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Analysis of axial compressive loaded beam under random support excitations
Journal of Sound and Vibration 410 (2017) 378e388
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Analysis of axial compressive loaded beam under random support excitations Wensheng Xiao, Fengde Wang*, Jian Liu Department of Mechanical and Electronic Engineering, China University of Petroleum (East China), Qingdao, 266580, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 December 2016 Received in revised form 13 August 2017 Accepted 24 August 2017
An analytical procedure to investigate the response spectrum of a uniform Bernoulli-Euler beam with axial compressive load subjected to random support excitations is implemented based on the Mindlin-Goodman method and the mode superposition method in the frequency domain. The random response spectrum of the simply supported beam subjected to white noise excitation and to Pierson-Moskowitz spectrum excitation is investigated, and the characteristics of the response spectrum are further explored. Moreover, the effect of axial compressive load is studied and a method to determine the axial load is proposed. The research results show that the response spectrum mainly consists of the beam's additional displacement response spectrum when the excitation is white noise; however, the quasi-static displacement response spectrum is the main component when the excitation is the Pierson-Moskowitz spectrum. Under white noise excitation, the amplitude of the power spectral density function decreased as the axial compressive load increased, while the frequency band of the vibration response spectrum increased with the increase of axial compressive load. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Bernoulli-Euler beam Axial compressive load Random support excitation The Mindlin-Goodman method
1. Introduction In offshore drilling, there are two kinds of jack-up riser: one with surface blowout preventers (BOP) and the other with subsea BOP. Due to lower cost, the jack-up platform often uses the riser with surface BOP to drill exploratory wells in shallow seas. However, the mainstream jack-up platform can be operated in water depths of 400 ft or more and the lateral rigidity of the platform decreases with increasing water depth, which makes the platform in random vibration status. The platform's vibration can reduce the drilling riser's service lifetime. Therefore, it is necessary to determine the riser's response to random vibrations before drilling an oil well. The jack-up riser can be modelled as a Bernoulli-Euler beam, the joint between the riser and the platform treated as a hinge, and the platform's vibration modelled as the riser's time-dependent boundary condition. In addition, the riser bears the axial compressive load resulting from the surface BOP. Considering these two kinds of riser, we chose the basic model of the beam as the research object in this study. The vibratory response of a beam with time-dependent boundary conditions can be obtained by the Laplace transform [1,2] and the Mindlin-Goodman methods [3e6]. In the Mindlin-Goodman method, the non-homogeneous boundary conditions are transformed into homogeneous ones. Therefore, the method of separation of variables can be used to solve the problem. The vibratory response of a non-uniform Bernoulli-Euler beam with time-dependent elastic boundary conditions
* Corresponding author. E-mail addresses: [email protected] (W. Xiao), [email protected] (F. Wang), [email protected] (J. Liu). http://dx.doi.org/10.1016/j.jsv.2017.08.045 0022-460X/© 2017 Elsevier Ltd. All rights reserved.
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was analysed by Lee and Lin [7], where they generalized the method of Mindlin-Goodman by introducing four shifting ez [8] investigated the axial vipolynomial functions with physical meanings. Josue' Aranda-Ruiz and Jose' Cerna'ndez-Sa brations of a rod with a time-dependent and non-harmonic force applied to the rod's free ends. Other scholars [9,10] have analysed the vibratory response of a Timoshenko beam with time-dependent boundary conditions by using the MindlinGoodman method without physical meanings in the shifting functions. The dynamic analysis of a non-uniform Timoshenko beam with general time-dependent boundary conditions was studied by S. Y. Lee and S. M. Lin [11], and the orthogonality conditions for the eigenfunctions of a non-uniform Timoshenko beam with elastic boundary conditions were explored. Furthermore, S. Y. Lee and S. M. Lin [12] investigated the vibration of a pre-twisted, non-uniform Timoshenko beam with time-dependent elastic boundary conditions. Yong-Woo Kim [13] presented an analytical solution procedure to solve the dynamic responses of a Timoshenko beam excited by support motions with the aid of the Mindlin-Goodman method and the eigenfunction expansion method. Pratiher [14] investigated the vibration control of a cantilever beam with tip mass under transverse base excitation by using the perturbation method. Isaac Elishakoff and David Livshits [15,16] derived closed-form solutions for both a Bernoulli-Euler and a Bresse-Timoshenko beam under stationary random excitation. Isaac Elishakoff and Eliezer Lubliner [17,18] studied the random vibration of a Bresse-Timoshenko beam subjected to spacewise white noise and concentrated point loading. The influence of axial load on the dynamics of structures has attracted much attention as a result of its wide application [19e23]. Most available literature on this topic treats support excitations as deterministic and little work can be found on the random response analysis of a beam with an axial load and time-dependent boundary conditions. Mingwu Li [24] presented the analytical solutions for the response of a uniform Bernoulli-Euler beam with axial force subjected to generalized support excitations by using the Mindlin-Goodman method in the time domain. However, in engineering fields, most excitations are random. There is no specific function to describe a random excitation, meaning the problem cannot be solved by the method proposed in the present literature. Therefore, we propose an analytical procedure to solve the random response of an axially loaded beam under random support excitations in the frequency domain. The current paper is organized as follows. First, the problem formulation is presented in Section 2. Then, the analytical procedure to solve the problem is carried out in Section 3. Subsequently, an investigation of the random displacement response spectrum of the simply supported beam with different excitations, the effect of axial compressive load, and a method for ascertaining the axial load is presented in Section 4. Finally, several important conclusions are summarized in Section 5.
2. Problem formulation Consider a simply supported beam of a linearly elastic material with length l, modulus of elasticity E, area moment of inertia I, and mass per unit length m. The beam, which is subjected to an axial compressive force p and random support excitations, is presented in Fig. 1. The supports' transverse motions, which are denoted as u1(t) and u2(t), are assumed to be stationary, random processes. The Euler-Bernoulli beam theory is adopted in this study, and the motion of the beam is governed by the following differential equation (without considering the material's internal damping):
EI
v4 yðx; tÞ v2 yðx; tÞ v2 yðx; tÞ þP þm ¼0 4 2 vx vx vt 2
(1)
In Eq. (1), t is the time, x is the coordinate measured along the beam axis, and y(x, t) is the transverse deflection of the beam axis. For the simply supported beam, the time-dependent boundary conditions at x ¼ 0 and x ¼ l are in the form of: 00
00
yð0; tÞ ¼ u1 ðtÞ; yðl; tÞ ¼ u2 ðtÞ; y ð0; tÞ ¼ 0; y ðl; tÞ ¼ 0
(2)
where, the prime denotes differentiation with respect to x. By using the Mindlin-Goodman method, the total transverse deflection y(x, t) is decomposed into two components as:
yðx; tÞ ¼ ys ðx; tÞ þ yd ðx; tÞ
(3)
Fig. 1. Beam with axial compressive load under random support excitations.
