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

Nội dung text: Theoretical and experimental analysis of the exact receptance function of a clamped-clamped beam with concentrated masses

1. Vietnam Journal of Mechanics, VAST, Vol.42, No. 1 (2020), pp. 29 – 42 DOI: THEORETICAL AND EXPERIMENTAL ANALYSIS OF THE EXACT RECEPTANCE FUNCTION OF A CLAMPED-CLAMPED BEAM WITH CONCENTRATED MASSES Nguyen Viet Khoa1,∗, Dao Thi Bich Thao1 1Institute 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 sim- ulations 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 recep- tance 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. 1. INTRODUCTION The receptance function is very important in vibration problems such as control de- sign, system identification or damage detection since it interrelates the harmonic excita- tion 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 non- proportionally damped dynamic systems. In this work an iteration procedure is devel- oped 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 mod- ification 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
2. 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 obtain the frequency response matrix of the constrained system. Gurg¨ oze¨ and Erol [7] 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 sub- jected 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 ex- act 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 an- other work, Burlon et al. [9] presented an exact frequency response of two-node coupled bending-torsional beam element with attachments. Karakas and Gurg¨ oze¨ [10] extended the work in [3] in which the receptance matrix was obtained directly without using the it- erations 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 clamped- clamped beam carrying an arbitrary number of concentrated masses. The proposed for- mula of receptance function is simple and can be applied easily to investigate the dy- namic 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. 2. 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 # 0000
   EIy + m + ∑ mkδ (x − xk) y¨ = δ 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 th the beam, µ is the mass density per unit length, mk is the k 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
   0 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 0000
   EIy + my¨ = δ x − x f f (t) − ∑ mkδ (x − xk)y¨. (2) k=1
3. 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 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. 2. TheoreticalTheoretical and background experimental analysis of the exact receptance function of a clamped-clamped beam with concentrated masses 31 f(t) y m1 m2 mn x xk xf Fig.Fig. 1. 1A. Acl clamped-clampedamped-clamped beam withwith concentratedconcentrated masses masses Considering the Euler-Bernoulli beam carrying concentrated masses subjected to a force as shown inEq. Fig. (2) 1 can, the be governing considered ben asding the motion equation equation of forced of vibrationthe beam ofcan a be beam extended without from con- [13] n as followscentrated: masses which is acted by the inertia forces of concentrated masses and the external force f (t). The solution of Eq. (2) can be expressed in the form éùn ∞ EIy¢¢¢¢ ++ m mdd x - x y= x- x f t (1) êúå kk( )y (!!x, t) =( ∑ φ fi )(x)(qi)(t), (3) ëûk=1 i=1 th th wherewhere E isφ itheis theYoung’si mode modulus, shape ofI is the the beam moment without of inertia concentrated of the cross masses sectional and qi isarea the ofi the th beam,generalized μ is the mass coordinate. density per unit length, mk is the k concentrated mass located at xk, y(x, t) is the bendingSubstituting deflection (3) into of the (2), beam yields at location x and time t, f(t) is the force acting at position ∞ ∞ n ∞   xf, d (xx- f0000) is the Dirac delta function. EI ∑ φi (x) qi (t) + m ∑ φi (x) q¨i (t) = − ∑ δ (x − xk) mk ∑ φi (x) q¨i (t) + δ x − x f f (t) . i=1 i=1 k=1 i=1 Eq. (1) can be rewritten in the form: (4) Multiplying Eq. (4) by φj(x) andn integrating from 0 to L and considering the defini- ¢¢¢¢ (2) tion ofEIy the+= Dirac my!! deltadd( x function, xfkk) f( t) oneå obtains m( x x) !! y L k=1 L Z ∞ Z ∞ 0000 Eq. (2) can be consideredEI ∑ φasi the(x )equationφj (x) qi (oft) dforcedx + vibrationm ∑ φi (x of) φ aj (beamx) q¨i ( twithout)dx concentrated = = masses which is acted0 byi 1 the inertia forces of n concentrated0 i 1 masses and the external force(5) f(t). n The solution of Eq. (2) can be expressed in the form:
