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On nonlinear frequency veering and mode localization of a beam with geometric imperfection resting on elastic foundation
Journal of Sound and Vibration 332 (2013) 4641–4655
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On nonlinear frequency veering and mode localization of a beam with geometric imperfection resting on elastic foundation A.A. Al-Qaisia a,n, M.N. Hamdan b a b
Mechanical Engineering Department, Faculty of Engineering and Technology, The University of Jordan, Amman, Jordan Mechanical Engineering Department, College of Engineering, King Faisal University, Ahsaa, Saudi Arabia
a r t i c l e i n f o
abstract
Article history: Received 28 May 2012 Received in revised form 24 March 2013 Accepted 30 March 2013 Handling Editor: W. Lacarbonara Available online 2 May 2013
This work presents an investigation on the effect of an initial geometric imperfection wavelength, amplitude and degree of localization on the in-plane nonlinear natural frequencies veering and mode localization of an elastic Euler–Bernoulli beam resting on a Winkler elastic foundation. The beam is assumed to be pinned–pinned with a linear torsional spring at one end. The effect of the axial force induced by mid-plane stretching is accounted for in the derivation of the mathematical model, due to its known importance and significant effect on the nonlinear dynamic behavior of the beam, as it was proved and presented in earlier investigations. The governing partial differential equation is discretized using the assumed mode method and the resulting nonlinear temporal equation was solved using the harmonic balance method to obtain results for the nonlinear natural frequencies and mode shapes. The results are presented in the form of characteristic curves which show the variations of the nonlinear natural frequencies of the first three modes of vibration, for a selected range of physical parameters like; torsional spring constant, elastic foundation stiffness and amplitude and wavelength of a localized and non-localized initial slack. & 2013 Elsevier Ltd. All rights reserved.
In memory of my friend and colleague Prof. M.N. Hamdan, to whom his sudden passing away is a grave personal and professional loss
1. Introduction The model of an imperfect elastic beam element resting on elastic foundation, which can exhibit frequency curve veering, is used to study the dynamic behavior of a wide range of engineering systems found in, for example, foundation and structural engineering, fluid–structural interaction problems, micro-switches, and sensing devices in MEMS. The free and forced responses of such a beam model, and other structures having similar frequency veering behavior, with various boundary conditions, vertical and axial loading conditions, types of elastic foundations, initial imperfections, and different assumptions about the effect of mid-plane stretching, have been the subject of numerous theoretical, numerical and experimental studies over the years, e.g. [1–23]. A review of the relevant literature can be found in, e.g. [24]. The phenomenon of natural frequency curve veering in elastic beam elements with an initial slack, e.g. shallow arches, have been the subject of many theoretical investigations. It is known that for some dynamical systems two of the system natural frequencies (usually the lowest ones) tend to approach each other as at least one of the physical system parameters
n
Corresponding author. Tel.: +962 798500056. E-mail addresses: [email protected], [email protected] (A.A. Al-Qaisia).
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.03.031
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is varied. Then, depending on other system parameters, the two frequencies may veer away with high local curvature or intersect transversely, e.g. “cross-over” [7]. Studies have shown that the frequency curve-veering can lead to mode localization [2,6,7,9], bifurcation instability [14], localized bucking, initiation of wrinkling, and significant changes of the associated mode shape functions. Due to its practical importance, the problem of dynamic behavior and dynamic stability of various beam systems attached to an elastic foundation with and without an initial rise (slack), e.g. straight beams and shallow arcs, has been the subject of numerous theoretical investigations: reviews of the relevant literature can be found, e.g. in [2,6,7,9,22]. These and other relevant studies indicate that the static and dynamic deflections and natural frequencies of beams resting on elastic foundation and shallow arcs can exhibit complex behavior uncovered by linear theory analysis. These studies also indicate that, in addition to initial slack, the foundation stiffness, lateral static loads, axial force, end support flexibility and differential settlement can have significant effects on the static and dynamic deflection and natural frequencies of such beam systems. Plaut and Johnson [14], studied the effects of elastic foundation, initial rise and initial thrust on the dynamic stability of a shallow arc having a half-sine initial shape, subjected to a lateral static load and hinged at both ends to rigid supports. They described the arc motion by an equation they obtained by modifying the Euler–Bernoulli beam equation of motion derived in [4,17] so as to account for an initial thrust and elastic foundation. They presented a closed form solution for the static deflection only for, and consequently limited their dynamic analysis to, the case where the transverse static load distribution has the same half-sine shape as the assumed initial elastic deflection. Then they used the assumed harmonic in time method to convert the nonlinear partial differential equation to an equivalent boundary nonlinear value problem. To simplify the problem, however, they ignored the effect of the nonlinear term on the system frequency and obtained a solution for the frequency of the linearized boundary value problem independent of motion amplitude. Lacarbonara et al. [7] studied the nonlinear interactions in a hinged–hinged uniform moderately curved beam with a torsional spring at one end. They explained the mechanism of veering using the fact that the beam possesses both symmetric and skew-symmetric modes where the degree of symmetry of a symmetric mode is affected by the magnitude of the attached torsional spring; i.e. the symmetry of a symmetric mode is full when the torsional spring stiffness (spring constant) is zero and is partial when the torsional spring stiffness is not zero. Also the stiffness associated with a symmetric mode is affected by both bending curvature and mid-plane stretching (i.e. is a stretching-bending mode) while the skewsymmetric modes are purely bending and are not affected by the mid-plane stretching. They showed that as the initial rise (i.e. control parameter) is increased and approaches a veering region value, the symmetric fist mode becomes a fundamentally bending mode and thus its frequency remains constant with further increase in the control parameter while at the same time the second skew-symmetric mode undergoes a reverse process whereby it becomes a bendingstretching mode with increasing frequency. They also investigated the one-to-one auto parametric resonance activated in the vicinity of veering of the frequencies of the lowest two modes and results from the nonlinear stretching of the beam centerline. The nonlinear responses and their stability were studied via bifurcation analysis. The sensitivity of the internal resonance detuning “the gap between the veering frequencies” and the linear modal structure are investigated by varying the rise of the beam half-sinusoidal rest configuration and the torsional spring constant. The obtained results have shown that the beam mixed-mode response experienced several bifurcations, including Hopf and homoclinic bifurcations, along with the phenomenon of frequency island generation and mode localization. It is noted that typically only sinusoidal, namely half-sine, imperfection has been considered in the available frequency veering studies. Al-Qaisia and Hamdan [24] studied the nonlinear frequency veering of an elastic Euler–Bernoulli beam resting on a Winkler elastic foundation subjected to a static lateral load and hinged–hinged, with a torsional spring at one end. The beam was assumed to have an initial 1/4 sine shape rise due to a constant differential edge settlement. The beam static deflection is obtained using a combined numerical–analytical procedure which accounts for the induced nonlinear axial force to mid-plane stretching. The authors also studied the influence of frequency curve veering on the primary resonance response of the same Euler–Bernoulli beam in [24] to a uniformly distributed vertical load which varies harmonically with time [25]. They used the assumed single-mode approach to obtain a reduced nonlinear temporal equation of motion about the static equilibrium deflection which contains quadratic and cubic nonlinear terms. They presented results of numerical simulation which indicate that the coefficients of the quadratic and cubic nonlinear terms in the temporal problem can, depending on the selected range of system parameters, vary widely and take positive and negative values, and thus change the number and stability of equilibrium positions as well as the system behavior which can be a hardening or softening type. Most of the available studies on frequency veering of a beam with an initial imperfection have considered the initial imperfection to be a 1/2 sine or a 1/4 sine wave extending over the full length of the beam. Also namely the effect of the amplitude of the initial rise was investigated and the effect of wavelength or localization of the rise was not ignored. From practical point of view periodic and non-periodic full beam length and localized initial imperfections (buckling) occur during, e.g., the manufacturing, handling and assembly of thin metal sheets. Kenny et al. [26] used finite difference and finite element methods to investigate dynamic buckling of an elastic slender Euler–Bernoulli beam with discrete and random initial imperfection subjected to an intense axial load. They expressed the random imperfections as a Fourier series with coefficients defined as Gaussian numbers based on a “preferred” mode analysis presented in [27], where a preferred mode is a dominant waveform which evolves through modal competition during the buckling event. Their analysis showed that unbounded growth of transverse deflection is initiated only when random geometric imperfections are included in the
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model. Wadee [28] investigated effects of generalized initial geometric imperfections on the static localized buckling in sandwich struts where the generalized imperfection was expressed as the product of sinusoidal and secant-hyperbolic functions with parameters defining the degree of imperfection and the number of half-waves in the sinusoidal component of the imperfection. The present work intends to extend two previous studies by the authors [24,25] on frequency veering by considering the effects of the wavelength ( different patterns) and localization of an initial rise on the nonlinear free vibration frequency veering and mode of a beam attached to a linear Winkler foundation described in the subsequent section. The present model formulation and analysis, although in some ways are similar to those in [24,25], consider the above effects and a different set of beam boundary conditions.