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In Eq. (3), ys(x, t) is the quasi-static displacement and yd(x, t) is the additional displacement due to the dynamic inertial force [25]; these displacements can be expressed as:
ys ðx; tÞ ¼
s X
gi ðxÞui ðtÞ
(4)
fn ðxÞqn ðtÞ
(5)
i¼1
yd ðx; tÞ ¼
∞ X n¼1
where, gi (x) is the static influence function, 4n (x) is the shape function of the beam, and qn(t) is the modal coordinate of nth mode. The corresponding boundary conditions for the simply supported beam are: 00
00
ys ð0; tÞ ¼ u1 ðtÞ; ys ðl; tÞ ¼ u2 ðtÞ; ys ð0; tÞ ¼ 0; ys ðl; tÞ ¼ 0 00
00
yd ð0; tÞ ¼ 0; yd ðl; tÞ ¼ 0; yd ð0; tÞ ¼ 0; yd ðl; tÞ ¼ 0
(6) (7)
Herein, the index s is determined by the boundary conditions of the beam; for the simply supported beam, s ¼ 2. The gi (x) [21] and 4n (x) of Eq. (4) and Eq. (5) are formulated as follows:
g1 ðxÞ ¼ 1 g2 ðxÞ ¼
x l
(8)
x l
(9)
4n ðxÞ ¼ sin
npx ðn ¼ 1; 2; :::∞Þ l
(10)
Natural frequencies of the simply supported beam with axial compressive load can be obtained by Eq. (11) [26]:
np un ¼ 2 l
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EIn2 p2 Pl2 ðn ¼ 1; 2; :::; ∞Þ m
(11)
3. Analysis Substituting Eq. (3) into Eq. (1) yields the equation:
EI
v4 yd v2 y v2 y v2 y s v2 ys þ P 2d þ m 2d ¼ P 2 m 2 4 vx vx vt vx vt
(12)
By using Eq. (4) and Eq. (5), Eq. (12) can be rewritten as follows:
EI
∞ X d4 4n n¼1
dx4
qn ðtÞ þ m
∞ X
4n ðxÞq€n ðtÞ þ P
n¼1
∞ X d2 4n n¼1
dx2
qn ðtÞ ¼ Feq
(13)
where, the dot denotes differentiation with respect to t. Feq, the equivalent load resulting from the support excitations, can be obtained by the following equation:
Feq ¼ m
2 X i¼1
gi ðxÞu€i ðtÞ P
2 X d2 gi ðxÞ i¼1
dx2
ui ðtÞ
(14)
Since the additional displacement has a homogeneous boundary condition, its mode shape functions conform to the following relationship.
EI
d4 4n ðxÞ d2 4n ðxÞ þ P mu2n 4n ðxÞ ¼ 0 dx4 dx2
The substitution of Eq. (15) into Eq. (13) generates Eq. (16):
(15)
W. Xiao et al. / Journal of Sound and Vibration 410 (2017) 378e388
m
∞ X
4n ðxÞq€n ðtÞ þ m
n¼1
∞ X
u2n 4n ðxÞqn ðtÞ ¼ Feq
381
(16)
n¼1
Substituting Eq. (8), Eq. (9) and Eq. (14) into Eq. (16) yields the equation: ∞ X
4n ðxÞq€n ðtÞ þ
n¼1
∞ X
u2n fn ðxÞqn ðtÞ ¼ 1
n¼1
x x u€ ðtÞ u€2 ðtÞ l 1 l
(17)
By applying the orthogonality conditions, Eq. (17) can be decoupled:
q€n ðtÞ þ u2n qn ðtÞ ¼ an u€1 ðtÞ bn u€2 ðtÞ
(18)
where, an and bn are weight coefficients of the inertial load.
Z l
an ¼
0
Z l x x fn ðxÞdx 1 fn ðxÞdx l 0 l bn ¼ Z l Z l f2n ðxÞdx f2n ðxÞdx 0
(19)
0
In this study, random support excitations are assumed to be uncorrelated. Therefore, the superposition method can be applied to calculate the dynamic response of the beam. Let S1(u) and S2(u) denote the auto-power spectral density function of u1(t) and u2(t) respectively and u be the vibration frequency of the beam's lateral displacement. The frequency response functions, H1(u) and H2(u), of the modal coordinates versus u1(t) and u2(t), are found to be:
H1 ðuÞ ¼
an u2 u2n u2
(20)
H2 ðuÞ ¼
bn u2 u2n u2
(21)
Then, the frequency response function of the additional displacement yd(x, t) versus u1(t) and u2(t) can be obtained by Eq. (22) and Eq. (23).
T1 ðuÞ ¼
∞ X
sin
npx H ðuÞ l 1
ðn ¼ 1; 2; :::∞Þ
(22)
sin
npx H ðuÞ l 2
ðn ¼ 1; 2; :::∞Þ
(23)
n¼1
T2 ð u Þ ¼
∞ X n¼1
As a result of the orthogonality of the mode shape, the fundamental modes are independent. Let Sd1(x, u) and Sd2(x, u) denote the power spectral density functions of the additional displacement yd(x, t) versus u1(t) and u2(t); then these spectra can be obtained by Eq. (24) and Eq. (25).
Sd1 ðx; uÞ ¼ S1 ðuÞ
2 ∞ X npx H1 ðuÞ sin l n¼1
ðn ¼ 1; 2; :::∞Þ
(24)
Sd2 ðx; uÞ ¼ S2 ðuÞ
2 ∞ X npx H2 ðuÞ sin l n¼1
ðn ¼ 1; 2; :::∞Þ
(25)
On the basis of the definition of the autocorrelation function and the Wiener-Khintchine principle, the power spectral density functions (Ss1(x, u) and Ss2(x, u),) of the quasi-static displacement can be derived as follows:
x2 Ss1 ðx; uÞ ¼ S1 ðuÞ½g1 ðxÞ2 ¼ S1 ðuÞ 1 l x2 Ss2 ðx; uÞ ¼ S2 ðuÞ½g2 ðxÞ2 ¼ S2 ðuÞ l
(26) (27)
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Let Sds1(x, u) denote the cross spectral density function between yd(x, t) and u1(t), and Sds2(x, u) denote the cross spectral density function between yd(x, t) and u2(t). Then, Sds1(x, u) and Sds2(x, u) can be calculated by Eq. (28) and Eq. (29).