   = − ∑ mkφi (xk) φj (xk) q¨i (t) + φj x f f (t). ¥ k=1 yxt, = f xq t (3) The( ) orthogonalityå ii( ) ( of) the normal mode shapes of the beam without concentrated i=1 masses can be addressed here th L th where fi is the i mode shapeZ of the beam without concentrated masses and qi is the i 0000 generalized coordinate. φi (x) EIφi (x) dx = 0 if i 6= j (6) Substituting (3) into (2), yields0: L Z φi (x) mφj (x) dx = 0 if i 6= j (7) 0
4. 32 Nguyen Viet Khoa, Dao Thi Bich Thao Integrating the first equation in Eq. (6) twice by parts, yields L Z ( ) 000 ( )L − 0 ( ) 00 ( )L + 00 ( ) 00 ( ) = 6= φi x EIφj x 0 φ i x EIφj x 0 φ i x EIφ j x dx 0 if i j. (8) 0 For general boundary conditions the first two terms in Eq. (8) vanish. Thus, from Eq. (8) we have  L 0 if i 6= j Z L 00 00  Z φ i (x) EIφ j (x) dx =  002 (9)  φi (x) EIdx if i = j 0 0 Applying Eqs. (6)–(9), Eq. (5) can be rewritten as  L   L  Z n Z 2 2 002
   m φi (x) dx + ∑ mkφi (xk) q¨i (t) + EI φi (x)dx qi (t) = φj x f f (t) . (10) = 0 k 1 0 By introducing notations  L  Z n n n 2 2  φ1 (x) dx + ∑ m¯ kφ1 (xk) ∑ m¯ kφ1 (xk) φ2 (xk) ∑ m¯ kφ1 (xk) φN (xk)   = = =   0 k 1 k 1 k 1   L   n Z n n   2 2   m¯ kφ2 (xk) φ1 (xk) φ2 (x) dx + m¯ kφ (xk) m¯ kφ2 (xk) φN (xk)  M = m  ∑ ∑ 2 ∑  ,  k=1 k=1 k=1   0     L   n n Z n   2 2   ∑ m¯ kφN (xk) φ1 (xk) ∑ m¯ kφN (xk) φ2 (xk) φN (x) dx + ∑ m¯ kφN (xk)  = = = k 1 k 1 0 k 1  L  Z 2  φ00 (x)dx 0 . . . 0   1   0   L   Z   2   0 φ00 (x)dx . . . 0  K = EI  2  ,    0     L   Z   002   0 0 . . . φ N (x)dx  0 T T Φ (x) = [φ1 (x) , , φN (x)] , ¨q (t) = [q¨1 (t) , q¨2 (t) , , q¨N (t)] , m q (t) = [q (t) , q (t) , , q (t)]T, m¯ = k . 1 2 N k m Eq. (10) can be expressed in matrix form as follows
   M ¨q (t) + Kq (t) = Φ x f f (t) . (11)
5. 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 − ω2M = 0. (12) If the force is harmonic f (t) = f¯ eiωt then the solution of Eq. (11) can be found in the form q (t) = ¯qeiωt. (13) Substituting Eq. (13) into Eq. (11) yields 2
   
   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 = T Φ (ξ) −1 displacement/force. Thus, left multiplying Eq. (14) with K − ω2M
    the re- f¯ ceptance at x due to the force at x f is obtained T Φ (x) ¯q −1 α x, x , ω
   = = ΦT (x) K − ω2M
   Φ x
   . (15) f f¯ f 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 φi (x) = (sin αix − sinh αix) + cos αix − cosh αix, cos αi L − cosh αi L  sin α L + sinh α L  φ00 (x) = −α2 i i (sin α x + sinh α x) + cos α x + cosh α x , i i cos α L − cosh α L i i i i i i (16) Z L 2 φi (x) dx = L, 0 Z L  00 2 4 φ i (x) dx = Lαi , 0 where αi is the solution of the frequency equation cos αL cosh αL − 1 = 0. From Eq. (16) the matrices M and K are derived    4  L + β11 β12 β1N α1 0 . . . 0 4  β L + β β N   0 α . . . 0  M = m  21 22 2  , K = EIL  2  , (17)     + 4 βN1 L βNN 0 . . . αN where n   sinα L + sinhα L  β = m¯ i i (sinα x − sinhα x ) + cosα x − coshα x ij ∑ k − i k i k i k i k k=1 cosαi L coshαi L   (18) sinαj L + sinhαj L
   × sinαjxk − sinhαjxk + cosαjxk − coshαjxk . cosαj L − coshαj L
6. 34 Nguyen Viet Khoa, Dao Thi Bich Thao The exact formula of the receptance of the clamped-clamped beam carrying concen- trated masses will be derived from Eqs. (16)–(18). 3. 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 with two concentrated masses m¯ 1 = m¯ 2 = 0.5 attached at 0.25L and 0.75L obtained by two methods are listed in Tab.1. 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 1. Frequency parameters of the clamped-clamped beam Frequency parameters Ref. [14] Present paper Error (%) α1L 4.0973 4.0976 0.00007 α2L 5.8984 5.8995 0.00019 α3L 9.1453 9.1534 0.00089 α4L 13.7527 13.7567 0.00029 α5L 16.9258 16.9399 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 = 1 m; b = 0.02 m; h = 0.01 m. Two equal con- centrated 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.2 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 fre- quency. The receptance of the beam is maximum at the positions of about 0.2L, 0.5L and
