Quasiperiodic galloping of a wind-excited tower near secondary resonances of order 2

Article

Quasiperiodic galloping of a wind-excited tower near secondary resonances

of order 2

Ilham Kirrou, Lahcen Mokni and Mohamed Belhaq

Abstract

Quasiperiodic galloping of a wind-excited tower under unsteady wind is investigated analytically near secondary (sub/superharmonic) resonances of order 2 considering a single degree-of-freedom model. The case where the unsteady wind develops multiharmonic excitations consisting of the two first harmonic terms is examined. We perform two successive multiple scale methods to obtain analytical expressions of a quasiperiodic solution and its modulation enve- lope near the secondary resonances. The influence of unsteady wind on the quasiperiodic galloping and on the frequency of its modulation is examined for different cases of wind excitation. The results show that the quasiperiodic galloping onset and its modulation envelope can be influenced, depending on the activated resonance and the harmonic compo- nent induced by the unsteady wind. It is also shown that the frequency of the quasiperiodic galloping is higher near the 2-superharmonic resonance in all cases of wind excitation.

Keywords

Quasiperiodic galloping, secondary resonances, wind effects, structural dynamics, perturbation analysis

1. Introduction

Wind-induced vibrations of tall buildings may cause galloping above a certain threshold of the wind speed (Parkinson and Smith, 1964; Novak, 1969; Nayfeh and Abdel-Rohman, 1990; Abdel-Rohman, 2001). This gal- loping is due to the aerodynamic self-excited forces that act in the direction of the transverse motion causing periodic oscillations near the resonance or quasiperio- dic (QP) response away from the resonance (Luongo and Zulli, 2011; Zulli and Luongo, 2012; Belhaq et al., 2013; Mokni et al., 2014; Kirrou et al., 2013) Such a dynamic response was examined in parametric and self-excited oscillators near primary and secondary resonance (Tondl, 1978; Szabelski and Warminski, 1995; Warminski, 2001; Belhaq et al., 1986; Belhaq, 1990). Considerable efforts have been made to develop methods for controlling and quenching the amplitude of such wind-induced vibrations. A review of the main classes of semi-active control devices and their full-scale implementation to civil infrastructure applications is given in Spencer and Nagarajaiah (2003).

The effect of external excitation on periodic gallop- ing of a tall structure has been studied near primary and

secondary resonances by Abdel-Rohman (2001), while the effect of parametric, external and self-induced exci- tation on galloping of a single tower and two towers linked by a nonlinear viscous device was examined by Luongo and Zulli (2011) and Zulli and Luongo (2012).

The periodic galloping was studied analytically using the multiple scale methods (MSMs), while the QP modulation envelope was approximated numerically.

Based on the single degree-of-freedom (SDOF) model (Luongo and Zulli, 2011), the effect of fast har- monic excitation on galloping onset of a tower under steady and unsteady wind was studied in Belhaq et al.

(2013) and Kirrou et al. (2013), and the influence of internal parametric damping on galloping onset was examined in Mokni et al. (2014). Analytical predictions (Belhaq et al., 2013; Kirrou et al., 2013) show that

Laboratory of Mechanics, University Hassan II-Casablanca, Morocco

Corresponding author:

Ilham Kirrou, Laboratory of Mechanics, University Hassan II-Casablanca, BP 5366 Maarif, Casablanca, Morocco.

Email: ilhamkirrou@gmail.com

Received: 19 September 2014; accepted: 2 March 2015

Journal of Vibration and Control 1–13

!The Author(s) 2015 Reprints and permissions:

sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546315581757 jvc.sagepub.com

wind-excited towers may perform QP galloping with amplitude of oscillations having the same order of mag- nitude as that of the periodic response, even for rela- tively small values of the wind velocity. Moreover, due to the frequency-locking phenomenon, the amplitude of the QP galloping may become higher than that of the periodic response just prior to synchronization. This QP galloping was reported using numerical simulations on a 3D tower-model indicating that the wind-excited large stress tower effectively develops QP responses rather than periodic oscillations (Qu et al., 2001).

Therefore, in order to enhance stability performance of tall buildings, the QP galloping should not be neg- lected and must be systematically taken into consider- ation in the dynamic analysis of such structures. In this context, QP galloping of a wind-excited tower was stu- died analytically near the primary resonance and the influence of different excitations on the QP modulation envelope was examined (Belhaq et al., 2013; Kirrou et al., 2013).

The purpose of this study is to investigate the QP galloping of a wind-excited tower near secondary res- onances of order 2. The study is motivated by the results given in Abdel-Rohman (2001), in which the effect of unsteady wind on periodic galloping was examined near primary and subharmonic resonances of order 2, 3 and 4. It was concluded, in particular, that while the subharmonic resonances of order 3 and 4 do not cause any significant reduction in the wind speed onset of periodic galloping, the secondary reson- ance of order 2 causes a nonnegligible reduction in the wind speed onset (but smaller than the case of primary resonance). However, the effect of unsteady wind on QP galloping has not been tackled near these reson- ances. In this study the influence of the unsteady wind on the QP galloping near the secondary resonances of order 2 is investigated.