2. System description and problem formulation A schematic of the beam under consideration with an arbitrarily localized initial 1/2-sine rise is shown in Fig. 1a, while Fig. 1b–d displays other rise shapes considered in this work, namely; 1/2-sine rise, full sine rise and 1/4 sine rise over the whole length of the beam “full beam span”. Here and without loss of generality, the derivation of the mathematical model will be based on a localized rise, i.e. when the rise is confined to a beam segment between two points x ¼xa and x¼ xb on the beam span. The case where the rise is over the full beam span is treated as a special case of the localized one with xa ¼ 0 and xb ¼ l. Regardless of the rise shape, the beam is assumed to be slender, uniform with length l, cross-sectional area A and area moment of inertia I. It has mass per unit length m, Young's modulus E and is resting on a linear Winkler type elastic foundation of stiffness K. It is assumed that the beam is hinged–hinged with a linear torsional spring Kt at one end. The equation governing moderately large vibrations w0 of the beam about the imperfect rest position can be written as [1,7] m
∂2 w ∂t
2
þ EI
∂4 w EA − l ∂x4
"Z
xb
xa
! ! # 2 1 ∂w 2 ∂w dw0 ∂2 w d w0 þ þ þ Kw ¼ 0 dx 2 ∂x ∂x dx ∂x2 dx2
(1)
4
Introducing the following non-dimensional parameters and variables: K f ¼ Kl =EI; w ¼ w=r, ξ ¼ x=l, ξa ¼ xa =l, ξb ¼ xb =l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 w0 ¼ w0 =r, t ¼ t EI=ml , where r is the radius of gyration of the beam cross-sectional area, Eq. (1) can be rewritten in the
Fig. 1. The schematic of beam resting on Winkler foundation with various types of imperfections (rise shapes): (a) beam with localized imperfection, (b) beam with half-sine imperfection over the whole length, (c) beam with full sine imperfection, and (d) beam with quarter sine imperfection.
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following dimensionless form: ∂2 w ∂4 w þ 4 − ∂t 2 ∂ξ
"Z
ξb
ξa
! ! # 2 1 ∂w 2 ∂w dw0 ∂2 w d w0 þ Kf w ¼ 0 dξ þ þ 2 ∂ξ ∂ξ dξ ∂ξ2 dξ2
(2)
In the present study, the initial rise or imperfection w0 is assumed to be a sinusoid with some degree of localization given by nπξ (3) w0 ¼ R sin ξb −ξa where R is a non-dimensional rise amplitude equals to the actual rise amplitude R divided by the radius of gyration r, i.e. R ¼ R=r, n is the number of half-waves in the sinusoid. Fig. 1 displays different rise shapes of a beam; arbitrarily localized rise, 1/2-sine rise, full sine rise and 1/4 sine rise. Note that in addition to integer values of n this work also considers, for illustrative purposes, the case n¼1/2 which corresponds to a beam with an initial differential edge displacement equal to R (e.g. edge up-left or settlement ) which was studied by the authors in [24] for a beam that has different boundary conditions than the present one. 3. Analysis and solutions 3.1. Nonlinear temporal model The nonlinear integral-partial differential equation (2), can be discretized by assuming w¼
∞
∑ ϕi ðξÞqi ðtÞ
(4)
i ¼ 1;2
where ϕi ðξÞ and qi ðtÞ are the associated linear mode shape of the beam and the generalized coordinates, respectively. Using a simplified single mode approximation w ¼ ϕðξÞqðtÞ and substituting w0 from Eq. (3) into Eq. (2), multiplying by ϕðξÞ integrating from 0 to 1, and for convenience using the abbreviations ϕðξÞ ¼ ϕ; qðtÞ ¼ q, one obtains the following reduced single-mode nonlinear temporal problem: nR hR i o nR hR i 2 o R1 R1 2 R1 1 11 1 ξb dw0 d w0 2 2 € dξ þ K f 0 ϕ2 qdξ ¼ 0 (5) 0 ϕ ϕ″″qdξ þ 0 ϕ qdξ− 0 ϕ 0 2 ϕ′ q dξ ðϕ″qÞdξ − 0 ϕ ξa ϕ′q dξ dξ dξ2 Eq. (5) can be re-arranged and written in the from β0 q€ þ β1 q þ β2 q2 þ β3 q3 ¼ 0;
(6)
where R1 β0 ¼ 0 ϕ2 dξ n h R1 R 1 R ξb dw0 i d2 w0 o R1 dξ þ K f 0 ϕ2 dξ β1 ¼ 0 ϕ ϕ″″dξ− 0 ξa ϕ dξ dξ ϕ dξ2 R 1 R ξb R1 Rξ 2 0 β2 ¼ − 0 ξa 12 ϕ′2 dξ ϕ ddξw20 dξ− 0 ξab ϕ′ dw dξ ϕϕ″dξ dξ R 1 R 1 β3 ¼ − 0 0 12 ϕ′2 dξ ϕϕ″dξ
(7)
3.2. Linear model mode shapes and natural frequencies The mode shapes and the associated natural frequencies can be obtained from the linearized version of the nonlinear integral-partial differential equation (2), which takes the form Z ξb
∂2 w ∂4 w ∂2 w0 ∂w ∂w0 dξ þ K f w ¼ 0 þ 4 − (8) 2 2 ∂t ∂ξ ∂ξ ξa ∂ξ ∂ξ By assuming wðξ; tÞ ¼ ϕ A sinðωtÞ, where ω is unknown natural frequency and using the expression given by Eq. (3) for w0 ðξÞ Eq. (8) leads to the following boundary value problem: 3 Z ξb nπ nπξ nπξ ϕ″″−ðω2 −K f Þ ϕ ¼ −R2 sin ϕ′cos dξ (9) ls ls ls ξa where ls ¼ ξb −ξa . Note that for a perfect beam the right side of Eq. (9) is zero and thus this equation has only a homogenous solution. However because of the initial imperfection the total solution to Eq. (9) has a particular part. It is noted that in previous works [24,25] the beam imperfection was over the full beam span and the analysis was based on the total solution of linear non-homogenous boundary value problem similar to Eq. (9). In the present work, for computational and representation of analysis conveniences, the present beam with localized initial imperfections is divided into initially perfect and initially imperfect segments where solutions of Eq. (9) corresponding to perfect and imperfect segments are combined
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using compatibility of deflection and slope at the segments common boundaries. Without loss of generality, the present beam with localized imperfection is divided into the following three segments: (1) from ξ ¼ 0 to ξa , (2) from ξ ¼ ξa to ξ ¼ ξb and (3) from ξ ¼ ξb to ξ ¼ 1. For the first and third segments Eq. (9) is homogeneous while for the second segment this equation is non-homogeneous, where the homogenous solution for both imperfect and perfect segments has the same general form. The total solution ϕðξÞ of the non-homogeneous linear boundary value ordinary differential equation (9) is given by ϕðξÞ ¼ ϕh ðξÞ þ ϕp ðξÞ, where ϕh ðξÞ is the homogenous part and ϕp ðξÞ is the particular solution. The homogeneous solution, after substituting ϕh ðξÞ ¼ esξ , and using a subscript i for segment identification, is given by ϕih ðξÞ ¼ Ai1 sin αξ þ Ai2 cos αξ þ Ai3 sinh αξ þ Ai4 cosh αξ
(10)
where α ¼ ω2 −K nf , i¼1,2,3 which corresponds to the three segments of the beam, Ai1 , Ai2 , Ai3 and Ai4 are arbitrary twelve constants for all cases of n including the case n¼ 1/2 which corresponds to an up-left as indicated in the previous section, The twelve constants Aij , i¼1,2,3; j¼ 1,2,3,4 are determined from the following twelve boundary conditions; which account for the full beam four boundary condition “at beams end conditions” and eight conditions for the beams two segments “common boundaries between the segments”, i.e. continuity of deflection, slope, bending moment and shear force: ϕ1 ð0Þ ¼ ϕ″1 ð0Þ ¼ 0
(11)
ϕ1 ðξa Þ−ϕ2 ðξa Þ ¼ 0;