7. frequencies of the beam-mass system, respectively. As can be seen from Fig. 1a 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. 1b 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. 1c 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 0.8L where the amplitude of the third mode shape is maximum. The receptance is minimum at positions of about 0.35L, 0.65L where the amplitude of the third mode shape is minimum. It frequencies of the beam-mass system, respectively. As can be seen from Fig. 1a when the can be concluded that, when the excitation frequency is equal to a natural frequency the forcing frequency is equal to the first natural frequency, the receptance is maximum at the frequencies of the beam-mass system, respectively. positions As can be of seenmaxima from and Fig. minima 1a when in the the receptance are the same with the positions of maxima middle of the beam which corresponds to the position where the amplitude of the first mode is forcing frequency is equal to the first natural frequency,and minima the receptance in the corresponding is maximum mode at the shape. Therefore, similar to the mode shape, we call the maximum. As can be observed from Fig. 1b that when the forcing frequency is equal to the middle of the beam which corresponds to the positionmaxima where the in amplitudethe receptance of the “peaks first mode of receptance” is and the minima in the receptance “nodes of second natural frequency, the receptance is maximum at position of about 0.3L and 0.7L from maximum. As can be observed from Fig. 1b that whenreceptance”. the forcing frequency is equal to the second naturalthe frequency,left end of the the receptanc beam whiche is maximum are the positions at position where of about the amplitude 0.3L and 0.7of Lthe from second mode shape the left end ofis maximum.the beam which Meanwhile, are the positions the receptance where the is amplitudesmallest at of the the middlesecond modeof beam shape which corresponds is maximum.to Meanwhile, the position t hewhe receptancere the amplitude is smallest of theat the second middle mode of beam shape which is minimum. correspond Fig.s 1c presents the to the positionreceptance where the matrix amplitude of theof the beam second when mode the shape frequency is minimum. of the Fig.force 1c ispresents equal theto the third natural receptance matrixfrequency. of the The beamTheoretical receptance when and experimentalthe of frequency the analysis beam ofof theis the exactmaximum force receptance is equal functionat the to ofpositions a the clamped-clamped third of natural about beam with 0.2 concentratedL, 0.5Lmasses and 35 frequency. The0.8L receptance where the of amplitude the beam of is themaximum third mode at the shape positions is maximum. of about 0.2TheL, receptance0.5L and is minimum at 0.8L where thepositions amplitude of0.8 about ofL wherethe 0.35third theL mode, amplitude0.65 shapeL where is of maximum. thethe thirdamplitude modeThe receptance of shape the third is maximum.is minimummode shape The at is receptance minimum. is min-It positions of canabout be 0.35 concludedimumL, 0.65 atL that,where positions when the ofamplitude the about excitation 0.35 of Lthe, 0.65 third frequencyL where mode theshape is equalamplitude is minimum. to a of nat theural It third frequency mode shape the is can be concluded that, when the excitation frequency is equal to a natural frequency the positions ofminimum. maxima and It can minima be concluded in the receptance that, when are the the excitation same with frequency the positions is equal of tomaxima a natural positions of maxima and minima in the receptance are the same with the positions of maxima and minimafrequency in the corresponding the positions mode of maxima shape. andTherefore, minima similar in the to receptance the mode are shape the, samewe call with the the and minima in the corresponding mode shape. Therefore, similar to the mode shape, we call the maxima inpositions the receptance of maxima “peaks and of minima receptance” in the and corresponding the minima modein the shape.receptance Therefore, “nodes similar of maxima in the receptanceto the“peaks mode of shape,receptance” we call and the the maxima minima in in the the receptance receptance “peaks “nodes ofof receptance” and the receptance”.receptance”. minima in the receptance “nodes of receptance”. a) ω=ω1 b) ω=ω2 position ofposition 0.4L when of 0.4 theL mass when is the located mass atis 0.25locatedL. T athe 0.25 receptanceL. The receptance seems to be seems “pulled’ to be toward “pulled’ toward the mass position. the mass(a) ωposition.