In Section 2, we present the equation of motion and perform the MSM to obtain the modulation equations near the secondary resonances of order 2. A second MSM is applied on the modulation equations to approximate the QP galloping and its modulation enve- lope. Results of various effects of parameters on the QP galloping and its wind speed onset are also reported and discussed. The final section concludes the work.

2. Equations of motion and QP galloping

According to Luongo and Zulli (2011), oscillations of a tower under turbulent wind flow can be modeled by the following dimensionless SDOF equation of motion

x € þ x þ ½c

a

ð1 UÞ b

1

uðtÞ x _ þ b

2

x _

²

þ b

₃₁

U þ b

₃₂

U

²

uðtÞ

_ x

³

¼

1

UuðtÞ þ

2

U

²

ð1Þ where the dot denotes differentiation with respect to the non-dimensional time t, and U represents the steady component of the wind velocity. Equation (1) includes the elastic, viscous, inertial linear terms, and quadratic and cubic components in the velocity. Usually, the nature of the unsteady wind is modeled as a random process, however, it is convenient to express the random phenomenon as a series of harmonic terms transforming its spectrum into the time domain. In this way, the unsteady wind can be approximated by a periodic force in the form u(t) ¼ u

1

sint þ u

2

sin2t, where u

1

, u

2

and are, respectively, the amplitudes and the fundamental frequency of the excitation. Note that in Abdel-Rohman (2001), only the first harmonic term was considered as a first approximation.

In our analysis, we shall consider the case u

1

6¼ 0, u

2

¼ 0, the case u

1

¼0, u

2

6¼ 0 and the case where both components u

1

, u

2

are present. The details of the derivation of equation (1) are given in Luongo and Zulli (2011), while the expressions of its dif- ferent coefficients and the numerical values of param- eters used here are provided in Appendix 1, for convenience.

To obtain the modulation equations of equation (1) near the secondary resonances, we perform the MSM (Nayfeh and Mook, 1979) by introducing a bookkeep- ing parameter ", scaling as x ¼ "

¹²

x, b

1

¼ "b

1

, b

2

¼ "

¹²

b

2

,

1

¼ "

¹²

1

,

2

¼ "

³²

2

and assuming that U ¼ 1 þ "V where V stands for the mean wind velocity (Luongo and Zulli, 2011).

A two-scale expansion of the solution is sought in the form

xðtÞ ¼ x

0

ðt

0

, t

1

Þ þ "x

1

ðt

0

, t

1

Þ þ Oð"

²

Þ ð2Þ

where t

i

¼ "

ⁱ

t (i ¼ 0,1). In terms of the variables t

i

, the time derivatives become

_dt^d

¼ d

0

þ "d

1

þ Oð"

²

Þ and

d²

dt²

¼ d

²₀

þ 2"d

0

d

1

þ Oð"

²

Þ, where d

^j_i

¼

_@^@jt^ji

. Substituting equation (2) into equation (1) and equating coefficients of the same power of ", we obtain the following two first-order equations

d

²₀

x

0

þ x

0

¼

1

uðt

0

Þ ð3Þ d

²₀

x

1

þ x

1

¼ 2d

0

d

1

x

0

þ ðc

a

V þ b

1

uðt

0

ÞÞðd

0

x

0

Þ

b

2

ðd

0

x

0

Þ

²

ðb

31

þ b

32

uðt

0

ÞÞðd

0

x

0

Þ

³

þ

2

ð4Þ

A solution of the first-order equation (3) is given by

x

0

¼ Aðt

1

Þe

^it0

i

1

u

1

2

²

e

^it0

þ u

2

8

²

e

^2it0

h i

þ cc ð5Þ

where i is the imaginary unit and A is an unknown complex amplitude. Equation (4) can be solved for the complex amplitude A by introducing its polar form as A ¼

¹₂

ae

ⁱ

.

2.1. Subharmonic resonance of order 2

In the case of subharmonic resonance of order 2, the resonance condition is expressed as ¼ 2 þ 2", where is a detuning parameter. Substituting the expression for A into equation (4) and eliminating the secular terms, the modulation equations of the amplitude a and the phase can be extracted as

_

a ¼ ½S

₁

S

₃

sinð2Þ S

₄

cosð2Þ a ½S

₂

þ S

₅

sinð4Þa

³

a _ ¼ ½ S

3

cosð2Þ þ S

4

sinð2Þ a ½S

5

cosð4Þa

³

8 >

> <

> >

:

ð6Þ

where S

₁

¼

¹₂

c

_a

V 3b

_{31 2}^1u2¹

þ

₄^1u2²

, S

₂

¼

³₈

b

₃₁

, S

₃

¼

¹₄

b

1

u

1

, S

4

¼

^b2₂¹2^u1

þ

³

2 1

4²

u

1

u

2

and S

5

¼

^b₁₆³²

u

2

.

