Yi-Ren Wang and Chia-Man Chang Department of … of Aerospace Engineering, Tamkang University, ......

ELASTIC BEAM WITH NONLINEAR SUSPENSION AND A DYNAMIC VIBRATIONABSORBER AT THE FREE END

Yi-Ren Wang and Chia-Man ChangDepartment of Aerospace Engineering, Tamkang University, New Taipei City, Tamsui Dist., Taiwan 25137

E-mail: [email protected]

Received April 2013, Accepted November 2013No. 13-CSME-130, E.I.C. Accession 3588

ABSTRACTThis study considered a slender hinged-free beam with suspension cables simulated using nonlinear springs.We added a time-dependent boundary dynamic vibration absorber (DVA) that was suspended at the freeend of the beam to reduce vibration and prevent internal resonance. DVA with various spring constantswere considered and the optimal mass range for the DVA to reduce vibration in the main structure was alsoproposed.

Keywords: beam vibration, dynamic vibration absorber, internal resonance, vibration reduction.

POUTRE ÉLASTIQUE AVEC SUSPENSION NON-LINÉAIRE ET AMORTISSEURDYNAMIQUE DE VIBRATION À L’EXTRÉMITÉ LIBRE

RÉSUMÉCette recherche porte sur l’examen d’une poutre fine à extrémité libre avec câbles de suspension à ressortsnon-linéaires. Nous avons ajouté un amortisseur dynamique de vibration aux limites dépendantes du temps(ADV), lequel est suspendu à l’extrémité libre de la poutre pour réduire la vibration et pour prévenir larésonance interne. Des ressorts aux constantes variées furent considérés, et la portée massique optimalepour l’ADV, dans le but de réduire la vibration de la structure principale, a été proposée.

Mots-clés : poutre vibrante ; amortisseur dynamique de vibration ; résonance interne ; réduction de la vi-bration.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014 107

1. INTRODUCTION

The issue of vibration has always been a concern for researchers and engineers and vibration within nonlin-ear systems is particularly problematic. Beams, the subject of this study, are essential elements of engineer-ing structures with a wide range of applications. This study considered a slender hinged-free elastic beamwith suspension cables simulated using nonlinear cubic springs. This also created multiple possibilities formode coupling and internal resonance. The phenomenon of internal resonance is often discussed in thevibration analysis of single nonlinear beams. Eftekhari et al. [1] investigated the performance of a mass-spring oscillator at the tip of a symmetrically cantilevered composite beam under chordwise base excitation.They selected oscillation parameters and the corresponding natural frequencies to produce a 2:1 and 1:1internal resonance conditions. Saturation phenomenon was detected in the force modulation response at 1:1internal resonance. Rossikhin and Shitikova [2] also examined nonlinear vibration in a suspension bridgewith internal resonances of 2:1 and 1:1. They then verified their qualitative hydrodynamic analysis using acase study of the Golden Gate Bridge in San Francisco. Palmeri and Adhikari [3] considered a double-beamsystem comprising two beams connected continuously by a Winkler type inner layer. Transverse vibra-tion in this beam system was studied using a state-space approach and the numerical results demonstratedthe accuracy and versatility of their approach. Van Horssen and Boertjens [4, 5] investigated a suspensionbridge subjected to nonlinear aerodynamics using linear springs to simulate the suspension cables. Theyproposed a brilliant mathematical analysis method that has contributed substantially to subsequent researchon complex fluid-structure coupled systems. Nayfeh and Mook [6] proposed a number of commonly usedapproaches to derive the analytical solutions of nonlinear problems, such as the Poincaré method, the Lind-stedt method, the average method, and method of multiple scales (MOMS). The suitability of MOMS forvibration systems with damping has led to its wide application. Sedighi et al. [7] studied the nonlinearvibration of cantilever beams under preloaded nonlinear cubic spring boundary conditions. The He’s Pa-rameter Expanding Method (HPEM) was used to obtain the exact solution for the dynamic behavior in thissystem. That study demonstrated that series expansions with one term are sufficient to obtain an accuratesolution. Nayfeh and Balachandran [8] explained the definitions of various stable conditions in nonlinearsystems and discussed a number of criteria to determine stability. This study was a crucial reference for usin the analysis of stability.

Adding vibration absorbers is the general approach to achieve damping, among which tuned mass damper(TMD) systems are the most common. Vakakis and Paipetis [9] investigated the influence of a TMD witha single degree of freedom on a primary system with multiple degrees of freedom. The research modelused by Hijmissen and Van Horssen [10] comprised a vertical Bernoulli-Euler beam fixed at the bottomwith a spring-mass-damper TMD installed at the top to minimize transverse vibration. Zou and Nayfeh[11] found that a modified TMD system with multiple-degrees-of-freedom could provide higher dampingeffect in the first three primary structural modes, compared with multiple single-degree-of-freedom TMDs.Addressing stability and nonlinear vibration, Wang and Lin [12] utilized a two-degree-of-freedom TMD toavoid internal resonance. Furthermore, by cross-referring the internal resonance contour plot (IRCP) andflutter speed contour plot (FSCP), they were able to identify the installation location capable of providingoptimal damping effects. Cai et al. [13] provided a theoretical foundation for suspended TMD systems.They suspended a TMD at the end of an elastic cable and considered the elastic parameters of the cable aswell as internal damping, inclination angle, the location of the damper, and the damping ratio in the motionequations to the influence of each factor. Cai and Wu [14] conducted numerical simulations to analyze theinfluence of the same parameters on system vibration. Samani and Pellicano [15] explored the vibrationdamping effects of linear and nonlinear DVAs in a beam. Clearly, the issue of vibration has attracted aconsiderable amount of attention.

108 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014

In this study, the DVA installed at the end of the beam is subject to time-dependent boundary conditions.To solve this type of problem, Mindlin and Goodman [16] proposed the use of shifting functions to con-vert non-homogeneous boundary conditions into homogeneous boundary conditions. Using the Mindlin–Goodman method, Lee and Lin [17] performed dynamic analysis on a non-uniform Bernoulli–Euler beamwith time dependent boundary conditions. Lin [18] investigated the vibration control of a pre-twistedbending-bending beam under elastic time dependent boundary conditions. The Mindlin-Goodman shift-ing function was used to transform this problem into a homogeneous boundary condition problem. Theuse of the shifting function proved to be a straightforward solution to overcoming time-dependent boundaryproblems.

The model in this study is applicable to the engineering of structures with nonlinear suspension systems.In addition, inverting the system would enable the simulation of a beam placed on a Winkler-type elasticfoundation in which the formulation of the problem and analysis methods would be the same. Thus, thissystem is applicable in a wide range of applications. In Section 2, the theoretical model of this hinged-freesuspension beam system was constructed. The time-dependent boundary problem was solved by Mindlin-Goodman method. In Section 3, we analyzed the internal resonance (I.R.) condition for the prime beamstructure without DVA. The 3:1 I.R. could happen when the ratio of the suspension stiffness to the beamYoung’s modulus is less than 15. In Section 4, the DVAs with various spring constants were considered andthe optimal mass range for the DVA to reduce vibration in the main structure was also proposed. In Section 5,we also considered the influence of simple aerodynamics on this system. The Floquet theory and the Floquetmultiplier (F.M.) were employed to determine the stability criteria and plot the basins of attraction for thesystem. This enabled us to observe the stability of the system at different wind speeds when equipped withvibration absorbers of different masses and moduli of elasticity. In Section 6, the results of this study wereverified using numerical simulation, which, in addition to confirming the accuracy by through comparison,established the applicability in this study.

2. THEORETICAL MODEL

2.1. Equation of MotionThis study considered a suspension or elastic foundation system with an elastic beam comprising a nonlinearspring and a linear damper. The illustration of this model is outlined in Fig. 1, in which the beam is aBernoulli–Euler beam; m denotes the mass per unit length, and l represents beam length; E and I are theYoung’s modulus and moment of inertia of the beam, respectively; k and β denote the linear and nonlinearspring constants; c is the damping coefficient, and m and k denote the mass and spring constant of theDVA, respectively. We supposed that the system is subject to the influence of a harmonic force and simpleaerodynamics, in which f and Ω are the magnitude and frequency of the harmonic excitation; ρa representsair density; U denotes wind speed; d is the width of the beam, and a0,a1,a2, . . . represent aerodynamiccoefficients.

