Theoretical and experimental analysis of the exact receptance function of a clamped clamped beam with concentrated masses

Vietnam Journal of Mechanics, VAST, Vol 42, No (2020), pp 29 – 42 DOI: https://doi.org/10.15625/0866-7136/14628 THEORETICAL AND EXPERIMENTAL ANALYSIS OF THE EXACT RECEPTANCE FUNCTION OF A CLAMPED-CLAMPED BEAM WITH CONCENTRATED MASSES Nguyen Viet Khoa1,∗ , Dao Thi Bich Thao1 Institute of Mechanics, VAST, Hanoi, Vietnam E-mail: nvkhoa@imech.vast.vn Received: 13 November 2019 / Published online: 01 March 2020 Abstract This paper establishes the exact receptance function of a clamped-clamped beam carrying concentrated masses The derivation of exact receptance and the numerical simulations are provided The proposed receptance function can be used as a convenient tool for predicting the dynamic response at arbitrary point of the beam acted by a harmonic force applied at arbitrary point The influence of the concentrated masses on the receptance is investigated The numerical simulations show that peak in the receptance will decrease when there is a mass located close to that peak position The numerical results have been compared to the experimental results to justify the theory Keywords: receptance, frequency response function, concentrated mass INTRODUCTION The receptance function is very important in vibration problems such as control design, system identification or damage detection since it interrelates the harmonic excitation and the response of a structure in the frequency domain The receptance method was first introduced by Bishop and Johnson [1] This method has been developed and applied widely in mechanical systems and structural dynamics Milne [2] proposed a general solution of the receptance function of uniform beams which can be applied for all combinations of beam end conditions Yang [3] derived the exact receptances of nonproportionally damped dynamic systems In this work an iteration procedure is developed based on a decomposition of the damping matrix, which does not require matrix inversion and eliminate the error caused by the undamped model data Lin and Lim [4] proposed the receptance sensitivity with respect to mass modification and stiffness modification from the limited vibration test data Mottershead [5] investigate the measured zeros from frequency response functions and its application to model assessment and updating Gurgoze [6] presented the receptance matrices of viscously damped systems c 2020 Vietnam Academy of Science and Technology 30 Nguyen Viet Khoa, Dao Thi Bich Thao subject to several constraint equations In this paper, the frequency response matrix of the unconstrained system and the coefficient vectors of the constraint equations was used to ă oze ă and Erol [7] obtain the frequency response matrix of the constrained system Gurg established the frequency response function of a damped cantilever simply supported beam carrying a tip mass In this paper, the frequency response function was derived by using a formula established for the receptance matrix of discrete linear systems subjected to linear constraint equations, in which the simple support was considered as a linear constraint imposed on generalized co-ordinates Burlon et al [8] derived an exact frequency response function of axially loaded beams with viscoelastic dampers The method relies on the theory of generalized functions to handle the discontinuities of the response variables, within a standard 1D formulation of the equation of motion In another work, Burlon et al [9] presented an exact frequency response of two-node coupled ă oze ă [10] extended bending-torsional beam element with attachments Karakas and Gurg the work in [3] in which the receptance matrix was obtained directly without using the iterations as presented in [3] to form the receptance matrix of non-proportionally damped dynamic systems Muscolino and Santoro [11] developed the explicit frequency response functions of discretized structures with uncertain parameters Recently, the authors of this paper [12] presented the exact formula of the receptance function of a cracked beam and its application for crack detection However, the exact form of frequency response function of a beam with concentrated masses has not been established yet The aim of the present paper is to present an exact receptance function of a clampedclamped beam carrying an arbitrary number of concentrated masses The proposed formula of receptance function is simple and can be applied easily to investigate the dynamic response of beam at an arbitrary point under a harmonic force applied at any point along the beam The influence of concentrated masses on the receptance of the clamped-clamped beam is investigated The comparison between numerical simulations and experimental results have been carried out to justify the proposed method THEORETICAL BACKGROUND Considering the Euler–Bernoulli beam carrying concentrated masses subjected to a force as shown in Fig 1, the governing bending motion equation of the beam can be extended from [13] as follows n EIy + m+ ∑ mk δ ( x xk ) yă = x x f f (t) , (1) k =1 where E is the Young’s modulus, I is the moment of inertia of the cross sectional area of the beam, µ is the mass density per unit length, mk is the kth concentrated mass located at xk , y( x, t) is the bending deflection of the beam at location x and time t, f (t) is the force acting at position x f , δ x − x f is the Dirac delta function Symbols “ ” and “ ˙ ” denote differentials with respect to x and t, respectively Eq (1) can be rewritten in the form n EIy + myă = δ x − x f f (t) − ∑ mk (x xk )y.