One observes from equation (1) that the u

1

cos t component of the unsteady wind produces a parametric resonance (ð ’ 2Þ). The component u

2

cos2t, on the other hand, does not produce a resonance here, but it produces a parametric resonance near the primary res- onance ð ’ 1Þ (Kirrou et al., 2013).

Periodic solutions of the original system in equation (1) correspond to equilibria of the slow flow in equation (6). In the case where only the first harmonic compo- nent is present (u

1

6¼ , 0 u

2

¼ 0), the amplitude-response equation obtained from the slow flow system in equa- tion (6) is given by

S

²₂

a

⁴

2S

₁

S

₂

a

²

þ ð

²

þ S

²₁

S

²₃

S

²₄

Þ ¼ 0 ð7Þ while in the case where the second harmonic compo- nent is activated (u

1

¼ 0, u

2

6¼ 0), the amplitude-response equation reads

ðS

²₅

S

²₂

Þa

⁴

þ 2S

1

S

2

a

²

ð

²

þ S

²₁

Þ ¼ 0 ð8Þ Next, we approximate QP responses of equation (1) corresponding to periodic solutions of the slow flow in equation (6) and we analyze the effect of different har- monic terms on the QP galloping near the 2- subharmonic resonance. To this end, we transform

the modulation equations from the polar form in equa- tion (6) to the following Cartesian system using the variable change u ¼ a cos and v ¼ a sin

du

dt ¼ ð þ S

3

Þv þ ðS

1

S

4

Þu ðS

2

u þ S

5

vÞ ðu

²

þ v

²

Þ 8S

5

u

²

v 4S

5

u

⁴

v þ 4S

5

u

²

v

³

u

²

þ v

²

dv

dt ¼ ð þ S

₃

Þu þ ðS

₁

S

₄

Þv ðS

₂

v þ S

₅

uÞ ðu

²

þ v

²

Þ 8S

₅

uv

²

4S

₅

uv

⁴

þ 4S

₅

u

³

v

²

u

²

þ v

²

8 >

> >

> >

> >

> >

> >

<

> >

> >

> >

> >

> >

> :

ð9Þ

where is a bookkeeping parameter introduced in damping and nonlinearity so that the unperturbed system admits a basic solution. Following the work of Belhaq and Houssni (1999) and Belhaq and Fahsi

−1.5 −1 −0.5 0 0.5 1 1.5

x 10⁻³ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

(a)

(b)

σ a

Periodic galloping

QP galloping

u₁=0.033

−1.5 −1 −0.5 0 0.5 1 1.5

x 10⁻³ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ a

u₁=0.1

Figure 1. Periodic and QP galloping versus for V ¼ 0.167;

(a) u

1

¼ 0.033, (b) u

1

¼ 0.1. The analytical approximation is shown

(solid lines for stable, red line for unstable), with circles repre-

senting the numerical simulation.

(2009), a periodic solution of the slow flow in equa- tion (9) can be sought in the form

uðtÞ ¼ u

0

ðT

1

, T

2

Þ þ u

1

ðT

1

, T

2

Þ þ Oð

²

Þ

vðtÞ ¼ v

0

ðT

1

, T

2

Þ þ v

1

ðT

1

, T

2

Þ þ Oð

²

Þ ð10Þ

where T

1

¼ t and T

2

¼ t. Introducing D

_i

¼

_@T^@

i

yields

d

dt

¼ D

₁

þ D

₂

þ Oð

²

Þ, substituting equation (10) into equation (9) and collecting terms, one obtains at

different orders of

D

²₁

u

₀

þ l

²

u

₀

¼ 0 v

0

¼ D

1

u

0

ð11Þ

D

²₁

u

₁

þ l

²

u

₁

¼

D

₂

v

₀

þ ðS

₁

þ S

₄

Þv

₀

ðS

2

v

0

þ S

5

u

0

Þðu

²₀

þ v

²₀

Þ 8S

₄

u

₀

v

²₀

4S

₅

u

₀

v

⁴₀

þ 4S

₅

u

³₀

v

²₀

u

²₀

þ v

²₀

2 3 4 5 6 7

x 10⁴

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02 0.03 0.04

t

x(t)

u1= 0.033

2 3 4 5 6 7

x 10⁴

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02 0.03 0.04

t

x(t)

u1= 0.1

(a) (b)

Figure 2. Time histories of equation (1) for V ¼ 0.167 and ¼ 0.001.

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

x 10⁴

−0.03

−0.02

−0.01 0 0.01 0.02 0.03

t

x(t)

σ = 0.0005

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

x 10⁴

−0.03

−0.02

−0.01 0 0.01 0.02 0.03

t

x(t)

σ = 0.001

(a) (b)

Figure 3. Time histories of equation (1) for the parameter values of Figure 1a.