= ω1 (b) ω = ω2 (c) ω = ω3 a) ω=ω1 b) ω=ω2 Fig. 4 presents thea) receptance ω=ω1 of beam carrying a concentrated mass b) ω =atω different2 c) positions ω=ω3 when Fig. 4 presents the receptanceFig. 2. Receptance of beam ofcarrying beam without a concentrated a masse mass at different positions when the forcingthe frequency forcing frequencyis equal to is the equal second to the natural second frequency. naturalFig. frequency. As 1 shownReceptance Asin Fig.shown of 4a, beam inwhen Fig. without the4a, when a masse the mass is locatedmassWhen isat therelocated0.3L, isthe at a peaksconcentrated0.3L, correspondingthe peaks mass, corresponding the to either receptance the to responseeither matrix the ofposition response the beam of position 0. is3 changed.L or ofthe 0. 3L or the force position of 0.3L decrease significantly.When there When is a concentratedthe mass is located mass, thethe receptancemiddle of thematrix beam, of the beam is changed. Fig. 2 presents Fig.force3 presents position the of receptance 0.3L decreasethe matrices rece significantly.ptance of the matrices beam When when of the the themass beam forcing is locatedwhen frequency the the forcing middle is equal frequency of the beam, is equal to the first natural the receptanceto the first shape natural is unchanged frequency as of shown the beam-mass in Fig. 4b. system.These results As can show be seen that fromwhen this the figure,mass the receptance shape is unchangedfrequency as of shown the beam in Fig.-mass 4b. system. These resultsAs can show be seen that from when this the figure, mass when the mass is located is attachedwhenis at attached thea peak mass ofat is thea locatedpeak receptance of 0.25the Lreceptance matrix,the position the matrix, peaks of the thecorresponding peak peaks of corresponding receptance to either “moves” the to reseitherponse to the the response positionleft or endforce of position beam. However,which is 0.25close whenL theto the the position mass mass is ofposition located the peak will at of the decrease.receptance middle ofMeanw “ themoves beam,hile,” to the the left end of beam. However, when position or force positionthe which mass is is close located to theat the mass middle position of the will beam, decrease. the shape Meanw of thehile, receptance the is unchanged. The shape ofshape receptanceshape of theof receptance receptanceis unchanged is is unchanged unchanged.when the mass when The is the changeattached mass ofis at positionattached the nod es ofat ofthe the the nod peak receptance.es ofof receptancethe receptance. The The changeis in depicted receptance clearer can inbe Fig.observed4 whenchange in the more of force position detail is fixed asof presentthe at positionpeaked of in receptance 0.5Fig.L .5 As when canis depicted the be observedforce clearer is in Fig. 3 when the force is fixed change in receptance can atbe position observed 0. 5inL .more As can detail be asobserved present edfrom in Fig.this figure,5 when the the peak force of is receptance moves to the fixed atfrom positionfixed this at figure,0.25 positionL. theAs 0.25can peak Lbe. of Asseen receptance can from be seenthis moves figure,from to this thethe figure, positionpeak of the receptance of peak 0.4L ofwhen receptance decreases. the mass decreases.In is In L addition,located theaddition, peak at 0.25 of the receptance. peak The receptanceof receptance moves seemsslightly moves to toward be slightly “pulled’ the toward mass toward position. the the mass mass position. position. c) ω=ω3 c) ω=ω Fig. 1 Receptance of beam without a masse3 When there is a concentrated mass, the receptanceFig. 1 Receptance matrix of ofthe beam beam without is changed. a masse Fig. 2 presents the receptanceWhen matrices there isof a the concentrated beam when mass, the forcing the receptance frequency matrix is equal of the to beamthe first is changed.natural Fig. 2 presents frequency ofthe the rece beamptance-mass matricessystem. Asof canthe bebeam seen when from thisthe figure,forcing w henfrequency the mass is isequal located to the first natural 0.25L the positionfrequency of the of peakthe beam of receptance-mass system. “moves As” to can the be left seen end from of beam this. figure,However, when when the mass is located the mass is 0.25locatedL the at theposition middle of of the the peak beam, of thereceptance shape of “ themoves receptance” to the isleft unchanged. end of beam The. However, when change of position of the peak of receptance is depicted clearer in Fig. 3 when the force is fixed the mass is located at the middle of the beam, the shape of the receptance is unchanged. The at 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. 