + sinh αk ξ 0i + sinh αk (2 − ξ 0i ) 2αk i∑ 2α 2α k k =1 1 cos αk (1 − ξ 0i ) sinh αk + sin αk (1 − ξ 0i ) cosh αk − αk αk + n 1 λi A5 (ξ 0i − 1) sinh αk ξ 0i + cosh αk ξ 0i + cosh αk (2 − ξ 0i ) 2αk i∑ 2α 2a k k =1 1 + sin αk (1 − ξ 0i ) sinh αk − cosαk (1 − ξ 0i ) cosh αk αk αk + + 1+ − − C12 C2 + C C sin 2αk + sinh 2αk + sin2 αk + sinh2 αk 4αk 4αk αk αk C2 − C12 + 2C1 sin αk cosh αk + cos αk sinh αk − sin αk sinh αk αk αk αk + C1 C2 sin2 αk sinh2 αk sin 2αk sinh 2αk + C3 C4 + C22 − C12 + C32 + C42 αk αk 4αk 4αk sin αk cosh αk cos αk sinh αk sin αk sinh αk + (C2 C4 − C1 C3 ) + (C1 C4 + C2 C3 ) αk αk αk cos αk cosh αk 1 + (C2 C3 − C1 C4 ) + C12 + C22 − C32 + C42 + (C C − C2 C3 ) , αk αk + (C1 C3 + C2 C4 ) (19) Exact receptance function and receptance curvature of a clamped-clamped continuous cracked beam 355 where A1 = C12 µi µ j + νi νj + C12 ζ i ζ j + ηi η j + 2C1 µi νj − 2C12 µi ζ j − 2C1 µi η j − 2C1 νi ζ j − 2νi η j + 2C1 ζ i η j , A2 = C12 µi + C1 νi − C12 ζ i − C1 ηi , A4 = −C12 µi − C1 νi + C12 ζ i + C1 ηi , A3 = C1 µi + νi − C1 ζ i − ηi , A5 = −C1 µi − νi + C1 ζ i + ηi (20) Substituting Eqs (15) to (20) into Eqs (7) and (8) the exact receptance and curvature receptance of the simply supported beam will be obtained Formulas of the receptance and receptance curvature of beam with other general boundary conditions can be obtained by the same procedure as the mode shapes of beams with general boundary conditions have been reported in [13] NUMERICAL SIMULATION AND DISCUSSIONS Numerical simulations of a clampedTable Five cases with cracks clamped beam with two cracks is presented in of varying depths this section Parameters of the beam are: Mass density ρ = 7800 kg/m3 ; modulus of elasticity Case Crack depth (%) E = 2.0 × 1011 N/m2 ; L = m; b = 0.02 m; h = 0.02 m Two cracks with the same depths are 10 made at arbitrary positions of 0.4L and 0.76L 20 from the left end of the beam Five levels of 30 the crack depth ranging from 0% to 20% have been applied These five cases are numbered in Tab The first ten mode shapes are used to calculate the receptance and receptance curvature The receptance and receptance curvature matrices are calculated at 100 points spaced equally on the beam while the force moves along these points The Dirac delta function is approximated by the following formula [17] ∆ξ 0ξ > ∆ξ ∆ξ − ≤x≤ (21) δ(ξ ) = ∆ξ 2 0ξ < − ∆ξ The value of the damage parameter λi is determined as follows [12] λi = where C ( β) = h C ( β ), L β (2 − β ) d and β = with d is the crack depth 0.9( β − 1) h (22) Table Five cases with cracks of varying Dx depths Case ï0 ỵ x