ϕ′1 ðξa Þ−ϕ′2 ðξa Þ ¼ 0;
ϕ″1 ðξa Þ−ϕ″2 ðξa Þ ¼ 0;
ϕ‴1 ðξa Þ−ϕ‴2 ðξa Þ ¼ 0
(12)
ϕ2 ðξb Þ−ϕ3 ðξb Þ ¼ 0;
ϕ′2 ðξb Þ−ϕ′3 ðξb Þ ¼ 0;
ϕ″2 ðξb Þ−ϕ″3 ðξb ¼ 0;
ϕ‴2 ðξb Þ−ϕ‴3 ðξb Þ ¼ 0
(13)
ϕ3 ð1Þ ¼ 0 and ϕ″3 ð1Þ þ K r ϕ′3 ð1Þ ¼ 0
(14)
where K r ¼ K t l=EI. The particular solution ϕp ðξÞ to Eq. (9), corresponding to the second beam's segment, takes the form nπξ ϕp ðξÞ ¼ D sin (15) ls where D is a constant obtained by substituting Eq. (15) into Eq. (9) whereby one obtains Rb ðnπ=ls Þ3 R2 a ϕ′2 cosðnπξ=ls Þdξ D¼ α4 −ðnπ=ls Þ4 Thus the total solution ϕ2 ðξÞ ¼ ϕ2h ðξÞ þ ϕp ðξÞ is thus given by
(16)
ϕ2 ðξÞ ¼ A21 sin αξ þ A22 cos αξ þ A23 sinh αξ þ A24 cosh αξ þ D sin
nπξ ls
Substituting Eq. (17) into Eq. (16), one obtains (Z ) ! ls αA21 cos αξ−αA22 sin αξ þ αA23 cosh αξþ ðnπ=ls Þ3 R2 D¼ Þdξ cosðnπξ=l s αA24 sinh αξ þ ðnπ=ls ÞD cosðnπξ=ls Þ α4 −ðnπ=ls Þ4 0
(17)
(18)
The constant D can be obtained and calculated from the following expression, after some mathematical manipulations, which takes the form a5 a6 a7 a8 A21 þ A22 þ A23 þ A24 (19) D¼ ad ad ad ad where
(
)Z
nπξ cosðαξÞcos dξ ls α4 −ðnπ=ls Þ ξa ( )Z ξb R2 ðnπ=ls Þ3 nπξ α sinðαξÞcos dξ a6 ¼ − ls α4 −ðnπ=ls Þ4 ξa ( )Z ξb R2 ðnπ=ls Þ3 nπξ a7 ¼ α coshðαξÞcos dξ ls α4 −ðnπ=ls Þ4 ξa ( )Z ξb R2 ðnπ=ls Þ3 nπξ dξ α sinhðαξÞcos a8 ¼ ls α4 −ðnπ=ls Þ4 ξa ( )Z ξb R2 ðnπ=ls Þ4 ad ¼ α ðcosðnπ=ls ÞÞ2 dξ 4 4 α −ðnπ=ls Þ ξa
a5 ¼
R2 ðnπ=ls Þ3
α 4
ξb
(20)
Applying the twelve boundary conditions given in (11)–(14), yields a system of linear equations for the twelve arbitrary constants; Ai1 , Ai2 , Ai3 and Ai4 . The obtained equation is a homogeneous 12 12 matrix equation where the system natural frequency associated with the calculated mode shape, for a given value of R, n, Kr and Kf, is obtained by setting to zero the
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determinant of the coefficient matrix. Once the linear natural frequency is calculated the corresponding mode shape is then obtained by using Eqs. (10) and (17). 3.3. Nonlinear natural frequencies The expressions of βi given in Eq. (7) contain all of the beam system physical parameters: beam elastic foundation Kf, torsional spring constant Kr, initial rise amplitude R, number of 1/2 waves in the sinusoidal rise n, and the parameters ξa and ξb defining the degree and center of localization of the rise. The calculation of βi for a given value or combination of the physical system parameters were carried out numerically using a developed code on MATLAB Software. For convenience, Eq. (6) can be scaled and rewritten in the following non-dimensional form: q€ þ q þ ε2 q2 þ ε3 q3 ¼ 0
(21)
1=2
where a dot denotes a derivative with respect to T ¼ ðβ1 =β0 Þ t, and ε2 and ε3 are dimensionless coefficients defined as ε2 ¼ β2 =β1 and ε3 ¼ β3 =β1 . It is noted from the definitions of ε2 and ε3 that they are functions of all of beam system physical parameters. In this study the nonlinear natural frequencies of the nonlinear oscillator given in Eq. (21) are obtained using the method of Harmonic Balance (HB). Since the oscillator includes asymmetric nonlinearity “quadratic term q2 ”, an approximate two terms HB takes the form: qðtÞ ¼ A0 þ A1 cosðωtÞ
(22)
where ω is the non-dimensional nonlinear natural frequency. The initial conditions are taken to be qð0Þ ¼ A0 þ A1 ¼ A and _ qð0Þ ¼ 0, where A is the amplitude of the motion. Substituting Eq. (22) and its derivatives into Eq. (21) and balancing coefficients of different harmonics one obtains 3 ε2 (23) A0 1 þ ε2 A0 þ ε3 A20 þ A21 þ A21 ¼ 0 2 2 A1 ð1 þ 2ε2 A0 þ 3ε3 A20 −ω2 Þ þ
3ε3 3 A ¼0 4 1
(24)
The above two coupled nonlinear algebraic equations were, for given physical parameters and amplitude of motion A, solved numerically to obtain results for the nonlinear natural frequency ω of a given mode of vibration. 4. Results and discussion Dynamic behaviors of the beam systems shown in Fig. 1 were analyzed for some selected values of the system physical parameters: beam elastic foundation Kf, torsional spring constant Kr, initial rise amplitude R, different types of imperfections localized and non-localized. Figs. 2–6 display results for the variation of non-dimensional natural frequencies ω versus the beam rise R for localized and non-localized imperfections/initial rise. Results for localized beam rise or imperfection are obtained for different values of Kr ¼0.5 and Kf ¼ 0.2, different locations and different spans, i.e. arbitrary values of ξa and ξb . Here it worth mentioning that many interesting results are obtained and some of which are presented in Figs. 2 and 3. As can be seen from Fig. 2, for (ξa ¼ 0:25, ξb ¼ 0:45), for the lower modes the veering does not occur regardless the degree of the beam rise. The veering phenomenon occur between the 5th and 6th natural frequencies at a beam rise around R≈1:5. Increasing the localization to (ξa ¼ 0:25, ξb ¼ 0:60), Fig. 3, different dynamic behavior is obtained and the veering occur between the 1st and 2nd natural frequencies at R≈1:83, followed by a separation between the two modes. On the other hand, veering between (3rd and 4th) and (4th and 5th) occur at R≈5:5 and R≈7, respectively. As can be seen from Fig. 4, for a localized but with symmetric rise about the beam's center, i.e. (ξa ¼ 0:2, ξb ¼ 0:8) and Kr ¼ 1 and Kf ¼ 0.2 the veering occurs between the 1st and 2nd at a rise of R≈2:0 and between the 2nd and 3rd natural frequencies at a rise of R≈10:3 and between 3rd and 4th natural at a rise of R≈11:8. Here it worth mentioning that the behavior is similar to that for non-localized 1/2 and full sine beam rise. Figs. 5 and 6, display obtained results for the variation of the linear non-dimensional natural frequencies with beam rise R for the 1/2 sine and 1/4 sine shapes, over the whole beams span. As can be seen from these results, for the 1/2 sine rise, the veering between the natural frequencies of the first and second modes occurs when R≈5:5 and those of the second and third modes when R≈12:8. Also, for the quarter sine shape rise, Fig. 6, the veering between the natural frequencies of the first and second modes occurs when R≈10:6, between second and third modes when R≈30:4 and between the third and fourth modes when R≈61:3. The presented results and others but not shown indicated that the veering between the first and second natural frequencies occurs at lower value of beam rise for the full sine shape rise. For comparison purposes, given the same values of Kr ¼1 and Kr ¼0.5, the veering between the first and second modes took place when R≈2:84, R≈3:5, R≈5:5 and R≈10:6, for the full, 3/4, 1/2 and 1/4 sine shape rise of the beam. This may indicate that the interaction between modes
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Fig. 2. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:45, with K r ¼ 0:5 and K f ¼ 0:2.