ă k =1 (2) The aim of the present paper is to present an exact receptance function of a clamped-clamped beam carrying an arbitrary number of concentrated masses The proposed formula of receptance function is simple and can be applied easily to investigate the dynamic response of beam at an arbitrary point under a harmonic force applied at any point along the beam The influence of concentrated masses on the receptance of the clamped-clamped beam is investigated The comparison between numerical simulations and experimental results have been carried out to justify the proposed method Theoretical and experimental analysis of the exact receptance function of a clamped-clamped beam with concentrated masses Theoretical background 31 f(t) y m1 x m2 … mn xk xf Fig 1.1.AAclamped-clamped withconcentrated concentratedmasses masses Fig clamped-clamped beam beam with Considering the Euler-Bernoulli beam carrying concentrated masses subjected to a force as Eq (2) can be considered as the equation of forced vibration of a beam without conshown in Fig 1, the governing bending motion equation of the beam can be extended from [13] centrated masses which is acted by the inertia forces of n concentrated masses and the as follows: external force f (t) The solution of Eq (2) can be expressed in the form n ∞ é ù EIy¢¢¢¢ + ê m + å mk d ( x - xk )yú(!!yx,=t)d=( x∑ - xφfi )( xf )(qti)(t), k =1 ë û i =1 (1) (3) th th where is the i mode shape of without concentrated masses and qi is theof i the where E isφithe Young’s modulus, I isthethebeam moment of inertia of the cross sectional area th generalized coordinate beam, μ is the mass density per unit length, mk is the k concentrated mass located at xk, y(x, t) Substituting (3) into (2), beam yieldsat location x and time t, f(t) is the force acting at position is the bending deflection of the ∞ ∞ n ∞ xf, dEI( x - φx f ) is( xthe δ (x − x ) m ( x ) qă (t) + δ x − x f (t) ) q Dirac (t) + mdeltafunction ( x ) qă (t) = ∑ i ∑ i i =1 i ∑ i i =1 k Eq (1) can be rewritten in the form: k ∑ i i f i =1 k =1 (4) Multiplying Eq (4) by φj ( x ) and n integrating from to L and considering the defini¢¢¢¢ +Dirac tion of function, (2) !! =delta EIythe my d xx f f ( t )one - obtains mk d ( x - xk )!!y ( L ) å k =1 ∞ L ∞ EI ∑ φas φj ( x ) qi (of t)dx + vibration m ∑ φi ( xof x ) qăi (twithout ( x )equation ) aj (beam )dx Eq (2) can be considered forced concentrated i the i = i = masses which is acted by the inertia forces of n concentrated masses and the external force 0 (5)f(t) n be expressed in the form: The solution of Eq (2) can ¥ = − ∑ mk φi ( xk ) φj ( xk ) qăi (t) + j x f f (t) (3) yThe = å fi ( x ) qi (of t ) the normal mode shapes of the beam without concentrated ( x, t )orthogonality i =1 k =1 masses can be addressed here L of the beam without concentrated masses and qi is the ith where fi is the ith mode shape generalized coordinate φi ( x ) EIφi ( x ) dx = if i = j (6) Substituting (3) into (2), yields: L φi ( x ) mφj ( x ) dx = 0 if i = j (7) 32 Nguyen Viet Khoa, Dao Thi Bich Thao Integrating the first equation in Eq (6) twice by parts, yields L L φi ( x ) EIφj ( x ) − φ i ( x ) EIφj ( x ) L + φ i ( x ) EIφ j ( x ) dx = if i = j (8) For general boundary conditions the first two terms in Eq (8) vanish Thus, from Eq (8) we have  if i = j L  L φ i ( x ) EIφ j ( x ) dx =  (9)  φi ( x ) EIdx if i = j  Applying Eqs (6)–(9), Eq (5) can be rewritten as  L mk i2 (xk ) qăi (t) + EI φi ( x ) dx + m k =1  L n φi ( x )dx  qi (t) = φj x f f (t) (10) By introducing notations L  n        M = m        φ1 ( x ) dx + ∑ ¯ k φ12 m ∑ m¯ k φ1 (xk ) φ2 (xk ) ( xk ) ∑ m¯ k φ1 (xk ) φN (xk )      n   ∑ m¯ k φ2 (xk ) φN (xk ) ,  k =1     L n  2 ¯ k φN ( xk )  φN ( x ) dx + ∑ m k =1 k =1 k =1 L n ∑ m¯ k φ2 (xk ) φ1 (xk ) φ2 ( x ) dx + k =1 n ∑ m¯ k φ22 (xk ) k =1 n n ∑ m¯ k φN (xk ) φ2 (xk ) ∑ m¯ k φN (xk ) φ1 (xk )         K = EI         k =1 k =1  n n k =1  L φ ( x )dx 0        ,     L   φ N ( x )dx  L φ2 ( x )dx 0 0 T Φ ( x ) = [φ1 ( x ) , , N ( x )] , qă (t) = [qă1 (t) , qă2 (t) , , qă N (t)] T , m ¯k = k q (t) = [q1 (t) , q2 (t) , , q N (t)] T , m m Eq (10) can be expressed in matrix form as follows Mqă (t) + Kq (t) = Φ x f f (t) (11) Theoretical and experimental analysis of the exact receptance function of a clamped-clamped beam with concentrated masses 33 The natural frequency of beam carrying concentrated masses can be obtained by solving the eigenvalue