Sds1 ðx; uÞ ¼ T1 ðuÞS1 ðuÞ ¼ S1 ðuÞ
∞ X
sin
npx H ðuÞ l 1
(28)
sin
npx H ðuÞ l 2
(29)
n¼1
Sds2 ðx; uÞ ¼ T2 ðuÞS2 ðuÞ ¼ S2 ðuÞ
∞ X n¼1
Based on the above analysis, the power spectral density function of the beam's displacement y(x, t) can be obtained by the following equations:
Sy1 ðx; uÞ ¼ Sd1 ðx; uÞ þ 2Re½Sds1 ðx; uÞ þ Ss1 ðx; uÞ
(30)
Sy2 ðx; uÞ ¼ Sd2 ðx; uÞ þ 2Re½Sds2 ðx; uÞ þ Ss2 ðx; uÞ
(31)
Sy ðx; uÞ ¼ Sy1 ðx; uÞ þ Sy2 ðx; uÞ
(32)
In Eq. (32), Sy(x, u) is the power spectral density function of the beam's total displacement, Sy1(x, u) the power spectral density function resulted from u1(t), and Sy2(x, u) the power spectral density function stimulated by u2(t). 4. Results and discussion In this research, the power spectral density functions of the random displacement responses of an axially, compressively loaded steel beam subjected to white noise excitation and Pierson-Moskowitz spectrum excitation are analysed, and the effect of the axial compressive load is investigated. In addition, a method for ascertaining the axial load is proposed. The beam's parameters are: EI ¼ 2746.7 N m2, m ¼ 3.14 kg m1, and l ¼ 1 m. 4.1. White noise excitation To simplify the analysis, white noise excitation at x ¼ l (S2(u) ¼ 1) is investigated in this study. By applying Eq. (21)and Eq. (25), the additional displacement's power spectral density function Sd(u) can be obtained by the following equation:
Sd ðx; uÞ ¼
2 ∞ X npx2 bn u2 sin u 2 u 2 l n n¼1
(33)
Using Eq. (21) and Eq. (29), the cross power spectral density function between the beam's additional displacement and the white noise excitation S2(u) ¼ 1 can be obtained as follows:
Sds ðx; uÞ ¼
∞ X
sin
n¼1
npx bn u2 l u2n u2
(34)
And the power spectral density function of the quasi-static displacement can be calculated by Eq. (27).
Ss ðx; uÞ ¼
x2 l
(35)
Based on the above calculations, the power spectral density function of the total displacement response can be obtained as follows:
Sy ðx; uÞ ¼
2 ∞ ∞ x2 X X npx2 bn u2 npx bn u2 sin þ2 sin þ 2 2 2 2 l l un u l un u n¼1 n¼1
(36)
Since the series in Eq. (36) has a fast convergence rate, taking the first five orders of the series is enough for the analysis. Consequently, the first five order natural frequencies of the beam and weight coefficients are calculated first, and are listed in Tables 1 and 2. By using Eq. (36), the power spectral density function of the beam stimulated by S2(u) ¼ 1 is calculated and depicted in Fig. 2. The results show that the vibratory response of the beam is a narrow-band random vibration, the principal oscillation is the beam's fundamental frequency; and the amplitude of the power spectral density function abides by sinusoidal regulation along the beam's axis, which can also be deduced from Eq. (23).
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Table 1 Natural frequencies of the simply supported beam with different axial compressive loads. P (N)
u1(rad/s)
u2(rad/s)
u3(rad/s)
u4(rad/s)
u5(rad/s)
0 200 400 600 800 1000
291.9043 290.8255 289.7428 288.6559 287.5649 286.4698
1167.6 1166.5 1165.5 1164.4 1163.3 1162.2
2627.1 2626.1 2625.0 2623.9 2622.8 2621.7
4670.5 4669.4 4668.3 4667.2 4666.2 4665.1
7297.6 7296.5 7295.5 7294.4 7293.3 7292.2
Table 2 The weight coefficients of the simply supported beam.
an bn
n¼1
n¼2
n¼3
n¼4
n¼5
2/p 2/p
1/p 1/p
2/3p 2/3p
1/2p 1/2p
2/5p 2/5p
Fig. 2. The power spectral density function of the beam's total lateral displacement under white noise excitation (p ¼ 1000 N).
Fig. 3. The cross power spectral density function between the beam's additional displacement and white noise excitation (p ¼ 1000 N).
Fig. 4. The power spectral density function of the beam's additional displacement under white noise excitation (p ¼ 1000 N).
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Fig. 3 shows the cross spectral density function between the additional displacement and the white noise excitation. The power spectral density function of the beam's additional displacement caused by white noise is calculated and plotted in Fig. 4, and the power spectral density function of the beam's quasi-static displacement under white noise excitation is presented in Fig. 5. The results show that the response spectrum of the additional displacement is the main component of the total response spectrum when the excitation is white noise, and both the response spectrum of the additional displacement and the cross spectrum have their maximum amplitudes at the beam's fundamental frequency. To explicitly illustrate the core characteristics of the response power spectral density function, the response spectrum at the midpoint of the beam is analyzed in detail. As shown in Fig. 6, the cross spectrum is positive when the vibration frequency is less than the beam's fundamental frequency, which can enhance the amplitude of the total response spectrum. The cross spectrum is negative when the vibration frequency exceeds the beam's fundamental frequency, which can reduce the amplitude of the total response spectrum. The amplitude of the spectrum has the maximum value when the vibration frequency is equal to the fundamental frequency. 4.2. Pierson-Moskowitz spectrum excitation The Pierson-Moskowitz spectrum, which is used to describe ocean waves, is a type of narrow-band random spectrum. In this study, the Pierson-Moskowitz spectrum of 10 m significant wave height, F(u), is taken as the power spectral density function of u2(t), which can be formulated by Eq. (37). F(u) is presented in Fig. 7.
FðuÞ ¼
0:78
u5
0:0311 exp 4
u
(37)
Therefore, the power spectral density function Sa(x,u) of the beam's additional displacement resulting from F(u) can be obtained by Eq. (38).
0:78
Sa ðx; uÞ ¼
u5
∞ npx bn u2 2 0:0311 X sin ; ðn ¼ 1; 2; :::∞Þ exp l u2n u2 u4 n¼1
(38)
The power spectral density function Sf(x,u) of the beam's quasi-static displacement resulting from F(u) can be calculated by the equation:
Sf ðx; uÞ ¼
0:78
u5
0:0311 x2 exp l u4
(39)
And we can get the cross power spectral density function Saf(x,u) between the additional displacement and the quasistatic displacement by Eq. (40).
Saf ðx; uÞ ¼
0:78
u5
∞ 0:0311 X npx bn u2 exp sin 4 l u2n u2 u n¼1
(40)
Subsequently, we can obtain the power spectral density function SF (x,u) of the simply supported beam's total displacement response caused by F(u).
Fig. 5. The power spectral density function of the beam's quasi-static displacement under white noise excitation (p ¼ 1000 N).
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Fig. 6. The cross power spectral density function between the additional displacement and white noise excitation at the beam's midpoint (p ¼ 1000 N).
Fig. 7. Pierson-Moskowitz spectrum of 10 m significant wave height.