Fig. 3. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6 , with K r ¼ 0:5 and K f ¼ 0:2.
may take place at moderate values of shape rise like the full sine shape rise, at which vibrational motion may lose its stability. It can be seen that for a given rise shape ( e.g. for a given n), either it is localized or not, as the rise amplitude R is increased, the first natural frequency approach the second one and further increase of the rise amplitude may lead to a second veering between the second and third natural frequencies. These frequency veering zones are called, respectively, the first and second veering zones. The fact that near the frequency veering zone the natural frequencies are almost equal each other and the associated modes shape tend to switch, is demonstrated in Table 1, which shows the associated modes shapes for a localized rise (ξa ¼ 0:25 and ξb ¼ 0:6), Kr ¼0.5, Kf ¼0.2 and for different values of rise amplitude R near the first veering zone in Fig. 3. Other results for the 1/2 sine, 1/4 sine and full sine rise (i.e. different values of the number of half-waves n in a nonlocalized initial rise), but not shown here, have indicated that the associated mode shapes near/at the veering zones have a drastic change also, depending of the rise level R. In general, analysis of the frequency veering phenomenon by a linearized version of the mathematical model (i.e., Eq. (9)) is not sufficient while it is more illuminating to study the dynamic behavior in the vicinity of veering employing the nonlinear theory. For example, the effect of vibration amplitude on the natural frequency is captured only by a nonlinear analysis.
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Fig. 4. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a localized imperfection between ξa ¼ 0:2 and ξb ¼ 0:8, with K r ¼ 1 and K f ¼ 0:2.
Fig. 5. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a half-sine rise with K r ¼ 1:0 and K f ¼ 0:2.
Fig. 6. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a quarter sine rise with K r ¼ 1:0 and K f ¼ 0:5.
Rise, R 1st Mode 0
1
2nd Mode
3rd Mode
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Table 1 The associated mode shapes for localized 1/2 sine shape rise between ξa ¼ 0:25 and ξb ¼ 0:6 and K r ¼ 0:5 and K f ¼ 0:2.
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1.83
1.5
Rise, R 1st Mode
Table 1 (continued )
2nd Mode
3rd Mode
4650
4
2.5
2.12
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4651
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Fig. 7. Variation of the nonlinear non-dimensional natural frequencies ω with the amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1, K r ¼ 0:5, and K f ¼ 0:2.
Fig. 8. Variation of the nonlinear non-dimensional natural frequencies ω with the amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1:5, K r ¼ 0:5, and K f ¼ 0:2.
To have a clear picture about the interaction between modes and natural frequencies in different veering zones, Figs. 7–11 display results for the nonlinear natural frequencies obtained for localized imperfection between (ξa ¼ 0:25 and ξb ¼ 0:6) and for Kr ¼0.5 and Kf ¼0.2. Results in these figures are presented as the variation of the nonlinear natural frequencies of the first three modes versus the amplitude of vibrational motion at a given beam rise R, near the veering zone. As it can be seen from these figures, the third mode of vibrations exhibits a hardening behavior, i.e. the natural frequency increases as the amplitude of motion increases, regardless the beam rise. On the hand, and for a moderate values of the amplitude A less than ≈0:3, the first mode exhibits a softening behavior and further increase of the vibration amplitude A the behavior switches to hardening behavior. These behavior become more obvious, as the beam rise R becomes large enough and approaches the first veering zone, as shown in Figs. 8–11. Also, it is clear that as the beam rise increases the nonlinear natural frequency of the second modes decreases and approaches the nonlinear natural frequency of the first mode, as shown in Fig. 11. This behavior is due to the nonlinear interaction between these modes near the veering zone, i.e. the softening and hardening nonlinearities introduced in the governing equation from the bending stiffness and stretching stiffness. Other results, but not shown, have indicated that in some cases the cross-over may occur between the first and the second nonlinear natural frequencies, but at large enough values of vibration amplitude. This behavior is due to the competing nonlinearities of the beam system, i.e. whether they are of softening or hardening type.
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Fig. 9. Variation of the nonlinear non-dimensional natural frequencies ω with the amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1:7, K r ¼ 0:5, and K f ¼ 0:2.
Fig. 10. Variation of the nonlinear non-dimensional natural frequencies ω with the Amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1:8, K r ¼ 0:5, and K f ¼ 0:2.
Fig. 11. Variation of the nonlinear non-dimensional natural frequencies ω with the amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1:83, K r ¼ 0:5, and K f ¼ 0:2.
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5. Conclusions The present analyses of the nonlinear free vibration of a beam element resting on elastic foundation with different rise shapes indicate that the curves of the nonlinear natural frequencies of the first three modes can show complicated behavior which is not observed using linear theory. In addition to frequency veering of these curves, depending on the system parameters, can exhibit crossovers. Results obtained have shown that, when two frequencies approach each other near the veering point predicted by linear analysis like, for example, the first and second natural frequencies in Fig. 3, these frequencies approach each other as the beam rise R increases toward the veering value, which is around R¼ 1.83. Also, the qualitative behavior of the two frequencies is almost the same as the vibration amplitude increases for a given value of the beam rise, like the first and second natural frequencies in Figs. 7–11. Also, the first and second modes have a softening-type behavior as the vibration amplitude increases to a certain value and the behavior becomes hardening for all modes. Results obtained for the localized beam rise have indicated that the dynamic behavior is complicated enough, depending on the degree of localization, and it needs further, in-depth analysis, which is beyond the scope of this paper. Also, it was shown that the veering between the first and second natural frequencies occurs at lower value of the beam rise for the full sine shape rise and the veering took place at larger values for 3/4, 1/2 and 1/4 sine shape rise of the beam Conflict of interest statement There is no conflict of interest.
Acknowledgment Prof. Al-Qaisia acknowledges the support of the University of Jordan. This research was conducted during a sabbatical leave for the academic year 2010/2011 granted by the University of Jordan. Prof. Hamdan acknowledges the support of King Faisal University. Authors are indebted to reviewers for their valuable comments during the review process of the manuscript. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jsv.2013. 03.031.