problem associated with Eq (11), that is det K − ω M = (12) If the force is harmonic f (t) = f¯eiωt then the solution of Eq (11) can be found in the form q (t) = qe ¯ iωt (13) Substituting Eq (13) into Eq (11) yields K − ω M q¯ = Φ x f f¯ (14) The receptance function is defined as the frequency response function in which the response is the displacement This means that in the frequency domain: receptance = Φ T (ξ ) −1 K − ω2 M the redisplacement/force Thus, left multiplying Eq (14) with ¯f ceptance at x due to the force at x f is obtained α x, x f , ω = Φ T ( x ) q¯ = ΦT ( x) K − ω2 M f¯ −1 Φ xf (15) It is noted that when infinite modes are applied, i.e N → ∞, Eq (15) becomes the exact formula of the receptance function For the clamped-clamped beam, following relations can be derived: sin αi L + sinh αi L (sin αi x − sinh αi x ) + cos αi x − cosh αi x, cos αi L − cosh αi L sin αi L + sinh αi L φ i ( x ) = −α2i (sin αi x + sinh αi x ) + cos αi x + cosh αi x , cos αi L − cosh αi L φi ( x ) = L L φi2 (16) ( x ) dx = L, φ i ( x ) dx = Lα4i , where αi is the solution of the frequency equation cos αL cosh αL − = From Eq (16) the matrices M and K are derived    α1 L + β 11 β 12 β 1N  β 21   L + β 22 β 2N  , K = EIL  α2 M = m    β N1 L + β NN α4N    , (17)  where n β ij = ∑ k =1 ¯k m sinαi L + sinhαi L (sinαi xk − sinhαi xk ) + cosαi xk − coshαi xk cosαi L − coshαi L sinα j L + sinhα j L × sinα j xk − sinhα j xk + cosα j xk − coshα j xk cosα j L − coshα j L (18) 34 Nguyen Viet Khoa, Dao Thi Bich Thao The exact formula of the receptance of the clamped-clamped beam carrying concentrated masses will be derived from Eqs (16)–(18) NUMERICAL SIMULATION 3.1 Reliability of the theory In order to check the reliability of the proposed receptance, frequency parameters αi L of a clamped-clamped beam carrying two masses are calculated from Eq (12) and compared to Ref [14] Five lowest frequency parameters of the clamped-clamped beam ¯1 = m ¯ = 0.5 attached at 0.25L and 0.75L obtained by with two concentrated masses m two methods are listed in Tab As can be seen from this table, the first five frequency parameters of the present work are in excellent agreement with Ref [14] This result justifies the reliability of the proposed receptance function Table Frequency parameters of the clamped-clamped beam Frequency parameters Ref [14] Present paper Error (%) α1 L α2 L α3 L α4 L α5 L 4.0973 5.8984 9.1453 13.7527 16.9258 4.0976 5.8995 9.1534 13.7567 16.9399 0.00007 0.00019 0.00089 0.00029 0.00083 3.2 Influence of location of the concentrated masses on the receptance In this paper, the numerical simulations of a clamped-clamped beam with two masses are presented Parameters of the beam are: Mass density ρ = 7800 kg/m3 ; modulus of elasticity E = 2.0 × 1011 N/m2 ; L = m; b = 0.02 m; h = 0.01 m Two equal concentrated masses of 0.6 kg are attached on the beam in different scenarios The first five mode shapes are used to calculate the receptance The receptance matrices are calculated at 50 points spaced equally on the beam while the force moves along these points The receptance of the clamped-clamped beam without masses is calculated first Fig presents the receptance matrices when the forcing frequencies equal to the first, second and third natural frequencies of the beam-mass system, respectively As can be seen from Fig 2(a) when the forcing frequency is equal to the first natural frequency, the receptance is maximum at the middle of the beam which corresponds to the position where the amplitude of the first mode is maximum As can be observed from Fig 2(b) that when the forcing frequency is equal to the second natural frequency, the receptance is maximum at position of about 0.3L and 0.7L from the left end of the beam which are the positions where the amplitude of the second mode shape is maximum Meanwhile, the receptance is smallest at the middle of beam which corresponds to the position where the amplitude of the second mode shape is minimum Fig 2(c) presents the receptance matrix of the beam when the frequency of the force is equal to the third natural frequency The receptance of the beam is maximum at the positions of about 0.2L, 0.5L and positions of maxima and minima in the receptance are the same with the positio middleisofequal the beam corresponds to thethe position whereis the amplitude of the first mode is cing frequency to thewhich first natural frequency, receptance maximum at the and minima in the corresponding mode shape Therefore, similar to the mode sha maximum As can be observed from Fig 1b that when the forcing frequency is equal to the ddle of the beam which corresponds to the positionmaxima where the amplitude of the first mode is in the receptance “peaks of receptance” and the minima in the recepta second natural frequency, the1b receptance isthe maximum at positionis of about 0.3L and 0.7L from ximum As can be observed from Fig that when forcing frequency equal to the receptance” thefrequency, left end ofthe thereceptance beam which are the positions