Analysis of the Forced Vibration of Geometrically Nonlinear Cantilever Beam with Lumping Mass by Multiple-Scales Lindstedt-Poincaré Method

Transcription

1 ENOC 17, June 5-3, 17, Budapest, Hungary Analysis of the Forced Vibration of Geometrically Nonlinear Cantilever Beam with Lumping Mass by Multiple-Scales Lindstedt-Poincaré Method Hai-En Du, Guo-Kang Er Vai Pan Iu Faculty of Science Technology, University of Macau, Macau SAR, China Summary. The forced vibration of cantilever beam with or without lumping mass can be found in many practical applications. With the variational approach based on extended Hamiltonian principle, the equation of motion boundary conditions governing a uniform cantilever beam carrying a lumping mass at the free end are formulated under the action of lateral tip concentrated load. After that, the multiple-scales method, multiple-scales Lindstedt-Poincaré method fourth-order Runge-Kutta method are employed to analyze the forced vibration of the considered cantilever. It is found that the first mode frequency response curve obtained by the multiplescales Lindstedt-Poincaré method agree well with the first mode frequency response curve obtained by the fourth-order Runge-Kutta method. However, the first mode frequency response curve obtained by multiple-scales method deviates from the first mode frequency response curve obtained by the fourth-order Runge-Kutta method when the beam deflection is large. When the excitation frequency is around second or third natural frequency, neither multiple-scales method nor multiple-scales Lindstedt-Poincaré method can give correct frequency response curves comparing to the frequency response curves obtained by the fourth-order Runge-Kutta method. Introduction Perturbation theory consists of the methods for obtaining the approximate analytical solutions to nonlinea equations. For instance, algebraic equations, differential equations, integrals equations integro-differential equations governing some physical systems can be approximately solved by perturbation methods. Some different perturbation methods such as multiple-scales (MS method, Lindstedt-Poincaré method, multiple-scales Lindstedt-Poincaré (MSLP method, the methods of matched composite asymptotic expansion, averaging method have been developed employed [1, ]. A small parameter must be introduced artificially to the equations when classic perturbation methods are employed. If the systems to be solved are strongly nonlinear, obtaining the approximate solutions to the problems becomes a challenge. The recent new theoretical results on the free forced vibrations of some strongly nonlinear equations oscillators are of great interest to the engineering community since their applications can be found in many areas. The new techniques can be summarized as 1 variational iteration method; linearized perturbation method; 3 parameter expansion perturbation method; 4 various modified Lindstedt-Poincaré methods. Each of these methods can be applied for obtaining the approximate solutions of a wide class of nonlinear systems without small perturbation parameter. Hu applied an iteration procedure for the solution of a quadratic nonlinear oscillator (QNO the obtained solution is improved in comparison with that obtained by the first-order harmonic balance method [4]. Marinca Herisanu extended the iteration method the obtained solution agrees well with exact solution [5]. Hu modified the equivalent linearization method for analyzing the nonlinear single-degree-of-freedom (SDOF systems with odd nonlinearity [6]. A modified expansion method proposed by He achieved a high accuracy even when the perturbation parameter is within ε < [7]. Xu applied He s parameter-exping method (PEM to determine the limit cycles of some strongly nonlinear oscillators [8]. Comparing with the exact solution, PEM shows its effectiveness accuracy for some nonlinear physical problems [9]. Cheung et al. introduced a new exping parameter to Lindstedt-Poincaré method by which A strongly nonlinear system with large perturbation parameter is transformed into a system with small parameter [1]. A modified Lindstedt-Poincaré method was first proposed the solution to a Duffing equation was obtained by Hu [11]. Hu Xiong applied the modified Lindstedt-Poincaré method to a Duffing equation compared the results with those from classical Lindstedt-Poincaré method [1]. They showed that the Lindstedt-Poincaré method is invalid when the perturbation parameter is large. The first-order second order analytical approximate solutions with the modified Lindstedt-Poincaré method are formulated for a two-degree-of-freedom (TDOF mass-spring system carrying quadratic nonlinearity by Lim et al [13]. An accurate result was achieved by them. In 9, Pakdemirli proposed a method named multiple-scales Lindstedt-Poincaré method by combining multiple-scales method Lindstedt-Poincaré method. This method has been applied to three oscillators, i.e., damped linear oscillator, Duffing oscillator, damped Duffing oscillator. The results obtained by the MSLP method were compared with those from conventional MS method numerical method. It is shown that the results from the MSLP method agree well with numerical simulation for some weakly strongly nonlinear systems. Meanwhile, the results of conventional MS method deviate a lot from numerical simulation as the system nonlinearity increases []. Moreover, the MSLP method was applied to a damped duffing oscillator, a quintic duffing oscillator with strong nonlinearity a quadratic nonlinear oscillator by Pakdemirli [14, 15, 16]. In this paper, the MS method, MSLP method fourth-order Runge-Kutta method are employed to analyze the forced motion of the cantilever beam with lumping mass. The responses obtained by these methods are compared the effectiveness of the MS MSLP methods is examined by the results of numerical simulation.