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[22] Y. Zhang, K.D. Murphy, Secondary buckling and tertiary states of a beam on a non-linear elastic foundation, International Journal of Non-Linear Mechanics 40 (2005) 795–805. [23] C. Xinzhong, K. Ahsan, Curve veering of eigenvalue loci of bridges with aeroelastic effects”, ASCE Journal of Engineering Mechanics 129 (2003) 146–159. [24] A.A. Al-Qaisia, M.N. Hamdan, Primary resonance response of a beam with a differential edge settlement attached to an elastic foundation, Journal of Vibration and Control 16 (2010) 853–877. [25] A.A. Al-Qaisia, M.N. Hamdan, Non-linear frequency veering in a beam resting on elastic foundation, Journal of Vibration and Control 15 (2009) 1627–1647. [26] K. Kenny, N. Pegg, F. Taheri, Dynamic buckling of a slender beam with geometric imperfections subject to an axial impulse, Finite Elements in Analysis and Design 35 (2000) 227–246. [27] H.E. Lindberg, A.L. Florence, in: H.E. Leipholz, G. Oravas (Eds.), Dynamic Pulse Buckling Theory and Experiment, Martinus Nijhoff Publishers, Dordrecht, Netherlands, 1987. [28] M.A. Wadee, Localized buckling in sandwich struts with pre-existing delaminations and geometrical imperfections, Journal of the Mechanics and Physics of Solids 50 (2002) 1767–1787.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
On nonlinear frequency veering and mode localization of a beam with geometric imperfection resting on elastic foundation A.A. Al-Qaisia a,n, M.N. Hamdan b a b
Mechanical Engineering Department, Faculty of Engineering and Technology, The University of Jordan, Amman, Jordan Mechanical Engineering Department, College of Engineering, King Faisal University, Ahsaa, Saudi Arabia
a r t i c l e i n f o
abstract
Article history: Received 28 May 2012 Received in revised form 24 March 2013 Accepted 30 March 2013 Handling Editor: W. Lacarbonara Available online 2 May 2013
This work presents an investigation on the effect of an initial geometric imperfection wavelength, amplitude and degree of localization on the in-plane nonlinear natural frequencies veering and mode localization of an elastic Euler–Bernoulli beam resting on a Winkler elastic foundation. The beam is assumed to be pinned–pinned with a linear torsional spring at one end. The effect of the axial force induced by mid-plane stretching is accounted for in the derivation of the mathematical model, due to its known importance and significant effect on the nonlinear dynamic behavior of the beam, as it was proved and presented in earlier investigations. The governing partial differential equation is discretized using the assumed mode method and the resulting nonlinear temporal equation was solved using the harmonic balance method to obtain results for the nonlinear natural frequencies and mode shapes. The results are presented in the form of characteristic curves which show the variations of the nonlinear natural frequencies of the first three modes of vibration, for a selected range of physical parameters like; torsional spring constant, elastic foundation stiffness and amplitude and wavelength of a localized and non-localized initial slack. & 2013 Elsevier Ltd. All rights reserved.
In memory of my friend and colleague Prof. M.N. Hamdan, to whom his sudden passing away is a grave personal and professional loss
1. Introduction The model of an imperfect elastic beam element resting on elastic foundation, which can exhibit frequency curve veering, is used to study the dynamic behavior of a wide range of engineering systems found in, for example, foundation and structural engineering, fluid–structural interaction problems, micro-switches, and sensing devices in MEMS. The free and forced responses of such a beam model, and other structures having similar frequency veering behavior, with various boundary conditions, vertical and axial loading conditions, types of elastic foundations, initial imperfections, and different assumptions about the effect of mid-plane stretching, have been the subject of numerous theoretical, numerical and experimental studies over the years, e.g. [1–23]. A review of the relevant literature can be found in, e.g. [24]. The phenomenon of natural frequency curve veering in elastic beam elements with an initial slack, e.g. shallow arches, have been the subject of many theoretical investigations. It is known that for some dynamical systems two of the system natural frequencies (usually the lowest ones) tend to approach each other as at least one of the physical system parameters
n
Corresponding author. Tel.: +962 798500056. E-mail addresses: [email protected], [email protected] (A.A. Al-Qaisia).
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.03.031
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is varied. Then, depending on other system parameters, the two frequencies may veer away with high local curvature or intersect transversely, e.g. “cross-over” [7]. Studies have shown that the frequency curve-veering can lead to mode localization [2,6,7,9], bifurcation instability [14], localized bucking, initiation of wrinkling, and significant changes of the associated mode shape functions. Due to its practical importance, the problem of dynamic behavior and dynamic stability of various beam systems attached to an elastic foundation with and without an initial rise (slack), e.g. straight beams and shallow arcs, has been the subject of numerous theoretical investigations: reviews of the relevant literature can be found, e.g. in [2,6,7,9,22]. These and other relevant studies indicate that the static and dynamic deflections and natural frequencies of beams resting on elastic foundation and shallow arcs can exhibit complex behavior uncovered by linear theory analysis. These studies also indicate that, in addition to initial slack, the foundation stiffness, lateral static loads, axial force, end support flexibility and differential settlement can have significant effects on the static and dynamic deflection and natural frequencies of such beam systems. Plaut and Johnson [14], studied the effects of elastic foundation, initial rise and initial thrust on the dynamic stability of a shallow arc having a half-sine initial shape, subjected to a lateral static load and hinged at both ends to rigid supports. They described the arc motion by an equation they obtained by modifying the Euler–Bernoulli beam equation of motion derived in [4,17] so as to account for an initial thrust and elastic foundation. They presented a closed form solution for the static deflection only for, and consequently limited their dynamic analysis to, the case where the transverse static load distribution has the same half-sine shape as the assumed initial elastic deflection. Then they used the assumed harmonic in time method to convert the nonlinear partial differential equation to an equivalent boundary nonlinear value problem. To simplify the problem, however, they ignored the effect of the nonlinear term on the system frequency and obtained a solution for the frequency of the linearized boundary value problem independent of motion amplitude. Lacarbonara et al. [7] studied the nonlinear interactions in a hinged–hinged uniform moderately curved beam with a torsional spring at one end. They explained the mechanism of veering using the fact that the beam possesses both symmetric and skew-symmetric modes where the degree of symmetry of a symmetric mode is affected by the magnitude of the attached torsional spring; i.e. the symmetry of a symmetric mode is full when the torsional spring stiffness (spring constant) is zero and is partial when the torsional spring stiffness is not zero. Also the stiffness associated with a symmetric mode is affected by both bending curvature and mid-plane stretching (i.e. is a stretching-bending mode) while the skewsymmetric modes are purely bending and are not affected by the mid-plane stretching. They showed that as the initial rise (i.e. control parameter) is increased and approaches a veering region value, the symmetric fist mode becomes a fundamentally bending mode and thus its frequency remains constant with further increase in the control parameter while at the same time the second skew-symmetric mode undergoes a reverse process whereby it becomes a bendingstretching mode with increasing frequency. They also investigated the one-to-one auto parametric resonance activated in the vicinity of veering of the frequencies of the lowest two modes and results from the nonlinear stretching of the beam centerline. The nonlinear responses and their stability were studied via bifurcation analysis. The sensitivity of the internal resonance detuning “the gap between the veering frequencies” and the linear modal structure are investigated by varying the rise of the beam half-sinusoidal rest configuration and the torsional spring constant. The obtained results have shown that the beam mixed-mode response experienced several bifurcations, including Hopf and homoclinic bifurcations, along with the phenomenon of frequency island generation and mode localization. It is noted that typically only sinusoidal, namely half-sine, imperfection has been considered in the available frequency veering studies. Al-Qaisia and Hamdan [24] studied the nonlinear frequency veering of an elastic Euler–Bernoulli beam resting on a Winkler elastic foundation subjected to a static lateral load and hinged–hinged, with a torsional spring at one end. The beam was assumed to have an initial 1/4 sine shape rise due to a constant differential edge settlement. The beam static deflection is obtained using a combined numerical–analytical procedure which accounts for the induced nonlinear axial force to mid-plane stretching. The authors also studied the influence of frequency curve veering on the primary resonance response of the same Euler–Bernoulli beam in [24] to a uniformly distributed vertical load which varies harmonically with time [25]. They used the assumed single-mode approach to obtain a reduced nonlinear temporal equation of motion about the static equilibrium deflection which contains quadratic and cubic nonlinear terms. They presented results of numerical simulation which indicate that the coefficients of the quadratic and cubic nonlinear terms in the temporal problem can, depending on the selected range of system parameters, vary widely and take positive and negative values, and thus change the number and stability of equilibrium positions as well as the system behavior which can be a hardening or softening type. Most of the available studies on frequency veering of a beam with an initial imperfection have considered the initial imperfection to be a 1/2 sine or a 1/4 sine wave extending over the full length of the beam. Also namely the effect of the amplitude of the initial rise was investigated and the effect of wavelength or localization of the rise was not ignored. From practical point of view periodic and non-periodic full beam length and localized initial imperfections (buckling) occur during, e.g., the manufacturing, handling and assembly of thin metal sheets. Kenny et al. [26] used finite difference and finite element methods to investigate dynamic buckling of an elastic slender Euler–Bernoulli beam with discrete and random initial imperfection subjected to an intense axial load. They expressed the random imperfections as a Fourier series with coefficients defined as Gaussian numbers based on a “preferred” mode analysis presented in [27], where a preferred mode is a dominant waveform which evolves through modal competition during the buckling event. Their analysis showed that unbounded growth of transverse deflection is initiated only when random geometric imperfections are included in the
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model. Wadee [28] investigated effects of generalized initial geometric imperfections on the static localized buckling in sandwich struts where the generalized imperfection was expressed as the product of sinusoidal and secant-hyperbolic functions with parameters defining the degree of imperfection and the number of half-waves in the sinusoidal component of the imperfection. The present work intends to extend two previous studies by the authors [24,25] on frequency veering by considering the effects of the wavelength ( different patterns) and localization of an initial rise on the nonlinear free vibration frequency veering and mode of a beam attached to a linear Winkler foundation described in the subsequent section. The present model formulation and analysis, although in some ways are similar to those in [24,25], consider the above effects and a different set of beam boundary conditions.