D

₁

D

₂

u

₀

þ ðS

₁

S

₄

ÞD

₁

u

₀

D

1

½ðS

2

u

0

þ S

5

v

0

Þðu

²₀

þ v

²₀

Þ þ S

5

u

²₀

v

0

þ 4S

₅

u

⁴₀

v

0

þ 4S

₅

u

²₀

v

³₀

u

²₀

þ v

²₀

v

₁

¼ D

₁

u

₁

þ D

₂

u

₀

S

₁

u

₀

þ ðS

₂

u

₀

þ S

₅

v

₀

Þðu

²₀

þ v

²₀

Þ þ 8S

₅

u

²₀

v

₀

þ 4S

5

u

0

v

⁴₀

þ 4S

5

u

³₀

v

²₀

u

²₀

þ v

²₀

ð12Þ where ¼ þ S

3

, and l ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

²

S

²₃

q

is the frequency of the periodic solution of the slow flow in equation (9) corresponding to the frequency of the QP modulation of the original system in equation (1). As it can be seen, the frequency l depends on the component u

1

via the coefficient S

3

, meaning that the modulation frequency l decreases with increasing u

1

and increases with increas- ing . Thus, the modulation frequency l of QP gallop- ing is influenced only by the first harmonic component u

1

near the 2-subharmonic resonance.

The solution of the first-order system in equation (11) can be written as

u

0

ðT

1

, T

2

Þ ¼ RðT

2

Þ cosðlT

1

þ ðT

2

Þ v

0

ðT

1

, T

2

Þ ¼ l

RðT

2

Þ sinðlT

1

þ ðT

2

ÞÞ ð13Þ Substituting equation (13) into equation (12) and removing secular terms, we obtain the following autonomous slow slow flow system on R and

dR

dt ¼ ðS

1

þ S

4

ÞR 1

2 S

2

þ l

²

2

²

S

2

R

³

R d dt ¼ l

4 S

4

l 4 S

5

R

³

þ 3

8l S

5

3l

³

8

³

S

4

R

³

ð14Þ Equilibria of this slow slow flow system corresponding to the periodic oscillations of the slow flow in equa- tion (9), determine the QP solutions of the original equation (1). The nontrivial equilibrium obtained by setting

^dR_dt

¼ 0 in equation (14) is given by

R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

²

ðS

1

þ S

4

Þ S

2

ð

²

þ l

²

Þ s

ð15Þ

and the approximate periodic solution of the slow flow in equation (9)

uðtÞ ¼ R cos t vðtÞ ¼ l

R sin t ð16Þ

Using equations (15) and (16), the amplitude of the QP oscillations reads

aðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2 þ l

²

2

²

R

²

1 2 l

²

2

²

R

²

cos 2t s

ð17Þ

and the modulation envelope is delimited by a

min

and a

max

given by

a

_min

¼ min

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2 þ l

²

2

²

R

²

1 2 l

²

2

²

R

²

8 s

<

:

9 =

; ð18Þ

a

max

¼ max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2 þ l

²

2

²

R

²

1 2 l

²

2

²

R

²

8 s

<

:

9 =

; ð19Þ

Figure 1 shows the amplitude of the periodic response along with the QP envelope versus , as given by equations (7), (18) and (19), for two different values of u

1

. The comparison between the analytical predictions (solid lines) and the numerical simulations obtained by using Runge-Kutta method (double circles) validates the analytical approach. One observes from Figure 1 a slight increase of the amplitudes of periodic and QP galloping as u

1

is increased. Moreover, the QP envelope becomes larger when approaching the peri- odic galloping (near the frequency locking) meaning that at the transition from QP to the periodic response, the amplitude of the QP galloping may reach large values just prior to the jump to periodic oscillations.

Figure 2 presents examples of time histories of equa- tion (1) obtained by numerical simulation for u

1

¼ 0.033

2 3 4 5 6 7

x 10⁴

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02 0.03 0.04

t

x(t)

σ =0.0004

Figure 4. Time history of equation (1) for the parameter values

of Figure 1b near to the jump between QP and periodic

responses.

(Figure 2a) and u

1

¼ 0.1 (Figure 2b), while Figure 3 shows time histories for ¼ 0.0005 (Figure 3a) and ¼ 0.001 (Figure 3b). These time histories confirm the analytical finding given above relating the compo- nent u

1

to the QP galloping envelope and the frequency l (Figure 2), and the detuning to the frequency l (Figure 3).

Figure 4 shows an example of time history of equa- tion (1) near synchronization, with the amplitude of the QP galloping having larger values than that of the peri- odic response.