3 when the force is fixed at position 0.5L. As can be observed from this figure, the peak of receptance moves to the (a) Mass is at 0.25L (b) Mass is at 0.5L a) Mass is a)at M0.25assL is at 0.25L b) Mass is b) at M0.5assL is at 0.5L Fig. 3 ω = ω Fig. 2. Receptance.Fig. Receptance 2. Receptance matrices matrices at matrices atω=ω1 at1 ω=ω1 Fig. 3. ReceptanceFig. 3. Receptance of beam with of beam force withposition force is position at L/2; ω is= ωat1 L /2; ω=ω1
8. position of 0.4L when the mass is located at 0.25L. The receptance seems to be “pulled’ toward the mass position. Fig. 4 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. 4a, 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. 4b. 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. 5 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 0.25L b) Mass is at 0.5L 36 Nguyen Viet Khoa, Dao Thi Bich Thao Fig. 2. Receptance matrices at ω=ω1 Fig. 4. Receptance of beam with force position is at L/2, ω = ω1 Fig. 3. Receptance of beam with force position is at L/2; ω=ω1 Fig.5 presents the receptance of beam carrying a concentrated mass at different po- sitions 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 lo- cated 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 ob- served in more detail as presented in Fig.6 when the force is fixed at position 0.25 L. 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 0.25L (b) Mass is at 0.5L a) Mass is at 0.25L b) Mass is at 0.5L a) Mass is at 0.25L b) Mass is at 0.5L Fig. 5.Fig. Receptance 4. Receptance of beam of atbeamω = atω 2ω=ω2 Fig. 4. Receptance of beam at ω=ω2 Fig. 5. MeasuredFig. 5. Measured receptance receptance with the withforce the acting force at acting0.25L, atω =0.25ω2 L, ω=ω2 a) Mass is at 0.2L b) mass is at 0.5L a) Mass is at 0.2L b) mass is at 0.5L
9. 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 the 4. exact Receptance receptance function of beam of a clamped-clamped at ω=ω2 beam with concentrated masses 37 Fig. 4. Receptance of beam at ω=ω2 a) Mass is at 0.25L b) Mass is at 0.5L Fig. 4. Receptance of beam at ω=ω2 Fig.Fig. 5. Measured 5. Measured receptance receptance with withthe force the force acting acting at 0.25 atL 0.25, ω=Lω,2 ω=ω2 Fig. 6. Measured receptance with the force acting at 0.25L, ω = ω2 Fig. 5. Measured receptance with the force acting at 0.25L, ω=ω2 (a) Mass is at 0.2L (b) Mass is at 0.5L a) Mass is at 0.2L b) mass is at 0.5L a) Massa) Mass is at is0.2 atL 0.2 L b) mass b) massis at 0.5 is Lat 0.5L (c) Masses are at 0.2L and 0.5L (d) Masses are at 0.2L and 0.8L d) M asses d)are M atasses 0.2L are and at 0.5 0.2LL and 0.5 L e) Massese) are M assesat 0.2 areL and at 0.20.8LL and 0.8L Fig 6. NormalizedFig.Fig 7. 6 Normalized. Normalized receptance receptance receptance at ω=ω at3 ωat =ω=ωω3 3 The changeThe in receptancechange in receptance can be seen can in bemore seen detail in more when detail the whenforce theis fixed force at is position fixed at position0.2L as 0.2L as depicted indepicted Fig.Fig. 7. 7Similar inpresents Fig. 7.conclusion Similar the receptance conclusion can be ofdrawn can beam befrom whendrawn this the fromfigure forcing this that figure when frequency that there when isis equal athere mass tois a the mass L L attached atthird attacheda peak natural, thisat a frequency.peak, willthis decreasepeak As shownwill significantly.decrease in Fig. significantly.7(a) ,When when there one When massis one there is mass located is one the mass atpeaks 0.2 the ofor peaks 0.8 , of receptance receptancemove toward move the toward mass position. the mass When position. there When are twothere masses are two attached masses attachedsymmetrically symmetrically at 0.2L andat 0.8 0.2LL the and peak 0.8L at the 0.2 peakL moves at 0.2 toL movesthe left to end the, whileleft end the, while peak theat 0.8 peakL moves at 0.8L to moves the to the right end. Whenright end. there When are two there masses are two attached masses at attached 0.2L and at 0.2L0.5L, and the 0.5L,receptance the receptance is “pulled” is “pulled”to to the left end.the In left this end. case, In the this receptance case, the receptance tends to “ movetends ”to toward “move the” toward heavier the side heavier of the side beam. of the beam. Figure 7. Normalized receptance when the force is fixed at 0.2L, ω=ω3 Figure 7. Normalized receptance when the force is fixed at 0.2L, ω=ω3 4. Experiment results 4. Experiment results The experimental setup is illustrated in Fig. 8. The clamped-clamped beam with the same The experimentalparameters setup presented is illustrated in Section in Fig. 3.1 8 has. The been clamped tested.