where the amplitude of thefrom second mode shape cond natural is maximum at position of about 0.3L and 0.7L left end of the beam which are the positions where the the middle second mode shape is maximum Meanwhile, the receptance is amplitude smallest atofthe of beam which corresponds maximum.toMeanwhile, receptance is smallest at the middle of beam which corresponds the positionthe where the amplitude of the second mode shape is minimum Fig 1c presents the the position where thematrix amplitude of the second mode is minimum 1c is presents receptance of the beam when theshape frequency of the Fig force equal the to the third natural eptance matrix of theThe beam when theoffrequency force is equal thirdofnatural Theoretical and experimental exact receptance function ofpositions a the clamped-clamped beam with concentrated frequency receptance theanalysis beamofofthe isthe maximum at theto about 0.2L, 0.5Lmasses and 35 quency The receptance of the beam is maximum at the positions of about 0.2L, 0.5L and 0.8L where the amplitude of the third mode shape is maximum The receptance is minimum at L where the amplitude of where the 0.35L, third shape isofmaximum The receptance is minimum at is themode amplitude the mode shape is maximum The receptance is minpositions of0.8L about 0.65L where the third amplitude of the third mode shape minimum It sitions ofcan about 0.35L, 0.65L where the amplitude of the third mode shape is minimum It imum at positions of about 0.35L, 0.65L where the amplitude of the third mode shape be concluded that, when the excitation frequency is equal to a natural frequency the is n be concluded that,minimum when the It excitation frequencythat, is equal tothe a natural frequency theis equal to a natural can be concluded whenare excitation frequency positions of maxima and minima in the receptance the same with the positions of maxima sitions of maxima and minima in the receptance are the same with the positions of maxima frequency the positions of maxima and minima in the receptance are the same with the and minima in the corresponding mode shape Therefore, similar to the mode shape, we call the d minima in the corresponding shape.and Therefore, the mode shape, we call the Therefore, similar positions mode of maxima minimasimilar in thetocorresponding mode shape maxima in the receptance “peaks of receptance” and the minima in the receptance “nodes of xima in the receptance “peaks ofshape, receptance” and minimaininthe thereceptance receptance“peaks “nodesofofreceptance” to the mode we call thethe maxima and the b) ω=ω2 eptance”.receptance” minima in the receptance “nodes of receptance” a) ω=ω1 position ofposition 0.4L when the mass located receptance seems to be “pulled’ of 0.4L whenisthe mass at is 0.25L locatedThe at 0.25L The receptance seems to betoward “pulled’ toward the mass position the mass position (a) ω = ω1 (b) ω = ω2 (c) ω = ω3 a) ω=ω b) ω=ω c)positions ω=ω3 when a)receptance ω=ω1the receptance b) ω=ω Fig presents of beam carrying concentrated mass at different Fig 4the presents of beamaof carrying a concentrated Fig Receptance beam without a masse mass at different positions when the forcingthe frequency is equal toisthe second natural frequency As1 shown in Fig when the forcing frequency equal to the second naturalFig frequency As shown in Fig 4a, when the Receptance of4a, beam without a masse mass is located at 0.3L, the peaks corresponding to either the response position of 0.3L or the mass is there located 0.3L, the peaks corresponding to either theofresponse position of 0.3L or the When is at a concentrated mass, the receptance matrix the beam is changed When there is a concentrated mass, the receptance matrix of the beam is changed force position ofposition 0.3L decrease When mass isthe located the middle of the Fig.force presents theofreceptance matrices of thethe beam when the forcing frequency isbeam, equal 0.3Lsignificantly decrease significantly When mass is located the middle of the beam, the receptance matrices of the beam when the forcing frequency is equal to th the receptance shape is unchanged asofshown in Fig 4b.system These results that from whenthis the figure, mass to the natural frequency the beam-mass AsThese canshow beresults seen thefirst receptance shape is unchanged as shown in Fig 4b show that when the mass frequency ofthe thepeaks beam-mass system Aseither can“moves” bethe seen from this figure, when the m is attached a peak receptance matrix, corresponding to when the massof the position of the peak of corresponding receptance to the the response isatattached atisthe a located peak of 0.25L the receptance matrix, the peaks to response either 0.25L the position of the peak of receptance “moves” to the of beam H end of position beam when mass isposition located atposition the middle of the beam, the left end positionleftorposition force which is close tothe the mass will decrease Meanwhile, the or forceHowever, position which is close to the mass will decrease Meanwhile, the the mass