2. System description and problem formulation A schematic of the beam under consideration with an arbitrarily localized initial 1/2-sine rise is shown in Fig. 1a, while Fig. 1b–d displays other rise shapes considered in this work, namely; 1/2-sine rise, full sine rise and 1/4 sine rise over the whole length of the beam “full beam span”. Here and without loss of generality, the derivation of the mathematical model will be based on a localized rise, i.e. when the rise is confined to a beam segment between two points x ¼xa and x¼ xb on the beam span. The case where the rise is over the full beam span is treated as a special case of the localized one with xa ¼ 0 and xb ¼ l. Regardless of the rise shape, the beam is assumed to be slender, uniform with length l, cross-sectional area A and area moment of inertia I. It has mass per unit length m, Young's modulus E and is resting on a linear Winkler type elastic foundation of stiffness K. It is assumed that the beam is hinged–hinged with a linear torsional spring Kt at one end. The equation governing moderately large vibrations w0 of the beam about the imperfect rest position can be written as [1,7] m
∂2 w ∂t
2
þ EI
∂4 w EA − l ∂x4
"Z
xb
xa
! ! # 2 1 ∂w 2 ∂w dw0 ∂2 w d w0 þ þ þ Kw ¼ 0 dx 2 ∂x ∂x dx ∂x2 dx2
(1)
4
Introducing the following non-dimensional parameters and variables: K f ¼ Kl =EI; w ¼ w=r, ξ ¼ x=l, ξa ¼ xa =l, ξb ¼ xb =l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 w0 ¼ w0 =r, t ¼ t EI=ml , where r is the radius of gyration of the beam cross-sectional area, Eq. (1) can be rewritten in the
Fig. 1. The schematic of beam resting on Winkler foundation with various types of imperfections (rise shapes): (a) beam with localized imperfection, (b) beam with half-sine imperfection over the whole length, (c) beam with full sine imperfection, and (d) beam with quarter sine imperfection.
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following dimensionless form: ∂2 w ∂4 w þ 4 − ∂t 2 ∂ξ
"Z
ξb
ξa
! ! # 2 1 ∂w 2 ∂w dw0 ∂2 w d w0 þ Kf w ¼ 0 dξ þ þ 2 ∂ξ ∂ξ dξ ∂ξ2 dξ2
(2)
In the present study, the initial rise or imperfection w0 is assumed to be a sinusoid with some degree of localization given by nπξ (3) w0 ¼ R sin ξb −ξa where R is a non-dimensional rise amplitude equals to the actual rise amplitude R divided by the radius of gyration r, i.e. R ¼ R=r, n is the number of half-waves in the sinusoid. Fig. 1 displays different rise shapes of a beam; arbitrarily localized rise, 1/2-sine rise, full sine rise and 1/4 sine rise. Note that in addition to integer values of n this work also considers, for illustrative purposes, the case n¼1/2 which corresponds to a beam with an initial differential edge displacement equal to R (e.g. edge up-left or settlement ) which was studied by the authors in [24] for a beam that has different boundary conditions than the present one. 3. Analysis and solutions 3.1. Nonlinear temporal model The nonlinear integral-partial differential equation (2), can be discretized by assuming w¼
∞
∑ ϕi ðξÞqi ðtÞ
(4)
i ¼ 1;2
where ϕi ðξÞ and qi ðtÞ are the associated linear mode shape of the beam and the generalized coordinates, respectively. Using a simplified single mode approximation w ¼ ϕðξÞqðtÞ and substituting w0 from Eq. (3) into Eq. (2), multiplying by ϕðξÞ integrating from 0 to 1, and for convenience using the abbreviations ϕðξÞ ¼ ϕ; qðtÞ ¼ q, one obtains the following reduced single-mode nonlinear temporal problem: nR hR i o nR hR i 2 o R1 R1 2 R1 1 11 1 ξb dw0 d w0 2 2 € dξ þ K f 0 ϕ2 qdξ ¼ 0 (5) 0 ϕ ϕ″″qdξ þ 0 ϕ qdξ− 0 ϕ 0 2 ϕ′ q dξ ðϕ″qÞdξ − 0 ϕ ξa ϕ′q dξ dξ dξ2 Eq. (5) can be re-arranged and written in the from β0 q€ þ β1 q þ β2 q2 þ β3 q3 ¼ 0;
(6)
where R1 β0 ¼ 0 ϕ2 dξ n h R1 R 1 R ξb dw0 i d2 w0 o R1 dξ þ K f 0 ϕ2 dξ β1 ¼ 0 ϕ ϕ″″dξ− 0 ξa ϕ dξ dξ ϕ dξ2 R 1 R ξb R1 Rξ 2 0 β2 ¼ − 0 ξa 12 ϕ′2 dξ ϕ ddξw20 dξ− 0 ξab ϕ′ dw dξ ϕϕ″dξ dξ R 1 R 1 β3 ¼ − 0 0 12 ϕ′2 dξ ϕϕ″dξ
(7)
3.2. Linear model mode shapes and natural frequencies The mode shapes and the associated natural frequencies can be obtained from the linearized version of the nonlinear integral-partial differential equation (2), which takes the form Z ξb
∂2 w ∂4 w ∂2 w0 ∂w ∂w0 dξ þ K f w ¼ 0 þ 4 − (8) 2 2 ∂t ∂ξ ∂ξ ξa ∂ξ ∂ξ By assuming wðξ; tÞ ¼ ϕ A sinðωtÞ, where ω is unknown natural frequency and using the expression given by Eq. (3) for w0 ðξÞ Eq. (8) leads to the following boundary value problem: 3 Z ξb nπ nπξ nπξ ϕ″″−ðω2 −K f Þ ϕ ¼ −R2 sin ϕ′cos dξ (9) ls ls ls ξa where ls ¼ ξb −ξa . Note that for a perfect beam the right side of Eq. (9) is zero and thus this equation has only a homogenous solution. However because of the initial imperfection the total solution to Eq. (9) has a particular part. It is noted that in previous works [24,25] the beam imperfection was over the full beam span and the analysis was based on the total solution of linear non-homogenous boundary value problem similar to Eq. (9). In the present work, for computational and representation of analysis conveniences, the present beam with localized initial imperfections is divided into initially perfect and initially imperfect segments where solutions of Eq. (9) corresponding to perfect and imperfect segments are combined
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using compatibility of deflection and slope at the segments common boundaries. Without loss of generality, the present beam with localized imperfection is divided into the following three segments: (1) from ξ ¼ 0 to ξa , (2) from ξ ¼ ξa to ξ ¼ ξb and (3) from ξ ¼ ξb to ξ ¼ 1. For the first and third segments Eq. (9) is homogeneous while for the second segment this equation is non-homogeneous, where the homogenous solution for both imperfect and perfect segments has the same general form. The total solution ϕðξÞ of the non-homogeneous linear boundary value ordinary differential equation (9) is given by ϕðξÞ ¼ ϕh ðξÞ þ ϕp ðξÞ, where ϕh ðξÞ is the homogenous part and ϕp ðξÞ is the particular solution. The homogeneous solution, after substituting ϕh ðξÞ ¼ esξ , and using a subscript i for segment identification, is given by ϕih ðξÞ ¼ Ai1 sin αξ þ Ai2 cos αξ þ Ai3 sinh αξ þ Ai4 cosh αξ
(10)
where α ¼ ω2 −K nf , i¼1,2,3 which corresponds to the three segments of the beam, Ai1 , Ai2 , Ai3 and Ai4 are arbitrary twelve constants for all cases of n including the case n¼ 1/2 which corresponds to an up-left as indicated in the previous section, The twelve constants Aij , i¼1,2,3; j¼ 1,2,3,4 are determined from the following twelve boundary conditions; which account for the full beam four boundary condition “at beams end conditions” and eight conditions for the beams two segments “common boundaries between the segments”, i.e. continuity of deflection, slope, bending moment and shear force: ϕ1 ð0Þ ¼ ϕ″1 ð0Þ ¼ 0
(11)
ϕ1 ðξa Þ−ϕ2 ðξa Þ ¼ 0;