2 ENOC 17, June 5-3, 17, Budapest, Hungary Vibrational analysis of the cantilever with large deformation $ FRV( ΩW W $ FRV( ΩW W, ( P Figure 1: The scheme of the cantilever beam energy harvester with lumped mass on the free end. Cantilever beam is widely used in many areas such as structural construction or energy harvester due to it can undertake strong forces it can remain elastic even undergo a large deflection. However, only MS method has been adopted to analyzed the cantilever beams in the past [, 1]. We consider the vibration of a cantilever with lumped tip mass large deformation. An example is the energy harvester designed to be a cantilever with tip lumped mass as shown in Fig. 1. The cantilever is assumed to be an isotropic inextensible Euler-Bernoulli beam. For flexible cantilevers, the large deformation can lead to obvious nonlinear behaviors. The MS method, with a small artificial parameter assumption, may be invalid in this case. In order to check the validity of each mentioned methods, the MS method, MSLP method fourth-order Runge-Kutta method are applied to study the response of the cantilever undergoing planar motion. The analytical solutions obtained with the MS MSLP methods are compared to the numerical solutions to figure out the effectiveness accuracy of the MS MSLP methods in various cases. The equation of motion of the cantilever with a lumped mass tip harmonic excitation is given in the following []. mÿ + M t δ(s Lÿ(L,t + c y ẏ + EIy iv = F A cos ( Ωt δ(s L EI [ y (y y ] 1 { s [ s ] ds} y y ds + mg[(s Ly + y ] + mg [(s L 3y y + y 3 ] L (1 The boundary conditions are y(,t =,y (,t =,y (L,t =,y (L,t = ( where M t is the lumped mass on the tip, m is the beam mass per unit length, L is the beam length, E is Young s modulus, I is the moment inertia of beam cross section, g is the ground acceleration, s is the arclength, t is time, y(s,t is the transverse displacement, c v is the coefficient of linear viscous damping per unit length, F A is the forcing amplitude Ω is the excitation frequency. The prime denotes the differentiation with respect to the arclength s. When the external excitation is nearby the ith natural frequency, the response is dominated by the ith mode. In this case, the approximate transverse displacement y(s,t of the beam is given by y(s,t = Φ i (sq i (t (3 where q i (t is the generalized coordinates corresponding to the i th mode, Φ i (s is the i th linear mode function of the cantilever, which is expressed as Φ i (s = cos(p i s cosh(p i s cos(p il + cosh(p i L [ sin(pi s + sinh(p i s ] (4 sin(p i L + sinh(p i L With Galerkin s method y(s,t = Φ i (sq i (t, the following system can be formulated. (1 + αε q i + uε q i + ω q i + βεq 3 i + αεq i q i = Fε cos(ωt (i = 1,,3 (5