The effect of the amplitude u

1

on the QP galloping envelope (bounded by a

max

and a

min

) versus the wind velocity V, as given by equations (18) and (19), is shown

in Figure 5. It can be seen that this envelope increases with u

1

while the QP galloping onset is not affected.

In the case where the unsteady wind activates only the component u

2

cos2t, no resonance is induced. In this case, the expression of the frequency l given above reduces to detuning , equation (17) yields a(t) ¼ R and thus equations (18) and (19) lead to

a

_min

þ a

max

¼ R ð20Þ In Figure 6 we show the amplitudes of periodic and QP galloping obtained numerically (circles) and analytic- ally (lines), as given by equations (8) and (20). The horizontal lines approximate, at the leading order, the small QP envelope (delimited by double circles).

Time histories of the original equation (1) obtained by numerical simulations are shown in Figure 7 for two different values of u

2

indicating that near the 2- subharmonic resonance, the influence of the term u

2

cos2t on the QP galloping is insignificant. Figure 8 illustrates the galloping amplitude versus the wind speed V confirming the smallness absence of QP modulation near this resonance. One can conclude that QP galloping occurs near the 2-subharmonic res- onance only when the first harmonic term u

1

cost is activated.

2.2. Superharmonic resonance of order 2

In this case the resonance condition reads ¼

¹₂

þ

¹₂

"

where is a detuning parameter. The modulation equa- tions are given by

a _ ¼ S

1

a S

2

a

³

þ S

4

sinðÞ þ ½S

₅

a

²

S

6

cosðÞ a _ ¼ S

₃

a þ S

₄

cosðÞ ½S

₅

a

²

S

₆

sinðÞ

ð21Þ

0 0.05 0.1 0.15 0.2

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

V

a

amin

amax

0 0.05 0.1 0.15 0.2

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

V a

amin

amax

(a) (b)

Figure 5. QP galloping envelope versus V for ¼ 0.001; (a) u

1

¼ 0.033, (b) u

1

¼ 0.15.

−2 −1 0 1 2

x 10⁻⁴ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ a

Figure 6. Periodic and QP galloping versus for V ¼ 0.167 and

u

2

¼ 0.1. The solid lines represent the analytical approximation

and the circles represent the numerical simulation.

where now S

1

¼

¹₂

c

a

V

₄³2

b

31

1

u

2

, S

2

¼

³₈

b

31

, S

3

¼ 2

₁₆³2

b

31

1

u

²₂

, S

4

¼ b

2

ð

₂^1u1

Þ

²

b

1

ð

₄^1u22

Þ, S

5

¼

₈³2

b

31

1

u

²₁

and S

6

¼

₁₆³4

b

31

²₁

u

1

u

2

.

Notice that in the case of 2-superharmonic reson- ance (

¹₂

), the component u

1

cost does not produce a resonance, while the component u

2

cos2t induces external resonance.

Periodic solutions of equation (1) are obtained by examining equilibria of the slow flow in equation (21).

In the case where the unsteady wind induces the com- ponent u

1

only (u

2

¼ 0), the amplitude-response

equation obtained from the slow flow system in equa- tion (21) is given by

S

²₂

a

⁶

ð2S

₁

S

₂

þ S

²₅

Þa

⁴

þ ðS

²₁

þ S

²₃

Þa

²

S

²₄

¼ 0 ð22Þ while in the case where the unsteady wind activates the component u

2

only, the amplitude-response equation reads

S

²₂

a

⁶

2S

1

S

2

a

⁴

þ ðS

²₁

þ S

²₃

Þa

²

S

²₄

¼ 0 ð23Þ As before, we transform the modulation equations from the polar form in equation (21) to the following Cartesian system using the variable change u ¼ acos and v ¼ a sin

du

dt ¼ S

3

v S

6

þ fS

1

u þ S

5

ðu

²

þ v

²

Þ S

2

uðu

²

þ v

²

Þg dv

dt ¼ S

3

u S

4

þ fS

1

v S

2

vðu

²

þ v

²

Þg 8 >

> <

> >

:

ð24Þ where is a bookkeeping parameter. A periodic solu- tion of the slow flow in equation (24) can be sought in the form

uðtÞ ¼ u

0

ðT

1

, T

2

Þ þ u

1

ðT

1

, T

2

Þ þ Oð

²

Þ

vðtÞ ¼ v

₀

ðT

₁

, T

₂

Þ þ v

₁

ðT

₁

, T

₂

Þ þ Oð

²

Þ ð25Þ

where T

1

¼ t and T

2

¼ t. Introducing D

_i

¼

_@T^@

i

yields

d

dt

¼ D

1

þ D

2

þ Oð

²

Þ, substituting equation (25) into equation (24) and collecting terms, one obtains at

1 2 3 4 5 6 7

x 10⁴

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02 0.03 0.04

t

x(t)

u1=0.2

1 2 3 4 5 6 7

x 10⁴

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02 0.03 0.04

t

x(t)

u2=0.3

(a) (b)

Figure 7. Time histories of equation (1) for V ¼ 0.167 and ¼ 0.0002.

0 0.05 0.1 0.15 0.2 0.25 0.3

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

V

a

^{amin, amax}

Figure 8. QP galloping versus V for the parameter values of

Figure 6, with ¼ 0.001, u

2

¼ 0.1.

different orders of

D

²₁

u

₀

þ S

²₃

u

₀

¼ S

₃

S

₄

S

3

v

0

¼ D

1

u

0

S

6

ð26Þ

D

²₁

u

1

þ S

²₃

u

1

¼ S

3

½D

2

v

0

þ S

1

v

0

S

2

v

0

ðu

²₀

þ v

²₀

Þ D

1

D

2

u

0

þ D

1

½S

1

u

0

þ ðS

5

S

2

u

0

Þðu

²₀

þ v

²₀

Þ

S

3

v

1

¼ D

1

u

1

þ D

2

u

0

½S

1

u

0

þ ðS

5

S

3

u

0

Þ ðu

²₀

þ v

²₀

Þ ð27Þ

where S

3

(¼ 2

₁₆³2

b

31

1

u

²₂

) is the frequency of the periodic solution of the slow flow in equation (24) cor- responding to the frequency of the QP modulation.

This frequency depends on detuning and u

2

. The solu- tion of the first-order system in equation (26) can be written as

u

0

ðT

1

, T

2

Þ ¼ RðT

2

Þ cosðS

3

T

1

þ ðT

2

ÞÞ

1

v

0

ðT

1

, T

2

Þ ¼ RðT

2

Þ sinðS

₃

lT

₁

þ ðT

2

ÞÞ

2

ð28Þ where

₁

¼

^S_S⁴

3

and

₂

¼

^S_S⁶

3

. Substituting equation (28) into equation (27) with the condition S

3

6¼ 0 and

2 3 4 5 6 7

x 10⁴

−0.03

−0.02

−0.01 0 0.01 0.02 0.03

t

x(t)

σ = 0.0005

2 3 4 5 6 7

x 10⁴

−0.03

−0.02

−0.01 0 0.01 0.02 0.03

t

x(t)

σ = 0.001

(a) (b)

Figure 10. Time histories of equation (1) for the parameter values of Figure 9a.

−1.5 −1 −0.5 0 0.5 1 1.5

x 10⁻³ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ a

u1=0.033

−1.5 −1 −0.5 0 0.5 1 1.5

x 10⁻³ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ a

u1=0.1

(a) (b)

Figure 9. Periodic and QP galloping versus for V ¼ 0.117; (a) u

1

¼ 0.033, (b) u

1

¼ 0.1. The analytical approximation is shown

(solid lines for stable, red lines for unstable), and the numerical simulation is represented by circles.

−1.5 −1 −0.5 0 0.5 1 1.5 x 10⁻³ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ a

u₂=0.033

−1.5 −1 −0.5 0 0.5 1 1.5

x 10⁻³ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ a

u₂=0.1

(a) (b)

Figure 12. Periodic and QP galloping versus for V ¼ 0.117; (a) u

2

¼ 0.033, (b) u

2

¼ 0.1. The analytical approximation is shown (solid lines for stable, red lines for unstable), and the numerical simulation is represented by circles.

2 3 4 5 6 7

x 10⁴

−0.03

−0.02

−0.01 0 0.01 0.02 0.03

t

x(t)

σ = 0.0005

2 3 4 5 6 7

x 10⁴

−0.03

−0.02

−0.01 0 0.01 0.02 0.03

t

x(t)

σ = 0.001

(a) (b)

0 0.05 0.1 0.15 0.2

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

V a

a_min a_max

0 0.05 0.1 0.15 0.2

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

V

a

amin amax

(a) (b)

Figure 11. QP envelope versus V for ¼ 0.001; (a) u

₁

¼ 0.033, (b) u

₁

¼ 0.15.

removing secular terms, we obtain the following autonomous slow slow flow system on R and

dR

dt ¼ ðS

1

2

₁

S

5

2

²₁

S

2

2

²₂

S

2

ÞR S

2

R

³

R d

dt ¼

2

S

5

R þ

1

2

S

3

R

²

ð29Þ

The nontrivial equilibrium obtained by setting

^dR_dt

¼ 0 in equation (29) is given by

R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S

1

2

1

S

5

2

²₁

S

2

2

²₂

S

2

S

₂

s

ð30Þ

and the approximate periodic solution of the slow flow in equation (24) is written as