- clamped The beam beam is excited with the by samethe Vibration parametersExciter presented Bruel in & Section Kjaer 3.14808 has and been the tested.response The is beammeasured is excited by the byinstrument the Vibration Polytec Laser Exciter BruelVibrometer & Kjaer PVD 4808-100. and Two the responseequal concentrated is measured masses by theof 0.6instrument kg are attached Polytec on Laser the beam in Vibrometerdifferent PVD-100. scenarios. Two equal The receptanceconcentrated is measuredmasses of along 0.6 k theg are beam attached when onthe theforcing beam frequency in is different scenarios.set to the The first receptance three natural is measured frequencies along of thethe beambeam- masswhen system. the forcing The frequencyreceptance is matrix is set to the firstobtained three at natural 50 points frequencies spaced equally of the on beam the beam.-mass system. The receptance matrix is obtained atAccording 50 points spacedto the simulationequally on results,the beam. when the forcing frequency is equal to the first natural According frequency,to the simulation the change results, in receptance when the is simpleforcing that frequency it has only is oneequal peak to atthe the first middle natural of the beam frequency, andthe changeit moves in toward receptance the position is simple of thatthe attachedit has only mass one. Meanwhile,peak at the middlewhen the of forcingthe beam frequency and it movesis hightoward the the change position in receptance of the attached becomes mass more. Meanwhile, complicated when with the different forcing configurations frequency of the is high the changeattached in masses. receptance Therefore, becomes when more the complicated forcing frequency with different is equal configurations to the first natural of the frequency attached masses. Therefore, when the forcing frequency is equal to the first natural frequency
10. 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 corre- sponding tod) either Masses the are response at 0.2L and positions 0.5L of 0.2 L, 0.5Le)or M theasses force are positions at 0.2L and of 0.20.8LL, 0.5L decrease significantly as depicted in Fig. 7(c). When two masses are located at 0.2L and Fig 6. Normalized receptance at ω=ω3 0.8L, the peaks corresponding to either the response positions of 0.2L, 0.8L or the force Thepositions change ofin 0.2receptanceL, 0.8L decrease can be seen significantly in more detail as shown when the in Fig. force 7(d) is fixed. The at change position in 0.2 recep-L as depictedtance can in Fig. be seen7. Similar in more conclusion detail when can be the drawn force from is fixed this atfigure position that when 0.2L as there depicted is a mass in attachedFig.8. Similarat a peak conclusion, this peak canwill be decrease drawn significantly. from this figure When that there when is one the massesmass the attached peaks of at peaks, these peaks will decrease significantly. When there is one mass the peaks of receptancereceptance move move toward toward the mass the mass position. position. When When there are there two are masses two massesattached attached symmetrically sym- at metrically0.2L and 0.8 atL 0.2 theL and peak 0.8 atL 0.2theL peakmoves at to 0.2 theL moves left end to, thewhile left the end, peak while at 0.8 theL peakmoves at to 0.8 theL rightmoves end. to When the right there end. are two When masses there attached are two at masses 0.2L and attached 0.5L, atthe 0.2 receptanceL and 0.5 Lis, “pulled” the recep- to thetance left end. is “pulled” In this case, to the the left receptance end. In this tend case,s to “ themove receptance” toward the tends heavier to “move” side of toward the beam. the heavier side of the beam. Fig. 8. Normalized receptance when the force is fixed at 0.2L, ω = ω3 Figure 7. Normalized receptance when the force is fixed at 0.2L, ω=ω3 4. Experiment results 4. EXPERIMENT RESULTS The experimental setup is illustrated in Fig.9. The clamped-clamped beam with Thethe experimental same parameters setup presented is illustrated in Sectionin Fig. 8 3.1. The has clamped been tested.