3 ENOC 17, June 5-3, 17, Budapest, Hungary where ε is perturbation parameter uε = ω ξ, ω = EI Φ i Φ iv i ds mg [ m m αε = m s s Φ i Φ i m L ( 3 ds + 4 βε = EI m mg m [ 3 Φ i Φ i (s LΦ i Φ i ds + Φ m i dsdsds + m Φ i Φ iφ i Φ i ds + (s LΦ i Φ i Φ i ds + 1 Φ i Φ i s ] (6 Φ i Φ ids, (7 Φ i dsds, (8 Φ i Φ i Φ iv i ds ] Φ i Φ 3 i ds, (9 Fε = F A Φ i (L, (1 m m = M t Φ i (L + m Φ i ds. (11 Eq. (5 is treated by the MS method the MSLP method in the following, respectively. Multiple-scales method With MS method, the oscillator response is expressed as q = q (T,T 1,T + εq 1 (T,T 1,T + ε q (T,T 1,T + O(ε 3 (1 where T, T 1 T are the fast slow time scales which are given by By chain rule, the operators of time derivatives are T = t, T 1 = εt, T = ε t. (13 d dt = D + εd 1 + ε D +..., (14 d dt = D + εd D 1 + ε (D 1 + D D +..., (15 where D n = / T n D n = / Tn. Substituting Eqs. (1, (14 (15 into Eq. (5 setting the coefficients of ε m (m =,1, to zero lead to the following equations. The solution to the O(ε equation is O(ε : D (q + ω q =, (16 O(ε 1 : D (q 1 + ω q 1 = D D 1 (q βq 3 αq [D (q ] αq D (q, (17 O(ε : D (q + ω q = D D 1 (q 1 D 1(q D D (q ud (q 3βq q 1 αq 1 [D (q ] αq D (q D (q 1 αq D (q D 1 (q αq D (q 1 αq D D 1 (q αq q 1 D (q + F cos(ωt. (18 q = Ae iω T + Āe iω T (19 where A is a function of time scales T 1 T which can be determined by omitting the secular terms in the higher-order equation. Substituting Eq. (19 into the righth side of the O(ε 1 equation eliminating the secular terms yield Then the solution of the O(ε 1 equation can be obtained to be where iω D 1 (A + 3βA Ā αω A Ā =. ( q 1 = Λe 3iω T + Λe 3iω T (1 Λ = βa3 8ω αa3 4. ( Substituting the expressions of q q 1 into the O(ε equation along with the assumption of Ω = ω + ε σ eliminating the secular terms yield D (A = FeiσT ua 9α ω A 3 Ā + 9αβA3 Ā + 15β A 3 Ā 4iω 4i 4iω 16iω 3 (3

4 ENOC 17, June 5-3, 17, Budapest, Hungary Then the solution to the O(ε equation can be obtained to be q = Γ 1 e 3iω T + Γ e 5iω T + Γ 1 e 3iω T + Γ e 5iω T, (4 where Express the time derivative of A as a polar form of A is assumed to be Γ 1 = 9α A 4 Ā 16 Γ = 3α A 5 16 αβa4 Ā 8ω αβa5 8ω 1β A 4 Ā 64ω 4, (5 + β A 5 64ω 4. (6 da dt = εd 1(A + ε D (A + O(ε 3 (7 A = 1 aeib. (8 Substituting Eqs. (, (3 (8 into Eq. (7 setting the real imaginary parts equal zero, respectively, yield ȧ = f ε ω sinγ uaε (9 ( a γ ( αω = Ω ω + ε 3a β F cosγ + ε 9a4 α ω + 9a4 αβ + 15a4 β 4 8ω aω 64 64ω 56ω 3. (3 In steady state, ȧ γ equal to zeros. The frequency response curve can be obtained by eliminating γ σ in Eq. (3 Ω = ω + ε σ. The relation between the excitation frequency steady-state response can be obtained to be ( ( Ω =ω + ε a αω + 3a β + ε F 1 4ω u a 4 8ω aω F + 9a4 α ω 64 9a4 αβ 15a4 β (31 64ω 56ω 3. The approximate response of the oscillator is finally obtained to be q = a{cos(ωt γ + X 1 cos[3(ωt γ] + X cos[5(ωt γ]}, (3 in which ( 9a X 1 = a α ε ε α a β ε 14ω 4 + β 3ω a αβε 18ω (33 ( 3α X = a 4 ε 56 + β 14ω 4 αβ 18ω. (34 Multiple-scales Lindstedt-Poincaré method The forced vibration of the same oscillator is analyzed with MSLP method in the following. With