This study incorporated the spring force of an elastic suspension system and simple aerodynamic forcesinto the equation of motion for Bernoulli–Euler beams, thereby deriving the equation of motion for the mainstructure used in this study:

m∂ 2w(x, t)

∂ t2 +EI∂ 4w(x, t)

∂x4 + c∂w(x, t)

∂ t+ k[w(x, t)+βw3 (x, t)

]= f sin(Ωt)

+ρaU2d

2

(a0 +

a1

U∂w(x, t)

∂ t+

a2

U2∂w2 (x, t)

∂ t+ · · ·

), (1)

where

m∂ 2w(x, t)

∂ t2 +EI∂ 4w(x, t)

∂x4


Fig. 1. Main structure with DVA of this system.

is the Bernoulli–Euler beam equation; c ∂w(x,t)∂ t denotes the damping term; k[w(x, t)+βw3 (x, t)] represents

the linear elastic and non-linear elastic terms of the suspension cable.

ρaU2d2

(a0 +

a1

U∂w(x, t)

∂ t+

a2

U2∂w2 (x, t)

∂ t+ · · ·

)is the aerodynamic term and f sin(Ωt) represents the harmonic force. The boundary conditions of the modelin this study can be expressed as follows:

w(0, t) = 0,∂ 2w(0, t)

∂x2 = 0

∂ 2w(l, t)∂x2 = 0, EI

∂ 3w(l, t)∂x3 = k [w(t)−w(l, t)] (2)

The equation of motion for the DVA is

m ¨w(t) = k [w(l, t)− w(t)] . (3)

After non-dimensionalizing and simplifying Eqs. (1) through (3), we derive

∗∗w (x,τ)+ w′′′′ (x,τ)+υ

∗w(x,τ)+ ω

2w(x,τ)+ Kw3 (x,τ) = f sin(rτ)+ a0U2a0 + a1U a1∗w . (4)

The boundary conditions are

w(0,τ) = 0, w′′(0,τ) = 0, w′′(l,τ) = 0, w′′′(l,τ) = mω2 [w(τ)−w(l,τ)

](5)

The equation of the dynamic vibration absorber is

∗∗w−ω

2[w(l,τ)− w(τ)] = 0, (6)

where

w =wl, x =

xl, (w) =

∂

∂x,

∗(w) =

∂

∂τ, τ = ωt, ω =

√EIl4m

,


ευ =l2c√mEI

, ω2 =

kω2m

, K =k

ω2mβ l2, f =

flω2m

, a0U2 =ρadU2

2mω2 ,

a1U =ρadlU2mω

, ω2 =

km, m =

mlm

and w =ww.

2.2. Solving Time-Dependent-Boundary ProblemsTo analyze this nonlinear equation, we adopted MOMS, and set T0 = ε0τ as the fast time-scale and T1 = ετ

as the slow time-scale. In addition,

w(x,τ,ε) = w0 (x,T0,T1,T2, . . .)+ εw1 (x,T0,T1,T2, . . .) (7)

Because ε is extremely small, we ignored the influence of higher-order terms such as ε2,ε3, . . . on thesystem. We also supposed that damping, nonlinear springs, and all external force terms are in the order ofε1; by substituting Eq. (7) into Eq. (4), we can derive the following:

∂ 2

∂T 20

w0 + ε∂ 2

∂T 20

w1 +2ε∂ 2

∂T0∂T1w0 +w′′′′0 + εw′′′′1 + ευ

∂

∂T0w0 +ω

2w0 + εω2w1 + εKw3

0

= ε f sin(rτ)+ ε a0U2a0 + ε a1U a1∂

∂T0w0, (8)

where the terms comprising the order of ε0 are

∂ 2

∂T 20

w0 +ω2w0 +w′′′′0 = 0 (9)

and the boundary conditions are

w0(0,τ) = 0, w′′0(0,τ) = 0

w′′0(l,τ) = 0, w′′′0 (l,τ) = mω2 [w0(τ)−w0(l,τ)

]. (10)

The terms comprising the order of ε1 are

∂ 2

∂T 20

w1 +ω2w1 +w′′′′1 = f sin(rτ)+ a0U2a0 + a1U a1

∂

∂T0w0−2

∂ 2

∂T0∂T1w0−υ

∂

∂T0w0− Kw3

0, (11)

and the boundary conditions are

w1(0,τ) = 0, w′′1(0,τ) = 0, w′′1(l,τ) = 0, w′′′1 (l,τ) = mω2 [w1(τ)−w1(l,τ)

]. (12)

DVA is linear, and its equation of motion is

∂ 2

∂T 20

w0 + ω2w0 = ω

2w0l. (13)

The boundary conditions in this model are time-varying and non-homogeneous. Thus, we can use themethod proposed by Mindlin and Goodman [16] to convert the non-homogeneous boundary conditions intohomogeneous boundary conditions using a shifting function before applying the separation of variables tosolve the problem. We hypothesized that v0 (x,τ) is the transformed displacement and defined h0 (x) as the


shifting function. Thus, the transverse displacement of the beam with the vibration absorber at the end canbe expressed as

w0 (x,T0,T1) = v0 (x,τ)+h0 (x) w0 (τ) , (14)

where the boundary conditions are the same as in Eq. (10).By substituting Eq. (14) into Eq. (10), we derive the following:

v0 (0,τ) =−h0(0)w0 (τ) , v′′0 (0,τ) =−h′′0 (0) w0 (τ) , v′′0(l,τ)=−h′′0

(l)

w0 (τ) ,

v′′′0(l,τ)+mω

2v0(l,τ) =−h′′′0(l)

w0 (τ)+mω2w0(τ)−mω

2h0(l)

w0 (τ) . (15)

In order to make Eq. (15) a homogeneous boundary condition, or in other words, to make the boundarycondition of v0 (x,τ) homogeneous, the right-hand side of Eq. (15) must be 0. Accordingly, we can obtainthe boundary conditions of the shifting function as below:

h0(0) = 0,h′′0 (0) = 0, h′′0(l)= 0, −h′′′0

(l)+mω

2−mω2h0(l)= 0. (16)

Since the highest derivative order of the boundary condition (Eq. 16) is the 3rd order, we assume the shiftingfunction h0 (x) as

h0 (x) = α00 +α01x+α02x2 +α03x3. (17)

By substituting Eq. (17) into Eqs. (16), we can derive that h0 (x) = x/l.Using the separation of variables, we assume that the general solution to v0(x,τ) is

v0(x,τ) =(A(T1)e−iς eiωnT0 +A(T1)eiς e−iωnT0

)Φ(x), (18)

the boundary conditions of which are

Φ(0) = 0, Φ′′ (0) = 0, Φ

′′ (l)= 0, Φ′′′ (l)+mω

2Φ(l) = 0. (19)

Since Φ(x) represents the spatial coordinate of the beam vibration shape, we choose the following assump-tion of Φ(x)to satisfy the boundary conditions.

Φ(x) = E1 sinγx+E2 cosγx+E3 sinhγx+E4 coshγx. (20)

By substituting Eq. (20) into Eq. (19), we can obtain

Φ(x) = E1 sinγx+E1sinγlsinhγl

sinhγx, (21)

and the characteristic equation

cotγl =2mω2

γ3 + cothγl. (22)

The eigen value γn can be numerically obtained from Eq. (22). Furthermore, substituting Eq. (21) intoEq. (18) will show that

v0(x,τ) =∞

∑n = 1

E1n(An(T1)e−iςneiωnT0 +An(T1)eiςne−iωnT0

)Φn(x)E1n

=∞

∑n = 1

(Bn(T1)e−iςneiωnT0 +Bn(T1)eiςne−iωnT0

)Φn(x) =

∞

∑n = 1

ξ0n (τ)Φn(x), (23)


and Φn(x) is a linear mode shape that can be expressed as

Φn(x) = sinγnx+sinγnlsinhγnl

sinhγnx, (24)

where, in the event that mω2 ≤ 0.01, we can use numerical methods to obtain the eigen value γn. Wesummarized that when γn ≈ 0.785+ nπ , its substitution into the characteristic equation (Eq. (22)) wouldresult in an error less than 0.07%. In mω2 ≤ 0.1, then the error would be less than 0.3%. Clearly, whenmω2 ≤ 0.1, under the hypothesis that γn ≈ 0.785+nπ is acceptable, mω2 represents the ratio of the moduliof elasticity in the vibration absorber and the beam.