-clamped The beam beam with is excited the same by parametersthe Vibration presented Exciter in Bruel Section & Kjaer 3.1 has 4808 been and tested. the response The beam is measured is excited by by the the instrument Vibration ExciterPolytec Bruel Laser & Vibrometer Kjaer 4808 PVD-100. and the response Two equal is concentratedmeasured by massesthe instrument of 0.6 kg Polytec are attached Laser Vibrometeron the beam PVD in- different100. Two scenarios. equal concentrated The receptance masses is of measured 0.6 kg are along attached the beam on the when beam the in differentforcing scenarios. frequency The is setreceptance to the first is measured three natural along frequencies the beam when of the the beam-massforcing frequency system. is setThe to receptancethe first three matrix natural is obtained frequencies at 50 of points the beam spaced-mass equally system. on The the beam.receptance matrix is According to the simulation results, when the forcing frequency is equal to the first obtainednatural at frequency, 50 points thespaced change equally in receptance on the beam. is simple that it has only one peak at the mid- Accordingdle of the to beam the andsimulation it moves results, toward when the positionthe forcing of thefrequency attached is mass.equal Meanwhile,to the first natural when 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
11. 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. 9. Comparing Figs. 3 and 9 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. 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 Fg. 4a and the simulation receptance matrix shown in Fig. 10a are in very good agreement. As can be seen from Fig. 10a, three peaks corresponding to the position of L/4 in the receptance matrix decrease significantly. When the mass is attached at L/2 and the forcing frequency is equal to the third natural frequency, five peaks of the receptance matrix corresponding to the position of L/2 decrease significantly as can be observed in Fig. 10b. This agrees with the simulation result depicted in 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. 5. 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 receptanceTheoretical andpeaks experimental decrease analysis significantly of the exact receptance at the function mass of a clamped-clamped positions. The beam experiment with concentratedal massesresults 39 presented in Fig. 12 are in very good agreement with the simulation results depicted in Fig. 7. Fig. 8. Experimental setup Fig. 9. 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.4 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. 1.2 Without mass With mass at 0.25L 1 0.8 0.6 0.4 0.2 Normalized receptance 0 0 0.2 0.4 0.6 0.8 1 Response position Fig. 9. Measured receptance curves of beam, force position=L/4, ω=ω1 Fig. 10. Measured receptance curves of beam, force position = L/4, ω = ω1 a) b) Fig. 10. Measured receptance matrices of beam: a) ω=ω2; b) ω=ω3 1.2 Without mass 1 With mass at 0.25L 0.8 0.6 0.4 0.2 Normalized receptance 0 0 0.2 0.4 0.6 0.8 1 Response position Figure 11. Measured receptance curves of beam, force position=L/4, ω=ω2
12. 40 Nguyen Viet Khoa, Dao Thi Bich Thao 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 re- 1.2 ceptance matrix decrease1.2 significantly.WithoutWithout Whenmass mass theWith massWith mass mass is at attached 0.25Lat 0.25L at L/2 and the forcing frequency is equal1 to1 the third natural frequency, five peaks of the receptance matrix cor- responding to the position of L/2 decrease significantly as can be observed in Fig. 11(b). This agrees with0.8 the0.8 simulation result depicted in Fig. 7(b). Fig. 12 presents the experi- mental receptance0.6 curves0.6 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,0.40.4 these receptance has two peaks at L/4 and 3L/4. These exper- 1.2 Without mass With mass at 0.25L imental results justify0.2 the correctness of the simulation results presented in Fig.6. When Normalized receptance 0.2 there1 is a mass attachedNormalized receptance