ϕ′1 ðξa Þ−ϕ′2 ðξa Þ ¼ 0;
ϕ″1 ðξa Þ−ϕ″2 ðξa Þ ¼ 0;
ϕ‴1 ðξa Þ−ϕ‴2 ðξa Þ ¼ 0
(12)
ϕ2 ðξb Þ−ϕ3 ðξb Þ ¼ 0;
ϕ′2 ðξb Þ−ϕ′3 ðξb Þ ¼ 0;
ϕ″2 ðξb Þ−ϕ″3 ðξb ¼ 0;
ϕ‴2 ðξb Þ−ϕ‴3 ðξb Þ ¼ 0
(13)
ϕ3 ð1Þ ¼ 0 and ϕ″3 ð1Þ þ K r ϕ′3 ð1Þ ¼ 0
(14)
where K r ¼ K t l=EI. The particular solution ϕp ðξÞ to Eq. (9), corresponding to the second beam's segment, takes the form nπξ ϕp ðξÞ ¼ D sin (15) ls where D is a constant obtained by substituting Eq. (15) into Eq. (9) whereby one obtains Rb ðnπ=ls Þ3 R2 a ϕ′2 cosðnπξ=ls Þdξ D¼ α4 −ðnπ=ls Þ4 Thus the total solution ϕ2 ðξÞ ¼ ϕ2h ðξÞ þ ϕp ðξÞ is thus given by
(16)
ϕ2 ðξÞ ¼ A21 sin αξ þ A22 cos αξ þ A23 sinh αξ þ A24 cosh αξ þ D sin
nπξ ls
Substituting Eq. (17) into Eq. (16), one obtains (Z ) ! ls αA21 cos αξ−αA22 sin αξ þ αA23 cosh αξþ ðnπ=ls Þ3 R2 D¼ Þdξ cosðnπξ=l s αA24 sinh αξ þ ðnπ=ls ÞD cosðnπξ=ls Þ α4 −ðnπ=ls Þ4 0
(17)
(18)
The constant D can be obtained and calculated from the following expression, after some mathematical manipulations, which takes the form a5 a6 a7 a8 A21 þ A22 þ A23 þ A24 (19) D¼ ad ad ad ad where
(
)Z
nπξ cosðαξÞcos dξ ls α4 −ðnπ=ls Þ ξa ( )Z ξb R2 ðnπ=ls Þ3 nπξ α sinðαξÞcos dξ a6 ¼ − ls α4 −ðnπ=ls Þ4 ξa ( )Z ξb R2 ðnπ=ls Þ3 nπξ a7 ¼ α coshðαξÞcos dξ ls α4 −ðnπ=ls Þ4 ξa ( )Z ξb R2 ðnπ=ls Þ3 nπξ dξ α sinhðαξÞcos a8 ¼ ls α4 −ðnπ=ls Þ4 ξa ( )Z ξb R2 ðnπ=ls Þ4 ad ¼ α ðcosðnπ=ls ÞÞ2 dξ 4 4 α −ðnπ=ls Þ ξa
a5 ¼
R2 ðnπ=ls Þ3
α 4
ξb
(20)
Applying the twelve boundary conditions given in (11)–(14), yields a system of linear equations for the twelve arbitrary constants; Ai1 , Ai2 , Ai3 and Ai4 . The obtained equation is a homogeneous 12 12 matrix equation where the system natural frequency associated with the calculated mode shape, for a given value of R, n, Kr and Kf, is obtained by setting to zero the
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determinant of the coefficient matrix. Once the linear natural frequency is calculated the corresponding mode shape is then obtained by using Eqs. (10) and (17). 3.3. Nonlinear natural frequencies The expressions of βi given in Eq. (7) contain all of the beam system physical parameters: beam elastic foundation Kf, torsional spring constant Kr, initial rise amplitude R, number of 1/2 waves in the sinusoidal rise n, and the parameters ξa and ξb defining the degree and center of localization of the rise. The calculation of βi for a given value or combination of the physical system parameters were carried out numerically using a developed code on MATLAB Software. For convenience, Eq. (6) can be scaled and rewritten in the following non-dimensional form: q€ þ q þ ε2 q2 þ ε3 q3 ¼ 0
(21)
1=2
where a dot denotes a derivative with respect to T ¼ ðβ1 =β0 Þ t, and ε2 and ε3 are dimensionless coefficients defined as ε2 ¼ β2 =β1 and ε3 ¼ β3 =β1 . It is noted from the definitions of ε2 and ε3 that they are functions of all of beam system physical parameters. In this study the nonlinear natural frequencies of the nonlinear oscillator given in Eq. (21) are obtained using the method of Harmonic Balance (HB). Since the oscillator includes asymmetric nonlinearity “quadratic term q2 ”, an approximate two terms HB takes the form: qðtÞ ¼ A0 þ A1 cosðωtÞ
(22)
where ω is the non-dimensional nonlinear natural frequency. The initial conditions are taken to be qð0Þ ¼ A0 þ A1 ¼ A and _ qð0Þ ¼ 0, where A is the amplitude of the motion. Substituting Eq. (22) and its derivatives into Eq. (21) and balancing coefficients of different harmonics one obtains 3 ε2 (23) A0 1 þ ε2 A0 þ ε3 A20 þ A21 þ A21 ¼ 0 2 2 A1 ð1 þ 2ε2 A0 þ 3ε3 A20 −ω2 Þ þ
3ε3 3 A ¼0 4 1
(24)
The above two coupled nonlinear algebraic equations were, for given physical parameters and amplitude of motion A, solved numerically to obtain results for the nonlinear natural frequency ω of a given mode of vibration. 4. Results and discussion Dynamic behaviors of the beam systems shown in Fig. 1 were analyzed for some selected values of the system physical parameters: beam elastic foundation Kf, torsional spring constant Kr, initial rise amplitude R, different types of imperfections localized and non-localized. Figs. 2–6 display results for the variation of non-dimensional natural frequencies ω versus the beam rise R for localized and non-localized imperfections/initial rise. Results for localized beam rise or imperfection are obtained for different values of Kr ¼0.5 and Kf ¼ 0.2, different locations and different spans, i.e. arbitrary values of ξa and ξb . Here it worth mentioning that many interesting results are obtained and some of which are presented in Figs. 2 and 3. As can be seen from Fig. 2, for (ξa ¼ 0:25, ξb ¼ 0:45), for the lower modes the veering does not occur regardless the degree of the beam rise. The veering phenomenon occur between the 5th and 6th natural frequencies at a beam rise around R≈1:5. Increasing the localization to (ξa ¼ 0:25, ξb ¼ 0:60), Fig. 3, different dynamic behavior is obtained and the veering occur between the 1st and 2nd natural frequencies at R≈1:83, followed by a separation between the two modes. On the other hand, veering between (3rd and 4th) and (4th and 5th) occur at R≈5:5 and R≈7, respectively. As can be seen from Fig. 4, for a localized but with symmetric rise about the beam's center, i.e. (ξa ¼ 0:2, ξb ¼ 0:8) and Kr ¼ 1 and Kf ¼ 0.2 the veering occurs between the 1st and 2nd at a rise of R≈2:0 and between the 2nd and 3rd natural frequencies at a rise of R≈10:3 and between 3rd and 4th natural at a rise of R≈11:8. Here it worth mentioning that the behavior is similar to that for non-localized 1/2 and full sine beam rise. Figs. 5 and 6, display obtained results for the variation of the linear non-dimensional natural frequencies with beam rise R for the 1/2 sine and 1/4 sine shapes, over the whole beams span. As can be seen from these results, for the 1/2 sine rise, the veering between the natural frequencies of the first and second modes occurs when R≈5:5 and those of the second and third modes when R≈12:8. Also, for the quarter sine shape rise, Fig. 6, the veering between the natural frequencies of the first and second modes occurs when R≈10:6, between second and third modes when R≈30:4 and between the third and fourth modes when R≈61:3. The presented results and others but not shown indicated that the veering between the first and second natural frequencies occurs at lower value of beam rise for the full sine shape rise. For comparison purposes, given the same values of Kr ¼1 and Kr ¼0.5, the veering between the first and second modes took place when R≈2:84, R≈3:5, R≈5:5 and R≈10:6, for the full, 3/4, 1/2 and 1/4 sine shape rise of the beam. This may indicate that the interaction between modes
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Fig. 2. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:45, with K r ¼ 0:5 and K f ¼ 0:2.