uðtÞ ¼ R cos t

₁

vðtÞ ¼ R sin t

2

ð31Þ

Using equations (30) and (29), the amplitude a(t) of the QP oscillations reads

aðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R

²

þ

²₁

þ

²₂

2R

₁

cos t 2R

₂

sin t q

ð32Þ and the modulation envelope is delimited by a

min

and a

max

given by

a

min

¼ min

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R

²

þ

²₁

þ

²₂

2Rð

1

þ

2

Þ q

ð33Þ

a

max

¼ max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R

²

þ

²₁

þ

²₂

2Rð

1

þ

2

Þ q

ð34Þ In the case where the first component is present (u

1

6¼ 0, u

2

¼ 0), Figure 9 shows for two different values of u

1

the amplitude of the periodic response and the QP envelope versus , as given by equations (23), (33) and (34). The comparison between the ana- lytical predictions (solid lines) and the numerical simu- lations obtained by using Runge-Kutta method (double circles) is reported in Figure 9a. One observes the small QP modulation envelope as compared to the case of 2- subharmonic resonance (see Figure 1).

Figure 10 presents examples of time histories of equation (1) obtained by numerical simulation for ¼ 0.0005 (Figure 10a) and ¼ 0.001 (Figure 10b).

0 0.05 0.1 0.15 0.2

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

V a

amin

amax

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

V a

2 4 6

x 10⁴

−0.02 0 0.02

2 4 6

x 10⁴

−0.02 0 0.02

amin

amax

(a) (b)

Figure 14. QP envelope versus V for ¼ 0.001; (a) u

₂

¼ 0.033, (b) u

₂

¼ 0.15.

−1.5 −1 −0.5 0 0.5 1 1.5

x 10⁻³ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ a

u₁=u₂=0.1

Figure 15. Periodic and QP galloping versus for V ¼ 0.167,

u

₁

¼ 0.1 and u

₂

¼ 0.1. The solid lines represent the analytical

approximation and the circles represent the numerical

simulation.

The effect of the amplitude u

1

on the QP galloping envelope versus the wind velocity V, as given by equa- tions (33) and (34), is shown in Figure 11 indicating that as u

1

is increased the QP galloping onset undergoes a small shift toward higher values of the wind velocity and its modulation envelope increases significantly.

Figure 11b also indicates that when the wind velocity V is increased, the QP galloping is activated directly from the rest.

In the case where the second component u

2

cos2t is activated, external resonance is produced ð

¹₂

Þ.

Figure 12a shows for two different values of u

2

the amplitude of periodic and QP galloping obtained numerically (circles) and analytically (solid lines), as given by equations (22), (33) and (34). The influence of the amplitude u

2

on the QP galloping envelope is reported in Figure 12b.

Time histories of the original equation (1) obtained via numerical simulations are shown in Figure 13 for ¼ 0.0005 (Figure 13a) and ¼ 0.001 (Figure 13b).

Notice the important difference in the frequency of the QP modulation between the case of superharmonic resonance and the case of subharmonic resonance (Figure 3).

The influence of the amplitude u

2

on the QP gallop- ing envelope versus the wind velocity V, as given by equations (33) and (34), is depicted in Figure 14.

Showing that the QP modulation envelope increases with u

2

and its onset shifts toward higher values of the wind velocity. Figure 14b indicates that the gallop- ing is first activated as periodic with small amplitude and then turns to QP for the small value of the wind velocity, as shown by the time histories insets the Figure 14.

0 0.05 0.1 0.15 0.2

0.005 0.01 0.015 0.02 0.025 0.03 0.035

V a

a_min amax

0 0.05 0.1 0.15 0.2

0.005 0.01 0.015 0.02 0.025 0.03 0.035

V a

amin

amax

(a) (b)

Figure 17. QP envelope versus V for ¼ 0.001; (a) u

1

¼ u

2

¼ 0.1, (b) u

1

¼ u

2

¼ 0.15.

2 3 4 5 6 7

x 10⁴

−0.03

−0.02

−0.01 0 0.01 0.02

(a)0.03

(b)

t

x(t)

σ = 0.0005

2 3 4 5 6 7

x 10⁴

−0.03

−0.02

−0.01 0 0.01 0.02 0.03

t

x(t)

σ = 0.001

Figure 16. Time histories of equation (1) for the parameter

values of Figure 15.

In the case where both components u

1

and u

2

are present, Figure 15 shows the periodic and QP galloping versus for a given value of u

1

and u

2

.

Time histories of the original equation (1) for ¼ 0.0005 and ¼ 0.001 are presented in Figure 16a and Figure 16b, respectively.

Figure 17 depicts the QP galloping amplitude versus the wind velocity showing that when the two harmonic terms of the unsteady wind are present, the QP gallop- ing increases and its onset shifts slightly toward smaller values of the wind velocity.

Finally, Figure 18 shows simultaneously the QP envelopes in the case where only the first component u

1

(Figure 18a) and only the second component u

2

(Figure 18b) is present. One observes that when the first component u

1

is present, the QP galloping envelope

is larger near the 2-subharmonic resonance (Sub: blue lines) than near the 2-superharmonic resonance (Sup:

red lines), while QP galloping occurs near the 2-super- harmonic only when u

2

is activated. To appreciate and compare the QP modulation envelopes near the second- ary resonances, the QP envelope near the primary res- onance (Kirrou et al., 2013) is also reported in Figure 18 (Pr: thick lines) (Kirrou et al., 2013).