2.3. Dynamic EquationAt this point, we have obtained the expressions of the eigen values and linear mode shape within the system.To derive the transverse displacement (w0) of the system, we must also solve the displacement of the DVA.We thus rewrite Eq. (13) as

∂ 2

∂T 20

w0 + ω2w0 = ω

2

[∞

∑n = 1


)Φn(l)

], (25)

and hypothesize that the general solution for the displacement of the DVA (w0) is

w0 =∞

∑n = 1

(D1neiωnT0 +D1ne−iωnT0

). (26)

By substituting Eq. (26) into Eq. (25), we can derive that

∞

∑n = 1

(ω2D1neiωnT0−ω2n D1neiωnT0 + ω

2D1ne−iωnT0−ω2n D1ne−iωnT0)

=∞

∑n = 1ω2[


)Φn(l)]. (27)

From Eq. (27), we can obtain the following relational expressions for the nth mode:

D1n =ω2Φn(l)ω2−ω2

nBn(T1)e−iςn , (28)

D1n =ω2Φn(l)ω2−ω2

nBn(T1)eiςn . (29)

The D1nand D1ncan be expressed as CtnBn(T1)e−iςn and CtnBn(T1)eiςn , respectively. Where we choose Ctn

as,

Ctn =ω2Φn(l)ω2−ω2

n. (30)

It is noted that the factors in Eq. (30) (Ctn) with the greatest influence include the natural frequency of themain structure ωn, represented by the ratio of the natural frequency of the vibration absorber and beam ω

and the linear mode shape Φn. Subscript n indicates the index of each mode. Equation (26) can be rewrittenas

w0 =∞

∑n = 1

Ctn(Bn(T1)e−iςneiωnT0 +Bn(T1)eiςne−iωnT0

)=

∞

∑n = 1

Ctnξ0n (τ) . (31)


We substitute h0 (x) and Eqs. (23) and (31) into Eq. (14) and assume that l=1, which reveals the w0 of theelastic beam as follows:

w0 =∞

∑n = 1

w0n =∞

∑n = 1

(Φn(x)+ xCtn

)ξ0n (τ) . (32)

By substituting Eq. (32) into Eq. (9), we can derive that∞

∑n = 1

[Φn(x)+ xCtn

] ∗∗ξ0n (τ)+ω

2 [Φn(x)+ xCtn

]ξ0n (τ)+ γ

4n Φn(x)ξ0n (τ)

= 0. (33)

Using the orthogonal property, we multiply both sides of Eq. (33) by Φm(Eq. (24)) and integrate from 0 tol. We obtain the following equation:

∗∗ξ0m

∫ l

0

(sin2

γmx+2sinγmlsinhγml

sinγmxsinhγmx+sin2

γmlsinh2

γmlsinh2

γmx)

dx

+Ctm

∗∗ξ0m

∫ l

0

(xsinγmx+ x

sinγmlsinhγml

sinhγmx)

dx

+ω2ξ0m

∫ l

0

(sin2

γmx+2sinγml

sinhγmlsinγmxsinhγmx+

sin2γml

sinh2γml

sinh2γmx)

dx

+ω2Ctmξ0m

∫ l

0

(xsinγmx+ x

sinγmlsinhγml

sinhγmx)

dx

+ γ4mξ0m

∫ l

0

(sin2

γmx+2sinγml

sinhγmlsinγmxsinhγmx+

sin2γml

sinh2γml

sinh2γmx)

dx = 0. (34)

The result can be simplified as∗∗

ξ0m+(ω

2 + γ4mCtm

)ξ0m = 0, (35)

where the natural frequency of the system is√

ω2 + γ4

mCtm, in which Ctm is a coefficient for integral values.The integrals of the coefficients in Eq.(34) are summarized and denoted as qm1 and qm2. For details pleaserefer to Appendix 1. The Ctm can be expressed as qm1/(qm1 +Ctmqm2). Similarly, we substitute Eq. (32)into Eq. (11) to obtain

∞

∑n = 1

[(Φn + xCtn

) ∗∗ξ1n+ω

2 (Φn + xCtn

)ξ1n + γ

4n Φnξ1n

]

=∞

∑n = 1

f sin(rτ)+ a0U2a0 + a1U a1

∂

∂T0

[(Φn(x)+ xCtn

)ξ0n (τ)

]−2 ∂ 2

∂T0∂T1

[(Φn(x)+ xCtn

)ξ0n (τ)

]−υ

∂

∂T0

[(Φn(x)+ xCtn

)ξ0n (τ)

]−K

[ξ0iξ0 jξ0k

(Φi(x)+ xCti

)(Φ j(x)+ xCt j

)(Φk(x)+ xCtk

)] . (36)

After orthogonalizing Eq. (36), we can write the dynamic equation of this system in the time domain asfollows:

∗∗ξ1m+

(ω

2 + γ4mCtm

)ξ1m = a1U a1

∗ξ0m (τ)−υ

∗ξ0m (τ)−2

∂

∗ξ0m (τ)

∂T1

+qm3

qm1 +Ctmqm2f eirτ +

qm3

qm1 +Ctmqm2a0U2a0

− K(qm1 +Ctmqm2)

ξ0iξ0 jξ0k

∫ l

0

(Φi(x)+ xCti

)(Φ j(x)+ xCt j

)(Φk(x)+ xCtk

)Φmdx. (37)


For details on qm1, qm2, and qm3, please refer to Appendix 1.

3. INTERNAL RESONANCE ANALYSIS

3.1. System without Vibration AbsorberTo determine the conditions of the internal resonance in the beam, we must first discuss the nonlinearvibration in the main structure without a vibration absorber, after which, methods for coping with vibrationreduction can be considered. Without a vibration absorber, the boundary condition for the right end of thesystem (Fig. 1) is that of a free end. The development process of the theoretical model is the same as theone before; therefore, we can simplify the descriptions below.

By eliminating the DVA equations, Eqs. (4) and (6), and nondimensionalizing the equation of motion, wederive the following:

∗∗w (x,τ)+w′′′′ (x,τ)+υ

∗w(x,τ)+ω

2w(x,τ)+ Kw3 (x,τ) = f sin(rτ)+ a0U2a0 + a1U a1∗w . (38)

In this research model, the two ends comprise a hinged end and a free end; therefore, the boundaryconditions are

w(0,τ) = 0, w′′(0,τ) = 0, w′′(l,τ) = 0, w′′′(l,τ) = 0. (39)

3.2. Conditions of System Internal ResonanceWe adopted MOMS to list the terms comprising ε0 and ε1, deriving the following:ε0:

∂ 2

∂T 20

w0 +ω2w0 +w′′′′0 = 0, (40)

with the following boundary conditions:

w0(0,τ) = 0, w′′0(0,τ) = 0, w′′0(l,τ) = 0, w′′′0 (l,τ) = 0, (41)

and ε1:

∂ 2

∂T 20

w1 +ω2w1 +w′′′′1 = f sin(rτ)+ a0U2a0 + a1U a1

∂

∂T0w0−2

∂ 2

∂T0∂T1w0−υ

∂

∂T0w0− Kw3

0, (42)

with the following boundary conditions:

w1(0,τ) = 0, w′′1(0,τ) = 0, w′′1(l,τ) = 0, w′′′1 (l,τ) = 0. (43)

Using classical structural dynamics, we obtain the linear mode shape, and by orthogonalizing Eqs. (40) and(42), we derive the following:

∗∗ξ0m+

(ω

2 + γ4m)

ξ0m = 0, (44)

∗∗ξ1m+

(ω

2 + γ4m)

ξ1m = a1U a1

∗ξ0m (τ)−υ

∗ξ0m (τ)−2

∂

∗ξ0m (τ)

∂T1+(

f eirτ + a0U2a0) pm3

pm1

− Kpm1

ξ0iξ0 jξ0k

∫ l

0Φi(x)Φ j(x)Φk(x)Φmdx, (45)

where

pm1 =l2− sin2γml

4γm+ sin2

γml(

cotγml2γm

− l2sinh2

γml

);


pm3 =1γm

(1− sinγml

sinhγml

),

and the natural vibration frequency is√

ω2 + γ4

m.To establish the conditions of internal resonance within the main structure, we must first derive the con-

ditions and determine the possibility that ω2 causes the phenomenon of internal resonance. In this respect,

Eq. (45) contains ξ0iξ0 jξ0k, a nonlinear term equal to the product of an infinite series. We employed themathematical technique proposed by Van Horssen [4] to derive the conditions in which ω

2 causes internalresonance. This method is briefly outlined below.