SF ðx; uÞ ¼ Sa ðx; uÞ þ 2Saf ðuÞ þ Sf ðuÞ
(41)
The power spectral density function of the beam's total displacement response stimulated by the Pierson-Moskowitz spectrum is calculated and plotted in Fig. 8. It is found that the vibratory response of the beam is also a narrow-band random vibration. Fig. 9 presents the power spectral density function of the beam's additional displacement caused by Pierson-Moskowitz spectrum, Fig. 10 shows the cross spectral density function, and the power spectral density function of the beam's quasi-static displacement under Pierson-Moskowitz spectrum excitation is depicted in Fig. 11. The results show that the response spectrum of the quasi-static displacement is the main component of the total response spectrum when the excitation is the Pierson-Moskowitz spectrum.
Fig. 8. The power spectral density function of the beam's total displacement under Pierson-Moskowitz spectrum excitation (p ¼ 1000 N).
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Fig. 9. The power spectral density function of the beam's additional displacement under Pierson-Moskowitz spectrum excitation (p ¼ 1000 N).
Fig. 10. The cross power spectral density function between the beam's additional displacement and Pierson-Moskowitz spectrum excitation (p ¼ 1000 N).
Fig. 11. The power spectral density function of the beam's quasi-static displacement under Pierson-Moskowitz spectrum excitation (p ¼ 1000 N).
4.3. Effect of the axial compressive load Based on these results, the axial compressive load's effect on the beam's response spectrum is studied. There are two differences between the axially loaded beam and the unloaded beam: one is the reduction of the natural frequencies resulting from the axial compressive load, and the other is the axial load factor in equivalent load presented in Eq. (14). In this study, the second term in Eq. (14) becomes zero because the static influence functions of the simply supported beam are linear polynomials. Therefore, the axial load affects the response spectrum by changing the simply supported beam's natural frequencies only. Under white noise excitation, the effect of axial compressive load is investigated by analysing the response spectrum of the additional displacement. Several cases of different axial compressive loads are analysed, and the results for the midpoint of the beam are plotted in Fig. 12. It is seen that the axial compressive load can reduce the amplitude of the beam's random displacement response spectrum, the amplitude of the power spectral density function decreased as the axial compressive load increased, and the frequency band of the vibrational response increased with increasing axial compressive load. When there is a narrow-band random excitation such as the Pierson-Moskowitz spectrum, the reduction of
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Fig. 12. The power spectral density function of the beam's additional displacement with different axial compressive loads under white noise excitation.
the beam's natural frequencies resulting from axial compressive load in Eq. (38) and Eq. (40) has little impact on the beam's total displacement response. In some cases, such as when the riser of a jack-up drilling platform is under working conditions, it can be modelled as a uniform Euler-Bernoulli beam with axial compressive load. The axial compressive force resulting from the blowout preventers cannot be determined accurately due to the complex environmental loadings. However, it is important to know the riser's stress state; therefore, it is necessary to find a method to determine the axial compressive force on the beam. The relationship between the natural frequencies of the beam and the axial compressive force is presented in Eq. (11). Motivated by the abovementioned engineering problem and Eq. (11), a formula can be derived as follows:
P ¼ PE
ml2 u21
p2
(42)
where, PE is the Euler critical load of a beam. Eq. (42) can be used to ascertain the axial load by testing the beam's fundamental frequency u1. 5. Conclusions Based on the Mindlin and Goodman method and mode superposition method, a method for studying the response of the axially, compressively loaded uniform Euler-Bernoulli beam subjected to random support excitations is proposed in this paper. The random displacement response spectra of the simply supported beam with axial compressive load under white noise excitation and Pierson-Moskowitz spectrum excitation were studied in the frequency domain, and a formula was derived to determine the axial compressive load. It is found that the power spectral density function of the additional displacement is the main component of the beam's response spectrum when the excitation is white noise, and the quasi-static displacement response spectrum is the main component of the total response spectrum when the excitation is the Pierson-Moskowitz spectrum. Under white noise excitation, the axial compressive load has a significant impact on the beam's response spectrum; the effect of the axial compressive load can be ignored when the excitation is the Pierson-Moskowitz spectrum. When the excitation is white noise, the amplitude of the beam's response spectrum decreases as the axial compressive load increases, but the frequency band of the response spectrum increases with the increase of axial compressive load. The method proposed in this paper can be extended to solve the response of a Bernoulli-Euler beam with other boundary conditions under random support excitation. When we solve the response of the jack-up riser, the support excitation is the platform's random lateral displacement. Furthermore, the damping resulting from sea water and drill fluid should be considered. In offshore field, this method mainly has two functions: firstly, it can solve the jack-up riser's random response which can be used to estimate the riser's fatigue life; secondly, it can determine whether a sea area and sea-state are appropriate for jack-up platform. Acknowledgements The authors would like to acknowledge the Ministry of Industry and Information Technology of the PR China for supporting this study through the project “Jack-up platform brand project (II)” with the grant number of 10200001-15-ZC06070018.
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G.A. Nothmann, Vibration of a cantilever beam with prescribed end motion, J. Appl. Mech-T. ASME 15 (1948) 327e334. T.C. Yen, S. Kao, Vibration of a beam-mass system with time-dependent boundary conditions, J. Appl. Mech-T. ASME. 26 (1959) 353e356. R.D. Mindlin, L.E. Goodman, Beam vibrations with time-dependent boundary conditions, J. Appl. Mech-T. ASME. 17 (1950) 377e380. D.A. Grant, Beam vibrations with time-dependent boundary conditions, J. Sound. Vib. 89 (4) (1983) 519e522. C.R. Edstrom, The vibrating beam with nonhomogeneous boundary conditions, J. Appl. Mech-T. ASME. 48 (1981) 669e670. P.F. Gou, K.K. Panahi, Beam with time dependent boundary conditions closed form solution versus response spectral method, in: ASME Pressure Vessels and Piping Conference, vol. 137, 1988. Pittsburgh, Pennsylvania. S.Y. Lee, S.M. Lin, Dynamic analysis of non-uniform beams with time dependent elastic boundary conditions, J. Appl. Mech-T. ASME. 63 (1996) 474e478. Josue Aranda-Ruiz, J. Fernandez-Saez, On the use of variable-separation method for the analysis of vibration problems with time-dependent boundary conditions, P. I. Mech. Eng. C-J. Mec. 1989-1996 226 (12) (2012) 2912e2924 vols. 203-210. G. Herrmann, Forced motions of Timoshenko beams, J. Appl. Mech-T. ASME. 22 (1955) 53e56. J.G. Berry, P.M. Naghdi, On the vibration of elastic bodies having time-dependent boundary conditions, Q. Appl. Math. 14 (1956) 43e50. S.Y. Lee, S.M. Lin, Non-uniform Timoshenko beams with time-dependent elastic boundary conditions, J. Sound. Vib. 217 (2) (1998) 223e238. S.M. Lin, S.Y. Lee, The forced vibration and boundary control of pre-twisted Timoshenko beams with general time dependent elastic boundary conditions, J. Sound. Vib. 254 (1) (2002) 69e90. Yong-Woo Kim, Dynamic analysis of Timoshenko beam subjected to support motions, J. Mech. Sci. Technol. 30 (9) (2016) 4167e4176. B. Pratiher, Vibration control of a transversely excited cantilever beam with tip mass, Arch. Appl. Mech. 82 (2012) 31e42. I. Elishakoff, D. Livshits, Some closed-form solutions in random vibration of Bernoulli-Euler beams, Int. J. Eng. Sci. 22 (11) (1984) 1291e1301. I. Elishakoff, D. Livshits, Some closed-form solutions in random vibration of Bresse-Timoshenko beams, Probabilist. Eng. Mech. 4 (1) (1984) 49e54. I. Elishakoff, E. Lubliner, Random Vibration of a Structure via Classical and Nonclassical Theories, Probabilistic Methods in the Mechanics of Solids and Structures, Springer, Heidelberg, Berlin, 1985, pp. 455e467. I. Elishakoff, E. Lubliner, Random Vibrations of Structures with Complicating Effects, Stochastic Structural Dynamics Progress: Theory and Applications, the Y.K. Lin Anniversary Volume, Elsevier, London, 1988, pp. 47e64. F.J. Shaker, Effect of Axial Load on Mode Shapes and Frequencies of Beams, NASA Technical Note (NASA-TN-8109), 1975. A. Bokaiant, Natural frequencies of beams under tensile axial loads, J. Sound. Vib. 142 (1990) 481e498. H.P. Lee, Dynamic response of a rotating Timoshenko shaft subject to axial forces and moving loads, J. Sound. Vib. 181 (1995) 169e177. Y. Yesilce, O. Demirdag, Effect of axial force on free vibration of Timoshenko multi-span beam carrying multiple spring-mass systems, Int. J. Mech. Sci. 50 (2008) 995e1003. J. Murin, M. Aminbaghai, V. Kutis, J. Hrabovsky, Modal analysis of the FGM beams with effect of axial force under longitudinal variable elastic Winkler foundation, Eng. Struct. 49 (2013) 234e247. Mingwu Li, Analytical study on the dynamic response of a beam with axial force subjected to generalized support excitations, J. Sound. Vib. 338 (2015) 199e216. R.W. Clough, J. Penzien, Dynamics of Structures, third ed., McGraw Hill Inc., New York, 1995. Isaac Elishakoff, Generalized Eringen problem: influence of axial force on random vibration response of simply supported beam, Struct. Saf. 4 (1987) 255e265.
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Analysis of axial compressive loaded beam under random support excitations Wensheng Xiao, Fengde Wang*, Jian Liu Department of Mechanical and Electronic Engineering, China University of Petroleum (East China), Qingdao, 266580, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 December 2016 Received in revised form 13 August 2017 Accepted 24 August 2017
An analytical procedure to investigate the response spectrum of a uniform Bernoulli-Euler beam with axial compressive load subjected to random support excitations is implemented based on the Mindlin-Goodman method and the mode superposition method in the frequency domain. The random response spectrum of the simply supported beam subjected to white noise excitation and to Pierson-Moskowitz spectrum excitation is investigated, and the characteristics of the response spectrum are further explored. Moreover, the effect of axial compressive load is studied and a method to determine the axial load is proposed. The research results show that the response spectrum mainly consists of the beam's additional displacement response spectrum when the excitation is white noise; however, the quasi-static displacement response spectrum is the main component when the excitation is the Pierson-Moskowitz spectrum. Under white noise excitation, the amplitude of the power spectral density function decreased as the axial compressive load increased, while the frequency band of the vibration response spectrum increased with the increase of axial compressive load. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Bernoulli-Euler beam Axial compressive load Random support excitation The Mindlin-Goodman method
1. Introduction In offshore drilling, there are two kinds of jack-up riser: one with surface blowout preventers (BOP) and the other with subsea BOP. Due to lower cost, the jack-up platform often uses the riser with surface BOP to drill exploratory wells in shallow seas. However, the mainstream jack-up platform can be operated in water depths of 400 ft or more and the lateral rigidity of the platform decreases with increasing water depth, which makes the platform in random vibration status. The platform's vibration can reduce the drilling riser's service lifetime. Therefore, it is necessary to determine the riser's response to random vibrations before drilling an oil well. The jack-up riser can be modelled as a Bernoulli-Euler beam, the joint between the riser and the platform treated as a hinge, and the platform's vibration modelled as the riser's time-dependent boundary condition. In addition, the riser bears the axial compressive load resulting from the surface BOP. Considering these two kinds of riser, we chose the basic model of the beam as the research object in this study. The vibratory response of a beam with time-dependent boundary conditions can be obtained by the Laplace transform [1,2] and the Mindlin-Goodman methods [3e6]. In the Mindlin-Goodman method, the non-homogeneous boundary conditions are transformed into homogeneous ones. Therefore, the method of separation of variables can be used to solve the problem. The vibratory response of a non-uniform Bernoulli-Euler beam with time-dependent elastic boundary conditions
* Corresponding author. E-mail addresses: [email protected] (W. Xiao), [email protected] (F. Wang), [email protected] (J. Liu). http://dx.doi.org/10.1016/j.jsv.2017.08.045 0022-460X/© 2017 Elsevier Ltd. All rights reserved.
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was analysed by Lee and Lin [7], where they generalized the method of Mindlin-Goodman by introducing four shifting ez [8] investigated the axial vipolynomial functions with physical meanings. Josue' Aranda-Ruiz and Jose' Cerna'ndez-Sa brations of a rod with a time-dependent and non-harmonic force applied to the rod's free ends. Other scholars [9,10] have analysed the vibratory response of a Timoshenko beam with time-dependent boundary conditions by using the MindlinGoodman method without physical meanings in the shifting functions. The dynamic analysis of a non-uniform Timoshenko beam with general time-dependent boundary conditions was studied by S. Y. Lee and S. M. Lin [11], and the orthogonality conditions for the eigenfunctions of a non-uniform Timoshenko beam with elastic boundary conditions were explored. Furthermore, S. Y. Lee and S. M. Lin [12] investigated the vibration of a pre-twisted, non-uniform Timoshenko beam with time-dependent elastic boundary conditions. Yong-Woo Kim [13] presented an analytical solution procedure to solve the dynamic responses of a Timoshenko beam excited by support motions with the aid of the Mindlin-Goodman method and the eigenfunction expansion method. Pratiher [14] investigated the vibration control of a cantilever beam with tip mass under transverse base excitation by using the perturbation method. Isaac Elishakoff and David Livshits [15,16] derived closed-form solutions for both a Bernoulli-Euler and a Bresse-Timoshenko beam under stationary random excitation. Isaac Elishakoff and Eliezer Lubliner [17,18] studied the random vibration of a Bresse-Timoshenko beam subjected to spacewise white noise and concentrated point loading. The influence of axial load on the dynamics of structures has attracted much attention as a result of its wide application [19e23]. Most available literature on this topic treats support excitations as deterministic and little work can be found on the random response analysis of a beam with an axial load and time-dependent boundary conditions. Mingwu Li [24] presented the analytical solutions for the response of a uniform Bernoulli-Euler beam with axial force subjected to generalized support excitations by using the Mindlin-Goodman method in the time domain. However, in engineering fields, most excitations are random. There is no specific function to describe a random excitation, meaning the problem cannot be solved by the method proposed in the present literature. Therefore, we propose an analytical procedure to solve the random response of an axially loaded beam under random support excitations in the frequency domain. The current paper is organized as follows. First, the problem formulation is presented in Section 2. Then, the analytical procedure to solve the problem is carried out in Section 3. Subsequently, an investigation of the random displacement response spectrum of the simply supported beam with different excitations, the effect of axial compressive load, and a method for ascertaining the axial load is presented in Section 4. Finally, several important conclusions are summarized in Section 5.
