Nonlinear Dynamics

, Volume 70, Issue 2, pp 1147–1172

Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

  • Ahmad Mamandi
  • Mohammad H. Kargarnovin
  • Salman Farsi
Open Access
Original Paper

DOI: 10.1007/s11071-012-0520-1

Cite this article as:
Mamandi, A., Kargarnovin, M.H. & Farsi, S. Nonlinear Dyn (2012) 70: 1147. doi:10.1007/s11071-012-0520-1

Abstract

In this paper, the Nonlinear Normal Modes (NNMs) analysis for the case of three-to-one (3:1) internal resonance of a slender simply supported beam in presence of compressive axial load resting on a nonlinear elastic foundation is studied. Using the Euler–Bernoulli beam model, the governing nonlinear PDE of the beam’s transverse vibration and also its associated boundary conditions are extracted. These nonlinear motion equation and boundary condition relations are solved simultaneously using four different approximate-analytical solution techniques, namely themethod ofMultiple Time Scales, themethod ofNormal Forms, themethod ofShaw and Pierre, and themethod ofKing and Vakakis. The obtained results at this stage using four different methods which are all in time–space domain are compared and it is concluded that all the methods result in a similar answer for the amplitude part of the transverse vibration. At the next step, the nonlinear normal modes are obtained. Furthermore, the effect of axial compressive force in the dynamic analysis of such a beam is studied. Finally, under three-to-one-internal resonance condition the NNMs of the beam and the steady-state stability analysis are performed. Then the effect of changing the values of different parameters on the beam’s dynamic response is also considered. Moreover, 3-D plots of stability analysis in the steady-state condition and the beam’s amplitude frequency response curves are presented.

Keywords

Beam’s nonlinear dynamics Nonlinear elastic foundation The 3:1 internal resonance Steady-state stability analysis Beam’s frequency response 

1 Introduction

The problem of dynamic analysis of uniform slender beam resting on elastic foundation subjected to axial loading is very common in structural systems under actual operating conditions. Hence, from the past decades the linear and nonlinear analysis of such systems is one of the important topics in many engineering fields, especially civil and railway engineering. More specifically, in the design of structural components in buildings, aircrafts, ships, buried pipes, concrete pavement slabs and bridges etc. One of the most essential types of external forces would be in-plane compressive forces due to either pre-stressing or changes in the environmental conditions such as temperature and moisture. For instance, when surrounding temperature increases, rails, beams, concrete slabs, etc. tend to expand and compressive forces are induced if there are some essential boundary constraints on these elements. Due to these induced in-plane compressive forces, these elements some times may experience buckling phenomenon. The in-plane compressive forces are also present in pre-stressed beams. Moreover, in working conditions these beams are almost resting on a foundation. In practice and from engineering point of view, the most well-known foundations are classified as elastic, viscoelastic, Winkler and Pasternak. In the last decades the combined effects of foundation stiffness and the in-plane compressive forces have attracted the attention of many researchers working in the field of structural analysis.

The cases of three-to-one and one-to-one internal resonances for the nonlinear free vibrations of a fixed-fixed buckled beam about its first post-buckling configuration is studied in Ref. [1]. In this study, the NNMs approach is used by implementing Multiple Time Scales method (MTS) directly to the governing PDE of motion as well as its BCs. In Ref. [2], nonlinear vibrations and instabilities of an elastic beam resting on a nonlinear elastic foundation are investigated using analytical and semi-analytical perturbation methods. Also, Melnikov’s method is applied to determine an algebraic expression for the boundary that separates the safe from the unsafe region in the force parameters space. Dynamics of nonlinear problem of non-uniform beams resting on a nonlinear triparametric elastic linear and nonlinear Winkler and also linear Pasternak foundation is presented in Ref. [3] where the solution is obtained using the analog equation method of Katsikadelis. In Ref. [4], by using the method of perturbation, the governing PDE of static deflection of a general elastically end-restrained non-uniform beam resting on a nonlinear elastic foundation subjected to axial and transverse loads is solved. A new method using the differential quadrature method (DQM) is applied in Ref. [5] for the dynamic analysis of non-uniform beams resting on nonlinear foundations. The obtained results solved using the DQM have excellent agreement with the results solved using the FEM analysis. In Ref. [6], the integro-differential equations of motion governing nonlinear vibrations of a slightly curved beam with immovable simply supported ends resting on a nonlinear elastic foundation are solved using the method of MTS. The amplitude and phase modulation equations are derived for the case of primary resonances. Both free and forced vibrations with damping are investigated. It is found that the effect of curvature is of softening type and the elastic foundation may suppress the softening behavior resulting in a hardening behavior of the nonlinearity. In Ref. [7], the differential equation describing the nonlinear dynamics of a simply supported beam resting on a nonlinear spring foundation with cubic stiffness is analyzed and then discretized using Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms (NFs method). The possibility of the model to exhibit primary, superharmonic, subharmonic and internal resonances has been investigated and the singular perturbation approach is used to study both the free and the forced oscillations of the beam. In Ref. [8], the differential transform method (DTM) is employed to predict induced vibration in pipelines resting on an elastic soil bed via the Euler–Bernoulli and Timoshenko beams models. In Ref. [9], the effects of nonlinear elastic foundation on free vibration of beams is investigated. In Ref. [10], energy-based NNMs methodologies are applied to a canonical set of equations and asymptotic solutions are obtained for computing the resonant nonlinear normal modes (NNMs) for discrete and continuous systems. This work was extended to study the 3:1 resonances in a two-degree-of-freedom system and 3:1 resonance in a hinged-clamped beam. In Ref. [11], a simply-supported beam lying on a nonlinear elastic foundation and a cantilever beam possessing geometric nonlinearities are considered and their partial differential equations are asymptotically solved using a perturbation method by computing the nonlinear normal modes. In Ref. [12], the concept of nonlinear normal mode is used to study localized oscillations for continuous systems of finite and infinite length. Moreover, the implications of nonlinear mode localization on the vibration and shock isolation of periodic flexible structures are discussed. In Ref. [13], an energy-based method is used to obtain the NNMs of a slender beam resting on an elastic foundation with cubic nonlinearity under an axial tensile load. Also, a double asymptotic expansion is performed to capture the boundary layer in the nonlinear mode shape due to the small bending stiffness of the beam. In Ref. [14], a numerical method based on invariant manifold approach is presented to construct NNMs for systems with internal resonances. The computationally-intensive solution procedure is used as a combination of finite difference schemes and Galerkin-based expansion approaches. Moreover, two examples are studied. In the first example an invariant manifold that captures two nonlinear normal modes is constructed for a simple three-degree-of-freedom system. The methodology is then applied to an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. In Ref. [15], systematic methods are developed for generating minimally sized reduced-order models that describe the vibrations of large-scale nonlinear engineering structures. The general approach in the phase space of the system model uses the NNMs that are defined in terms of invariant manifolds. To construct NNMs that are accurate to large amplitudes of vibration, the Galerkin projection method is developed. This approach is extended to construct the NNMs for systems with internal resonance subjected to external excitation. Furthermore, the Galerkin-based construction of the nonlinear normal modes is also applied for a rotating beam. A derivation of nonlinear equations governing the dynamics of an axially loaded beam is studied in Ref. [16] where two load cases are considered; primarily a structure is subjected to a uniformly distributed axial load and in the other case to a thrust force for modeling an offshore riser. The NNMs and nonlinear multi-modes (NMMs) have been constructed by using the method of multiple time scales. The dynamic transverse response of the beam has been calculated by monitoring the modal responses and beam’s mode interactions. Moreover, the FEM analysis has been performed. The comparisons of the dynamical responses are made in terms of time histories, phase portraits and mode shapes. The evidence of the nonlinear normal modes in the forced response of a non-smooth piecewise two-degree-of-freedom system and bifurcations characterized by the onset of superabundant modes in internal resonance conditions is investigated in Ref. [17]. In Ref. [18], the classical Lindstedt–Poincare method for evaluating nonlinear normal modes of a piecewise linear two-degree-of-freedom system is adapted to analyze the nonlinear normal modes of a simple piecewise linear two-degree-of-freedom system representing a beam with a breathing crack. In this study, numerical results obtained by a Poincare map approach show the existence of superabundant normal modes which arise in the unstable interval of the first mode and can be predicted. Moreover, in Ref. [19], the nonlinear normal modes of an oscillating 2-DOF system with non-smooth piecewise linear characteristics which are considered to model a beam with a breathing crack or a system colliding with an elastic obstacle are studied. The considered model having two discontinuity boundaries is non-linearizable and exhibits the peculiar feature of a number of nonlinear normal modes (NNMs) that are greater than the degrees of freedom of the system.

The main objective in the first part of this paper is to derive primarily the PDE of the beam’s motion such that the concept of nonlinear normal modes is implemented. The obtained equations for the system under investigation are analyzed and solved using four different approximate-analytical methods known as the method ofMultiple Time Scales, the method of Normal Forms, the method ofShaw and Pierre and the method ofKing and Vakakis [20, 21, 22, 23, 24]. A parametric study is carried out, using MTS method. So, the effect of the various physical and geometrical parameters of the mathematical model on the nonlinear response of the beam is evaluated. In particular, the relation between the nonlinear natural frequencies and the vibration amplitude is scrutinized. Moreover, the effect of changes on stiffness values of linear and cubic nonlinear parts of elastic foundation and the value of the compressive axial load on the linear and nonlinear dynamic response of the beam are studied.

In the second part of this study, the dynamic behavior and steady-state stability analyses of the system under investigation in the case of three-to-one internal resonance are investigated. The deflection time histories in the case of 3:1 internal resonance condition under variation of different parameters are obtained. Moreover, the surface plot of stable/unstable points of the results in the steady-state condition and the frequency response analysis are accordingly extracted.

Based on the above reviews, for a beam under compressive axial load resting on a nonlinear elastic foundation, it was noticed that so far no study has been distinctly reported on the nonlinear dynamic analysis using different nonlinear normal modes techniques and moreover, the study of steady-state stability analysis in the case of 3:1 internal resonance.

2 Derivation of governing equation of motion

A one-dimensional simply supported Euler–Bernoulli beam of finite length l resting on a nonlinear elastic foundation is shown in Fig. 1. As is seen from the figure, the beam is under a compressive axial load of P at its both ends. The nonlinear governing PDE of motion (EOM) of the beam is [13]:
$$ \rho Aw_{,tt} + EIw_{,xxxx} + Pw_{,xx} + k_{1}w + k_{3}w^{3} = 0 $$
(1)
in which w=w(x,t) is the time-dependent transverse deflection of the beam measured upwards from its equilibrium position, the subscripts (,t) and (,x) stand for the derivative with respect to the time (t) and spatial coordinate (x) to the related order, respectively. Also, ρ is the beam density, A is the cross-sectional area of the beam, I is the beam’s cross-sectional second moment of inertia, E is Young’s modulus, EI is the beam’s flexural rigidity which is constant along the beam’s length, and ρA is the beam’s mass per unit length. Moreover, k1 and k3 are the linear and cubic nonlinear parts of the elastic foundation stiffness (please see [13]). Moreover, boundary conditions (BCs) at both ends of the beam are
$$ w = w_{,xx} = 0\quad \mathrm{at}\ x = 0\ \mathrm{and}\ x = l. $$
(2)
Fig. 1

A uniform hinged-hinged axially loaded beam resting on a nonlinear elastic foundation

To make upcoming analysis more convenient, the following dimensionless variables are defined:
$$ \begin{aligned} \lefteqn{x^{ *} = \frac{x}{l},\qquad t^{ *} = \frac{r_{g}}{l^{2}}\sqrt{\frac{E}{\rho}} t,\qquad w^{ *} = \frac{w}{l},} \\ \lefteqn{r^{ *} = \frac{r_{g}}{l},\qquad P^{ *} = \frac{Pl^{2}}{r_{g}^{2}EA},\qquad k_{1}^{*} = \frac{k_{1}l^{4}}{r_{g}^{2}EA},} \\ \lefteqn{k_{3}^{*} = \frac{k_{3}l^{6}}{r_{g}^{2}EA}} \end{aligned} $$
(3)
where rg is the radius of gyration of the beam’s cross section. Substituting these dimensionless quantities into Eqs. (1) and (2) we obtain By dropping the asterisks, one can rewrite Eqs. (4a) and (4b) as In order to study the effect of nonlinear stiffness term, i.e., k3 in response of the system, we introduce a non-dimensional multiplying parameter ε into Eq. (5), as [13, 20, 21]
$$ w_{,tt} + w_{,xxxx} + Pw_{,xx} + k_{1}w + \varepsilon k_{3}w^{3} = 0. $$
(7)

It is clear that, if ε=0, one deals with linear system and in the case of ε=1, the system is behaving as fully nonlinear. In subsequent sections, we try to solve Eq. (7) under boundary conditions of Eq. (6), using four different nonlinear normal modes (NNMs) methods.

3 Dynamic response with no 3:1 internal resonance [20, 21, 22, 23, 24]

3.1 Solution method 1: the method of Multiple Time Scales (MTS method)

In this method we seek a two-term two-time expansion of the solution for both Eqs. (6) and (7) using the following form:
$$ w(x,t;\varepsilon) = w_{0}(x,T_{0},T_{1}) + \varepsilon w_{1}(x,T_{0},T_{1}) + O\bigl( \varepsilon^{2}\bigr) $$
(8)
where T0=t,T1=εt and Tn=εnt (n=0,1,…). Moreover, the first and second time derivatives in the dimensionless time domain become in which \(D_{n} = \frac{\partial}{\partial T_{n}}\) with n=0,1,…. Substituting Eqs. (8) and (9) into Eqs. (6) and (7), then equating coefficients of the like powers of ε, we obtain: in which the prime over any parameter denotes the derivative with respect to dimensionless variable parameter x to the related order. Now, to obtain the solution for nonlinear mode, we primarily begin with a linear solution for the mth linear mode by setting ε→0. It is obvious that at this stage one obtains the solution of zeroth-order linear problem (ε0) as in which \(i = \sqrt{ - 1}, A_{m}(T_{1})\) and \(\bar{A}_{m}(T_{1})\) are two constant complex conjugates and expression \(\sqrt{2} \sin(m\pi x)\) is the mode shape of vibration for a simply supported beam. Moreover, the corresponding linear circular frequency is given by
$$ \omega_{m}^{2} = k_{1} + m^{4} \pi^{4} - Pm^{2}\pi^{2}\quad \mathrm{with}\ m = 1,2,3,\ldots . $$
(15)
Substituting Eq. (14) into the first-order Eq. (12) yields where C.C. stands for a complex term in conjunction with other terms in Eq. (16). In addition, the prime on the RHS of Eq. (16) indicates the derivative with respect to T1. To determine the solution for the case of the first-order (ε1) problem, we consider By substituting Eq. (17) into Eq. (16), one can then equate the coefficients of exp(3mT0) and \(\exp(i\omega_{m}\*T_{0})\), respectively, on both sides to get
the coefficient of exp(3mT0):
and the coefficient of exp(mT0):
The solution of Eqs. (18) and (19) is given by It is clear that the homogeneous part of Eqs. (20) and (21) has a nontrivial solution. On the other hand, the particular solution of Eqs. (20) and (21) exists only if a solvability condition is satisfied. In this case, the solvability condition demands that the right-hand side of Eq. (20) be orthogonal to the sin(mπx) which is the solution for the homogeneous problem. Hence, the solvability condition yields
$$ 2i\omega_{m}A'_{m}(T_{1}) = - \frac{9}{2}k_{3}A_{m}^{2}(T_{1}) \bar{A}_{m}(T_{1}), $$
(23)
where the amplitude Am(T1) in polar form is defined by
$$ A_{m}(T_{1}) = \frac{1}{2}a_{m}(T_{1})e^{i\beta _{m}(T_{1})}. $$
(24)
Substituting the above into Eq. (23) and separating real and imaginary parts we obtain Then, the final solution of Eqs. (20) and (21) yields By substituting g1 and g2 into Eq. (17) one will obtain the solution for w1. Having on hand the solution for w0 and w1, one can substitute them into Eq. (8) and then, by setting ε=1, the expression for the transverse vibration deflection of the beam to the second order becomes It follows from Eqs. (23) and (27) that the nonlinear natural frequencies (ωNL)m of the beam’s oscillation of the mth nonlinear mode can be given by
$$ (\omega_\mathrm{NL})_{m} = \omega_{m}\biggl( 1 + \frac{9k_{3}}{16\omega_{m}^{2}}a_{m}^{2} \biggr) + \cdots . $$
(28)
In the next step we try to determine the values of critical compressive axial loads out of w(x,t) expression, i.e. Eq. (27). It becomes clear that if the denominator of the second term on the RHS in w(x,t) expression approaches zero, i.e. w(x,t)→∞, this leads us to determine the value of critical compressive axial load of Pcr=10m2π2 with m=1,2,…,n. Furthermore, we have the similar condition if the denominator of the fourth term on the RHS in w(x,t) expression approaches zero, that is, if 9m4π4k1→0, or k1=9m4π4 (m=1,2,…,n). This condition corresponds to the case of 3:1 internal resonance; in other words, in this condition it can be easily proven that ω3m=3ωm which is called a three-to-one internal resonance. In this case, there is a strong coupling between both the mth and 3mth modes and neither of them can be activated without activating the other one.

3.2 Solution method 2: the method of Normal Forms (NFs method)

The general form of the nonlinear partial-differential equation of the beam’s motion, i.e., Eq. (5), can be expressed in the following form:
$$ \ddot{w} + L(w) + N(w, \dot{w}, \ddot{w}) = 0 $$
(29)
where in our case L(w)=wiv+Pw″+k1w and \(N(w, \dot{w}, \ddot{w}) = k_{3}w^{3}\). In this way the boundary conditions, i.e., Eq. (6), are
$$ B(w) + \mathit{BN}(w, \dot{w}, \ddot{w}) = 0 $$
(30)
where the overdot indicates the derivative with respect to t, and in general L and B are linear spatial operators, and N and BN are nonlinear spatial and temporal operators of degree three, respectively. It should be noted that in our problem, BN≡0. We implement an invariant manifold for Eqs. (29) and (30) that suggests the following solution for the motion mth normal mode where in this way the nonlinear nature of equations are converted to the linear behavior: It should be noted that the first and second terms in the RHS of the above relation correspond to the solution of the linear and nonlinear parts, respectively. It further should be noted that the ϕ(x) in Eq. (31) represents the normal mode shape of simply supported beam, that is, \(\phi_{m}(x) = \sqrt{2} \sin(m\pi x)\). It has been proved that the nature of the second term in Eq. (31) is cubic, therefore \(W(x, \zeta_{m}, \bar{\zeta}_{m})\) can be approximated by where H.O.T. stands for terms that are of the fourth or higher order in ζm(t) and \(\bar{\zeta}_{m}(t)\). Moreover, it has to be emphasized that in any way W in Eq. (32) should yield a real value. In addition, the dynamics on this manifold is governed by
$$ \dot{\zeta}_{m}(t) = i\omega_{m}\zeta_{m}(t) + h_{m}\bigl( \zeta_{m}(t),\bar{\zeta}_{m}(t) \bigr) $$
(33)
with
$$ h_{m}\bigl( \zeta_{m}(t),\bar{\zeta}_{m}(t) \bigr) = \varGamma_{m}\zeta_{m}^{2}(t)\bar{ \zeta}_{m}(t), $$
(34)
in which Γm is an unknown coefficient to be determined.
Having on hand the general solution for W, its second derivative can be calculated as Substituting Eq. (35) into Eq. (5) and noting that ωm=k1+m4π4Pm2π2, after some mathematical simplifications one gets Similarly for the boundary condition, that is, Eq. (6) in conjunction with Eq. (31), we have
$$ W = W'' = 0\quad \mathrm{at}\ x = 0\ \mathrm{and}\ x = 1. $$
(37)
Now, by substituting Eq. (32) into Eqs. (36) and (37) and equating the corresponding coefficients of \(\zeta_{m}^{3}(t)\) and \(\zeta_{m}^{2}(t)\bar{\zeta}_{m}(t)\) on both sides, we obtain:
the coefficient of \(\zeta_{m}^{3}(t)\):
the coefficient of \(\zeta_{m}^{2}(t)\bar{\zeta}_{m}(t)\):
The solution of Eqs. (38) and (39) yields It can be easily verified that the nontrivial solution for Eqs. (40) and (41) exists if the following condition prevails:
$$ 2i\omega_{m}\varGamma_{m} = - \frac{9}{2}k_{3}. $$
(43)
Then, the solution for f2(x) accordingly yields
$$ f_{2}(x) = \frac{3\sqrt{2} k_{3}}{16(10m^{4}\pi^{4} - Pm^{2}\pi^{2})}\sin(3m\pi x). $$
(44)
Now, having on hand the solutions for f1(x) and f2(x), one can obtain the final solution for the nonlinear normal mode: To determine the dynamics on the nonlinear normal mode, we substitute the calculated value for the Γm into Eq. (34), and the obtained result into Eq. (33):
$$ \dot{\zeta}_{m}(t) = i\omega_{m}\zeta_{m}(t) + \frac{9ik_{3}}{4\omega_{m}}\zeta_{m}^{2}(t)\bar{\zeta}_{m}(t). $$
(46)
By a close inspection of Eqs. (45) and (46) it becomes clear that if we replace ζm(t) with Amexp(mt), the results yield exactly Eqs. (23) and (27), in which the MTS method has been utilized. Again, Eq. (45) becomes independently singular if k1=9m4π4 or Pcr=10m2π2 with m=1,2,…,n, indicating the cases of the three-to-one internal resonance and the critical compressive axial load of the beam, respectively.

3.3 Solution method 3: the method of Shaw and Pierre (S&P method)

Based on the methodology presented by Shaw and Pierre, one can express Eq. (29) as a system of two partial-differential equations; that is, In their proposed method, if w0(t) and v0(t) represent the displacement and velocity of the system at an arbitrary reference point of x=x0, then for an equilibrium point one can establish [w(x0,t),v(x0,t)]=[0,0]. Furthermore, this reference position should not be taken over any nodal point associated with the nonlinear mode. Then, the entire displacement and velocity fields can be expressed in terms of the dynamics of the reference point x0 as [20, 21]: Based on this technique, the set of independent variables (x,t) is changed to (x,w0(t),v0(t)) and, accordingly, the explicit dependence on t will be eliminated. Consequently, in the new domain and at equilibrium condition, Eqs. (49a) and (49b) will change to In general, Eqs. (50a) and (50b) are known as the compatibility conditions. Moreover, it follows from Eqs. (47)–(50a), (50b) that the dynamics of the nonlinear normal mode is In order to convert the general form of the beam’s PDE of motion, i.e., Eq. (29), we begin with differentiation of Eqs. (49a), (49b) with respect to t: By combining Eqs. (47), (48), (51) and (52), one would get the functional equations as Similarly, for the boundary conditions we have
$$ B(W) + \mathit{BN}\bigl[ W, V, - L(W) - N(W, V) \bigr] = 0. $$
(55)
Now, in our problem we can substitute the form of L(w)=wiv+Pw″+k1w and N(w,v)=k3w3 operators into Eqs. (53) to (54), which yields the following functional equations: and the corresponding boundary conditions become
$$ W = W'' = 0\quad \mathrm{at}\ x = 0\ \mathrm{and}\ x = 1. $$
(58)
To solve Eqs. (56)–(58) subject to the compatibility conditions, i.e., Eqs. (50a) and (50b), we use the expanded form of W and V in terms of w0(t) and v0(t) as following: Noting that the above solutions should satisfy the compatibility conditions, Eqs. (50a) and (50b), thus
$$ \begin{aligned} \lefteqn{a_{1}(x_{0},x_{0}) = 1,\qquad b_{1}(x_{0},x_{0}) = 1,}\\ \lefteqn{a_{i}(x_{0},x_{0}) = 0,\qquad b_{i}(x_{0},x_{0}) = 0\quad \mathrm{for}\ i \ge 2.} \end{aligned} $$
(61)
And by forcing the boundary conditions equation (58) on Eqs. (59) and (60), one gets
$$ a_{i}(x,x_{0}) = a''_{i}(x,x_{0}) = 0\quad \mathrm{at}\ x = 0\ \mathrm{and}\ x = 1. $$
(62)
The rest of the unknown variable coefficients of ai’s and bi’s can be obtained by equating the coefficients of the like powers of \(w_{0}^{m}v_{0}^{n}\), in Eqs. (56)–(57) and using the compatibility conditions, Eq. (61), which yields:
the coefficient of v0:
$$ b_{1}(x,x_{0}) = a_{1}(x,x_{0}), $$
(63)
the coefficient of w0:
the coefficient of \(w_{0}^{2}v_{0}\):
the coefficient of \(v_{0}^{3}\):
$$ b_{3}(x,x_{0}) = a_{3}(x,x_{0}), $$
(66)
the coefficient of \(w_{0}v_{0}^{2}\):
the coefficient of \(w_{0}^{3}\):
The solution of Eq. (64) subject to the compatibility and boundary conditions equations (61) and (62) is
$$ a_{1}(x,x_{0}) = \frac{\sin (m\pi x)}{\sin (m\pi x_{0})}\quad \mathrm{with}\ m = 1, 2, 3, \ldots. $$
(69)
Having on hand a1(x,x0), one can easily eliminate b2(x,x0) and a1(x,x0) from Eqs. (67) and (68) to obtain Again, we seek the solution of Eqs. (70) and (71) that satisfies the compatibility and boundary conditions equations (61) and (62) in the form [20, 21]: Differentiating Eq. (72) up to four times with respect to x, one can find that Substituting Eq. (72) into Eqs. (70) and (71) and using Eq. (73), we obtain
$$ \begin{aligned} \lefteqn{\bigl(78m^{4}\pi^{4} - 2k_{1} - 6m^{2}\pi^{2}P\bigr)c_{1} }\\ \lefteqn{\quad{}+ 2 \bigl(k_{1} + m^{4}\pi^{4} - m^{2} \pi^{2}P\bigr)^{2}c_{2} = - k_{3}, } \\ \lefteqn{6c_{1} + \bigl(74m^{4}\pi^{4} - 6k_{1} - 2m^{2}\pi^{2}P\bigr)c_{2} = 0,} \end{aligned} $$
(74)
whose solution yields
$$ \begin{aligned} \lefteqn{c_{1} = \frac{(3k_{1} - 37m^{4}\pi^{4} + m^{2}\pi^{2}P)k_{3}}{32(9m^{4}\pi^{4} - k_{1})(10m^{4}\pi^{4} - m^{2}\pi^{2}P)} ,} \\ \lefteqn{c_{2} = \frac{3k_{3}}{32(9m^{4}\pi^{4} - k_{1})(10m^{4}\pi^{4} - m^{2}\pi^{2}P)} .} \end{aligned} $$
(75)
After some mathematical manipulation and simplifications one can easily find the solutions for a2(x,x0) and a3(x,x0). Then, having on hand the final answer for a1(x,x0),a2(x,x0) and a3(x,x0), the final solution for the nonlinear normal mode becomes Moreover, in order to have the solution of V, by back-substitution of obtained values of a1(x,x0),a2(x,x0) and a3(x,x0) from Eqs. (69) and (72) into Eq. (65) and using the values of c1 and c2, one gets Substituting for b1(x,x0),b2(x,x0) and b3(x,x0) into Eq. (60) yields It emerges from Eqs. (51), (52) and (5) that the dynamics of the nonlinear mode is given by
$$ \ddot{w}_{0} + \bigl[ W^{iv} + PW'' + k_{1}W + k_{3}W^{3} \bigr]_{x = x_{0}} = 0 $$
(79)
which, upon using Eq. (76), becomes After imposing the method of MTS mentioned in Sect. 3.1 to Eq. (80), we primarily have to begin with an initial solution for w0(T0,T1), leading us to the final solution. It can be proven that this initial solution will be in the form of w0(T0,T1)=εw1(T0,T1)+ε3w3(T0,T1)+⋯. Having on hand this primary solution, we then can substitute it into Eq. (80) and then, by separating the first-order (ε1) and third-order (ε3) of perturbation levels, one respectively yields \(w_{1}(T_{0},T_{1}) = B_{m}(T_{1})e^{i\omega _{m}T_{0}} + \mathrm{C.C.}\) and \(w_{3}(T_{0},T_{1}) = K_{m}(T_{1})e^{3i\omega _{m}T_{0}} + \mathrm{C.C.}\) By back-substitution of the obtained results for w1(T0,T1) and w3(T0,T1) into w0(T0,T1) relation and the result into Eq. (80), now we can impose condition of ε=1. In the next step we equate the coefficient of exp(3mT0) on both sides of the result to obtain the final solution for the amplitude Km(T1) in terms of Bm(T1). Then finally w0(T0,T1) becomes where the solvability condition is given by
$$ 2i\omega_{m}B'_{m}(T_{1}) = - \frac{9k_{3}}{4\sin^{2}(m\pi x_{0})}B_{m}^{2}(T_{1}) \bar{B}_{m}(T_{1}), $$
(82)
in which the amplitude Bm(T1) in polar form is defined by \(B_{m}(T_{1}) = \frac{1}{2}b_{m}(T_{1})e^{i\gamma _{m}(T_{1})}\). Substituting this relation for Bm(T1) into Eq. (82) and separating real and imaginary parts, one obtains \(b_{m}(T_{1}) = \mathrm{const.}\) and \(\gamma_{m}(T_{1}) = \frac{9k_{3}}{32\omega_{m}\sin^{2}(m\pi x_{0})}b_{m}^{2}(T_{1})T_{1}\), with m=1,2,…,n.
Substituting Eq. (81) into Eq. (76) and using Eq. (51), one gets Now, if we take the expression given in the first bracket at the RHS of Eq. (83), i.e., \(\frac{B_{m}(T_{1})}{\sin (m\pi x_{0})} - \frac{3k_{3}[ 3 - 4\sin^{2}(m\pi x_{0}) ]}{320m^{4}\pi^{4}\sin^{3}(m\pi x_{0})}B_{m}^{2}(T_{1})\bar{B}_{m}(T_{1})\), and equal it to \(\sqrt{2} A_{m}(T_{1})\), then Eqs. (82) and (83) yield exactly the same form as was in Eqs. (23) and (27), respectively, in which the MTS method has been used. Again, it is clear that Eqs. (76) and (78) become independently singular if k1=9m4π4 or Pcr=10m2π2 with m=1,2,…,n, indicating the cases of the three-to-one internal resonance and the critical compressive axial load of the beam, respectively.

3.4 Solution method 4: the method of King and Vakakis (K&V method)

Based on the methodology presented by King and Vakakis, by specifying the displacement w0 at the reference point x0, one can express the displacement field w(x,t) in terms of w0(t) only because v0(t) can be expressed in terms of w0(t); that is,
$$ w(x, t) = W\bigl( x, x_{0}, w_{0}(t) \bigr). $$
(84)
In this method, the set of independent variables (x,t) will change to (x,w0(t)), where w0(t) can be thought of as a nonlinear time scale. Therefore, at a specific fixed point x0 it follows from Eq. (84) that
$$ w_{0}(x_{0}, t) = W\bigl( x_{0}, x_{0}, w_{0}(t) \bigr). $$
(85)
Equation (85) is usually called a compatibility condition. Once W(x0,x0,w0(t)) is computed, the dynamics of the nonlinear normal mode can be found from Eq. (29) as
$$ \ddot{w}_{0} = - \bigl[ L(W) + N(W) \bigr]_{x = x_{0}}. $$
(86)
Again, in order to convert the general form of the beam’s PDE of motion, Eq. (29), we begin with differentiation of Eq. (84) with respect to t to yield
$$ \dot{w}(x, t) = \frac{\partial W}{\partial w_{0}}\dot{w}_{0}, $$
(87)
which, upon differentiation with respect to t, yields
$$ \ddot{w} = \frac{\partial W}{\partial w_{0}}\ddot{w}_{0} + \frac{\partial^{2}W}{\partial w_{0}^{2}} \dot{w}_{0}^{2}. $$
(88)
In this method, by employing the principle of total energy and doing some mathematical simplifications, one gets [20, 21]:
$$ \ddot{w}_{0} + \omega_{m}^{2}w_{0} + \varepsilon f(w_{0}) = 0, $$
(89)
in which ωm is the linear frequency of the mth normal mode and ε and f(w0)=N[W(x0,x0,w0(t))] are a non-dimensional parameter and a nonlinear function, respectively defined as before. Hence [20, 21],
$$ \frac{1}{2}\dot{w}_{0}^{2} + \frac{1}{2} \omega_{m}^{2}w_{0}^{2} + \varepsilon F(w_{0}) = \frac{1}{2}\omega_{m}^{2}w_{0}^{ * 2} + \varepsilon F\bigl(w_{0}^{ *} \bigr), $$
(90)
in which \(w_{0}^{ *}\) is the maximum of w0 and F′(w0)=f(w0). From Eq. (90), \(\dot{w}_{0}^{2}\) can be easily found:
$$ \dot{w}_{0}^{2} = \omega_{m}^{2} \bigl(w_{0}^{ * 2} - w_{0}^{2}\bigr) + 2 \varepsilon F\bigl(w_{0}^{ *} \bigr) - 2\varepsilon F(w_{0}). $$
(91)
By combining Eqs. (29), (86), (88) and (91), one yields the singular functional equation: Now, we can replace the expression L(W)+N(W) in Eq. (92) by recalling Eq. (7). The result becomes subject to the boundary conditions given by Eq. (37). The solution of Eq. (93) using this method is given by an approximation to W in a power series of ε as [20, 21]:
$$ W = a_{1}(x,x_{0})w_{0}(t) + \varepsilon W_{1}\bigl( x,x_{0},w_{0}(t) \bigr) + \cdots, $$
(94)
in which a1(x,x0) and W1(x,x0,w0) are unknown functions to be determined later. By applying the compatibility conditions on Eq. (94), we obtain
$$ a_{1}(x_{0},x_{0}) = 1\quad \mathrm{and}\quad W_{1}\bigl( x_{0},x_{0},w_{0}(t) \bigr) = 0. $$
(95)
Furthermore, by applying the boundary conditions given in Eq. (37) to Eq. (94), one gets Now, in order to obtain a solution for W, we substitute Eq. (94) into Eqs. (93) and (37), out of which a general relation is derived. Then, in the first step of deriving complete solution for a1 we equate the coefficient of ε0 on both sides of this general relation, and then an ODE relation is obtained which can be solved simultaneously with the compatibility condition, i.e., Eq. (95). The final result at this stage becomes exactly the same as the one given in Eq. (64). The solution of Eq. (64) subject to the boundary and compatibility conditions in Eqs. (95) and (96) is given by Eq. (69) where ωm is given by Eq. (15).
In the next step of deriving complete set of solutions for W1(x,x0,w0(t)) , by equating the coefficients of ε on both sides of the above general relation and using a similar technique as before, we obtain Solution of Eq. (97) for W1(x,x0,w0) is given by [20, 21]:
$$ W_{1}\bigl( x,x_{0},w_{0}(t) \bigr) = a_{2}(x,x_{0})w_{0}(t) + a_{3}(x,x_{0})w_{0}^{3}(t). $$
(98)
Equation (98), subject to constraint relations specified by Eqs. (95) and (96), and after equating coefficients of the like powers of w0(t) from both sides, yields the following results: Moreover, by substituting Eq. (98) into Eq. (97) and equating coefficients of the like powers of w0(t) from both sides, we have:
the coefficient of \(w_{0}^{3}\):
the coefficient of w0:
We note that Eqs. (100) and (101) are uncoupled. Thus, one can solve for a3(x,x0) first and then solve for a2(x,x0). Note that the uncoupled Eqs. (100) and (101) can be obtained separately by considering the transformations ψ3=a3 and \(\psi_{2} = a_{2} - \omega_{m}^{2}\psi_{3}\) imposed on coupled Eqs. (70) and (71) which were derived by using the method of Shaw and Pierre. Still, one can transform the coupled Eqs. (70) and (71) to uncoupled equations other than the above method if instead of \(a_{2} - \omega_{m}^{2}a_{3}\) one uses a new dependent variable, say ψ1. In this way the final results become exactly the same as the ones obtained by the method of King and Vakakis.
To obtain the final solution for W, we seek for an answer to the Eqs. (100) and (101). To do this, we follow the same steps as described in deriving Eq. (76). In other words, by considering Eq. (99) and substituting Eqs. (72) and (69) into Eqs. (100) and (101) along with Eq. (73), one primarily gets
$$ \begin{aligned} \lefteqn{\bigl(72m^{4}\pi^{4} - 8k_{1}\bigr)c_{2} = - k_{3},} \\ \lefteqn{\bigl(80m^{4}\pi^{4} - 8m^{2} \pi^{2}P\bigr)c_{1} = - 6\omega_{m}^{2}w_{0}^{ * 2}c_{2} } \end{aligned} $$
(102)
yielding the following answers for c1 and c2:
$$ \begin{aligned} \lefteqn{c_{1} = \frac{3\omega_{m}^{2}k_{3}w_{0}^{ * 2}}{32(9m^{4}\pi^{4} - k_{1})(10m^{4}\pi^{4} - m^{2}\pi^{2}P)}\quad \mathrm{and}}\\ \lefteqn{c_{2} = - \frac{k_{3}}{8(9m^{4}\pi^{4} - k_{1})}.} \end{aligned} $$
(103)
Then one gets the final answer for W as Substituting Eq. (104) into Eq. (15), evaluating the result at x=x0 and setting ε=1, one gets the dynamics of the nonlinear mode as Again, Eq. (104) becomes independently singular if k1=9m4π4 or Pcr=10m2π2 with m=1,2,…,n, indicating the cases of the three-to-one internal resonance and the critical compressive axial load, respectively.
Now, in deriving nonlinear normal modes (NNMs), we can qualitatively compare the outcomes of the above four different methods related to Eqs. (5) and (6). By comparison of the above four methods, the following preliminary conclusions are drawn:
  1. (i)

    If in Eqs. (45) and (46) we replace ζm(t) with Am(T1)exp(mt), by a close inspection of obtained results using the method of Normal Forms (NFsmethod) it becomes clear that the outcome yields exactly those obtained from the method ofMultiple Time Scales (see Eqs. (23) and (27)).

     
  2. (ii)

    If we replace the first bracket of RHS of Eq. (83) with \(\sqrt{2} A_{m}(T_{1})\), the obtained results in Eqs. (82) and (83) using the method of Shaw and Pierre will turn exactly to the same expression as given in the method of Multiple Time Scales (MTS) (see Eqs. (23) and (27)).

     
  3. (iii)
    Comparison of the S&P and K&V methods:
    1. (iii-1)

      It can be proven that solution of Eq. (105) using the MTS method up to the second approximation yields the same result as in Eq. (81) derived by the method ofShaw and Pierre if \(B'_{m}(T_{1})\) is replaced by Eq. (82).

       
    2. (iii-2)

      Substituting Eq. (81) obtained by the method ofShaw and Pierre into Eq. (104) obtained by the method of King and Vakakis, considering \(w_{0}^{ * 2}(t) = 4B_{m}(T_{1})\*\bar{B}_{m}(T_{1})\) yields the answer for the W which is the same as given in Eq. (83) obtained by the method of Shaw and Pierre. That is, by following the above steps the answer out of the fourth method yields the answer out of the third method.

       
    3. (iii-3)

      Conversely, we can replace \(w_{0}^{ * 2}(t)\) with \(w_{0}^{2}(t) + \omega_{m}^{ - 2}\dot{w}_{0}^{2}(t)\) in Eqs. (104) and (105) and arrive at the results given in Eqs. (76) and (80). That is, by following the above steps the answer out of the third method yields the answer out of the fourth method.

       
     
  4. (iv)

    Interconnections of all the NNMs methods:

    Referred to all the above findings, it can be observed that the results obtained individually by each method ofK&V, S&P, MTS and NFs become exactly the same if proper replacements or conversions between these methods are imposed.

     

It has to be observed that the final solution of the governing ODE of the beam’s vibration, i.e., Eq. (5) under given boundary conditions in Eq. (6), using the above four nonlinear normal modes techniques yields the same answer. Now, in order to continue further in-depth investigation of the effect of different parameters on the beam’s dynamics, we prefer to use the solution technique given by the MTS method.

4 Behavioral analysis in the case of 3:1 internal resonance

4.1 Dynamic response

In this section, to have a better insight of what happens in the presence of 3:1 internal resonance, i.e. k1=9m4π4 (m=1,2,…,n), and to construct the nonlinear normal modes in this case using the MTS method, Eqs. (5) and (6) will be considered. Having on hand the results given in Eqs. (10) and (11), the zeroth-order solution in the case of 3:1 internal resonance with n=3m will be taken as Substituting w0 from Eq. (106) into Eq. (12) yields Now, we consider the simply supported boundary condition:
$$ w_{1} = w_{1}^{\prime \prime} = 0\quad \mathrm{at}\ x = 0\ \mathrm{and}\ x = 1. $$
(108)
Based on the methodology given in [24], the solvability conditions of Eq. (107) will be and where in Eqs. (109) and (110) the amplitudes Am(T1), An(T1) in polar form are defined by \(A_{m}(T_{1}) = \frac{1}{2}a_{m}(T_{1})e^{i\beta _{m}(T_{1})}, A_{n}(T_{1}) = \frac{1}{2}a_{n}(T_{1})e^{i\beta _{n}(T_{1})}\). Moreover, the symbol σ known as internal detuning parameter is defined by Solution of Eq. (107) subject to the boundary conditions given in Eq. (108) will be
Substituting Eq. (112) into Eq. (107), using modulation Eqs. (109) and (110), equating the coefficients of the like harmonics of T0, noting that n=3m and replacing ωn with 3ωm, one gets Moreover, substituting Eq. (112) into Eq. (108) and equating the coefficients of the like harmonics of T0 yields It should be noted that the functions f1 to f10 can be analytically obtained out of the above ten uncoupled ODEs. However, for brevity, the detailed calculations are not included here. Having on hand the values of f1 to f10, the nonlinear normal mode w(x,T0,T1) (w=w0+εw1+⋯with ε=1) in the internal resonance condition that tends to the mth and 3mth linear modes can be expressed as

4.2 Steady-state stability analysis

To study the steady-state stability of the solutions in Eqs. (109) and (110) the method of eigenvalues-eigenvectors has to be employed. To do this, we take the amplitudes \(A_{m}(T_{1}) = \frac{1}{2}a_{m}(T_{1})e^{i\beta _{m}(T_{1})}\), \(A_{n}(T_{1}) = \frac{1}{2}a_{n}(T_{1})e^{i\beta _{n}(T_{1})}\) and insert them into Eqs. (109) and (110). Then, by separating real and imaginary parts in the obtained relations, one yields the following differential algebraic equations (DAEs): where
$$ \gamma = \beta_{n} - 3\beta_{m} + \sigma T_{1} \quad \mathrm{where}\ n = 3m, $$
(128)
in which where g1, g2 and g3 are functions of am, an and γ. Again, it is emphasized that the prime over any parameter denotes the first derivative with respect to T1. For the steady-state response, we set \(a'_{m} = a'_{n} = \gamma' = 0\). To check the stability condition of the steady-state solution, we linearize Eqs. (125)–(127) near singular (or steady-state) points. This will lead to a set of linear equations having constant coefficients multiplied by unknown disturbance terms. In other words, this is a typical case known as eigenvalue problem in the form {X′}=[A]{X} in which X=[am,an,γ]T and \(A_{ij} = \frac{\partial g_{i}(a_{m},a_{n},\gamma )}{\partial X_{j}}\) with i,j=1,2 and 3, and Aij is known as the Jacobian matrix. In the next step, \(\operatorname{det} ([\mathbf{A}]-\lambda [\mathbf{I}]) = 0\) is solved, out of which the eigenvalues (λi) of complex nature can be obtained. Now, the stability near singular points can be checked using these eigenvalues.

5 Results and case studies

5.1 Dynamic analysis in the non-internal resonance condition

Based on extracted equation (27) in conjunction with Eq. (28), a computer program has been written using MATLAB software out of which dimensionless linear and nonlinear results for the frequency and beam’s deflection, beam’s midspan time history can be calculated.

Figure 2 illustrates the first outcome of our computer program in which the variation of dimensionless beam nonlinear frequencies, i.e., (ωNL)1, (ωNL)2 and (ωNL)3, are plotted against dimensionless modal amplitude of vibration, a, in which k1=9π4, P=π2 and the k3/k1 ratio is selected to be 10, 50 and 100. It can be seen from the figure that by increasing the value of k3, the value of nonlinear frequency increases accordingly. Moreover, the rate of variation of (ωNL)1 is greater than the other two frequencies. Referring to Eq. (28), if k3=0, one deals with a linear problem in which it is clear that (ωL)i+1>(ωL)i (i=1,2,3,…). However, this not always can be true for a nonlinear problem (k3≠0) in which some modal amplitude range for example (ωNL)1 is greater than (ωNL)2 and (ωNL)3 (see Figs. 2b and c).
Fig. 2

Variation of dimensionless frequencies ωNL ((ωNL)1, (ωNL)2 and (ωNL)3) vs. modal amplitude of vibration a with k1=9π4 and P=π2; (ak3=90π4, (bk3=450π4, (ck3=900π4

In the next step of our approximate-analytical solution, we try to obtain results for the beam’s deflection and time history of beam’s midspan. To do this, we primarily consider a general form for the beam’s deflection at t=0 such as w(x,0)=H(x) yet to be obtained. In other words, H(x) represents the amplitude of the beam deflection at t=0. In order to comply with the solution technique, instead of using H(x) directly we prefer to employ the Fourier Transform Series of such a function in our solution methodology. Hence, after some mathematical simplifications on the Fourier TransformSeries of Eq. (27) with t=0, one gets the following general form: with i=1,2,3,…. For example, in the case of a beam resting on a linear elastic foundation (k3=0) for the first three modes of vibration one gets: \(a_{1} |_{k_{3} = 0, t = 0} = h_{1}/\sqrt{2}\), \(a_{2} |_{k_{3} = 0, t = 0} = h_{2}/\sqrt{2}\) and \(a_{3} |_{k_{3} = 0, t = 0} = h_{3}/\sqrt{2}\), whereas in the case of a beam on a nonlinear foundation (k3≠0), ai (i=1,2,3) can be obtained by solving the following nonlinear coupled algebraic equations:
$$\everymath{\displaystyle} \left\{ \begin{array}{l} \sqrt{2} a_{1} \vert _{k_{3} \ne 0, t = 0} + \frac{3\sqrt{2} k_{3}}{64\omega_{1}^{2}} a_{1}^{3} \bigg \vert _{k_{3} \ne 0, t = 0} = h_{1}, \\[15pt] \sqrt{2} a_{2} \vert _{k_{3} \ne 0, t = 0} + \frac{3\sqrt{2} k_{3}}{64\omega_{2}^{2}} a_{2}^{3} \bigg \vert _{k_{3} \ne 0, t = 0} = h_{2}, \\[15pt] \sqrt{2} a_{3} \vert _{k_{3} \ne 0, t = 0} + \frac{3\sqrt{2} k_{3}}{64\omega_{3}^{2}} a_{3}^{3} \bigg \vert _{k_{3} \ne 0, t = 0}\\[15pt] \quad = h_{3} - \frac{\sqrt{2} k_{3}}{64} \biggl[ \frac{1}{9\pi^{4} - k_{1}} \\[15pt] \qquad{}+ \frac{3}{10\pi^{4} - \pi^{2}P} \biggr] a_{1}^{3} \bigg \vert _{k_{3} \ne 0, t = 0} \end{array} \right. $$
out of which ai (i=1,2,3) are derived in terms of hi as an arbitrary inputs. Moreover, when one wants to compare the mode shapes, it is customary to normalize all the higher modes with respect to the mode number one. Now, with no restriction on the generality of the solution method, we assume some arbitrary values for hi, such as h2/h1=0.1 and h3/h1=0.001 and hi/h1|i>3=0. Under these conditions the ai are calculated. Therefore the dynamic response of the beam at t=0 will be on hand to get through solution for t>0. Figure 3 shows the deflection of the beam w(x,t) at an instant t=0 for the above-mentioned values of hi, with arbitrary values of k3 while k1=0.18π4 and P=π2. Note that this result for the beam’s deflection is obtained by summation of the nine first modes of vibration.
Fig. 3

Variation of deflection vs. x at an instant t=0 for the known values of hi, arbitrary values of k3, while k1=0.18π4 and P=π2 using nine modes of vibration

Using first nine modes of vibration, Fig. 4 shows the general shape of deflection of the beam over entire length of the beam at an instant t=0.25 for different values of k3=0,10,25,50,100 and 250, while k1=0.18π4 and P=π2. The figure illustrates that when introducing an additional nonlinearity into the system, the absolute value of transverse dynamic deflection of the beam becomes smaller than those obtained for a linear elastic foundation, i.e., when k3=0. Furthermore, it can be seen that by increasing the value of k3 (nonlinear stiffness part) of elastic foundation, the peak value of deflection curve generally decreases. This means that an increase in the value of k3 will cause an increase in the system stiffness.
Fig. 4

Variation of deflection along length of the beam at t=0.25 (w(x,t=0.25)) for different values of k3=0,10,25,50,100 and 250, with k1=0.18π4 and P=π2

Figure 5a shows the time history for deflection of a point at the middle of the beam resting on a nonlinear foundation w(0.5,t) vs. time t (0<t<0.5) for different values of k3=0,10,25,50,100 and 250 while k1=0.18π4 and P=π2, using mode summation of the first nine modes. From Fig. 5a it can be seen that by increasing the value of nonlinear stiffness of the elastic foundation the maximum dynamic deflection of the beam remains almost the same but at a different time corresponding to the specified k3. In addition, referring to Fig. 5b it can be said that an increase in the value of k3 will induce a higher vibration frequency (or smaller period). It should be noted that the same trend for variation of the beam deflection holds for t>0.5.
Fig. 5

Time history of the middle-point deflection of the beam w(0.5,t) vs. time t for different values of k3=10,25,50,100 and 250, while k1=0.18π4 and P=π2; (a) Time history for interval 0 < t<0.5, (b) Part (a) magnified for time interval of 0 < t<0.1

Figure 6 shows the general deflection shape of the beam using its first nine modes over its entire length at an instant t=0.25 with P=π2, for different values of k1=(0.18,0.36,0.54,0.72)π4 while k3=0 and 90π4. A close inspection of the figure reveals that for the case of linear foundation (k3=0) by increasing the value of k1, somehow the trend of beam curvature changes its sign. Moreover, for the case of k3=90π4 the peak value of the deflection increases by increasing the values of k1. However, it is worth to mention that this trend might change at other instants.
Fig. 6

Variation of deflection of the beam at t=0.25 where P=π2, for different values of k1 and k3; (- - - - -) k3=0 and (____) k3=90π4

Figure 7 shows deflection of the central point of the beam resting on a foundation, i.e. w(0.5,t) of first nine modes vs. time, with different values of k1=(0.18,0.36,0.54,0.72)π4 and k3=0 and 90π4 while P=π2. It can be seen that by increasing the value of linear stiffness of elastic foundation, i.e. k1, the vibration period decreases, as it could be expected.
Fig. 7

Time history of deflection of the middle-point of the beam w(0.5,t) vs. time t where P=π2, for different values of k1 and k3; (- - - - -) k3=0 and (____) k3=90π4

Figure 8 shows the deflection of the beam resting on an elastic foundation using nine modes of vibration at t=0.25 with k1=0.18π4, in which compressive axial force and nonlinear stiffness values of foundation can vary such as P=(0,0.5,1,1.5,1.9,2)π2 and k3=0 and 90π4, respectively. It can be observed that by increasing the value of k3 the peak values of deflection curve decrease for the same values of compressive axial load P.
Fig. 8

Deflection of the beam resting on an elastic foundation using nine modes at t=0.25 with k1=0.18π4, for different values of P and k3; (- - - - -) k3=0 and (____) k3=90π4

Figure 9 shows a midpoint deflection time history of the nine first modes of vibration of the beam resting on a linear foundation with k1=0.18π4 for different values of P=(0,0.5,1,1.5,1.9,2)π2 and k3=0 and 90π4. It can be seen that by increasing the value of nonlinear stiffness of elastic foundation, i.e. k3, the peak values of deflection curve slightly decrease for the same values of compressive axial load P, where by increasing the value of nonlinear stiffness of elastic foundation, i.e. k3, the frequency of vibrations increases.
Fig. 9

Time history of deflection of the middle-point of the beam w(0.5,t) vs. time t where k1=0.18π4, for different values of P and k3; (- - - - -) k3=0 and (____) k3=90π4

5.2 Results of dynamic and steady-state stability analyses in the case of 3:1 internal resonance

5.2.1 Dynamic response

Based on the dynamic response analysis in the case of 3:1 internal resonance mentioned in Sect. 4.1, a computer program is developed using MATLAB solver package, out of which different results can be obtained by changing different parameters.

Figure 10 shows the variation of k1 vs. σ for different values of P/Pcr=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1 (Pcr=10m2π2 with m=1) with k3=10. It should be mentioned that σ has an upper limit which can be obtained from relation ωm=0 for every value of P. This limiting condition for different values of P is shown by a dashed curve in Fig. 10. On the other hand, k1 should always be physically greater than zero, which corresponds to \(\sigma_{\max} = \min( \sigma|_{\omega _{m} = 0}, \sigma|_{k_{1} = 0})\), where
$$\begin{aligned} \lefteqn{\sigma|_{\omega _{m} = 0} = \sqrt{8m^{2}\pi^{2}( 10m^{2}\pi^{2} - P )}\quad \mathrm{and}}\\ \lefteqn{\sigma|_{k_{1} = 0} = \sqrt{( 3m\pi )^{4} - P( 3m\pi )^{2}}}\\ \lefteqn{\phantom{\sigma|_{k_{1} = 0} =}{} - \sqrt{( m\pi )^{4} - P( m\pi )^{2}}.} \end{aligned} $$
Fig. 10

Variation of k1 vs. σ for different values of P with k3=10 and m=1

Note that in this figure k1res=876.7, which can be obtained from relation k1=9m4π4 with m=1 in the case of 3:1 internal resonance.

Based on the aforementioned developed computer program and by incorporating outcome results extracted from Fig. 10, the midpoint deflection time history of the beam resting on a foundation in time interval 0 ≤tt1, where t1=4π/ω1res (ω1res=k1+π4+2), for different values of k3 when P=0,σ=0 and m=1, is shown in Fig. 11. From the figure it can be seen that by increasing the value of nonlinear stiffness of the elastic foundation, i.e. k3, the peak values of deflection curve slightly increase.
Fig. 11

Deflection of the middle-point of the beam w(0.5,t) vs. time t for different values of k3=0,100 and 1000 with P=0,σ=0 and m=1

Figure 12 shows midpoint deflection time history of the beam resting on a foundation in time interval 0≤tt1 for different values of P/Pcr with k3=100, σ=0,m=1. From this figure it can be observed that by increasing the value of P/Pcr, the peak values of deflection curve shift a bit toward the right.
Fig. 12

Deflection of the middle-point of the beam w(0.5,t) vs. time t for different values of P/Pcr=0,0.1 and 0.2 with k3=100,σ=0,m=1

Figure 13 shows midpoint deflection of the beam resting on a nonlinear foundation in time interval 0≤tt1 for different values of σ with P=0,k3=10 and m=1. From the figure one can see that the effect of detuning parameter σ on the dynamic response is quite significant.
Fig. 13

Deflection of the middle-point of the beam w(0.5,t) vs. time t for different values of σ=−10,−5,0,5,10 with P=0,k3=10 and m=1

Figure 14 shows the general deflection shape of the beam at a time t=4π/ω1res (w(x,t=4π/ω1res)) for different values of k3 while σ=0,P=0 and m=1.
Fig. 14

Variation of deflection along length of the beam at an instant t=4π/ω1res for different values of k3=0,100 and 1000 with σ=0,P=0 and m=1

Figure 15 shows the deflection of the beam resting on a foundation with variable compressive axial force at an instant t=4π/ω1res with σ=0,k3=100 and m=1.
Fig. 15

Variation of deflection along length of the beam at an instant t=4π/ω1res for different values of P/Pcr=0,0.1 and 0.2 with σ=0,k3=100 and m=1

Figure 16 shows the deflection of the beam resting on an elastic foundation at an instant t=4π/ω1res for different values of σ with P=0,k3=10 and m=1. It can be seen that by increasing the absolute value of detuning parameter σ the peak values of the deflection curve increase where the negative values of σ correspond to the higher values for the defection of similar positive σ.
Fig. 16

Variation of deflection along the beam length at an instant t=4π/ω1res for different values of σ=−10,−5,0,5 and 10 with P=0,k3=10 and m=1

5.2.2 Results of steady-state stability analysis

Based on the stability analysis for the steady-state responses conducted in Sect. 4.2 and by incorporating outcome results extracted from Fig. 10, a 3-D variation of amσan for different P/Pcr values with k3=10 and m=1 is shown in Fig. 17. In this figure it should be noted that by increasing the value of compressive axial load P the 3-D surface will tend to shrink where for brevity only surfaces for two values of P/Pcr are shown. Moreover, it should be mentioned that these surfaces are plotted for an>0 and for an<0 similar surfaces exist. In addition, the green areas on these 3-D surface plots represent the unstable regions.
Fig. 17

Variations of the amplitudes an and am vs. detuning parameter σ for different compressive axial load \(\frac{P}{P_{\mathrm{cr}}}= 0\) and 0.2 with k3=10 and m=1. (a) The reference view; (b) View-A w.r.t. the reference view

The three-dimensional surface plot of variation of amplitudes an and am vs. frequency detuning parameter σ for \(\frac{P}{P_{\mathrm{cr}}}= 0.5\), k3=10 and m=1 is depicted in Fig 18. In this figure green regions represent the unstable regions.

Figure 19 is a two-dimensional representation of Fig. 18 in which the variation of nonlinear normal mode amplitude am is shown while detuning parameter σ is changing from −25 to 55 and the nonlinear normal mode amplitude an takes the values of 0.5, 1.16, 3 and 5.6. It should be noted that for the points with specific values of am,an and σ within the green region the unstable condition is prevailing; for the points outside this region the system is in the stable condition.
Fig. 18

Variations of the amplitudes an and am vs. detuning parameter σ with \(\frac{P}{P_{\mathrm{cr}}}= 0.5, k_{3} = 10\) and m=1. (a) The reference view; (b) 180° rotated view w.r.t. the reference view

Similarly to what has been done in Fig. 19, in the next step the variation of nonlinear normal mode amplitude an vs. nonlinear normal mode amplitude am for different values of σ=0,5,10,20,30,40,50 and 60 with k3=10 and m=1 is illustrated in Fig. 20. Again, in this figure it should be noted that for the points with specific values of am,an and σ within the green region the unstable condition is prevailing; for the points outside this region the system is in the stable condition.
Fig. 19

Variation of amplitude am vs. frequency detuning parameter σ for an=0.5,1.16,3 and 5.6, \(\frac{P}{P_{\mathrm{cr}}}= 0.5, k_{3} = 10\) and m=1

Fig. 20

Variation of nonlinear normal mode amplitude an vs. nonlinear normal mode amplitude am for σ=0,5,10,20,30,40,50 and 60 with \(\frac{P}{P_{\mathrm{cr}}}= 0.5, k_{3} = 10\) and m=1

6 Conclusions

The nonlinear PDE governing dynamics of an Euler–Bernoulli beam resting on a nonlinear elastic foundation under axial compressive load is solved by using four different approximate-analytical nonlinear normal modes methods and the outcome results are as follows:
  1. 1.

    For the system under investigation it can be seen that there is the three-to-one internal resonance between the mth and 3mth modes and simultaneously the mth mode critical compressive load is seen accompanied with this internal resonance which is not seen when linear analysis is performed.

     
  2. 2.

    It can be seen that by increasing the values of k3, the values of nonlinear frequencies increase accordingly. Moreover, the trend of (ωNL)1 variation is greater than the other \((\omega_{\mathrm{NL}})_{n| _{n > 1}}\).

     
  3. 3.

    In the no 3:1 internal resonant condition, when one introduces an additional nonlinearity into the system, for example k3, the absolute value of the dynamic deflection of the beam generally becomes smaller than those obtained if k3=0.

     
  4. 4.

    Any increases in the value of k3 will produce higher vibration frequency (or smaller period) for the case of no 3:1 internal resonant condition. In other words, as the nonlinearity k3 increases, the contributions of 3mth mode become pronounced.

     
  5. 5.

    In the case of no 3:1 internal resonant condition and at the same level of compressive axial load P, it is observed that by increasing the value of k3 the peak value of deflection curve slightly decreases.

     
  6. 6.

    In the 3:1 internal resonant condition it can be observed that by increasing the absolute value of detuning parameter, σ, the peak value of the deflection curves increases where the similar negative values of σ yield to the higher values of the deflection.

     
  7. 7.

    In the case of 3:1 internal resonance condition from NNMs steady-state analysis, the (σ,am,an) 3-D surface will tend to shrink in a nonlinear fashion as the value of compressive axial load P increases.

     

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Copyright information

© The Author(s) 2012

Authors and Affiliations

  • Ahmad Mamandi
    • 1
  • Mohammad H. Kargarnovin
    • 2
  • Salman Farsi
    • 3
  1. 1.Department of Mechanical Engineering, Parand BranchIslamic Azad UniversityTehranIran
  2. 2.Department of Mechanical EngineeringSharif University of TechnologyTehranIran
  3. 3.Department of Mechanical EngineeringTarbiat Modares UniversityTehranIran

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