isThe located at the middle of the beam, the shape is un shape of the receptance unchanged change of position of thethe peak of shape of receptance is unchanged when the mass isthe attached the nodes receptance Theof the receptance shape of receptance isisunchanged when mass isatpeak attached at of the nodes of receptance the receptance The change of position of the of receptance is depicted clearer in Fig when the clearer when the force ismore fixed at position can be5observed changeisindepicted receptance can in beFig observed in more detail as presented in 0.5L Fig 5As when the force isthe force is change in receptance can be observed in detail as presented in Fig when at position 0.5L As can be observed from this figure, the this figure, the peakbeofseen moves the position of 0.4L when the massIn is peak of receptance fixed atfrom position As 0.25L can this figure, thefigure, peak ofthereceptance decreases fixed at0.25L position Asreceptance canfrom be seen fromtothis peak of receptance decreases In located at 0.25L The receptance seems to be “pulled’ toward the mass position addition, the peak ofthe receptance moves slightly toward thetoward mass position addition, peak of receptance moves slightly the mass position c) ω=ω3 c) ω=ω Fig Receptance of beam without a masse Fig Receptance a masse hen there is a concentrated mass, the receptance matrix ofof thebeam beamwithout is changed Fig presents receptance matrices ofathe beam whenmass, the forcing frequency is equal to beam the first natural Fig presents When there is concentrated the receptance matrix of the is changed quency ofthe thereceptance beam-mass matrices system As can be seen from this figure, when the mass is located of the beam when the forcing frequency is equal to the first natural 5L the position of the peak of receptance “moves” thebe left endfrom of beam However, when frequency of the beam-mass system Astocan seen this figure, when the mass is located mass is located at the middle of the beam, the shape of the receptance is unchanged The 0.25L the position of the peak of receptance “moves” to the left end of beam However, when ange of position of the peak of receptance is depicted clearer in Fig when the force is fixed the mass is located at the middle of the beam, the shape of the receptance is unchanged The position 0.5L As can be observed from this figure, the peak of receptance moves to the change of position of the peak of receptance is depicted clearer in Fig when the force is fixed at position 0.5L As can be observed from this figure, the peak of receptance moves to the is at 0.25L (b) Mass is at 0.5L a) Mass(a)isMass at Mass 0.25L b) Mass isb) at 0.5L a) is at 0.25L Mass is at 0.5L Fig 3.Fig Receptance matrices at ω =1 ωat Fig Receptance matrices atmatrices ω=ω ω=ω1 Receptance a) Mass is at 0.25L 36 b) Mass is at 0.5L Viet Khoa, Dao Thi Bich Thao Fig Nguyen Receptance matrices at ω=ω1 Fig Receptance of beam with force position is at L/2, ω = ω Fig Receptance of beam with force position is at L/2; ω=ω1 Fig presents the receptance of beam carrying a concentrated mass at different positions when the forcing frequency is equal to the second natural frequency As shown in Fig 5(a), when the mass is located at 0.3L, the peaks corresponding to either the response position of 0.3L or the force position of 0.3L decrease significantly When the mass is located the middle of the beam, the receptance shape is unchanged as shown in Fig 5(b) These results show that when the mass is attached at a peak of the receptance matrix, the peaks corresponding to either the response position or force position which is close to the mass position will decrease Meanwhile, the shape of receptance is unchanged when the mass is attached at the nodes of the receptance The change in receptance can be observed in more detail as presented in Fig when the force is fixed at position 0.25L As can be seen from this figure, the peak of receptance decreases In addition, the peak of receptance moves slightly toward the mass position (a) Mass is at is 0.25L (b) Mass is at 0.5L at 0.25L b)atMass a) Massa) is Mass at 0.25L b) Mass is 0.5Lis at 0.5L Fig 5.Fig Receptance of beam at ω =at ω2ω=ω2 Receptance beam Fig Receptance of beam of at ω=ω a) Mass is at 0.25L b) Mass is at 0.5L a) Mass is at 0.25L b) Mass is at 0.5L Theoretical and experimental analysisFig of the4 exact receptance function of a clamped-clamped beam with concentrated masses Receptance of beam at ω=ω Fig Receptance of beam at 2ω=ω a) Mass is at 0.25L b) Mass is at 0.5L2 37 Fig Receptance of beam at ω=ω2 Fig Measured Measured receptance withthe theforce force acting at 0.25L, Measured receptance with theacting force at Fig Fig receptance with atacting 0.25L, ω 0.25L, =ω=ω ω2 ω=ω2 Fig Measured receptance with the force acting at 0.25L, ω=ω2 (a) Mass is at 0.2L a) Mass atis0.2L a)isMass at 0.2L a) Mass at is0.2L (c) Masses are at 0.2L and 0.5L (b) Mass is at 0.5L b) mass isb) at mass 0.5L is at 0.5L b) mass is at 0.5L (d) Masses are at 0.2L