First, in observing the integral term∫ l

0 Φi(x)Φ j(x)Φk(x)Φmdx on the right-hand side of Eq. (45), it isclear that this is actually an orthogonal multiplication of Φm(x) and Φi(x)Φ j(x)Φk(x). According to theorthogonal property, when m =±i± j±k, the integral term within Eq. (45) is not 0. Thus, in the multipliedterms within

∫ l0 Φi(x)Φ j(x)Φk(x)Φmdx, we can only confirm its orthogonality by observing whether ±γi±

γ j± γk and γm are equal. Due to the fact that γm = 0.785+mπ > 0, both −γm = γi + γ j− γk and −γm =γi− γ j + γk do not exist. Furthermore, the combination of γi− γ j− γk and γi + γ j + γk does not equal γm,and because γm = γi + γ j− γk, −γm = γi− γ j− γk, and γm = γi− γ j + γk are similar, we need only discussγm = γi + γ j− γk.

Among the factor terms in ξ0iξ0 jξ0k on the right side of Eq. (45), only the combination of ei(±ωi±ω j±ωk)To

could possibly be equal to the harmonic vibration of the system on the left of the equation. Theseare secular terms; therefore, we need only consider the combination of ωm = ±ωi ±ω j ±ωk. We arelooking for the constraint of 0 < ω < 15. Furthermore, because ωm = ±ωi±ω j±ωk can be rewritten as±(γ4

m +ω2)1/2 =±(γ4

i +ω2)1/2±(γ4

j +ω2)1/2± (γ4

k +ω2)1/2, the various conditions considered above can

be discussed in the following combinations:

Combination A: (γ

4m +ω

2)1/2=(γ

4i +ω

2)1/2+(γ

4j +ω

2)1/2+(γ

4k +ω

2)1/2.

Let γ2m = γ2

i + γ2j + γ2

k +λ , where λ = 2(γiγ j− γ jγk− γiγk), and using the inequality

h2 <(h4 +ω

2)1/2 ≤ h2−a2 +(a4 +ω

2)1/2. (46)

We can derive the inequality criterion of the elastic beam system as follows:

γ2m <

(γ

4m +ω

2)1/2 ≤ γ2i + γ

2j + γ

2k −3(0.785+π)2 +3[(0.785+π)4 +ω

2]1/2

and derive that2(γiγ j− γ jγk− γiγk)<−3(0.785+π)2 +3[(0.785+π)4 +ω

2]1/2. (47)

In addition, we can deduce from Eq. (46) that

γ2m− (0.785+π)2 +[(0.785+π)4 +ω

2]1/2 ≥ (γ4m +ω

2)1/2 > γ2i + γ

2j + γ

2k

and, therefore,2(γiγ j− γ jγk− γiγk)> (0.785+π)2− [(0.785+π)4 +ω

2]1/2. (48)

Eqs. (47) and (48) can be combined into

(0.785+π)2− [(0.785+π)4 +ω2]1/2 < λ <−3(0.785+π)2 +3[(0.785+π)4 +ω

2]1/2. (49)


Because i, j are symmetric, we can assume that γi ≥ γ j. If γi = γ j and λ = 2γi (γi−2γk), then the constraintis 0 < ω < 15, for which no solution exists. We additionally considered γi > γ j, where i > j. The definitionof λ indicates that γi (γ j− γk) = (λ/2)− γ jγk > 0. Therefore, (γ j− γk) > 0, where j > k, and as a result,i > j > k ≥ 1. We can then derive that k ≥ 1, j ≥ 2, i≥ 3, and m≥ 4.

Using Eq. (46), we can rewrite the inequality as

γ2m <

(γ

4m +ω

2)1/2 ≤ γ2i − (0.785+ iπ)2 +[(0.785+ iπ)4 +ω

2]1/2

+ γ2j − (0.785+ jπ)2 +[(0.785+ jπ)4 +ω

2]1/2 + γ2k − (0.785+ kπ)2 +[(0.785+ kπ)4 +ω

2]1/2

which becomes

λ < −(0.785+3π)2 +[(0.785+3π)4 +ω2]1/2− (0.785+2π)2 +[(0.785+2π)4 +ω

2]1/2

− (0.785+π)2 +[(0.785+π)4 +ω2]1/2.

In other words, the constraints are 0 < ω < 15, and λ < 9.36974. If we assume that k = 1, then there isonly one solution: i = 3, j = 2; if k≥ 2, such that to derive a solution, the constraint must be ω > 15. Thus,within the constraint of ω < 15, only a single solution exists: i = 3, j = 2, k = 1, m = 4, from which we candeduce that ω is 14.90629. Furthermore, due to the symmetry of i and j, they can be switched, resulting ini = 2, j = 3, k = 1, m = 4.

Combination B: (γ

4m +ω

2)1/2=(γ

4i +ω

2)1/2+(γ

4j +ω

2)1/2−(γ

4k +ω

2)1/2

which has a solution of m = j, i = k. Because i and j are symmetric, they can be switched.

Combination C: (γ

4m +ω

2)1/2=(γ

4i +ω

2)1/2−(γ

4j +ω

2)1/2+(γ

4k +ω

2)1/2

which has a solution of m = j, i = k.

Combination D: (γ

4m +ω

2)1/2=−

(γ

4i +ω

2)1/2+(γ

4j +ω

2)1/2+(γ

4k +ω

2)1/2

in which, similar to Combination C, i and j can be switched.

Combination E: (γ

4m +ω

2)1/2=−

(γ

4i +ω

2)1/2−(γ

4j +ω

2)1/2+(γ

4k +ω

2)1/2

in which, similar to Combination A, m and k can be switched. Therefore, i = 3, j = 2, k = 4, m = 1.Furthermore, because i and j are symmetric, they can be switched, from which we can derive i = 2, j = 3,k = 4, m = 1.

Combination F: (γ

4m +ω

2)1/2=(γ

4i +ω

2)1/2−(γ

4j +ω

2)1/2−(γ

4k +ω

2)1/2

in which, similar to Combination A, m and i can be switched with j and k. Therefore, i = 4, j = 1, k = 2,m = 3. Furthermore, because m and k can be switched, we can derive that i = 4, j = 1, k = 3, m = 2.


Combination G: (γ

4m +ω

2)1/2=−

(γ

4i +ω

2)1/2+(γ

4j +ω

2)1/2−(γ

4k +ω

2)1/2

in which, similar to Combination F, i and j can be switched. Therefore, i = 1, j = 4, k = 2, m = 3. Further-more, because m and k can be switched, i = 1, j = 4, k = 3, m = 2.

In comprehensive view of the combinations above, despite the existence of solutions that conform to theconstraint 0 < ω < 15 when m = 1–4 in Combinations A, E, F, and G, internal resonance cannot be createdin any two modes. In Combinations B, C, and D, however, those that conform to the condition of 0<ω < 15incur a 3:1 internal resonance only when m = 1, m = 2, and ω = 6.6751166. When m = 1,

ω1 =

√γ4

1 +ω2 =

√(0.785+π)4 +(6.6751166)2 = 16.801

and when m = 2,

ω2 =

√γ4

2 +ω2 =

√(0.785+2π)4 +(6.6751166)2 = 50.403.

In other words, 3ω1 = ω2. Accordingly, we determined that 3:1 internal resonance occurs with the combi-nation of m = 1, m = 2, and ω = 6.6751166.

3.3. Frequency ResponseTo consider the frequency response of the system, we assumed that the external force term is

f eirτ = f ei(ωm+εσ)T0 = f(eiεσT0eiωmT0

)= f eiσT1eiωmT0 .

From the discussion in the previous section, we determined that in Combinations B, C, and D, 3:1 internalresonance occurs when m = 1, m = 2, and ω = 6.6751166, which were the only conditions satisfying0 < ω < 15. To analyze this phenomenon, we rewrite Eq. (45) as

∗∗ξ1m+

(ω

2 + γ4m)

ξ1m

= (a1U a1−υ)(iωmBm(T1)e−iςmeiωmT0− iωmBm(T1)eiςme−iωmT0

)−2(

iωmB′m(T1)e−iςmeiωmT0− iωmB′m(T1)eiςme−iωmT0 +ωmς′mξ0m

)+ Bm

(f eirτ + a0U2a0

)−Cm

∫ l

0Φi(x)Φ j(x)Φk(x)Φmdx

[(Bi(T1)e−iςieiωiT0 +Bi(T1)eiςie−iωiT0

)×(B j(T1)e−iς j eiω jT0 +B j(T1)eiς j e−iω jT0

)(Bk(T1)e−iςk eiωkT0 +Bk(T1)eiςk e−iωkT0

)], (50)

where Bm = pm3/pm1 and Cm = K/pm1. For details on pm1 and pm3, please refer to Appendix 2.To obtain solvability conditions, we must single out the secular terms in Eq. (50). Regarding the 1st

mode (m = 1), the harmonic secular terms include the terms of ω1 and ω2 − 2ω1. Regarding the 2ndmode (m = 2), the harmonic secular terms include the terms of ω2 and 3ω1. Once selected, we let thesecular terms be 0 to obtain the solvability conditions. Supposing that external forces excite the 1st mode( f eirτ = f eiσT1eiω1T0), we multiply the secular terms of the 1st mode by eiς1 . For the secular terms of the2nd mode, we multiply them by eiς2 and let ΓA = σT1 + ς1, ΓB = 3ς1− ς2, Γ′A = σ + ς ′1 = 0 ⇒ ς ′1 = −σ ,Γ′B = 3ς ′1− ς ′2 = 0⇒ ς ′2 = −3σ , and ∂B1/∂T1 = ∂B2/∂T1 = 0, to obtain the solvability condition for the1st mode. The real part is

2ω1σB1−C1

(6B1B2

2

∫ l

0Φ

21Φ

22dx+3B3

1

∫ l

0Φ

41dx+3B2

1B2 cosΓB

∫ l

0Φ

31Φ2dx

)=−B1 f cosΓA. (51)


The imaginary portion is

a1U a1ω1B1−υω1B1−C1

(3B2

1B2 sinΓB

∫ l

0Φ

31Φ2dx

)=−B1 f sinΓA. (52)

The real and imaginary portions of the solvability condition in the 2nd mode are

6ω2σB2−C2

[6B2B2

1

∫ l

0Φ

22Φ

21dx+3B3

2

∫ l

0Φ

42dx+B3

1 cos(−ΓB)∫ l

0Φ

31Φ2dx

]= 0, (53)

a1U a1ω2B2−υω2B2−C2

(B3

1 sin(−ΓB)∫ l

0Φ

31Φ2dx

)= 0. (54)

Next, we square Eqs. (51) and (52) and add them together. After eliminating the terms associated with time,we can obtain[

2ω1σB1− K1

(6B1B2

2

∫ l

0Φ

21Φ

22dx+3B3

1

∫ l

0Φ

41dx+3B2

1B2 cosΓB

∫ l

0Φ

31Φ2dx

)]2

+

[a1U a1ω1B1−υω1B1− K1

(3B2

1B2 sinΓB

∫ l

0Φ

31Φ2dx

)]2

= A21 f 2. (55)

Using numerical methods, we solve Eqs. (53–55) and plot the frequency responses of the amplitudes B1and B2 in the system and the detuned frequency, σ , to observe the nonlinear internal resonance. If externalforces excite the 2nd mode ( f eirτ = f eiσT1eiω2T0), we can also use the above approach to obtain the solvabilityconditions.

4. SYSTEM WITH VIBRATION ABSORBER

In this section, we analyze the effectiveness of a TDB DVA installed at the end of a suspended beam ora support beam with an elastic foundation. We analyze the internal resonance conditions and frequencyresponse throughout the entire vibration system, the results of which are presented in Section 6.

4.1. Internal ResonanceWe have already established that 3:1 internal resonance occurs in the 1st and 2nd modes of the primestructure when ω = 6.6751166. We therefore placed a TDB DVA at the end of the beam to reduce thevibration in the system and prevent internal resonance. With the addition of the vibration absorber, thenatural frequency of the prime structure changes as follows

ωnNA =

√ω

2 + γ4m, (56)

ωnWA =

√ω

2 + γ4mCtm, (57)

where ωnNA and ωnWA are the natural frequencies of the prime structure without and with a vibration absorber,respectively. In Ctm = qm1/(qm1 +Ctmqm2) of Eq. (57), qm1 and qm2 are constants, which are explained inAppendix 1, and Ctm = ω2Φn(l)/(ω2 −ω2

nNA). As ω is the ratio of the natural vibration frequency of

the vibration absorber and the vibration frequency of the beam, we consider the vibration absorber to benonexistent when the natural vibration frequency of the vibration absorber equals 0. For this reason, Ctm


does not equal 0, and thus, Ctm does not equal 1. As a result, the natural vibration frequency of the entiresystem no promotes internal resonance when ω = 6.6751166. Thus, we established that the addition of thevibration absorber can prevent internal resonance. For further verification of the damping effects of the TDBDVA on the system, we investigated the frequency responses.

4.2. Frequency ResponseSimilar to the analysis of the system without a vibration absorber, we suppose that the external force term is

f eirτ = f ei(ωm+εσ)T0 = f(eiεσT0eiωmT0

)= f eiσT1eiωmT0 .

Using the expressionξ0m (τ) = Bm(T1)e−iςmeiωmT0 +Bm(T1)eiςme−iωmT0

we rewrite Eq. (37) into the general form∗∗

ξ1m+(ω

2 + γ4mCtm

)ξ1m = (a1U a1−υ)

(iωmBm(T1)e−iςmeiωmT0− iωmBm(T1)eiςme−iωmT0

)−2(

iωmB′m(T1)e−iςmeiωmT0− iωmB′m(T1)eiςme−iωmT0 +ωmς′mξ0m

)+ Am

(f eirτ + a0U2a0

)− Km

∫ l

0

(Φi(x)+ xCti

)(Φ j(x)+ xCt j

)(Φk(x)+ xCtk

)Φmdx

×[(

Bi(T1)e−iςieiωiT0 +Bi(T1)eiςie−iωiT0)(

B j(T1)e−iς j eiω jT0 +B j(T1)eiς j e−iω jT0)

×(Bk(T1)e−iςk eiωkT0 +Bk(T1)eiςk e−iωkT0

)], (58)

where Am = qm3/(qm1 +Ctmqm2) and Km = K/(qm1 +Ctmqm2). After singling out the secular terms in Eq.(58), we rearrange the terms and derive the solvability condition for cases in which external forces excitethe 1st mode:

2ω1σB1− K1

[6B1B2

2

∫ l

0

(Φ1 + xCt1

)(Φ2 + xCt2

)2Φ1dx

+ 3B31

∫ l

0

(Φ1 + xCt1

)3Φ1dx+3B2

1B2 cosΓB

∫ l

0

(Φ1 + xCt1

)2 (Φ2 + xCt2

)Φ1dx

]2

+

a1U a1ω1B1−υω1B1− K1

[3B2

1B2 sinΓB

∫ l

0

(Φ1 + xCt1

)2 (Φ2 + xCt2

)Φ1dx

]2

= A21 f 2.