2. Problem formulation Consider a simply supported beam of a linearly elastic material with length l, modulus of elasticity E, area moment of inertia I, and mass per unit length m. The beam, which is subjected to an axial compressive force p and random support excitations, is presented in Fig. 1. The supports' transverse motions, which are denoted as u1(t) and u2(t), are assumed to be stationary, random processes. The Euler-Bernoulli beam theory is adopted in this study, and the motion of the beam is governed by the following differential equation (without considering the material's internal damping):
EI
v4 yðx; tÞ v2 yðx; tÞ v2 yðx; tÞ þP þm ¼0 4 2 vx vx vt 2
(1)
In Eq. (1), t is the time, x is the coordinate measured along the beam axis, and y(x, t) is the transverse deflection of the beam axis. For the simply supported beam, the time-dependent boundary conditions at x ¼ 0 and x ¼ l are in the form of: 00
00
yð0; tÞ ¼ u1 ðtÞ; yðl; tÞ ¼ u2 ðtÞ; y ð0; tÞ ¼ 0; y ðl; tÞ ¼ 0
(2)
where, the prime denotes differentiation with respect to x. By using the Mindlin-Goodman method, the total transverse deflection y(x, t) is decomposed into two components as:
yðx; tÞ ¼ ys ðx; tÞ þ yd ðx; tÞ
(3)
Fig. 1. Beam with axial compressive load under random support excitations.
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In Eq. (3), ys(x, t) is the quasi-static displacement and yd(x, t) is the additional displacement due to the dynamic inertial force [25]; these displacements can be expressed as:
ys ðx; tÞ ¼
s X
gi ðxÞui ðtÞ
(4)
fn ðxÞqn ðtÞ
(5)
i¼1
yd ðx; tÞ ¼
∞ X n¼1
where, gi (x) is the static influence function, 4n (x) is the shape function of the beam, and qn(t) is the modal coordinate of nth mode. The corresponding boundary conditions for the simply supported beam are: 00
00
ys ð0; tÞ ¼ u1 ðtÞ; ys ðl; tÞ ¼ u2 ðtÞ; ys ð0; tÞ ¼ 0; ys ðl; tÞ ¼ 0 00
00
yd ð0; tÞ ¼ 0; yd ðl; tÞ ¼ 0; yd ð0; tÞ ¼ 0; yd ðl; tÞ ¼ 0
(6) (7)
Herein, the index s is determined by the boundary conditions of the beam; for the simply supported beam, s ¼ 2. The gi (x) [21] and 4n (x) of Eq. (4) and Eq. (5) are formulated as follows:
g1 ðxÞ ¼ 1 g2 ðxÞ ¼
x l
(8)
x l
(9)
4n ðxÞ ¼ sin
npx ðn ¼ 1; 2; :::∞Þ l
(10)
Natural frequencies of the simply supported beam with axial compressive load can be obtained by Eq. (11) [26]:
np un ¼ 2 l
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EIn2 p2 Pl2 ðn ¼ 1; 2; :::; ∞Þ m
(11)
3. Analysis Substituting Eq. (3) into Eq. (1) yields the equation:
EI
v4 yd v2 y v2 y v2 y s v2 ys þ P 2d þ m 2d ¼ P 2 m 2 4 vx vx vt vx vt
(12)
By using Eq. (4) and Eq. (5), Eq. (12) can be rewritten as follows:
EI
∞ X d4 4n n¼1
dx4
qn ðtÞ þ m
∞ X
4n ðxÞq€n ðtÞ þ P
n¼1
∞ X d2 4n n¼1
dx2
qn ðtÞ ¼ Feq
(13)
where, the dot denotes differentiation with respect to t. Feq, the equivalent load resulting from the support excitations, can be obtained by the following equation:
Feq ¼ m
2 X i¼1
gi ðxÞu€i ðtÞ P
2 X d2 gi ðxÞ i¼1
dx2
ui ðtÞ
(14)
Since the additional displacement has a homogeneous boundary condition, its mode shape functions conform to the following relationship.
EI
d4 4n ðxÞ d2 4n ðxÞ þ P mu2n 4n ðxÞ ¼ 0 dx4 dx2
The substitution of Eq. (15) into Eq. (13) generates Eq. (16):
(15)
W. Xiao et al. / Journal of Sound and Vibration 410 (2017) 378e388
m
∞ X
4n ðxÞq€n ðtÞ þ m
n¼1
∞ X
u2n 4n ðxÞqn ðtÞ ¼ Feq
381
(16)
n¼1
Substituting Eq. (8), Eq. (9) and Eq. (14) into Eq. (16) yields the equation: ∞ X
4n ðxÞq€n ðtÞ þ
n¼1
∞ X
u2n fn ðxÞqn ðtÞ ¼ 1
n¼1
x x u€ ðtÞ u€2 ðtÞ l 1 l
(17)
By applying the orthogonality conditions, Eq. (17) can be decoupled:
q€n ðtÞ þ u2n qn ðtÞ ¼ an u€1 ðtÞ bn u€2 ðtÞ
(18)
where, an and bn are weight coefficients of the inertial load.
Z l
an ¼
0
Z l x x fn ðxÞdx 1 fn ðxÞdx l 0 l bn ¼ Z l Z l f2n ðxÞdx f2n ðxÞdx 0
(19)
0
In this study, random support excitations are assumed to be uncorrelated. Therefore, the superposition method can be applied to calculate the dynamic response of the beam. Let S1(u) and S2(u) denote the auto-power spectral density function of u1(t) and u2(t) respectively and u be the vibration frequency of the beam's lateral displacement. The frequency response functions, H1(u) and H2(u), of the modal coordinates versus u1(t) and u2(t), are found to be:
H1 ðuÞ ¼
an u2 u2n u2
(20)
H2 ðuÞ ¼
bn u2 u2n u2
(21)
Then, the frequency response function of the additional displacement yd(x, t) versus u1(t) and u2(t) can be obtained by Eq. (22) and Eq. (23).