at the position of L/4, the peak at the mass position decreases clearly.0.8 Fig. 13 presents0 0 the receptance measured when the force frequency is equal to the 0 0 0.20.2 0.40.4 0.60.6 0.80.8 1 1 third0.6 natural frequency. As can be seen from this figure, when there is no mass attached ResponseResponse position position the0.4 receptance has three peaks at L/6, L/2 and 5L/6. When there are masses attached, Fig.Fig. 9. Measured9. Measured receptance receptance curves curves of ofbeam, beam, force force position= position=L/4L/4, ω, =ωω=ω1 1 0.2 Normalized receptance 0 0 0.2 0.4 0.6 0.8 1 Response position Fig. 9. Measured receptance curves of beam, force position=L/4, ω=ω1 (a) ω = ω2 (b) ω = ω3 a) a) b)b) a) b) Fig.Fig. 10 .1 Measured0. MeasuredFig. 11 .receptance Measured receptance receptancematrices matrices of matrices ofbeam beam: a) of: a)ω beam =ωω=2ω; 2b); b) ω =ωω=ω3 3 Fig. 10. Measured receptance matrices of beam: a) ω=ω2; b) ω=ω3 1.21.2 1.2 WithoutWithout mass mass Without mass 1.2 Without mass WithWith mass mass at 0.25Lat 0.25L 1 1 With1 mass at 0.25L Mass is at 0.5L 1 0.8 0.80.8 0.8 0.6 0.60.6 0.6 0.4 0.4 0.4 0.2 0.4 Normalized receptance 0.2 Normalized receptance 0 0.2 0.2Normalized receptance Normalized receptance 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Response0 0 position Response position 0 0.2 0.4 0.6 0.8 1 Figure 11. Measured receptance curves0 of beam0.2, force position=0.4FigureL/4, ω 1=2ω. 0.6Measured2 receptance0.8 curves1 of beam, force position=L/6, ω=ω3 Fig. 12. Measured receptance curves of beam,ResponseFig. position 13. Measured receptance curves of beam, 5. ConclusionResponse position force position = L/4, ω = ω2 force position = L/6, ω = ω3 FigureFigure 11 .1 Measured1. Measured receptance receptanceIn this curves paper, curves the of exact ofbeam beam receptance, force, force functionposition= position= of clampedL/4L/4, ω,- clampedω=ω=ω 2 beam carrying concentrated 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 of the 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, D.C. Johnson, The Mechanics of Vibration, Cambridge University Press, Cambridge, 1960. [2] H.K. Milne, The receptance functions of uniform beams. Journal of Soundand Vibration (1989) 131(3), 353-365. [3] B. Yang, Exact receptances of nonproportionally damped systems, Transactions of American Society of Mechanical Engineers Journal of Vibration and Acoustics 115 (1993) 47– 52. [4] R.M. Lin, M.K. Lim, Derivation of structural design sensitivities from vibration test data, Journal of Sound and Vibration 201 (1997) 613–631. [5] J.E. Mottershead, On the zeros of structural frequency response functions and their sensitivities, Mechanical Systems and Signal Processing 12 (1998) 591–597. [6] M. Gurgoze, Receptance matrices of viscously damped systems subject to several constraint equations, Journal of Sound and Vibration (2000) 230(5), 1185-1190.
13. 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 re- sults presented in Fig. 13 are in very good agreement with the simulation results depicted in Fig.8. 5. CONCLUSIONS In this paper, the exact receptance function of clamped-clamped beam carrying con- centrated 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 ex- citation 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 correspond- ing 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 sig- nificantly. 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. [3] B. Yang. Exact receptances of nonproportionally damped dynamic systems. Journal of Vibra- tion and Acoustics, 115, (1), (1993), pp. 47–52. [4] R. M. Lin and M. K. Lim. Derivation of structural design sensitivities from vi- bration test data. Journal of Sound and Vibration, 201, (5), (1997), pp. 613–631. [5] J. E. Mottershead. On the zeros of structural frequency response functions and their sensitivities. Mechanical Systems and Signal Processing, 12, (5), (1998), pp. 591–597. [6] M. Gurgoze. Receptance matrices of viscously damped systems subject to several constraint equations. Journal of Sound and Vibration, 230, (5), (2000), pp. 1185–1190. [7]M.G urg¨ oze¨ and H. Erol. On the frequency response function of a damped cantilever simply supported in-span and carrying a tip mass. Journal of Sound and Vibration, 255, (3), (2002), pp. 489–500.
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