and 0.8L 0.2L and 0.5L Masses areand at 0.2L d) Masses d) areMasses at 0.2Lare andat0.5L e) Massese)are at 0.2L 0.8Land 0.8L Fig Normalized receptance at = ω=ω Fig Normalized receptance ω3 Fig Normalized receptance at ω=ωat3 ω change in receptance can seendetail in more detail forceatisposition fixed at position The changeThe in receptance can be seen in be more when the when force the is fixed 0.2L as 0.2L as Fig 7inpresents the receptance ofcan beam when from the forcing frequency is equal to the depicted Fig Similar conclusion be drawn this figure that when depicted in Fig Similar conclusion can be drawn from this figure that when there is athere massis a mass third natural frequency As shown in Fig 7(a), when one mass is located at 0.2L or 0.8L, at a peak peak, will this decrease peak will significantly decrease significantly When there is one attached at attached a peak, this When there is one mass themass peaksthe ofpeaks of move the massWhen position When theremasses are twoattached masses attached symmetrically receptance receptance move toward thetoward mass position there are two symmetrically at 0.2L and 0.8L the peak at 0.2L moves to the left end, while the peak at 0.8L at 0.2L and 0.8L the peak at 0.2L moves to the left end, while the peak at 0.8L moves tomoves the to the right end there are two massesatattached at 0.2L the receptance is “pulled” to right end When thereWhen are two masses attached 0.2L and 0.5L,and the 0.5L, receptance is “pulled” to the left end In this case, the receptance tends to “move” toward the heavier side of the beam the left end In this case, the receptance tends to “move” toward the heavier side of the beam 38 Nguyen Viet Khoa, Dao Thi Bich Thao the peaks corresponding to either the response position of 0.2L or the force positions of 0.2L decrease significantly When one mass is located at 0.5L, the peaks corresponding to either the response position of 0.5L or the force position of 0.5L decrease significantly as shown in Fig 7(b) When two masses are located at 0.2L and 0.5L, the peaks corred)either Masses at 0.2L and 0.5L of 0.2L, 0.5L e) arepositions at 0.2L and 0.8L 0.5L sponding to theare response positions orMasses the force of 0.2L, decrease significantly as depicted in Fig 7(c) When two masses are located at 0.2L and Fig Normalized receptance at ω=ω 0.8L, the peaks corresponding to either the response positions of 0.2L, 0.8L or the force positions 0.8L decrease significantly as shown in Fig The at change in 0.2L recepThe change of in 0.2L, receptance can be seen in more detail when the force7(d) is fixed position as tance can be seen in more detail when the force is fixed at position 0.2L as depicted in depicted in Fig Similar conclusion can be drawn from this figure that when there is a mass Fig Similar conclusion can be drawn from this figure that when the masses attached attached at a peak, this peak will decrease significantly When there is one mass the peaks of at peaks, these peaks will decrease significantly When there is one mass the peaks of receptance move toward the mass position WhenWhen there are twoare masses attachedattached symmetrically receptance move toward the mass position there two masses symatmetrically 0.2L and 0.8L the peak at 0.2L moves to the left end, while the peak at 0.8L moves the at 0.2L and 0.8L the peak at 0.2L moves to the left end, while the peak atto 0.8L right end.toWhen thereend are two masses 0.2L and 0.5L, at the0.2L receptance is “pulled” to moves the right When thereattached are two at masses attached and 0.5L, the receptance is “pulled” to the left end In this case, the receptance tends to “move” toward the the left end In this case, the receptance tends to “move” toward the heavier side of the beam heavier side of the beam Fig Normalized receptance when the force is fixed at 0.2L, ω = ω3 Figure Normalized receptance when the force is fixed at 0.2L, ω=ω3 Experiment results EXPERIMENT RESULTS The experimental setup is illustrated in Fig The clamped-clamped beam with The setup presented is illustrated in Fig the same theexperimental same parameters in Section 3.1The has clamped-clamped been tested The beam beam with is excited by parameters presented Section 3.1 has The beam is excitedbyby Vibration the Vibration ExciterinBruel & Kjaer 4808been and tested the response is measured thethe instrument Exciter Bruel Kjaer 4808PVD-100 and the response is concentrated measured by masses the instrument Laser Polytec Laser&Vibrometer Two equal of 0.6 kgPolytec are attached on the beam in different The receptance measured along the beam Vibrometer PVD-100 Twoscenarios equal concentrated massesisof 0.6 kg are attached on thewhen beamthein forcing frequency is set to the first three natural frequencies of the beam-mass system different scenarios The receptance is measured along the beam when the forcing frequency is matrix is obtained at 50 points spaced equally on the beam setThe to receptance the first three natural frequencies of the beam-mass system The receptance