T1 ðuÞ ¼
∞ X
sin
npx H ðuÞ l 1
ðn ¼ 1; 2; :::∞Þ
(22)
sin
npx H ðuÞ l 2
ðn ¼ 1; 2; :::∞Þ
(23)
n¼1
T2 ð u Þ ¼
∞ X n¼1
As a result of the orthogonality of the mode shape, the fundamental modes are independent. Let Sd1(x, u) and Sd2(x, u) denote the power spectral density functions of the additional displacement yd(x, t) versus u1(t) and u2(t); then these spectra can be obtained by Eq. (24) and Eq. (25).
Sd1 ðx; uÞ ¼ S1 ðuÞ
2 ∞ X npx H1 ðuÞ sin l n¼1
ðn ¼ 1; 2; :::∞Þ
(24)
Sd2 ðx; uÞ ¼ S2 ðuÞ
2 ∞ X npx H2 ðuÞ sin l n¼1
ðn ¼ 1; 2; :::∞Þ
(25)
On the basis of the definition of the autocorrelation function and the Wiener-Khintchine principle, the power spectral density functions (Ss1(x, u) and Ss2(x, u),) of the quasi-static displacement can be derived as follows:
x2 Ss1 ðx; uÞ ¼ S1 ðuÞ½g1 ðxÞ2 ¼ S1 ðuÞ 1 l x2 Ss2 ðx; uÞ ¼ S2 ðuÞ½g2 ðxÞ2 ¼ S2 ðuÞ l
(26) (27)
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Let Sds1(x, u) denote the cross spectral density function between yd(x, t) and u1(t), and Sds2(x, u) denote the cross spectral density function between yd(x, t) and u2(t). Then, Sds1(x, u) and Sds2(x, u) can be calculated by Eq. (28) and Eq. (29).
Sds1 ðx; uÞ ¼ T1 ðuÞS1 ðuÞ ¼ S1 ðuÞ
∞ X
sin
npx H ðuÞ l 1
(28)
sin
npx H ðuÞ l 2
(29)
n¼1
Sds2 ðx; uÞ ¼ T2 ðuÞS2 ðuÞ ¼ S2 ðuÞ
∞ X n¼1
Based on the above analysis, the power spectral density function of the beam's displacement y(x, t) can be obtained by the following equations:
Sy1 ðx; uÞ ¼ Sd1 ðx; uÞ þ 2Re½Sds1 ðx; uÞ þ Ss1 ðx; uÞ
(30)
Sy2 ðx; uÞ ¼ Sd2 ðx; uÞ þ 2Re½Sds2 ðx; uÞ þ Ss2 ðx; uÞ
(31)
Sy ðx; uÞ ¼ Sy1 ðx; uÞ þ Sy2 ðx; uÞ
(32)
In Eq. (32), Sy(x, u) is the power spectral density function of the beam's total displacement, Sy1(x, u) the power spectral density function resulted from u1(t), and Sy2(x, u) the power spectral density function stimulated by u2(t). 4. Results and discussion In this research, the power spectral density functions of the random displacement responses of an axially, compressively loaded steel beam subjected to white noise excitation and Pierson-Moskowitz spectrum excitation are analysed, and the effect of the axial compressive load is investigated. In addition, a method for ascertaining the axial load is proposed. The beam's parameters are: EI ¼ 2746.7 N m2, m ¼ 3.14 kg m1, and l ¼ 1 m. 4.1. White noise excitation To simplify the analysis, white noise excitation at x ¼ l (S2(u) ¼ 1) is investigated in this study. By applying Eq. (21)and Eq. (25), the additional displacement's power spectral density function Sd(u) can be obtained by the following equation:
Sd ðx; uÞ ¼
2 ∞ X npx2 bn u2 sin u 2 u 2 l n n¼1
(33)
Using Eq. (21) and Eq. (29), the cross power spectral density function between the beam's additional displacement and the white noise excitation S2(u) ¼ 1 can be obtained as follows:
Sds ðx; uÞ ¼
∞ X
sin
n¼1
npx bn u2 l u2n u2
(34)
And the power spectral density function of the quasi-static displacement can be calculated by Eq. (27).
Ss ðx; uÞ ¼
x2 l
(35)
Based on the above calculations, the power spectral density function of the total displacement response can be obtained as follows:
Sy ðx; uÞ ¼
2 ∞ ∞ x2 X X npx2 bn u2 npx bn u2 sin þ2 sin þ 2 2 2 2 l l un u l un u n¼1 n¼1
(36)
Since the series in Eq. (36) has a fast convergence rate, taking the first five orders of the series is enough for the analysis. Consequently, the first five order natural frequencies of the beam and weight coefficients are calculated first, and are listed in Tables 1 and 2. By using Eq. (36), the power spectral density function of the beam stimulated by S2(u) ¼ 1 is calculated and depicted in Fig. 2. The results show that the vibratory response of the beam is a narrow-band random vibration, the principal oscillation is the beam's fundamental frequency; and the amplitude of the power spectral density function abides by sinusoidal regulation along the beam's axis, which can also be deduced from Eq. (23).
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Table 1 Natural frequencies of the simply supported beam with different axial compressive loads. P (N)
u1(rad/s)
u2(rad/s)
u3(rad/s)
u4(rad/s)
u5(rad/s)
0 200 400 600 800 1000
291.9043 290.8255 289.7428 288.6559 287.5649 286.4698
1167.6 1166.5 1165.5 1164.4 1163.3 1162.2
2627.1 2626.1 2625.0 2623.9 2622.8 2621.7
4670.5 4669.4 4668.3 4667.2 4666.2 4665.1
7297.6 7296.5 7295.5 7294.4 7293.3 7292.2
Table 2 The weight coefficients of the simply supported beam.
an bn
n¼1
n¼2
n¼3
n¼4
n¼5
2/p 2/p
1/p 1/p
2/3p 2/3p
1/2p 1/2p
2/5p 2/5p
Fig. 2. The power spectral density function of the beam's total lateral displacement under white noise excitation (p ¼ 1000 N).
Fig. 3. The cross power spectral density function between the beam's additional displacement and white noise excitation (p ¼ 1000 N).
Fig. 4. The power spectral density function of the beam's additional displacement under white noise excitation (p ¼ 1000 N).
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Fig. 3 shows the cross spectral density function between the additional displacement and the white noise excitation. The power spectral density function of the beam's additional displacement caused by white noise is calculated and plotted in Fig. 4, and the power spectral density function of the beam's quasi-static displacement under white noise excitation is presented in Fig. 5. The results show that the response spectrum of the additional displacement is the main component of the total response spectrum when the excitation is white noise, and both the response spectrum of the additional displacement and the cross spectrum have their maximum amplitudes at the beam's fundamental frequency. To explicitly illustrate the core characteristics of the response power spectral density function, the response spectrum at the midpoint of the beam is analyzed in detail. As shown in Fig. 6, the cross spectrum is positive when the vibration frequency is less than the beam's fundamental frequency, which can enhance the amplitude of the total response spectrum. The cross spectrum is negative when the vibration frequency exceeds the beam's fundamental frequency, which can reduce the amplitude of the total response spectrum. The amplitude of the spectrum has the maximum value when the vibration frequency is equal to the fundamental frequency. 4.2. Pierson-Moskowitz spectrum excitation The Pierson-Moskowitz spectrum, which is used to describe ocean waves, is a type of narrow-band random spectrum. In this study, the Pierson-Moskowitz spectrum of 10 m significant wave height, F(u), is taken as the power spectral density function of u2(t), which can be formulated by Eq. (37). F(u) is presented in Fig. 7.
FðuÞ ¼
0:78
u5
0:0311 exp 4
u
(37)
Therefore, the power spectral density function Sa(x,u) of the beam's additional displacement resulting from F(u) can be obtained by Eq. (38).
0:78
Sa ðx; uÞ ¼
u5
∞ npx bn u2 2 0:0311 X sin ; ðn ¼ 1; 2; :::∞Þ exp l u2n u2 u4 n¼1
(38)
The power spectral density function Sf(x,u) of the beam's quasi-static displacement resulting from F(u) can be calculated by the equation:
Sf ðx; uÞ ¼
0:78
u5
0:0311 x2 exp l u4
(39)
And we can get the cross power spectral density function Saf(x,u) between the additional displacement and the quasistatic displacement by Eq. (40).
Saf ðx; uÞ ¼
0:78
u5
∞ 0:0311 X npx bn u2 exp sin 4 l u2n u2 u n¼1
(40)
Subsequently, we can obtain the power spectral density function SF (x,u) of the simply supported beam's total displacement response caused by F(u).
Fig. 5. The power spectral density function of the beam's quasi-static displacement under white noise excitation (p ¼ 1000 N).
W. Xiao et al. / Journal of Sound and Vibration 410 (2017) 378e388
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Fig. 6. The cross power spectral density function between the additional displacement and white noise excitation at the beam's midpoint (p ¼ 1000 N).
Fig. 7. Pierson-Moskowitz spectrum of 10 m significant wave height.
SF ðx; uÞ ¼ Sa ðx; uÞ þ 2Saf ðuÞ þ Sf ðuÞ
(41)
The power spectral density function of the beam's total displacement response stimulated by the Pierson-Moskowitz spectrum is calculated and plotted in Fig. 8. It is found that the vibratory response of the beam is also a narrow-band random vibration. Fig. 9 presents the power spectral density function of the beam's additional displacement caused by Pierson-Moskowitz spectrum, Fig. 10 shows the cross spectral density function, and the power spectral density function of the beam's quasi-static displacement under Pierson-Moskowitz spectrum excitation is depicted in Fig. 11. The results show that the response spectrum of the quasi-static displacement is the main component of the total response spectrum when the excitation is the Pierson-Moskowitz spectrum.
Fig. 8. The power spectral density function of the beam's total displacement under Pierson-Moskowitz spectrum excitation (p ¼ 1000 N).
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Fig. 9. The power spectral density function of the beam's additional displacement under Pierson-Moskowitz spectrum excitation (p ¼ 1000 N).
Fig. 10. The cross power spectral density function between the beam's additional displacement and Pierson-Moskowitz spectrum excitation (p ¼ 1000 N).
Fig. 11. The power spectral density function of the beam's quasi-static displacement under Pierson-Moskowitz spectrum excitation (p ¼ 1000 N).
4.3. Effect of the axial compressive load Based on these results, the axial compressive load's effect on the beam's response spectrum is studied. There are two differences between the axially loaded beam and the unloaded beam: one is the reduction of the natural frequencies resulting from the axial compressive load, and the other is the axial load factor in equivalent load presented in Eq. (14). In this study, the second term in Eq. (14) becomes zero because the static influence functions of the simply supported beam are linear polynomials. Therefore, the axial load affects the response spectrum by changing the simply supported beam's natural frequencies only. Under white noise excitation, the effect of axial compressive load is investigated by analysing the response spectrum of the additional displacement. Several cases of different axial compressive loads are analysed, and the results for the midpoint of the beam are plotted in Fig. 12. It is seen that the axial compressive load can reduce the amplitude of the beam's random displacement response spectrum, the amplitude of the power spectral density function decreased as the axial compressive load increased, and the frequency band of the vibrational response increased with increasing axial compressive load. When there is a narrow-band random excitation such as the Pierson-Moskowitz spectrum, the reduction of
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387
Fig. 12. The power spectral density function of the beam's additional displacement with different axial compressive loads under white noise excitation.
the beam's natural frequencies resulting from axial compressive load in Eq. (38) and Eq. (40) has little impact on the beam's total displacement response. In some cases, such as when the riser of a jack-up drilling platform is under working conditions, it can be modelled as a uniform Euler-Bernoulli beam with axial compressive load. The axial compressive force resulting from the blowout preventers cannot be determined accurately due to the complex environmental loadings. However, it is important to know the riser's stress state; therefore, it is necessary to find a method to determine the axial compressive force on the beam. The relationship between the natural frequencies of the beam and the axial compressive force is presented in Eq. (11). Motivated by the abovementioned engineering problem and Eq. (11), a formula can be derived as follows:
P ¼ PE
ml2 u21
p2
(42)
where, PE is the Euler critical load of a beam. Eq. (42) can be used to ascertain the axial load by testing the beam's fundamental frequency u1. 5. Conclusions Based on the Mindlin and Goodman method and mode superposition method, a method for studying the response of the axially, compressively loaded uniform Euler-Bernoulli beam subjected to random support excitations is proposed in this paper. The random displacement response spectra of the simply supported beam with axial compressive load under white noise excitation and Pierson-Moskowitz spectrum excitation were studied in the frequency domain, and a formula was derived to determine the axial compressive load. It is found that the power spectral density function of the additional displacement is the main component of the beam's response spectrum when the excitation is white noise, and the quasi-static displacement response spectrum is the main component of the total response spectrum when the excitation is the Pierson-Moskowitz spectrum. Under white noise excitation, the axial compressive load has a significant impact on the beam's response spectrum; the effect of the axial compressive load can be ignored when the excitation is the Pierson-Moskowitz spectrum. When the excitation is white noise, the amplitude of the beam's response spectrum decreases as the axial compressive load increases, but the frequency band of the response spectrum increases with the increase of axial compressive load. The method proposed in this paper can be extended to solve the response of a Bernoulli-Euler beam with other boundary conditions under random support excitation. When we solve the response of the jack-up riser, the support excitation is the platform's random lateral displacement. Furthermore, the damping resulting from sea water and drill fluid should be considered. In offshore field, this method mainly has two functions: firstly, it can solve the jack-up riser's random response which can be used to estimate the riser's fatigue life; secondly, it can determine whether a sea area and sea-state are appropriate for jack-up platform. Acknowledgements The authors would like to acknowledge the Ministry of Industry and Information Technology of the PR China for supporting this study through the project “Jack-up platform brand project (II)” with the grant number of 10200001-15-ZC06070018.
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