the MSLP method, a dimensionless parameter τ for the time transformation τ = ωt is introduced to oscillator (5 first, which leads to ( Ω ω y + uωε y + ω y + αεω y y + αεω y y + βεy 3 = Fε cos ω t, (35 in which y(τ = q(ωt the prime sts for derivative with respect to the new time variable τ. The fast slow time scales with respect to τ are T = τ, T 1 = ετ, T = ε τ. (36 By chain rule, the operators of time derivatives are given by d dτ = D + εd 1 + ε D +..., (37 d dτ = D + εd D 1 + ε D 1 + ε D + D +..., (38

5 ENOC 17, June 5-3, 17, Budapest, Hungary where D n = / T n D n = / T n. The approximate solution is assumed to be y = y (T,T 1,T + εy 1 (T,T 1,T + ε y (T,T 1,T + O(ε 3. (39 The following expression is used in the following analysis [] ω = ω εω 1 ε ω. (4 Substituting Eqs. (37-(4 into Eq. (35 equating coefficients of ε m, (m =,1,, to zero lead to the following equations. O(ε : ω D (y + ω y = (41 O(ε 1 : ω D (y 1 + ω y 1 = ω D D 1 (y + ω 1 y βy 3 αω y [D (y ] αω y D (y O(ε : ω D (y + ω y = ω D D 1 (y 1 ω D 1(y ω D D (y The solution to the O(ε equation is obtained as uωd (y + ω y + ω 1 y 1 αω y 1 [D (y ] αω y D (y 1 αω y D (y D (y 1 αy ω D (y D 1 (y αω y D D 1 (y ( Ω αy y 1 ω D (y 3βy y 1 + F cos ω τ (4 (43 y = A(T 1,T e it + Ā(T 1,T e it. (44 Eq. (44 is substituted into the righth side the O(ε 1 equation the secular term is eliminated by ω id 1 (A + ω 1 A + αω A Ā 3βA Ā =. (45 It is assumed that D 1 (A = since ω 1 is real []. The frequency parameter ω 1 is then expressed as ω 1 = 3βAĀ αω AĀ. (46 With the secular term eliminated, the solution to the O(ε 1 equation can be obtained as y 1 = Λe 3iT + Λe 3iT, (47 where Λ is given by Λ = βa3 8ω αa3 4 For the solution to the O(ε equation, the excitation frequency is related to the transformed frequency by (48 Ω = ω(1 + ε σ. (49 Substituting Eqs. (44, (47 (49 into the righth side of the O(ε equation eliminating the secular term, it gives Aω D 1(Aω id (Aω 4iαAω ĀD 1 (A 3A3 α Ā ω 3A3 Ā β 8ω iuωa + 3A3 αā β + F eiσt =. (5 D (A may not equal zero since ω would be complex. Therefore the possible choice is ω =. Substituting A = 1 aeib into Eq. (5 along with γ = σt b setting the real imaginary parts equal zero, respectively, yield D (γ = σ 3α a 4 The solution to the O(ε equation is given as D (a = 64 F ua sinγ ω ω (51 + 3αβa4 64ω 3β a 4 56ω 4 + F cos(γ. (5 aω y = Γ 1 e 3iT + Γ e 5iT +Γ 1 e 3iT +Γ e 5iT, (53

6 ENOC 17, June 5-3, 17, Budapest, Hungary where ( Γ 1 = A 3 αω1 3ω αβaā ω βω 1 64ω 4 + 3βAĀ 3ω 4 + 5α AĀ 8 (54 ( 3α Γ = A 5 16 αβ 8ω + β 64ω 4. (55 In steady-state, D (a = D (γ =, which can make γ eliminated in Eqs. (51 (5. Then the following expression of Ω about the frequency response curve can be obtained with Eq. (49. ( 3α Ω = ω [1 + ε a 4 3αβa ω + 3β a 4 56ω 4 F aω 1 4u ω a ] F (56 4ω where ω = +3εβa. The approximate response of the oscillator is finally obtained to be 4+εαa { ( [ ( ] [ ( ]} Ω Ω Ω y = a cos ω t γ + X 1 cos 3 ω t γ + X