matrix is According to the simulation results, when the forcing frequency is equal to the first obtained 50 points the spaced equally on the beam naturalatfrequency, change in receptance is simple that it has only one peak at the middle of theto beam it moves toward the position of the attachedismass when According the and simulation results, when the forcing frequency equal Meanwhile, to the first natural frequency, the change in receptance is simple that it has only one peak at the middle of the beam and it moves toward the position of the attached mass Meanwhile, when the forcing frequency is high the change in receptance becomes more complicated with different configurations of the attached masses Therefore, when the forcing frequency is equal to the first natural frequency Fig 6b Fig 11 presents the experimental receptance curves of beam without and with an attached mass at L/4 which was measured when the forcing frequency is equal to the second natural frequency When there is no mass attached, these receptance has two peaks at L/4 and 3L/4 These experimental results justify the correctness of the simulation results presented in Fig When there is a mass attached at the position of L/4, the peak at the mass position decreases clearly Fig 12 presents the receptance measured when the force frequency is equal to the third natural frequency As can be seen from this figure, when there is no mass attached the receptance has three peaks at L/6, L/2 and 5L/6 When there are masses attached, the receptance peaks decrease at the mass positions The results39 Theoretical and experimental analysissignificantly of the exact receptance function of a clamped-clamped beam experimental with concentrated masses presented in Fig 12 are in very good agreement with the simulation results depicted in Fig Fig Experimental setup Fig Experimental setup the forcing frequency is high the change in receptance becomes more complicated with different configurations of the attached masses Therefore, when the forcing frequency is equal to the first natural frequency only the receptance curve extracted with the force fixed at one position is measured, while the whole receptance matrices are measured at the second and third natural frequencies When the mass is attached at the position of L/4, the force is fixed at L/2 and the forcing frequency is equal to the first natural frequency, the measured receptance moves to the left end as presented in Fig 10 Comparing Figs and 10 it is concluded that the measured receptance and the simulation results are in very good agreement in both cases without and with an attached mass Normalized receptance 1.2 Without mass With mass at 0.25L 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 Response position 0.8 Measuredreceptance receptancecurves curvesofofbeam, beam,force force position=L/4, ω=ω Fig.Fig 10 9.Measured position = L/4, ω =ω 40 Nguyen Viet Khoa, Dao Thi Bich Thao Normalized receptance Normalized receptance Normalized receptance When the excitation frequency is equal to the second natural frequency and the mas is attached at the position of L/4, the measured receptance matrix presented in Fig 5(a) and the simulation receptance matrix shown in Fig 11(a) are in very good agreement As can be seen from Fig 11(a), three peaks corresponding to the position of L/4 in the re1.2 1.2 Without mass mass 0.25L Without mass theWith With mass at 0.25L at L/2 and the forcing ceptance matrix decrease significantly When mass is atattached frequency is equal 1to1the third natural frequency, five peaks of the receptance matrix corresponding to the position of L/2 decrease significantly as can be observed in Fig 11(b) 0.8simulation result depicted in Fig 7(b) Fig 12 presents the experiThis agrees with 0.8 the mental receptance of beam without and with an attached mass at L/4 which was 0.6curves 0.6 measured when the forcing frequency is equal to the second natural frequency When 0.4 0.4 there is no mass attached, these receptance has two peaks at L/4 and 3L/4 These exper1.2 mass the correctness With mass at 0.25L of the simulation results presented in Fig When imental resultsWithout justify 0.2 0.2 there is a mass attached at the position of L/4, the peak at the mass position decreases 0 the receptance measured when the force frequency is equal to the 0.8 clearly Fig 13 presents 0 0.20.2 0.4 0.80.8there is no 0.4 this0.6 0.6 mass attached third from figure, when 0.6 natural frequency As can be seen Response position Response position the receptance has three peaks at L/6, L/2 and 5L/6 When there are masses attached, 0.4 Fig Measured receptance curves ofof beam, force position=L/4, Fig Measured receptance curves beam, force position=L/4,ω=ω ω=ω 1 0.2 0 0.2 0.4 0.6 Response position 0.8 Fig Measured receptance curves of beam, force position=L/4, ω=ω1 (a) ω = ω2 (b) ω = ω3 a) a) b)b) a) Fig 10 Measured b) receptance receptance matrices of beam: ω=ω ; b)ω=ω ω=ω Fig 11.receptance Measured ofa) beam Fig 10 Measured matrices ofmatrices beam: a) ω=ω 2; 2b) 3 Fig 10 Measured receptance matrices of beam: a) ω=ω2; b) ω=ω3 1.2 1.2 Without mass Without mass Normalized receptance Normalized receptance Without mass 1.2 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0 0.2 0.2 0.2 0.4 0.6 Response 0 position Without mass Mass is at 0.5L With mass at 0.25L With mass at 0.25L mass at 0.25L With Normalized receptance Normalized receptance 1.2 0.8 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 Response position 0.8 0.2 0.4 0.6 0of beam,0.2 0.4Figure 0.80.8 curves1 of 12 0.6 Measured receptance beam, force position=L/6, ω=ω3 Figure 11 Measured receptance curves force position=L/4, ω=ω Fig 12 Measured receptance curves of beam, Fig 13 Measured receptance curves of beam, Response position Response position force position = L/4, ω = ω25 Conclusion force position = L/6, ω = ω3 In this paper, the exact receptance function of clamped-clamped Figure Measured receptance curves beam, force position=L/4, ω=ω2 beam carrying concentrated Figure 11.11 Measured receptance curves ofof beam, force position=L/4, ω=ω masses is derived The proposed receptance function can be 2applied easily for predicting the response of the beam under a harmonic excitation The influence of the concentrated masses on the receptance of beam is also investigated When the excitation frequency is equal to a natural frequency, the peaks and nodes positions of the receptance are the same with the maximum and minimum positions of the corresponding mode shape When there are concentrated masses the shape of receptance is changed When the mass positions are close to peaks of receptance, these peaks will decrease significantly When the masses are located at the nodes of receptance, the receptance is not influenced The influence Theoretical and experimental analysis of the exact receptance function of a clamped-clamped beam with concentrated masses 41 the receptance peaks decrease significantly at the mass positions The experimental results presented in Fig 13 are in very good agreement with the simulation results depicted in Fig CONCLUSIONS In this paper, the exact receptance function of clamped-clamped beam carrying concentrated masses is derived The proposed receptance function can be applied easily for predicting the response of the beam under a harmonic excitation The influence of the concentrated masses on the receptance of beam is also investigated When the excitation frequency is equal to a natural frequency, the peaks and nodes positions of the receptance are the same with the maximum and minimum positions of the corresponding mode shape When there are concentrated masses the shape of receptance is changed When the mass positions are close to peaks of receptance, these peaks will decrease significantly When the masses are located at the nodes of receptance, the receptance is not influenced The influence of masses on the receptance matrices can be used to control the vibration amplitudes at some specific positions at given forcing frequencies The experiment has been carried out when the forcing frequency is set to the first three natural frequencies of the beam carrying concentrated masses The experimental and simulation results are in very good agreement which justifies the proposed method ACKNOWLEDGEMENTS This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.300 REFERENCES [1] R E D Bishop and D C Johnson The mechanics of vibration Cambridge University Press, (2011) [2] H K Milne The receptance functions of uniform beams Journal of Sound and Vibration, 131, (3), (1989), pp 353–365 https://doi.org/10.1016/0022-460X(89)90998-X [3] B Yang Exact receptances of nonproportionally damped dynamic systems Journal of 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continuous cracked beam Vietnam Journal of Mechanics, 41, (4), (2019), pp 349–361 https://doi.org/10.15625/0866-7136/14566 [13] J S Wu and T L Lin Free vibration analysis of a uniform cantilever beam with point masses by an analytical-and-numerical-combined method Journal of Sound and Vibration, 136, (2), (1990), pp 201–213 https://doi.org/10.1016/0022-460X(90)90851-P [14] S Maiz, D V Bambill, C A Rossit, and P A A Laura Transverse vibration of Bernoulli–Euler beams carrying point masses and taking into account their rotatory inertia: Exact solution Journal of Sound and Vibration, 303, (3-5), (2007), pp 895–908 https://doi.org/10.1016/j.jsv.2006.12.028  ... 0.5L Theoretical and experimental analysisFig of the4 exact receptance function of a clamped- clamped beam with concentrated masses Receptance of beam at ω=ω Fig Receptance of beam at 2ω=ω a) Mass... frequency receptance theanalysis beamofofthe isthe maximum at theto about 0.2L, 0.5Lmasses and 35 quency The receptance of the beam is maximum at the positions of about 0.2L, 0.5L and 0.8L where the amplitude... eptance matrix of theThe beam when theoffrequency force is equal thirdofnatural Theoretical and experimental exact receptance function ofpositions a the clamped- clamped beam with concentrated frequency