(59)

The real part of the solvability condition in the 2nd mode is

6ω2σB2− K2

[6B2B2

1

∫ l

0

(Φ2 + xCt2

)(Φ1 + xCt1

)2Φ2dx+3B3

2

∫ l

0

(Φ2 + xCt2

)3Φ2dx

+ B31 cos(−ΓB)

∫ l

0

(Φ1 + xCt1

)3Φ2dx

]= 0 (60)

and the imaginary part is


[B3

1 sin(−ΓB)∫ l

0

(Φ1 + xCt1

)3Φ2dx

]= 0. (61)


Likewise, if the external forces excite the 2nd mode, then the real portion of the solvability condition inthe 1st mode can be expressed as

23

ω1σB1− K1

[6B1B2

2

∫ l

0

(Φ1 + xCt1

)(Φ2 + xCt2

)2Φ1dx+3B3

1

∫ l

0

(Φ1 + xCt1

)3Φ1dx

+ 3B21B2 cosΓB

∫ l

0

(Φ1 + xCt1

)2 (Φ2 + xCt2

)Φ1dx

]= 0 (62)

and the imaginary portion as


(3B2

1B2 sinΓB

∫ l

0

(Φ1 + xCt1

)2 (Φ2 + xCt2

)Φ1dx

)= 0. (63)

The 2nd mode is thus2ω2σB2− K2

[6B2B2

1

∫ l

0

(Φ2 + xCt2

)(Φ1 + xCt1

)2Φ2dx

+ 3B32

∫ l

0

(Φ2 + xCt2

)3Φ2dx+B3

1 cos(−ΓB)∫ l

0

(Φ1 + xCt1

)3Φ2dx

]2

+


[B3

1 sin(−ΓB)∫ l

0

(Φ1 + xCt1

)3Φ2dx

]2

= A22 f 2. (64)

Using numerical methods, we plotted the frequency responses of amplitudes B1 and B2 and the detunedfrequency, σ . In addition to observing whether nonlinear internal resonance exists, we can also analyze thedamping effects of the vibration absorber on the main structure.

5. STABILITY ANALYSIS

To analyze the applicability of the vibration absorber with regard to stability, we intentionally included theinfluence of simple aerodynamics in the elastic beam system. In the original model Eq. (1) of this study, wechanged the magnitude of wind speed U in the aerodynamic term

ρaU2d2

(a0 +

a1

U∂w(x, t)

∂ t+

a2

U2∂w(x, t)

∂ t

2

+ · · ·

)and used the Floquet theory to analyze the system. We adopted the fourth-order Runge–Kutta method toobtain the Floquet transition matrix of the system with vibration absorber. Using the derived eigen values ofthe matrix, we obtained the Floquet multipliers (F.M.) [8] to determine the stability of the system.

We first express the modes using the following assumptions:

ξ1 = ξ10 +ξ 1, (65)

ξ2 = ξ20 +ξ 2, (66)

where ξ10 and ξ20 are the terms of periodic solutions, and ξ 1 and ξ 2 are the disturbance terms. By substi-tuting the equations above into Eq. (37), we can derive that( ∗∗

ξ10+∗∗ξ 1)+(υ− a1U a1)

( ∗ξ10+

∗ξ 1)+(ω2 + γ

41Ct1)(ξ10 +ξ 1)+ K1Γ1 = A1

(a0U2a0 + f eirτ

), (67)


( ∗∗ξ20+

∗∗ξ 2)+(υ− a1U a1)

( ∗ξ20+

∗ξ 2)+(ω

2 + γ42Ct2

)(ξ20 +ξ 2

)+ K2Γ2 = A2

(a0U2a0 + f eirτ

), (68)

and the equations for the disturbance terms are

∗∗ξ 1+(υ− a1U a1)

∗ξ 1+

(ω

2 + γ41Ct1

)ξ 1 + K1Γ1 = A1

(a0U2a0 + f eirτ

), (69)

∗∗ξ 2+(υ− a1U a1)

∗ξ 2+

(ω

2 + γ42Ct2

)ξ 2 + K2Γ2 = A2

(a0U2a0 + f eirτ

). (70)

For details on the respective coefficients, please refer to Appendix 3.From Eqs. (69) and (70), we can see that when the damping coefficient υ in the damping term (υ− a1U a1)

is 0.01 and the aerodynamic coefficient a1 equals 0.5, the damping term (υ− a1U a1) may change frompositive to negative and divergence may occur in the system if the dimensionless wind speed a1U is greaterthan 0.02. From the perspective of physics, when the system is subjected to this wind speed, the constantexcitation from the wind will deplete the damping effects on the system, which will diverge and generatestructural damage. For this reason, the divergent wind speed in this model is a1U = 0.02. To verify theaccuracy of this value, we employed the fourth-order Runge–Kutta method to derive the Floquet transitionmatrix of the damped system, which further enabled us to obtain the eigen values. By incorporating differentinitial conditions and F.M. criteria, we can create the basins of attraction for the various conditions andobserve the stability of the system under various combinations of wind speed and the masses and moduli ofelasticity for the vibration absorber installed at the end of the beam.

6. RESULTS AND DISCUSSION

The issue of internal resonance in nonlinear systems is difficult to analyze and easily overlooked. However,it has the capacity to cause system divergence and structural instability. For these reasons, we endeavoredto increase the stability of a system with nonlinear internal resonance under the influences of harmonicforces and aerodynamics to avoid internal resonance and reduce the amplitude of vibrations in the systemby changing the mass and modulus of elasticity of the vibration absorber.

6.1. Internal ResonanceHaving completed theoretical analysis of the nonlinear systems, we used the relationship between the am-plitude and the frequency of the nonlinear system without a vibration absorber to show system internalresonance. To verify the accuracy of the fixed-points graphs, we created the time domain response andPoincaré map for a specific frequency and confirmed whether its amplitudes were identical to those in thefixed-points graphs. Figure 2 exhibits the fixed-points graphs and time responses of the 1st mode and 2ndmode when a harmonic force excites the 1st mode (in other words, the lower mode). The horizontal axisrepresents the detuned frequency (σ ) near the natural frequency of the mode and the vertical axis presentsthe dimensionless amplitude of the mode. The figure shows that within a specific frequency range in the1st mode, there may be more than one corresponding amplitude. If the system operates within this unstableregion for long periods of time, fatigue and damage may occur. This is a classic nonlinear phenomenon.Figure 2 also presents the verification graphs for the excitations of the 1st and 2nd modes at a frequencyof σ = −5, 0 and 1.5. Both time response plot and Poincaré map for each σ are presented in Fig. 2. Ob-servation showed that the time responses converge within a stable region (σ = −5, 0, and the 2nd modewhen σ = 1.5), the Poincaré maps indicate limit cycle oscillation (L.C.O.). Moreover, the magnitude of theamplitudes matched those of the fixed-points graphs. In the unstable region when σ = 1.5; the 1st modepresents chaos. Figure 3 presents the fixed-points graphs when the 2nd mode is excited. Despite being thehigher mode of excitation, the corresponding amplitudes were significantly smaller than those in the lower


Fig. 2. Verification graph for time response and Poincaré map of excitation to 1st mode when σ =−5, 0 and 1.5.

mode. This proves the existence of nonlinear 3:1 internal resonance in the 1st and 2nd modes of the system.Figure 3 also presents the verification graphs for the excitations of the 1st and 2nd modes at a frequency ofσ = −5, 0 and 1.5. The results also correspond to those for the fixed points, and we can see from the timeresponse and Poincaré maps that the excitations of the 1st and 2nd modes are L.C.O. when σ = –5 and 0.In the unstable region when σ = 1.5; the 1st and 2nd modes both display chaos. The verification graphsconsistently matched the fixed points, indicating the influence of internal resonance on the system.

6.2. DVA EffectsThe damping effects of the vibration absorber on the system are analyzed in this section. According toEqs. (99) and (100), we can see a significant difference in the form of Ctm between the natural vibration

frequency of the main structure with a vibration absorber, (ωnWA =√

ω2 + γ4

mCtm) and that of the main


Fig. 3. Verification graph for time response and Poincaré map of excitation to 2nd mode when σ =−5, 0 and 1.5.

structure without a vibration absorber, (ωnNA =√

ω2 + γ4

m). In Ctm = qm1/(qm1 +Ctmqm2), qm1 and qm2 areconstants, which are explained in Appendix 1.