cos 5 ω t γ (57 where ( β X 1 = a ε 3ω α 16 + αω 1ε 18ω βω 1ε 56ω 4 a αβε 3ω + 3a β ε 51ω 4 + 5a α ε 18 (58 ( 3α X = a 4 ε 56 αβ 18ω + β 14ω 4. (59 Comparison analysis Eq. (5 is analyzed in the following by MS method, MSLP method fourth-order Runge-Kutta method, respectively, to examine the effectiveness of MS MSLP methods in different cases. The parameter values of a piezoelectric vibration energy harvester being a vertical cantilever with tip mass are taken from [3]. They are listed in Table 1. Corresponding Table 1: Parameters of the cantilever energy harvester Symbol Meaning Value L Length 3 mm E Young s modulus 1 Gpa b Width of cross section 16 mm h Height of cross section.54 mm ξ damping ratio. I Moment of inertia bh 3 /1 ρ Material density 785 kg/m 3 m Mass per unit length ρa p i Characteristic value of beam /L, 4.694/L or 7.855/L ω Natural frequency p i EI/m µ Tip self mass ratio M t /ml.5 to each mode, the frequency response curves obtained by MS method, MSLP method Runge-Kutta method are shown in Fig.. In the case of i = 1, the nonlinearity of restoring force dominates system nonlinearity when the excitation frequency is around the first natural frequency. In this case, the solution obtained by MSLP method agree well with the numerical solution while the results obtained by MS method deviate a lot from numerical simulation when the response amplitude is large as shown in Fig. (a. However, In the case of i = or 3, the nonlinearity of inertial force dominates the system nonlinearity when the excitation frequency is around the second or third natural frequency. In this case, neither MS method nor MSLP method can give acceptable results in comparison to the results from numerical simulation as shown in Figs. (b (c. Conclusions The responses of the forced vibration of the cantilever beam with lumping mass are investigated with MS method, MSLP method numerical simulation, respectively, to examine the effectiveness of the MS method MSLP method in various cases. The single mode frequency response curves of the cantilever obtained by MS method, MSLP method

7 ENOC 17, June 5-3, 17, Budapest, Hungary.8.7 MS method MSLP method 4 th order Runge Kutta method.6.5 y tip /L Ω/ω (a Comparison of the first mode FRCs obtained by MS method, MSLP method numerical simulation with parameters being ω = rad/s F A =.4N..8.7 MS method MSLP method 4 th order Runge Kutta method.6 y tip /L Ω/ω (b Comparison of the second mode FRCs obtained by MS method, MSLP method numerical simulation with parameters being ω = 5.7rad/s F =.N..3.5 MS method MSLP method 4 th order Runge Kutta method. y tip /L Ω/ω (c Comparison of the third mode FRCs obtained by MS method, MSLP method numerical simulation with forcing parameters being ω = rad/s F =.4N. Figure : Comparison of the first three mode frequency response curves obtained by MS method, MSLP method numerical simulations of a vertical cantilever beam energy harvester.