3. Conclusion

The QP galloping of a tower subjected to unsteady wind was studied near the 2-subharmonic and 2-super- harmonic resonances. The case where the unsteady wind develops multiharmonic excitations consisting of the two first harmonic terms was assumed and a lumped mass SDOF model was considered. The double MSM was performed to obtain analytical expressions of periodic and QP solutions as well as the QP modulation envelope. The effect of unsteady wind on the QP galloping and on the frequency of its modulation was reported near the two resonances for different cases of harmonic excitation. Numerical simu- lations were carried out to support the analytical predictions.

The results show that near the 2-subharmonic reson- ance, QP galloping occurs only when the first harmonic component u

1

cost is activated while its onset is not influenced. Instead, near the 2-superharmonic reson- ance, the QP galloping occurs for whichever harmonic component is activated, and its onset shifts toward higher values of the wind velocity as the unsteady wind is increased. The results also revealed that the frequency of the QP modulation is much higher near the 2-superharmonic than near the 2-subharmonic res- onance in the two cases of excitation. When the two harmonic components u

1

cost and u

2

cos2t are pre- sent simultaneously, the QP galloping occurs with large modulation envelope and its onset shifts slightly toward lower values of the wind velocity.

One can conclude that near secondary resonances, QP galloping occurs with amplitudes having the same order of magnitude as that near the primary resonance but with higher frequency and smaller modulation envelope. Consequently, as near primary resonance, the QP galloping should not be neglected in the analysis of the dynamics and stability of tall buildings near the secondary resonances of order 2, except near the 2- subharmonic resonance when the unsteady wind acti- vates only the component u

2

cos2t.

Funding

This funding received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

−1.5 −1 −0.5 0 0.5 1 1.5

x 10⁻³ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ a

Pr Sub Sup

−1.5 −1 −0.5 0 0.5 1 1.5

x 10⁻³ 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ a

Pr Sub Sup

(a)

(b)

Figure 18. QP envelopes versus . Sub: Subharmonic reson- ance (blue color); Sup: Superharmonic resonance (red color); Pr:

Primary resonance (thick line picked from Kirrou et al. (2013)).

For V ¼ 0.117; (a) u

1

¼ 0.033, u

2

¼ 0, (b) u

1

¼ 0, u

2

¼ 0.033.

References

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Appendix 1

The expressions of the coefficients of equation (1) are

! ¼ ffiffiffiffiffiffiffiffi p 3EI h‘ ffiffiffiffi

p m , c

_a

¼ A

1

bh‘ U

c

2 ffiffiffiffiffiffiffiffiffiffiffiffi

p 3EIm , b

1

¼ c

_a

, b

2

¼ 4 A

2

b‘

3m , b

31

¼ 3 A

3

b‘ ffiffiffiffiffiffiffiffi

p 3EI 8h U

c

ffiffiffiffiffiffi m

³

p

b

₃₂

¼ b

₃₁

,

₁

¼ 4 A

₀

bh

²

‘ U

²_c

3

³

EI ,

₂

¼

₁

2 , UðtÞ ¼ U þ uðtÞ

where ‘ is the height of the tower, b the cross-section width, EI the total stiffness of the single story, m the mass longitudinal density, h the interstory height, and the air mass density. A

i

, i ¼ 0, . . . , 3 are the aerody- namic coefficients for the squared cross-section. The dimensional critical velocity is given by

U

c

¼ 4 ffiffiffiffiffiffiffiffiffiffiffiffi p 3EIm bA

₁

h‘

where is the modal damping ratio, depending on both the external and internal damping according to

¼ h

²

24EI ! þ c 2m!

where and c are the external and internal damping coefficients, respectively.

Appendix 2

The following numerical values are used for a case study: the height of the tower is ‘ ¼ 36 m, the cross- section is b ¼ 16 m wide, the total stiffness of the single story is EI ¼ 115,318,000 Nm

²

, the mass longitu- dinal density is m ¼ 4737 kg/m, ¼ 128,513 Ns and c ¼ 34.8675 Ns/m

²

. The interstory height is assumed h ¼ 4 m. The aerodynamic coefficients A

i

, i ¼ 0, . . . , 3 are taken from Abdel-Rohman (2001) for the squared cross-section: A

0

¼ 0.0297, A

1

¼ 0.9298, A

2

¼0.2400, A

3

¼ 7.6770. The air mass density is ¼ 1.25 kg/m

³

. The (dimensional) natural frequency of the rod is

! ¼ 5.89 rad/s. The (dimensional) critical wind velocity

assumes the value U

c

¼ 30 m/s.