Moreover, because qm2 is a very small value, Ctm must be great in magnitude in order for the naturalvibration frequency of the vibration absorber to exert an influence. Thus, in Ctm = ω2Φn(l)/(ω2−ω2

nNA),

the vibration absorber is only effective when ω2 ≈ω2nNA

.To obtain specific parameters, we fixed the modulusof elasticity of the vibration absorber (k/EI = mω2). In other words, when mω2 is fixed at 0.1, 0.05, and0.005, we derive the mass of the vibration absorber as m≈mω2m/ω2

nNAusing ω2 ≈ω2

nNA. Subsequently, we

can plot the relationships between the Ct1,2 of the 1st and 2nd modes and the mass of the vibration absorber,as shown in Figs. 4 through 6. From these graphs, we can see the optimal range for the natural vibrationfrequency of the vibration absorber in the 1st and 2nd modes.


Fig. 4. Relationship between Ctn and m when k/EI = 0.1.


Using these optimal values, we can then create amplitude tables for different k/EI values when the 1st and2nd modes of the system are excited, as shown in Tables 1 through 3. These tables compare the amplitudesof the two modes resulting from different combinations of m and k/EI. We divided these amplitudes by theamplitudes without the vibration absorber to normalize them. We then totaled and averaged the normalizedamplitudes of the 1st and 2nd modes, listing them in the last column (Norm Amp.) for comparison. Fromthe top half of Table 1, we can see that when k/EI = 0.1, the optimal mass of the Ct1 vibration absorberis 0.003542633. Under excitation, the amplitude of the 1st mode equals 1.4E-05, whereas the amplitudeof the 2nd mode is 9.5E-07. Once we nondimensionalized the amplitudes without the vibration absorber,



Table 1. Influence of DVA mass on system amplitude when k/EI = 0.1.m 1st Mode Amp. 2nd Mode Amp. Ct1 Ct2 Norm Amp.No DVA 0.025000 0.02000000 0 0 10.0035425000 0.024000 0.00150000 –37489 –0.176732 0.51750.0035426000 0.005500 0.00037000 –151691 –0.176726 0.119250.0035426330 0.000014 0.00000095 –66700000 –0.176724 0.00030380.0035426340 0.000185 0.00001500 4600000 –0.176724 0.0040750.0035426400 0.001150 0.00007800 723565 –0.176724 0.024950.0035426600 0.004300 0.00029000 185242 –0.176723 0.093250.0003936250 0.380000 0.77000000 –1.58781 616205 26.850.0003936258 0.380000 0.35000000 –1.58781 5400000 16.350.0003936259 0.380000 0.34000000 –1.58781 18500000 16.10.0003936262 0.380000 0.30000000 –1.58781 –2130000 15.10.0003937000 0.380000 0.09000000 –1.58785 –7512.67 9.850.0005000000 0.380000 0.06500000 –1.64332 –5.23 9.225


Table 2. Influence of DVA mass on system amplitude when k/EI = 0.05.m 1st Mode Amp. 2nd Mode Amp. Ct1 Ct2 Norm Amp.No DVA 0.025000 0.02000000 0 0 10.0017712900 0.018500 0.00118000 –21485 –0.176772 0.39950.0017713000 0.015000 0.00072000 –147895 –0.176761 0.3180.0017713164 0.000028 0.00000210 –67022440 –0.176759 0.0006130.0017713170 0.000350 0.00005100 2549689 –0.176759 0.0082750.0017713200 0.002200 0.00015000 828876 –0.176759 0.047750.0017713300 0.008400 0.00470000 1852390 –0.176758 0.28550.0001968127 0.380000 1.10000000 –1.58908 139314 35.10.0001968128 0.380000 0.50000000 –1.58908 284259 20.10.0001968129 0.380000 0.36000000 –1.58908 4000352 16.60.0001968150 0.380000 0.18000000 –1.58909 –134006 12.10.0001968200 0.380000 0.11500000 –1.58909 –39415 10.4750.0005000000 0.380000 0.06500000 –1.96805 –0.917942 9.225

we derived a normalized amplitude of 3.038E-04. This shows that when the mass of the vibration absorberis 0.003542633, it exerts optimal damping effects on the system. The trend in the lower left graph of thetable shows that the mass range for optimal damping is m = 0.0035426–0.00354266. In other words, thedamping effects of the vibration absorber are best when the mass of the vibration absorber is within thisrange. With regard to Ct2, the lower half of Table 1 shows that the mass of the vibration absorber has anoptimal value of 0.0003936259. However, the trend in the lower right graph of the table also indicates thatwhile an optimal value has been derived for the vibration absorber, inadequate mass renders its dampingeffects useless in the system. Conversely, the addition of a vibration absorber considerably increased theamplitudes of the system. The results in Tables 2 and 3 are similar, respectively presenting that whenk/EI = 0.05 and k/EI = 0.005, the optimal mass ranges for the vibration absorber are m = 0.0017713–0.00177133 and m = 0.000177134–0.0001771319. The data in Tables 1–3 with very high precision confirmthe possibility of the optimal mass ratio (m) to diminish the beam vibration amplitudes theoretically. Theeffects of the DVA to reduce the beam vibration can be seen in comparing with the “No DVA” case inTables 1–3. Practically we can choose the m in a suitable range of order to have a better damping effect.For example, in the case of k/EI = 0.1, we can see that when m is about the order of 0.35%, it can achievethe purpose of this work. However, a quite smaller order of m (0.039%) gives worse result. The vibrationamplitudes can reach more than 10 times of the “No DVA” case.

Tables 2 and 3 give similar results. This result discovers that a smaller mass ratio of DVA does not givedesirable results. The optimal range of m does exist in this system. Furthermore, a comparison of Tables 1,


Table 3. Influence of DVA mass on system amplitude when k/EI = 0.005.m 1st Mode Amp. 2nd Mode Amp. Ct1 Ct2 Norm Amp.No DVA 0.025000 0.02000000 0 0 10.000177131400 0.016000 0.00105000 –1055371 –0.176777 0.346250.000177131500 0.009700 0.00065000 –1711495 –0.176777 0.210250.000177131640 0.000052 0.00000390 –324148700 –0.176777 0.0011380.000177131700 0.003400 0.00024000 4654088 –0.176777 0.0740.000177131800 0.010000 0.00070000 1536654 –0.176777 0.21750.00017713900 0.015300 0.00104000 986179 –0.176777 0.3320.000019681292 0.380000 1.30000000 –1.58902 8763500 40.10.000019681293 0.380000 1.28000000 –1.58902 10763500 39.60.000019681294 0.380000 0.35000000 –1.58902 89156390 16.350.000019681500 0.380000 0.08500000 –1.58902 –137756 9.7250.000019682000 0.380000 0.07200000 –1.59021 –39633 9.40.000050000000 0.380000 0.06500000 –1.96943 –0.918034 9.225

2, and 3 reveals that the smaller the modulus of elasticity in the vibration absorber is, the smaller the optimalmass range is. As a result, when the modulus of elasticity of the vibration absorber is exceedingly small,the mass must be extremely close to the optimal value in order for the system to have the best dampingeffects. Figures 7–10 display the time response and Poincaré maps created from the best and worst vibrationabsorber masses from Table 1 in conjunction with excitation to the 1st and 2nd modes. Again, we verifiedwhether the amplitudes within were consistent with those of the fixed-points graphs. For example, Figs. 7and 8 respectively display excitations of the 1st and 2nd modes when k/EI = 0.1 and the mass of thevibration absorber is optimal. In the four aforementioned figures, the left side shows the time responseand Poincaré map for the 1st mode and the right side exhibits the time response and Poincaré map for the2nd mode. As can be seen, all of the time responses ultimately converge to within a stable range and thePoincaré maps present L.C.O., while the amplitudes match those of the fixed-points graphs. According tothe numerical simulations in Figs. 7 through 10, we can confirm that the amplitudes of the system equippedwith a vibration absorber shown in Table 1 are accurate, and that the damping effects are significant.

6.3. Stability AnalysisTo observe the stability of the system under various initial conditions, we applied the Floquet theory anddetermined that the dimensionless divergence speed is a1U = 0.02. Using F.M. as judgment criteria, weused a symbol to represent instability and no marks for stability in the basins of attraction that were created.This enabled us to observe the stability of the system when subjected to various wind speeds and equipped


Fig. 7. Verification graph for time response and Poincaré map of excitation to 1st mode when k/EI = 0.1 and m =0.003542633.

Fig. 8. Verification graph for time response and Poincaré map of excitation to 2nd mode when k/EI = 0.1 andm = 0.003542633.


Fig. 9. Verification graph for time response and Poincaré map of excitation to 1st mode when k/EI = 0.1 and m =0.0005.

Fig. 10. Verification graph for time response and Poincaré map of excitation to 2nd mode when k/EI = 0.1 andm = 0.0005.


Fig. 11. Basin of attraction when a1U = 0.0001, k/EI = 0.1, m = 0.003542633 (O), and m = 0.0003936259 (•).


with vibration absorbers of various masses and moduli of elasticity. First, with almost no consideration fordimensionless wind speed a1U , we constructed Fig. 11 with a modulus of elasticity of k/EI = 0.1 combinedwith vibration absorbers with the best and worst masses m. The plots for m = 0.003542633 are markedas circles (O). The plots for m = 0.0003936259 are marked as black dots (•). As can be seen, the stableregion in m = 0.003542633 is substantially larger than that in the case of m = 0.0003936259. This clearlyshows that when the wind speed and modulus of elasticity are the same, the mass of the vibration absorberinfluences the damping effects in the system, leading to significant differences in the size of the stable



regions. Figure 12 displays the best and worst vibration absorber masses m combined with an a1U = 0.019approaching dimensionless divergent wind speed and k/EI equaling 0.1. The stable regions in Fig. 12 areboth smaller than those in Fig. 11, indicating that the unstable region increases in size with wind speed. Inaddition, when the dimensionless wind speed a1U reaches the divergent wind speed of 0.02, the stable regionnarrows considerably, as shown in Fig. 13. Although a1U is only 0.001 greater than that in Fig. 12, the stableregions differ significantly. Furthermore, the lack of a stable region in Fig. 13 indicates that the systemhas diverged completely, thereby demonstrating the associated risk. Theoretically speaking, the systemshould diverge completely when the divergent wind speed is exceeded, as in the case of m = 0.0003936259in Fig. 13; nevertheless, the case of m = 0.003542633 shows that if the mass of the vibration absorberapproaches the optimal value and exerts outstanding damping effects on the system, it can only increasethe scope of the stable region to a limited degree, thereby increasing the speed of divergence. Figures 14and 15 were employed to verify the accuracy of the basins of attraction. In Fig. 14, for example, the initialconditions of the stable region are (−1.0,0.4), and the time response and Poincaré map for the 1st and 2ndmodes present L.C.O. (as shown on the left of Fig. 14), verifying that the white area is indeed the stableregion. In contrast, the initial conditions for the unstable region in Fig. 14 are (−1.0,0.5), and the timeresponse and Poincaré map for the 1st and 2nd modes display chaos (as shown on the right of Fig. 14), thusconfirming that the black area is the unstable region. Fig. 15 presents the verification graph for the initialconditions of the stable region and the unstable region when a1U reaches the divergence speed. On the leftof Fig. 15, the initial conditions are (−0.8,−0.8), the time response and Poincaré map for the 1st and 2ndmodes indicate L.C.O., thereby verifying that the white area is the region of stability. On the right of Fig. 15is the verification graph for the unstable region (the initial conditions are (−0.8,−0.9)) when a1U reachesthe divergence speed; the 1st mode diverges while the 2nd mode is stable, indicating the existence of asaddle node in the system. Figure 16 is the verification graph for the region of stability when a1U reachesthe divergence speed and m is less optimal. As can be seen, both the 1st and 2nd modes diverge. Finally, theresults of the numerical simulations in Figs. 14–16 conform to the basins of attraction, thus demonstratingthe accuracy of the basins of attraction in this study.


Fig. 14. Verification graph for time response and Poincaré map of stable region when a1U = 0.0001, k/EI = 0.1 andm = 0.0003936259.

7. CONCLUSION

This study investigated the stability of an elastic beam subjected to external aerodynamic forces. The beamwas suspended by cables simulated using nonlinear cubic springs and linear dampers as well as a time-dependent boundary dynamic vibration absorber (TDB DVA) installed at the free end. This system presentsmultiple mode couplings and potential internal resonance. This study used the MOMS method, fixed pointsgraphs, time response graphs, Poincaré maps, and the Floquet theory to compare and verify the results ofour theoretical stability analysis. In addition to providing the vibration characteristics of nonlinear systems,this study analyzed the influence of vibration absorber mass and modulus of elasticity on the overall system.


Fig. 15. Verification graph for time response and Poincaré map of stable region when a1U = 0.0020, k/EI = 0.1 andm = 0.003542633.

The impact of aerodynamic forces was also considered in the stability analysis to provide a reference foracademics and practitioners. We present our conclusions as follows:

1. The possibility for internal resonance within the system without vibration absorbers was elucidated.Using mathematical analysis and fixed points graphs aided by time response and Poincaré maps,we verified the existence of 3:1 nonlinear internal resonance in the 1st and 2nd modes of the mainstructure.

2. The approach proposed in this study is used to derive optimal combinations of mass and modulus ofelasticity for the TDB DVA to avoid internal resonance and provide effective damping. Under various


Fig. 16. Verification graph for time response and Poincaré map of unstable region when a1U = 0.0020, k/EI = 0.1and m = 0.0003936259.

moduli of elasticity, we used Ct1 to obtain the range of masses for the vibration absorber capable ofproviding optimal damping effects. However, the mass value obtained using Ct2 was too small toprovide effective damping. Moreover, a smaller modulus of elasticity corresponded to a smaller rangeof mass values for the vibration absorber.

3. Stability analysis of the system indicated that in circumstances with fixed wind speeds and moduliof elasticity, the mass of the vibration absorber may still influence the damping effects in the system,leading to significant differences in the region of stability. When the mass of the vibration absorberapproaches the optimal value derived in this study, the range of stability can be increased moderately,thereby increasing the divergence speed sufficiently to be withstood by the system.

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APPENDIX 1

qm1 =l2− sin2γml

4γm− 2mω2 sin2

γmlγ4

m+ sin2

γml(

cotγml2γm

− mω2

γ4m− l

2sinh2γml

),

qm2 =−2mω2l sin

(γml)

γ4m

,qm3 =1γm

(1− sinγml

sinhγml

)− 2mω2 sinγml

γ4m

.

APPENDIX 2

pm1 =l2− sin2γml

4γm+

sinγmlsinhγml

[(coshγml sinγml− sinhγml cosγml

)γm

]

+sin2

γmlsinh2

γml

[sinhγml coshγml

2γm− l

2

]=

l2− sin2γml

4γm+ sin2

γml(

cotγml2γm

− l2sinh2

γml

)

pm3 =−cosγml +1

γm+

sinγmlsinhγml

(coshγml−1

γm

)=

1γm

(1− sinγml

sinhγml

)APPENDIX 3

A1 =q13

q11 +Ct1q12, A2 =

q23

q21 +Ct2q22, K1 =

Kq11 +Ct1q12

, K2 =K

q21 +Ct2q22,


Γ1 =∫ l

0

[Φ1(Φ1 + xCt1

)3(

3ξ210ξ 1

)+ Φ1

(Φ2 + xCt2

)3(

3ξ220ξ 2

)+3Φ1

(Φ1 + xCt1

)2 (Φ2 + xCt2

)(2ξ20ξ10ξ 1 +ξ 2ξ

210

)+3Φ1

(Φ1 + xCt1

)(Φ2 + xCt2

)2(

2ξ10ξ20ξ 2 +ξ 1ξ220

)]dx,

Γ2 =∫ l

0

[Φ2(Φ1 + xCt1

)3(

3ξ210ξ 1

)+ Φ2

(Φ2 + xCt2

)3(

3ξ220ξ 2

)+3Φ2

(Φ1 + xCt1

)2 (Φ2 + xCt2

)(2ξ20ξ10ξ 1 +ξ 2ξ

210

)+3Φ2

(Φ1 + xCt1

)(Φ2 + xCt2

)2(

2ξ10ξ20ξ 2 +ξ 1ξ220

)]dx.


Yi-Ren Wang and Chia-Man Chang Department of … of Aerospace Engineering, Tamkang University, ......

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