Fig. 3. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6 , with K r ¼ 0:5 and K f ¼ 0:2.
may take place at moderate values of shape rise like the full sine shape rise, at which vibrational motion may lose its stability. It can be seen that for a given rise shape ( e.g. for a given n), either it is localized or not, as the rise amplitude R is increased, the first natural frequency approach the second one and further increase of the rise amplitude may lead to a second veering between the second and third natural frequencies. These frequency veering zones are called, respectively, the first and second veering zones. The fact that near the frequency veering zone the natural frequencies are almost equal each other and the associated modes shape tend to switch, is demonstrated in Table 1, which shows the associated modes shapes for a localized rise (ξa ¼ 0:25 and ξb ¼ 0:6), Kr ¼0.5, Kf ¼0.2 and for different values of rise amplitude R near the first veering zone in Fig. 3. Other results for the 1/2 sine, 1/4 sine and full sine rise (i.e. different values of the number of half-waves n in a nonlocalized initial rise), but not shown here, have indicated that the associated mode shapes near/at the veering zones have a drastic change also, depending of the rise level R. In general, analysis of the frequency veering phenomenon by a linearized version of the mathematical model (i.e., Eq. (9)) is not sufficient while it is more illuminating to study the dynamic behavior in the vicinity of veering employing the nonlinear theory. For example, the effect of vibration amplitude on the natural frequency is captured only by a nonlinear analysis.
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Fig. 4. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a localized imperfection between ξa ¼ 0:2 and ξb ¼ 0:8, with K r ¼ 1 and K f ¼ 0:2.
Fig. 5. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a half-sine rise with K r ¼ 1:0 and K f ¼ 0:2.
Fig. 6. Variation of the non-dimensional natural frequencies ω with the non-dimensional rise R for a quarter sine rise with K r ¼ 1:0 and K f ¼ 0:5.
Rise, R 1st Mode 0
1
2nd Mode
3rd Mode
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Table 1 The associated mode shapes for localized 1/2 sine shape rise between ξa ¼ 0:25 and ξb ¼ 0:6 and K r ¼ 0:5 and K f ¼ 0:2.
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1.83
1.5
Rise, R 1st Mode
Table 1 (continued )
2nd Mode
3rd Mode
4650
4
2.5
2.12
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4651
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Fig. 7. Variation of the nonlinear non-dimensional natural frequencies ω with the amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1, K r ¼ 0:5, and K f ¼ 0:2.
Fig. 8. Variation of the nonlinear non-dimensional natural frequencies ω with the amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1:5, K r ¼ 0:5, and K f ¼ 0:2.
To have a clear picture about the interaction between modes and natural frequencies in different veering zones, Figs. 7–11 display results for the nonlinear natural frequencies obtained for localized imperfection between (ξa ¼ 0:25 and ξb ¼ 0:6) and for Kr ¼0.5 and Kf ¼0.2. Results in these figures are presented as the variation of the nonlinear natural frequencies of the first three modes versus the amplitude of vibrational motion at a given beam rise R, near the veering zone. As it can be seen from these figures, the third mode of vibrations exhibits a hardening behavior, i.e. the natural frequency increases as the amplitude of motion increases, regardless the beam rise. On the hand, and for a moderate values of the amplitude A less than ≈0:3, the first mode exhibits a softening behavior and further increase of the vibration amplitude A the behavior switches to hardening behavior. These behavior become more obvious, as the beam rise R becomes large enough and approaches the first veering zone, as shown in Figs. 8–11. Also, it is clear that as the beam rise increases the nonlinear natural frequency of the second modes decreases and approaches the nonlinear natural frequency of the first mode, as shown in Fig. 11. This behavior is due to the nonlinear interaction between these modes near the veering zone, i.e. the softening and hardening nonlinearities introduced in the governing equation from the bending stiffness and stretching stiffness. Other results, but not shown, have indicated that in some cases the cross-over may occur between the first and the second nonlinear natural frequencies, but at large enough values of vibration amplitude. This behavior is due to the competing nonlinearities of the beam system, i.e. whether they are of softening or hardening type.
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Fig. 9. Variation of the nonlinear non-dimensional natural frequencies ω with the amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1:7, K r ¼ 0:5, and K f ¼ 0:2.
Fig. 10. Variation of the nonlinear non-dimensional natural frequencies ω with the Amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1:8, K r ¼ 0:5, and K f ¼ 0:2.
Fig. 11. Variation of the nonlinear non-dimensional natural frequencies ω with the amplitude for a localized imperfection between ξa ¼ 0:25 and ξb ¼ 0:6, with R ¼ 1:83, K r ¼ 0:5, and K f ¼ 0:2.
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5. Conclusions The present analyses of the nonlinear free vibration of a beam element resting on elastic foundation with different rise shapes indicate that the curves of the nonlinear natural frequencies of the first three modes can show complicated behavior which is not observed using linear theory. In addition to frequency veering of these curves, depending on the system parameters, can exhibit crossovers. Results obtained have shown that, when two frequencies approach each other near the veering point predicted by linear analysis like, for example, the first and second natural frequencies in Fig. 3, these frequencies approach each other as the beam rise R increases toward the veering value, which is around R¼ 1.83. Also, the qualitative behavior of the two frequencies is almost the same as the vibration amplitude increases for a given value of the beam rise, like the first and second natural frequencies in Figs. 7–11. Also, the first and second modes have a softening-type behavior as the vibration amplitude increases to a certain value and the behavior becomes hardening for all modes. Results obtained for the localized beam rise have indicated that the dynamic behavior is complicated enough, depending on the degree of localization, and it needs further, in-depth analysis, which is beyond the scope of this paper. Also, it was shown that the veering between the first and second natural frequencies occurs at lower value of the beam rise for the full sine shape rise and the veering took place at larger values for 3/4, 1/2 and 1/4 sine shape rise of the beam Conflict of interest statement There is no conflict of interest.
Acknowledgment Prof. Al-Qaisia acknowledges the support of the University of Jordan. This research was conducted during a sabbatical leave for the academic year 2010/2011 granted by the University of Jordan. Prof. Hamdan acknowledges the support of King Faisal University. Authors are indebted to reviewers for their valuable comments during the review process of the manuscript. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jsv.2013. 03.031.
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