8 ENOC 17, June 5-3, 17, Budapest, Hungary numerical simulation, respectively, are presented compared to show the effectiveness of the MS method MSLP method with different excitation frequencies. Generally speaking, for the forced vibrations of cantilever beams, nonlinear restoring force dominate the response of the beam when the excitation frequency is around first natural frequency. In the contrary, when the excitation frequency is around second, third or higher natural frequency, the nonlinear inertial force dominate the response of beam. Same phenomenon can be found that the first mode frequency response curve obtained by the multiple-scales Lindstedt-Poincaré method agree well with the first mode frequency response curve obtained by the numerical method. However, the first mode frequency response curve obtained by multiple-scales method deviates from the first mode frequency response curve obtained by numerical method when the beam deflection is large. When the excitation frequency is around second third natural frequency, neither multiple-scales method nor multiple-scales Lindstedt-Poincaré method can give correct frequency response curves comparing to the frequency response curves obtained by the numerical method. Acknowledgement The results presented in this paper were obtained under the supports of the Research Committee of University of Macau (Grant No. MYRG14-84-FST the Science Technology Development Fund of Macau (Grant No. 43/13/A. References [1] Nayfeh A. H. (4 Perturbation Mehtods. Wiley-VCH Verlag GmbH Co. KGaA, Weinheim. [] Pakdemirli M., Karahan M. F., Boyac H. (9 A new perturbation algorithm with better convergence properties: Multiple Scales Lindstedt Poincaré method. Math. Comput. Appl. 14(1: [3] Nayfeh A. H., Pai P. F. (4 Linear Nonlinear Structural Mechanics. Wiley-VCH Verlag GmbH Co. KGaA, Weinheim. [4] Hu H. (6 Solutions of a quadratic nonlinear oscillator: Iteration procedure. J. Sound Vib. 98: [5] Marinca V., Herisanu N. (6 A modified iteration perturbation method for some nonlinear oscillation problems. Acta Mech. 184: [6] Hu H. (4 A modified method of equavalent linearization that works even when the non-linearity is not small. J. Sound Vib. 76: [7] He J. H. (1999 Modified straightforward expansion. Meccanica 34: [8] Xu L. (7 Determination of limit cycle by He s parameter-exping method for strongly nonlinear oscillators. J. Sound Vib. 3: [9] He J. H. (6 Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B (1: [1] Cheung Y. K., Chen S. H., Lau S. L. (1991 A modified Lindstedt-Poincaré method for certain strongly non-linear oscillators. Int. J. Nonlinear Mech. 6(3/4: [11] Hu H. (4 A classical perturbation technique that works even when the linear part of restoring force is zero. J. Sound Vib. 71: [1] Hu H., Xiong Z. G. (4 Comparison of two Lindstedt-Poincaré-type perturbation methods. J. Sound Vib. 78: [13] Lim C. W., Lai S. K., Wu B. S., Sun W. P., Yang Y., Wang C. (9 Application of a modified Lindstedt-Poincaré method in coupled TDOF systems with quadratic nonlinearity a constant external excitation. Arch. Appl. Mech. 79: [14] Pakdemirli M., Karahan M. M. F., BoyacÄś H. (11 Forced vibrations of strongly nonlinear systems with Multiple Scales Lindstedt Poincaré method. Math. Comput. Appl. 16(4: [15] Pakdemirli M., Sarí G. (15 Perturbation solutions of the quintic duffing equation with strong nonlinearities. Comm. Num. Anal. 15(1: [16] Pakdemirli M., Sarí G. (15 Solution of quadratic nonlinear problems with Multiple Scales Lindstedt Poincaré method. Math. Comput. Appl. (: [17] Hegazy U. H.. (9 Single-mode response of control of a hinged-hinged flexible beam. Arch. Appl. Mech. 79: [18] Ma J. J., Peng J., Gao X. J., Xie L.. (15 Effect of soil-structure interaction on nonlinear response of an inextensional beam on elastic foundation. Arch. Appl. Mech. 85: [19] Tang D., Zhao M. H. Dowell E. H. (14 Inextensible beam plate theory: computational analysis comparison with experiment. J. Appl. Mech. 81(6: [] Pratiher B. (1 Vibration control of a transversely excited cantilever beam with tip mass. Arch. Appl. Mech. 8: [1] Huo Y. L., Wang Z. M. (16 Dynamic analysis of a rotating double-tapered cantilever Timoshenko beam. Arch. Appl. Mech. 86: [] Malatkar P. (3 Nonlinear vibrations of beams plates. Virginia Polytechnic Institute State University, PhD Dissertation. [3] Friswell M. I., Ali S. F., Bilgen O., Adhikari S., Lees A. W., Litak G. (1 Non-linear piezoelectric vibration energy harvesting from a vertical cantilver beam with tip mass. J. Intel. Mat. Syst. Str. 3(13: