# Temperature Effects on Nonlinear Vibration Behaviors of Euler-Bernoulli Beams with Different Boundary Conditions

Authors: Yaobing Zhao; Chaohui Huang
Journal: Shock and Vibration
 (2018)

---

## Abstract

This paper is concerned with temperature effects on the modeling and vibration characteristics of Euler-Bernoulli beams with symmetric and nonsymmetric boundary conditions. It is assumed that in the considered model the temperature increases/decreases instantly, and the temperature variation is uniformly distributed along the length and the cross-section. By using the extended Hamilton’s principle, the mathematical model which takes into account thermal and mechanical loadings, represented by partial differential equations (PDEs), is established. The PDEs of the planar motion are discretized to a set of second-order ordinary differential equations by using the Galerkin method. As to three different boundary conditions, eigenvalue analyses are performed to obtain the close-form eigenvalue solutions. First four natural frequencies with thermal effects are investigated. By using the Lindstedt-Poincaré method and multiple scales method, the approximate solutions of the nonlinear free and forced vibrations (primary, super, and subharmonic resonances) are obtained. The influences of temperature variations on response amplitudes, the localisation of the resonance zones, and the stability of the steady-state solutions are investigated, through examining frequency response curves and excitation response curves. Numerical results show that response amplitudes, the number and the stability of nontrivial solutions, and the hardening-spring characteristics are all closely related to temperature changes. As to temperature effects on vibration behaviors of structures, different boundary conditions should be paid more attention.


---

## Body

## 1. Introduction

Due to the importance in many applications in many fields, such as the industrial, civil, mechanical, automotive, aerospace, and other structural systems, a flexible beam with nonlinear characteristics has attracted more attention in the past few years. The linear and nonlinear vibration characteristics have been investigated for many years and were reviewed, e.g., by Nayfeh and Mook [1], Nayfeh and Balachandran [2], Nayfeh and Pai [3], Luongo and Zulli [4], and Lacarbonara et al. [5].On the one hand, it is noted that beams in compression are subjected to buckle, and once the beam buckles, the beam exhibits an initially deflected static equilibrium position and looks like an arch. Due to the initial curvature, the nonlinear vibration behaviors become much more complicated [6–8]. The nonlinear dynamics of buckled beams have been investigated by many researchers. As to these buckled and postbuckling beams, natural frequencies, mode shapes, nonlinear normal modes, and nonlinear resonance responses were investigated through theoretical and experimental methods [9–16]. The geometrically nonlinear vibration of an aluminium beam hinged at both ends was investigated experimentally by Ribeiro and Carneiro [17]. The nonlinear vibration analysis of a curved beam subjected to the uniform base harmonic excitation with both quadratic and cubic nonlinearities was investigated by Huang et al. [18].On the other hand, in many applications, these structures are often subjected to vibration under thermal and dynamic loadings [19]. Temperature fields develop thermal stress due to the thermal expansion or contraction which influences the dynamic behavior of mechanical systems. The literature on thermal vibrations of mechanical structures is quite abundant; here, only a few papers are reviewed. The dynamic instability of a pinned beam subjected to an alternating magnetic field and thermal loads with the nonlinear strain has been studied by Wu [20, 21], and effects of frequency ratio, loading factor and amplitude, damping factor, and temperature changes on the vibration behaviors have been illustrated. Manoach and Ribeiro [22, 23] studied geometrically nonlinear vibrations of moderately thick and curved beams under the combined action of mechanical and thermal loads. The coupled thermoelastic vibration characteristics of the axially moving beam were investigated by Guo et al. [24]. Treyssède [25] proposed an analytical model for the vibration analysis of horizontal beams that are self-weighted and thermally stressed. Moreover, due to the relatively high possible temperature variations in beams, Avsec and Oblak [26] have developed the mathematical model where fundamental thermomechanical properties of state are functions of temperature. Thermomechanical vibrations of a simply supported spring-mass-beam system were investigated analytically by Ghayesh et al. [27]. Three-dimensional nonlinear motion characteristics of the perfect and imperfect Timoshenko microbeams under mechanical and thermal forces have been examined numerically by Farokhi and Ghayesh [28]. Large amplitude vibrations and regular and chaotic oscillations of a Timoshenko beam under the influence of temperature were analysed by Warminska et al. [29, 30], and mechanical and thermal loadings have been discussed. Moreover, nonlinear vibration characteristics of cross-ply composite plates in thermal environments were investigated by Settimi et al. [31] and Saetta et al. [32]. Recently, based on Hamilton’s principle, temperature effects on the vibration behaviors of the cable-stayed-beam by introducing two nondimensional factors were investigated by Zhao et al. [33]. The nonlinear free and forced vibration characteristics of the suspended cable were studied by Zhao et al. [34, 35].Furthermore, the nonlinear vibration behavior of the system is closely related to its boundary conditions. For example, the nonlinear free and forced vibrations of a beam-mass system under five different boundary conditions were investigated by Ozkaya et al. [36]. Large amplitude vibrations of rectangular plates subjected to the radical harmonic excitation were investigated by Amabili [37], and three boundary conditions were considered. The nonlinear dynamic response of an inclined Timonshenko beam with different boundary conditions subjected to a varying mass with variable velocity was investigated by Mamandi et al. [38]. The nonlinear free vibrations of a two-layer elastic composite beam were studied by Lenci et al. [39], and different boundary conditions, both symmetric (e.g., free-free, fixed-fixed, and hinged-hinged) and nonsymmetric (e.g., free-fixed and hinged-fixed) with respect to the beam midpoint, were investigated. Furthermore, by using a unified approach, the planar nonlinear vibrations of the shear indeformable beams with either movable or immovable supports were investigated by Luongo et al. [40].One of the motivations of this study is that changes in the vibration characteristics of the structure due to the damage may be smaller than changes in ones due to variations in temperature [41]. Therefore, an accurate mathematical model of the beam with thermal effects is necessary to predict its vibration characteristics. To the best of our knowledge, no specific study has addressed temperature effects on the nonlinear free and forced oscillations of the beam with different boundary conditions. The structure of the paper is organized as follows: in Section 2, the governing equations of an Euler-Bernoulli beam by using the extended Hamilton’s principle are derived. In addition, the PDEs of the planar motion are discretized to the second-order ordinary equations via the Galerkin method. Considering three different boundary conditions, eigenvalue analyses are performed to obtain close-form eigenvalue solutions. Then, the Lindstedt-Poincaré method and multiple scales method are adopted to obtain the approximate solutions of the nonlinear free and forced resonance responses in Section 3. Some numerical results and discussions are given to illustrate the influences of temperature variations on response amplitudes, localisation of the resonance zones, and the stability of the nontrivial solutions in Section 4. Finally, at the end of the paper (Section 5), some conclusions are drawn.


## 2. Mathematics for Nonlinear Modeling

### 2.1. Equations of Motion

In this study,u and v denote the axial and transverse deflections of the beam, respectively, x is the distance along the undeflected beam, t is the time, L is the beam length, m=ρA is the beam’s mass per unit, E is Young’s modulus, A is the area of cross-section, I is the area moment of inertia, P is the applied axial compressive load, cu and cv are the viscous damping coefficients per unit length, ΔT is the temperature variation, and α is the thermal expansion coefficient.The linear relation between the temperature variation and the stress-temperature coefficientγ is(1)γΔT=EαΔT,and the nonlinear relation is expressed as [21](2)γΔT=EαΔT+h~α2ΔT2.Here, the first linear relation is adopted for the sake of simplicity. By assuming the Lagrangian strain as the strain measure and introducing temperature variations, the strain in the beam motion is obtained(3)ε=∂u∂x+12∂v∂x2-γE.The extended Hamilton’s principle is given by(4)∫t1t2δWe+δT-δUdt=0,where T is the kinetic energy, U is the elastic energy, and We is the work done by external loads on the system.By using Hamilton’s principle, one obtains [1, 20, 21](5)ρA∂2u∂t2+cu∂u∂t-A∂∂xE∂u∂x+12∂v∂x2-γ=0,(6)ρA∂2v∂t2+cv∂v∂t+EI∂4v∂x4-A∂∂x∂v∂xE∂u∂x+12∂v∂x2-γ+P∂2v∂x2=Fxcos⁡Ωt,where F(x) is the spatial distribution of the harmonic load in the v direction and Ω is the excitation frequency.Introducing the quasistatic assumptions, the acceleration and velocity terms in (5) are neglected [20, 21], and we obtain(7)A∂∂xE∂u∂x+12∂v∂x2-γ=0,where(8)∂u∂x+12∂v∂x2-γE=constant=δ,where δ is the average strain of the systems, and one obtains(9)δ=1L∫0L∂u∂x+12∂v∂x2-γEdx=12L∫0L∂v∂x2dx-γE.Therefore, substituting (9) into (6), we obtain the following nonlinear partial differential equation of motion without considering the torsion and shear rigidities:(10)EI∂4v∂x4+ρA∂2v∂t2+cv∂v∂t+Aγ∂2v∂x2+P∂2v∂x2=EA2L∂2v∂x2∫0L∂v∂x2dx+Fxcos⁡Ωt.For the sake of simplicity, the following nondimensional quantities are introduced:(11)v∗=vL,x∗=xL,t∗=tPρAL2,vf=EIPL2,γ∗=AγP,v1=EAP,F∗=FLP,Ω∗=ΩρAL2P,cv∗=cvL2ρAP.Substituting (11) into (10) and neglecting the asterisks, one obtains(12)vf2∂4v∂x4+∂2v∂t2+cv∂v∂t+∂2v∂x2+γ∂2v∂x2=12v12∂2v∂x2∫01∂v∂x2dx+Fxcos⁡Ωt.


### 2.2. Linear Analysis

Dropping the damping, excitation, and nonlinear terms in (12), the following linear equation is obtained:(13)vf2∂4v∂x4+∂2v∂t2+γ+1∂2v∂x2=0,and one lets(14)vx,t=ϕxeiωt,where i denotes the imaginary unit, ϕ(x) is the normalized undamped mode shape, and ω is the natural frequency.Substituting (14) into (13), one obtains(15)vf2∂4v∂x4+γ+1∂2v∂x2-ω2ϕ=0.The corresponding eigenvalue equation is expressed as follows:(16)vf2D4+γ+1D2-ω2=0.Accordingly, one lets(17)θ=1+γ2+4vf2ω2+1+γ2vf2,β=1+γ2+4vf2ω2-1+γ2vf2.The homogeneous solution of (16) is given by(18)ϕx=c1cos⁡θx+c2sin⁡θx+c3cosh⁡βx+c4sinh⁡βx.In the following sections, three different symmetric and nonsymmetric boundary conditions are chosen and investigated: hinged-hinged, hinged-fixed, and fixed-fixed.Case A (hinged-hinged).
As to the hinged-hinged boundary conditions, we have(19)ϕ0=ϕ1=0,∂2ϕ∂x2x=0=∂2ϕ∂x2x=1=0.
Substituting (19) into (18), we obtain the following coefficient matrix:(20)1010cos⁡θsin⁡θcosh⁡βsinh⁡β-θ20β20-θ2cos⁡θ-θ2sin⁡θβ3cosh⁡ββ3sinh⁡βwhere the determinant of the coefficient matrix in (20) equals zero; we have(21)θ2+β3sin⁡θsinh⁡β=0.
Hence, the natural frequency of the beam with hinged-hinged boundary conditions could be obtained by solving the following equation:(22)sin⁡θ=0,and the corresponding mode shapes are(23)ϕix=Cisin⁡θi,where Ci is determined by normalization conditions.Case B (hinged-fixed).
As to the hinged-fixed boundary conditions, we have(24)ϕ0=ϕ1=0,∂2ϕ∂x2x=0=∂ϕ∂xx=1=0.
Substituting (24) into (18), the coefficient matrix is(25)1010cos⁡θsin⁡θcosh⁡βsinh⁡β-θ20β20-θsin⁡θθcos⁡θβsinh⁡ββcosh⁡β
Similarly, the following equation is obtained:(26)-θ2+β2-βcosh⁡βsin⁡θ+θcos⁡θsinh⁡β=0.
Hence, the natural frequency could be obtained by solving the following transcendental equation:(27)-βcosh⁡βsin⁡θ+θcos⁡θsinh⁡β=0,with the following mode shapes:(28)ϕix=Cisinh⁡βisin⁡θix-sin⁡θisinh⁡βixsin⁡θi,where Ci is also determined by normalization conditions.Case C (fixed-fixed).
As to the fixed-fixed boundary conditions, we have(29)ϕ0=ϕ1=0,∂ϕ∂xx=0=∂ϕ∂xx=1=0.
Similarly, the coefficient matrix is(30)1010cos⁡θsin⁡θcosh⁡βsinh⁡β0θ0β-θsin⁡θθcos⁡θβsinh⁡ββcosh⁡βand we obtain the transcendental equation as(31)
-
2
θ
β
+
2
θ
β
cos
⁡
θ
cosh
⁡
β
+
θ
2
-
β
2
sin
⁡
θ
sinh
⁡
β
=
0
,where the natural frequency could be obtained, and the mode shapes are(32)ϕix=Cicosh⁡βix-cos⁡θixβisin⁡θi-θisinh⁡βiθicosh⁡βi-θicos⁡θi+sinh⁡βix-βiθisin⁡θi,where Ci is also determined by normalization conditions.


### 2.3. Discrete Model

The Galerkin procedure is used to discretize (12) into a set of second-order ordinary differential equations(33)vx,t=∑n=1∞ϕnxqnt,where qn(t) is the generalized coordinate and ϕn(x) is the linear vibration mode shape.Substituting (33) into (12), multiplying the result by ϕn(x), integrating the outcome over the domain, and finally using the orthonormality condition yield(34)q¨nt+ωn2qnt+2μnq˙nt+∑i=1∞∑j=1∞∑k=1∞Γnijkqitqjtqkt=fncos⁡Ωt,where(35)ωn2=γ+1∫01ϕn′′xϕnxdx+vf2∫01ϕn′′′′xϕnxdx,(36)Γnijk=-12vf2∫01ϕi′′xϕnx∫01ϕjxϕkxdxdx,(37)fn=∫01Fxϕnxdx,(38)μn=12∫01cvϕnxdx.


## 2.1. Equations of Motion

In this study,u and v denote the axial and transverse deflections of the beam, respectively, x is the distance along the undeflected beam, t is the time, L is the beam length, m=ρA is the beam’s mass per unit, E is Young’s modulus, A is the area of cross-section, I is the area moment of inertia, P is the applied axial compressive load, cu and cv are the viscous damping coefficients per unit length, ΔT is the temperature variation, and α is the thermal expansion coefficient.The linear relation between the temperature variation and the stress-temperature coefficientγ is(1)γΔT=EαΔT,and the nonlinear relation is expressed as [21](2)γΔT=EαΔT+h~α2ΔT2.Here, the first linear relation is adopted for the sake of simplicity. By assuming the Lagrangian strain as the strain measure and introducing temperature variations, the strain in the beam motion is obtained(3)ε=∂u∂x+12∂v∂x2-γE.The extended Hamilton’s principle is given by(4)∫t1t2δWe+δT-δUdt=0,where T is the kinetic energy, U is the elastic energy, and We is the work done by external loads on the system.By using Hamilton’s principle, one obtains [1, 20, 21](5)ρA∂2u∂t2+cu∂u∂t-A∂∂xE∂u∂x+12∂v∂x2-γ=0,(6)ρA∂2v∂t2+cv∂v∂t+EI∂4v∂x4-A∂∂x∂v∂xE∂u∂x+12∂v∂x2-γ+P∂2v∂x2=Fxcos⁡Ωt,where F(x) is the spatial distribution of the harmonic load in the v direction and Ω is the excitation frequency.Introducing the quasistatic assumptions, the acceleration and velocity terms in (5) are neglected [20, 21], and we obtain(7)A∂∂xE∂u∂x+12∂v∂x2-γ=0,where(8)∂u∂x+12∂v∂x2-γE=constant=δ,where δ is the average strain of the systems, and one obtains(9)δ=1L∫0L∂u∂x+12∂v∂x2-γEdx=12L∫0L∂v∂x2dx-γE.Therefore, substituting (9) into (6), we obtain the following nonlinear partial differential equation of motion without considering the torsion and shear rigidities:(10)EI∂4v∂x4+ρA∂2v∂t2+cv∂v∂t+Aγ∂2v∂x2+P∂2v∂x2=EA2L∂2v∂x2∫0L∂v∂x2dx+Fxcos⁡Ωt.For the sake of simplicity, the following nondimensional quantities are introduced:(11)v∗=vL,x∗=xL,t∗=tPρAL2,vf=EIPL2,γ∗=AγP,v1=EAP,F∗=FLP,Ω∗=ΩρAL2P,cv∗=cvL2ρAP.Substituting (11) into (10) and neglecting the asterisks, one obtains(12)vf2∂4v∂x4+∂2v∂t2+cv∂v∂t+∂2v∂x2+γ∂2v∂x2=12v12∂2v∂x2∫01∂v∂x2dx+Fxcos⁡Ωt.


## 2.2. Linear Analysis

Dropping the damping, excitation, and nonlinear terms in (12), the following linear equation is obtained:(13)vf2∂4v∂x4+∂2v∂t2+γ+1∂2v∂x2=0,and one lets(14)vx,t=ϕxeiωt,where i denotes the imaginary unit, ϕ(x) is the normalized undamped mode shape, and ω is the natural frequency.Substituting (14) into (13), one obtains(15)vf2∂4v∂x4+γ+1∂2v∂x2-ω2ϕ=0.The corresponding eigenvalue equation is expressed as follows:(16)vf2D4+γ+1D2-ω2=0.Accordingly, one lets(17)θ=1+γ2+4vf2ω2+1+γ2vf2,β=1+γ2+4vf2ω2-1+γ2vf2.The homogeneous solution of (16) is given by(18)ϕx=c1cos⁡θx+c2sin⁡θx+c3cosh⁡βx+c4sinh⁡βx.In the following sections, three different symmetric and nonsymmetric boundary conditions are chosen and investigated: hinged-hinged, hinged-fixed, and fixed-fixed.Case A (hinged-hinged).
As to the hinged-hinged boundary conditions, we have(19)ϕ0=ϕ1=0,∂2ϕ∂x2x=0=∂2ϕ∂x2x=1=0.
Substituting (19) into (18), we obtain the following coefficient matrix:(20)1010cos⁡θsin⁡θcosh⁡βsinh⁡β-θ20β20-θ2cos⁡θ-θ2sin⁡θβ3cosh⁡ββ3sinh⁡βwhere the determinant of the coefficient matrix in (20) equals zero; we have(21)θ2+β3sin⁡θsinh⁡β=0.
Hence, the natural frequency of the beam with hinged-hinged boundary conditions could be obtained by solving the following equation:(22)sin⁡θ=0,and the corresponding mode shapes are(23)ϕix=Cisin⁡θi,where Ci is determined by normalization conditions.Case B (hinged-fixed).
As to the hinged-fixed boundary conditions, we have(24)ϕ0=ϕ1=0,∂2ϕ∂x2x=0=∂ϕ∂xx=1=0.
Substituting (24) into (18), the coefficient matrix is(25)1010cos⁡θsin⁡θcosh⁡βsinh⁡β-θ20β20-θsin⁡θθcos⁡θβsinh⁡ββcosh⁡β
Similarly, the following equation is obtained:(26)-θ2+β2-βcosh⁡βsin⁡θ+θcos⁡θsinh⁡β=0.
Hence, the natural frequency could be obtained by solving the following transcendental equation:(27)-βcosh⁡βsin⁡θ+θcos⁡θsinh⁡β=0,with the following mode shapes:(28)ϕix=Cisinh⁡βisin⁡θix-sin⁡θisinh⁡βixsin⁡θi,where Ci is also determined by normalization conditions.Case C (fixed-fixed).
As to the fixed-fixed boundary conditions, we have(29)ϕ0=ϕ1=0,∂ϕ∂xx=0=∂ϕ∂xx=1=0.
Similarly, the coefficient matrix is(30)1010cos⁡θsin⁡θcosh⁡βsinh⁡β0θ0β-θsin⁡θθcos⁡θβsinh⁡ββcosh⁡βand we obtain the transcendental equation as(31)
-
2
θ
β
+
2
θ
β
cos
⁡
θ
cosh
⁡
β
+
θ
2
-
β
2
sin
⁡
θ
sinh
⁡
β
=
0
,where the natural frequency could be obtained, and the mode shapes are(32)ϕix=Cicosh⁡βix-cos⁡θixβisin⁡θi-θisinh⁡βiθicosh⁡βi-θicos⁡θi+sinh⁡βix-βiθisin⁡θi,where Ci is also determined by normalization conditions.


## 2.3. Discrete Model

The Galerkin procedure is used to discretize (12) into a set of second-order ordinary differential equations(33)vx,t=∑n=1∞ϕnxqnt,where qn(t) is the generalized coordinate and ϕn(x) is the linear vibration mode shape.Substituting (33) into (12), multiplying the result by ϕn(x), integrating the outcome over the domain, and finally using the orthonormality condition yield(34)q¨nt+ωn2qnt+2μnq˙nt+∑i=1∞∑j=1∞∑k=1∞Γnijkqitqjtqkt=fncos⁡Ωt,where(35)ωn2=γ+1∫01ϕn′′xϕnxdx+vf2∫01ϕn′′′′xϕnxdx,(36)Γnijk=-12vf2∫01ϕi′′xϕnx∫01ϕjxϕkxdxdx,(37)fn=∫01Fxϕnxdx,(38)μn=12∫01cvϕnxdx.


## 3. Perturbation Analysis

Perturbation methods are used to investigate the nonlinear free, primary, super, and subharmonic resonance responses of the beam with thermal effects subjected to the planar excitation, and no internal resonance is considered in this study. For the sake of simplicity, only the single-mode discretization equation is considered(39)q¨nt+ωn2qnt+2μnq˙nt+Γnnnnqn3t=fncos⁡Ωt.


### 3.1. Nonlinear Free Vibration

This section begins with the approximate series solutions for the nonlinear free vibrations obtained with the Lindstedt-Poincaré method [42].Firstly, a new independent variablet~ is introduced, which is(40)t~=ωnt,where ωn is the natural frequency.Therefore, by neglecting the damping and forcing terms, (39) is transformed into(41)q¨n+qn+Γnnnnωn2qn3=0,where the coefficient of the linear term is equal to unity since the time is nondimensionalized with respect to the linear vibration frequency.By assuming an expansion forqn=εqn~ (where ε is a small finite nondimensional parameter) and omitting the tilde, we obtain(42)q¨n+qn+ε2Γnnnnωn2qn3=0.Following the method of Lindstedt-Poincaré, we seek the fourth-order approximate solution to (42) by letting(43)qnt~;ε=∑m=04εmqnmt,and a strained time coordinate τ is introduced(44)τ=1+∑m=14εmαmt~.Then, we could obtain the relation between the nonlinear frequencyΩn and the linear one ωn as follows:(45)Ωn-ωnωn=∑m=14εmαm.Substituting (43)-(44) into (42) and equating the coefficients εm on both sides, the nonlinear ordinary equations are reduced to a set of linearized equations. Then, the polar form is introduced and the secular terms are set to zero. Finally, the series solutions of qnm and αm are obtained, based on which the fourth-order series solutions of the frequency amplitude relationship and displacement are as follows:(46)Ωn=ωn1+38Γnnnnωn2a2-15256Γnnnn2ωn4a4,where a is the actual nondimensional response amplitude.


### 3.2. Primary Resonances

First of all, we order the damping, excitation, and nonlinearity term as follows [43]:(47)2μnq˙nt=2εμnq˙nt,fncos⁡Ωnt=εfncos⁡Ωnt,Γnnnnq˙n3t=εΓnnnnq˙n3t.Substituting (47) into (39), the governing equation becomes(48)q¨nt+ωn2qnt+2εμnq˙nt+εΓnnnnqn3t=εfncos⁡Ωt.In the case of the primary resonance, we introduce a detuning parameterσn, which quantitatively describes the nearness of Ωn to ωn(49)Ωn=ωn+εσn,where ε is a small finite parameter.We express the solution in terms of different time scales as(50)qnt;ε=un0T0,T1+εun1T0,T1+⋯where T0=t and T1=εt.Substituting (49)-(50) into (48), equating the coefficients of ε0 and ε1 on both sides, following the method of multiple scales, finally the frequency response equation is obtained(51)σn=38Γnnnnωnan2±fn24ωn2an2-μn2,where an is the actual nondimensional response amplitude.Therefore, the first approximation to the steady-state solution is obtained(52)qnt=ancos⁡Ωnt-γn+Oε,where t is the actual time scale and γn is the phase.


### 3.3. Subharmonic Resonances

In order to investigate the subharmonic resonance, a detuning parameterσn is introduced(53)Ωn=3ωn+εσn,where ε is a small but finite parameter.Similarly, the frequency response equation can be obtained by using the multiple scales method(54)9μn2+σn-9ΓnnnnΛn2ωn-9Γnnnn8ωnan22=81Γnnnn2Λn216ωn2an2,where(55)Λn=12fnωn2-Ωn2.The first approximation to the steady-state solution is given by(56)qnt=ancos⁡13Ωnt-γn+fnωn2-Ωn2cos⁡Ωnt+Oε,where an is the actual nondimensional response amplitude, t is the actual time scale, and γn is the phase.


### 3.4. Super Harmonic Resonances

In this case, to express the nearness ofΩn to (1/3)ωn, we also introduce the detuning parameter σn defined by(57)3Ωn=ωn+εσn,where ε is a small but finite parameter.Following the step of the multiple scales method, a frequency response equation can be obtained(58)σn=3ΓnnnnΛn2ωn+3Γnnnn8ωnan2±Γnnnn2Λn6ωn2an2-μn2,where(59)Λn=12fnωn2-Ωn2.Therefore, the first approximation in this case is obtained(60)qnt=ancos⁡3Ωnt-γn+fnωn2-Ωn2cos⁡Ωnt+Oε,where an is the actual nondimensional response amplitude, t is the actual time scale, and γn is the phase.


## 3.1. Nonlinear Free Vibration

This section begins with the approximate series solutions for the nonlinear free vibrations obtained with the Lindstedt-Poincaré method [42].Firstly, a new independent variablet~ is introduced, which is(40)t~=ωnt,where ωn is the natural frequency.Therefore, by neglecting the damping and forcing terms, (39) is transformed into(41)q¨n+qn+Γnnnnωn2qn3=0,where the coefficient of the linear term is equal to unity since the time is nondimensionalized with respect to the linear vibration frequency.By assuming an expansion forqn=εqn~ (where ε is a small finite nondimensional parameter) and omitting the tilde, we obtain(42)q¨n+qn+ε2Γnnnnωn2qn3=0.Following the method of Lindstedt-Poincaré, we seek the fourth-order approximate solution to (42) by letting(43)qnt~;ε=∑m=04εmqnmt,and a strained time coordinate τ is introduced(44)τ=1+∑m=14εmαmt~.Then, we could obtain the relation between the nonlinear frequencyΩn and the linear one ωn as follows:(45)Ωn-ωnωn=∑m=14εmαm.Substituting (43)-(44) into (42) and equating the coefficients εm on both sides, the nonlinear ordinary equations are reduced to a set of linearized equations. Then, the polar form is introduced and the secular terms are set to zero. Finally, the series solutions of qnm and αm are obtained, based on which the fourth-order series solutions of the frequency amplitude relationship and displacement are as follows:(46)Ωn=ωn1+38Γnnnnωn2a2-15256Γnnnn2ωn4a4,where a is the actual nondimensional response amplitude.


## 3.2. Primary Resonances

First of all, we order the damping, excitation, and nonlinearity term as follows [43]:(47)2μnq˙nt=2εμnq˙nt,fncos⁡Ωnt=εfncos⁡Ωnt,Γnnnnq˙n3t=εΓnnnnq˙n3t.Substituting (47) into (39), the governing equation becomes(48)q¨nt+ωn2qnt+2εμnq˙nt+εΓnnnnqn3t=εfncos⁡Ωt.In the case of the primary resonance, we introduce a detuning parameterσn, which quantitatively describes the nearness of Ωn to ωn(49)Ωn=ωn+εσn,where ε is a small finite parameter.We express the solution in terms of different time scales as(50)qnt;ε=un0T0,T1+εun1T0,T1+⋯where T0=t and T1=εt.Substituting (49)-(50) into (48), equating the coefficients of ε0 and ε1 on both sides, following the method of multiple scales, finally the frequency response equation is obtained(51)σn=38Γnnnnωnan2±fn24ωn2an2-μn2,where an is the actual nondimensional response amplitude.Therefore, the first approximation to the steady-state solution is obtained(52)qnt=ancos⁡Ωnt-γn+Oε,where t is the actual time scale and γn is the phase.


## 3.3. Subharmonic Resonances

In order to investigate the subharmonic resonance, a detuning parameterσn is introduced(53)Ωn=3ωn+εσn,where ε is a small but finite parameter.Similarly, the frequency response equation can be obtained by using the multiple scales method(54)9μn2+σn-9ΓnnnnΛn2ωn-9Γnnnn8ωnan22=81Γnnnn2Λn216ωn2an2,where(55)Λn=12fnωn2-Ωn2.The first approximation to the steady-state solution is given by(56)qnt=ancos⁡13Ωnt-γn+fnωn2-Ωn2cos⁡Ωnt+Oε,where an is the actual nondimensional response amplitude, t is the actual time scale, and γn is the phase.


## 3.4. Super Harmonic Resonances

In this case, to express the nearness ofΩn to (1/3)ωn, we also introduce the detuning parameter σn defined by(57)3Ωn=ωn+εσn,where ε is a small but finite parameter.Following the step of the multiple scales method, a frequency response equation can be obtained(58)σn=3ΓnnnnΛn2ωn+3Γnnnn8ωnan2±Γnnnn2Λn6ωn2an2-μn2,where(59)Λn=12fnωn2-Ωn2.Therefore, the first approximation in this case is obtained(60)qnt=ancos⁡3Ωnt-γn+fnωn2-Ωn2cos⁡Ωnt+Oε,where an is the actual nondimensional response amplitude, t is the actual time scale, and γn is the phase.


## 4. Numerical Examples and Discussions

In the following numerical analysis, material properties and physical parameters of the beam are chosen in Table1. The nondimensional temperature variation factor γ is chosen as [-1.0,+1.0], and the corresponding dimensional temperature variation ΔT is [-20.83∘C,+20.83∘C]. Moreover, the nondimensional factor vf in (11) is 0.5 in this study.Table 1
Physical parameters and material properties.
Parameter Physical meaning Value Unit ρ density 7781 kg/m3 L length 2.032 m E Young’s modulus 207 GPa A area of cross-section 2.581 × 1 0 - 3 m 2 α thermal expansion coefficient 1.0 × 1 0 - 5 / ∘C P axial load 11.281 kN I area moment of inertia 5.550 × 1 0 - 7 m 4


### 4.1. Natural Frequencies

By using the numerical method, the transcendental equations (see (22), (27), and (31)) can be solved, and the corresponding natural frequencies are obtained. In order to investigate temperature effects on natural frequencies with different boundary conditions, a frequency variation factor Rω is defined(61)Rω=ωΔT-ω0ω0,where ωΔT and ω0 are the natural frequencies with and without thermal effects, respectively.Figure1 shows effects of temperature variations on the first four mode frequencies in the case of three different boundary conditions. As shown in these figures, with the increase (decrease) in the temperature, the mode frequencies all decrease (increase). In previous studies, it is shown that an increase in temperature leads to a decrease in natural frequencies [41], and the conclusion in this study is in good agreement with some of the previous findings.Figure 1
Temperature effects on the first four natural frequencies: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Furthermore, the changing of natural frequencies with temperature was found to be dependent on the order of the mode and boundary conditions. With the increase of the order, the natural frequency becomes less sensitive to the temperature variation. To be more specific, in the case ofγ=+1.0, as to three different boundary conditions, the variation factors Rω of these first order mode frequencies are 43.6% (hinged-hinged), 13.0% (hinged-fixed), and 5.7% (fixed-fixed), respectively. However, as to the fourth-order mode frequencies, the corresponding variation ratios Rω are 1.3%, 1.1%, and 0.9%, respectively. Moreover, we can observe a slightly higher sensitivity to warming than cooling in the case of the hinged-hinged boundary condition, as shown in Figure 1(a). However, in the case of the fixed-fixed boundary condition (Figure 1(c)), the asymmetry phenomenon between the cooling and warming conditions almost disappears. It is concluded that temperature effects are very closely related to the boundary conditions, and the natural frequency decreases with an increase in temperature. In addition, it should be pointed out that, as to these three cases, temperature effects on mode shapes can be negligible.


### 4.2. Nonlinear Free and Forced Vibrations

Neglecting the damping and excitation terms, the nonlinear free vibrations of the beam with thermal effects are investigated. Figure2 illustrates temperature effects on the nonlinear free vibration behaviors in the case of three different boundary conditions. Frequency response curves all show the hardening-spring characteristics due to the cubic nonlinearity term. The hardening behavior tends to increase for a positive temperature change and to decrease for a negative one. Moreover, with the enhancement of the constraint conditions (from hinged-hinged, hinged-fixed to fixed-fixed), temperature effects on the vibration behaviors are reduced.Figure 2
Frequency response curves of nonlinear free vibrations: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Temperature effects on the nonlinear responses to the primary, super, and subharmonic excitations are studied in the following in terms of frequency response curves and excitation response curves, respectively. Moreover, by using the multiple scales method, the stability of these solutions is determined by examining the eigenvalues of the Jacobian matrix evaluated at the equilibrium point.In the nonlinear resonant cases, the nondimensional excitation amplitude is chosen as0.05, and the nondimensional damping coefficient is assumed to be independent of the temperature variation (μn=0.01). Figure 3 shows temperature effects on frequency response curves of the primary resonance, and three different boundary conditions are considered. There are almost three steady-state solutions: one unstable, plotted in dashed lines, and two stable, of low and high amplitudes, plotted in continuous lines, respectively. The stable solution is physically determined by the initial condition of the system finally. In the primary resonance, the system exhibits the hardening-spring behavior due to the cubic nonlinearity term, and all these curves bend to the right side. As to the hinged-hinged boundary condition as shown in Figure 3(a), temperature effects are obvious. Specifically, in the range of the large response amplitude, the response amplitude decreases with the increase in the temperature. On the contrary, as to the small response amplitude, the response amplitude increases with the increase in the temperature. As to the hinged-fixed and fixed-fixed boundary conditions, the temperature effect on the frequency response curves is negligible in the range of small response amplitudes. However, as to the large response amplitudes, the curve bends to the right more in the warming condition. As to the boundary conditions from hinged-hinged to fixed-fixed, it is noted that the temperature effect on the resonance characters is reduced.Figure 3
Frequency response curves of primary resonances: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Figure4 shows the variation of the response amplitude with the excitation amplitude in the case of the primary resonance under three boundary conditions. Actually, it is known that, depending on the value of detuning parameter σ, some curves are multivalued while others are single-value. In this study, only the case of σ=2.0 is investigated. As shown in these figures, the multivaluedness of frequency response curves due to the nonlinearity leads to the jump phenomenon in the excitation response curves. There are significant quantitative changes of these curves due to temperature variations. Moreover, it is noted that, in the range of large and small response amplitudes, temperature effects are different. Specifically, as to the small response amplitude, with the increase in the temperature, the response amplitude increases slightly. On the contrary, as to the large one, the response amplitude decreases significantly in the same condition.Figure 4
Excitation response curves of primary resonances(σ=2.0): (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Temperature effects on the subharmonic resonance are shown in Figure5. In this case, the stable trivial solution (an≡0) is not shown, and only two nontrivial solutions are studied, in which one is stable and another is unstable. It is noted that there is no jump phenomenon in the case of the subharmonic resonance. As shown in Figure 5(a), as to the hinged-hinged boundary condition, the temperature effect is the most visible, and the threshold response amplitude decreases with an increase temperature and increases with a decrease one. The same conclusion could be observed from the Figure 5(b), in the case of the hinged-fixed boundary condition. In the last case (Figure 5(c)), it is shown that the temperature effect on the frequency response curve is not obvious.Figure 5
Frequency response curves of subharmonic resonances: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Moreover, excitation response curves of the subharmonic resonance in the case of the hinged-hinged boundary condition whenσ=2.0 are illustrated in Figure 6. It is noted that there are significant quantitative changes of these curves due to temperature variations. Considering the temperature variations, the same response amplitude a is induced by very different excitation amplitude f. Moreover, under the excitation with the same amplitude f, very different response amplitudes are obtained due to the temperature variations.Figure 6
Excitation response curves of subharmonic resonances in the case of hinged-hinged boundary condition(σ=2.0).Figure7 illustrates temperature effects on frequency response curves in the case of the super harmonic resonance, and three boundary conditions are also included. Due to the effect of the cubic nonlinearity, the system exhibits the hardening behavior. Generally, as same as the previous resonant cases, the hardening behavior also increases with an increase in temperature and decreases with a decrease in temperature. However, it should be pointed out that, in the case of the super harmonic resonance (Figure 7(a)), no unstable solution is obtained and no jump phenomenon is observed in the warming condition (γ=+1.0). As shown in Figure 7(b), with the increase in the temperature, the curve bends to the right more and the system exhibits stronger hardening-spring behavior. It should be pointed out that, in the case of the cooling condition (γ=-1.0), the peak value of the frequency response curve decreases significantly. In the last case as shown in Figure 7(c), as to the system with fixed-fixed boundary condition, temperature effects on the super harmonic resonance are much more obvious than the ones on the primary resonance, and frequency response curves are shifted to the right more with the increase in temperature.Figure 7
Frequency response curves of super harmonic resonances: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)


## 4.1. Natural Frequencies

By using the numerical method, the transcendental equations (see (22), (27), and (31)) can be solved, and the corresponding natural frequencies are obtained. In order to investigate temperature effects on natural frequencies with different boundary conditions, a frequency variation factor Rω is defined(61)Rω=ωΔT-ω0ω0,where ωΔT and ω0 are the natural frequencies with and without thermal effects, respectively.Figure1 shows effects of temperature variations on the first four mode frequencies in the case of three different boundary conditions. As shown in these figures, with the increase (decrease) in the temperature, the mode frequencies all decrease (increase). In previous studies, it is shown that an increase in temperature leads to a decrease in natural frequencies [41], and the conclusion in this study is in good agreement with some of the previous findings.Figure 1
Temperature effects on the first four natural frequencies: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Furthermore, the changing of natural frequencies with temperature was found to be dependent on the order of the mode and boundary conditions. With the increase of the order, the natural frequency becomes less sensitive to the temperature variation. To be more specific, in the case ofγ=+1.0, as to three different boundary conditions, the variation factors Rω of these first order mode frequencies are 43.6% (hinged-hinged), 13.0% (hinged-fixed), and 5.7% (fixed-fixed), respectively. However, as to the fourth-order mode frequencies, the corresponding variation ratios Rω are 1.3%, 1.1%, and 0.9%, respectively. Moreover, we can observe a slightly higher sensitivity to warming than cooling in the case of the hinged-hinged boundary condition, as shown in Figure 1(a). However, in the case of the fixed-fixed boundary condition (Figure 1(c)), the asymmetry phenomenon between the cooling and warming conditions almost disappears. It is concluded that temperature effects are very closely related to the boundary conditions, and the natural frequency decreases with an increase in temperature. In addition, it should be pointed out that, as to these three cases, temperature effects on mode shapes can be negligible.


## 4.2. Nonlinear Free and Forced Vibrations

Neglecting the damping and excitation terms, the nonlinear free vibrations of the beam with thermal effects are investigated. Figure2 illustrates temperature effects on the nonlinear free vibration behaviors in the case of three different boundary conditions. Frequency response curves all show the hardening-spring characteristics due to the cubic nonlinearity term. The hardening behavior tends to increase for a positive temperature change and to decrease for a negative one. Moreover, with the enhancement of the constraint conditions (from hinged-hinged, hinged-fixed to fixed-fixed), temperature effects on the vibration behaviors are reduced.Figure 2
Frequency response curves of nonlinear free vibrations: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Temperature effects on the nonlinear responses to the primary, super, and subharmonic excitations are studied in the following in terms of frequency response curves and excitation response curves, respectively. Moreover, by using the multiple scales method, the stability of these solutions is determined by examining the eigenvalues of the Jacobian matrix evaluated at the equilibrium point.In the nonlinear resonant cases, the nondimensional excitation amplitude is chosen as0.05, and the nondimensional damping coefficient is assumed to be independent of the temperature variation (μn=0.01). Figure 3 shows temperature effects on frequency response curves of the primary resonance, and three different boundary conditions are considered. There are almost three steady-state solutions: one unstable, plotted in dashed lines, and two stable, of low and high amplitudes, plotted in continuous lines, respectively. The stable solution is physically determined by the initial condition of the system finally. In the primary resonance, the system exhibits the hardening-spring behavior due to the cubic nonlinearity term, and all these curves bend to the right side. As to the hinged-hinged boundary condition as shown in Figure 3(a), temperature effects are obvious. Specifically, in the range of the large response amplitude, the response amplitude decreases with the increase in the temperature. On the contrary, as to the small response amplitude, the response amplitude increases with the increase in the temperature. As to the hinged-fixed and fixed-fixed boundary conditions, the temperature effect on the frequency response curves is negligible in the range of small response amplitudes. However, as to the large response amplitudes, the curve bends to the right more in the warming condition. As to the boundary conditions from hinged-hinged to fixed-fixed, it is noted that the temperature effect on the resonance characters is reduced.Figure 3
Frequency response curves of primary resonances: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Figure4 shows the variation of the response amplitude with the excitation amplitude in the case of the primary resonance under three boundary conditions. Actually, it is known that, depending on the value of detuning parameter σ, some curves are multivalued while others are single-value. In this study, only the case of σ=2.0 is investigated. As shown in these figures, the multivaluedness of frequency response curves due to the nonlinearity leads to the jump phenomenon in the excitation response curves. There are significant quantitative changes of these curves due to temperature variations. Moreover, it is noted that, in the range of large and small response amplitudes, temperature effects are different. Specifically, as to the small response amplitude, with the increase in the temperature, the response amplitude increases slightly. On the contrary, as to the large one, the response amplitude decreases significantly in the same condition.Figure 4
Excitation response curves of primary resonances(σ=2.0): (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Temperature effects on the subharmonic resonance are shown in Figure5. In this case, the stable trivial solution (an≡0) is not shown, and only two nontrivial solutions are studied, in which one is stable and another is unstable. It is noted that there is no jump phenomenon in the case of the subharmonic resonance. As shown in Figure 5(a), as to the hinged-hinged boundary condition, the temperature effect is the most visible, and the threshold response amplitude decreases with an increase temperature and increases with a decrease one. The same conclusion could be observed from the Figure 5(b), in the case of the hinged-fixed boundary condition. In the last case (Figure 5(c)), it is shown that the temperature effect on the frequency response curve is not obvious.Figure 5
Frequency response curves of subharmonic resonances: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)Moreover, excitation response curves of the subharmonic resonance in the case of the hinged-hinged boundary condition whenσ=2.0 are illustrated in Figure 6. It is noted that there are significant quantitative changes of these curves due to temperature variations. Considering the temperature variations, the same response amplitude a is induced by very different excitation amplitude f. Moreover, under the excitation with the same amplitude f, very different response amplitudes are obtained due to the temperature variations.Figure 6
Excitation response curves of subharmonic resonances in the case of hinged-hinged boundary condition(σ=2.0).Figure7 illustrates temperature effects on frequency response curves in the case of the super harmonic resonance, and three boundary conditions are also included. Due to the effect of the cubic nonlinearity, the system exhibits the hardening behavior. Generally, as same as the previous resonant cases, the hardening behavior also increases with an increase in temperature and decreases with a decrease in temperature. However, it should be pointed out that, in the case of the super harmonic resonance (Figure 7(a)), no unstable solution is obtained and no jump phenomenon is observed in the warming condition (γ=+1.0). As shown in Figure 7(b), with the increase in the temperature, the curve bends to the right more and the system exhibits stronger hardening-spring behavior. It should be pointed out that, in the case of the cooling condition (γ=-1.0), the peak value of the frequency response curve decreases significantly. In the last case as shown in Figure 7(c), as to the system with fixed-fixed boundary condition, temperature effects on the super harmonic resonance are much more obvious than the ones on the primary resonance, and frequency response curves are shifted to the right more with the increase in temperature.Figure 7
Frequency response curves of super harmonic resonances: (a) hinged-hinged; (b) hinged-fixed; (c) fixed-fixed.
(a) (b) (c)


## 5. Summary and Conclusions

A positive (resp., negative) temperature change does decrease (resp., increase) natural frequencies, and temperature effects on mode shapes can be negligible. Temperature effects on the vibration behaviors are closely related to the mode order and boundary conditions, and one can observe a slightly higher sensitivity to warming than cooling. As to the nonlinear vibration characteristics with thermal effects, no matter the primary, super, or subharmonic resonant cases, the system exhibits the hardening-spring behavior due to the cubic nonlinearity term. The hardening behavior increases with an increase in temperature, and it decreases with a decrease in temperature. Only some quantitative changes are observed due to the temperature changes by examining the frequency response curves and excitation response curves. The response amplitude, the stability, and the number of the nontrivial solutions and the range of the resonance response are all closely related to temperature variations. Moreover, as to temperature effects on vibration characteristics of the beam, different symmetric and nonsymmetric boundary conditions should be paid more attention.



---

*Source: 9834629-2018-06-27.xml*