Improved complex mode theory and truncation and ... - SAGE Journals

Research Article

Improved complex mode theory and truncation and acceleration of complex mode superposition

Advances in Mechanical Engineering 2016, Vol. 8(10) 1–16 Ó The Author(s) 2016 DOI: 10.1177/1687814016671510 aime.sagepub.com

Yaping Zhao and Yimin Zhang

Abstract The traditional complex mode theory is substantially improved. By means of the conjugacy of the complex characteristic value and vector, the formulae to compute the accuracy solution to the time-domain dynamic response for a symmetrical system with generalized linear viscous damping are established. Based on the formulation constructed, the truncation of the complex mode superposition is investigated. By employing the complex mode expansion of the system flexibility matrix discovered in this work, an accelerating method for the complex mode superposition after truncation is proposed. According to such improved complex mode theory, the system dynamical parameters, for example, the natural frequency and the complex vibration shape, are all the functions with respect to the mass, stiffness, and damping. However, the influence of the damping on the natural frequency may always be relatively slight. The numerical simulating investigation shows that the precise dynamic response can be obtained simply and conveniently in line with the formulae established. Meanwhile, the truncating and accelerating methods are effective for both the narrow or wide band external excitation and can provide the approximate solution to the dynamic response with better precision even when the frequency of the excitation is higher. Under certain conditions, the accelerating approach can be used to boost the accuracy of the truncated response. Keywords Complex mode, linear viscous damping, truncation of mode superposition, accelerating method, complicated excitation

Date received: 24 August 2015; accepted: 29 August 2016 Academic Editor: Elsa de Sa Caetano

Introduction Generally speaking, if the damping of a symmetrical multi-degree-of-freedom system is generalized linear viscous, the dynamic response of the system commonly needs to be worked out by means of the complex mode method in time domain. The reason essentially is that the damping matrix of such a system cannot be represented as the linear combination of its mass and stiffness matrices. In the late 1950s, the study on the complex mode theory started from the work of Foss1 on the state space approach to solve the dynamic response of a damped linear system. Afterward, Schmitz2 developed the thought of Foss and preliminarily constructed the fundamental formulation for the complex mode theory.

Nevertheless, he did not provide any numerical example. In the methods proposed by Foss and Schmitz, the n governing differential equations of motion of second order should be transformed to the 2n differential equations of first order. In order to avoid increasing the number of equation, some authors3,4 straightforwardly established the orthogonal relationship to uncouple the

School of Mechanical Engineering and Automation, Northeastern University, Shenyang, China Corresponding author: Yaping Zhao, School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China. Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

2 differential equations of motion by virtue of solving a quadratic eigenvalue problem. After coming into 1980s, Li5 studied the relation between the complex and real modes. Meanwhile, the complex mode method was included in some monographs6–8 and textbooks9,10 for postgraduate students on mechanical vibration. In addition, the idea of the complex mode has been embedded into the component mode synthesis successfully.11 On the other hand, the complex mode theory has been applied in the experimental modal testing widely.12,13 All these seem to show that the complex mode science has been mature. However, in the opinion of the author, it is not so at all. The deficiency mainly exists at least in three aspects. First, the quadratic eigenvalue problem generally cannot be solved using full-blown commercial computing software easily and directly. Second, the arithmetic of real number inextricably commingles with complex operation and this often results in an outcome with bad precision. Third, unlike the status of the real mode, the truncation and acceleration of the complex mode superposition have not been studied until recently. Unfortunately, in the past decade or so, there is almost no progress in such fields from the perspective of the literature in the mainstream journals. In this work, the traditional complex mode theory is ameliorated considerably. The complex eigenvalue and eigenvector are proposed to be obtained by means of solving the generalized characteristic value problem in respect of the augmented matrices, which are the partitioned matrices made up to by the mass, stiffness, and damping matrices of the system. Hence, the trouble to solve the quadratic eigenvalue problem in the traditional complex mode theory can be kept away completely. In accordance with the classical complex mode philosophy, the dynamical response is generally attained by virtue of the integration transformation technique after knowing the complex characteristic value and vector. In the methodology suggested, the response is directly attained in time domain by making use of the artifice to separate the real and imaginary parts. Clearly, the new approach is more simple and convenient. On the basis of the orthogonal relation between the complex modes, the system flexibility matrix can be expanded as the superimposing summation of the complex modes. On the strength of this, the truncation and acceleration of the complex mode superposition are discussed in detail. An accelerating technique is put forward for the complex mode superposition after truncation. In addition, the utilization of the trick to separate the real and imaginary parts leads up to the avoidance of the mixture of the real and complex operations. This obviously is beneficial to improve the computing precision of the dynamical response.

Advances in Mechanical Engineering

Improved complex mode theory Complex eigenvalue and complex mode of vibration The governing differential equation of motion for a N-degree-of-freedom linear system can be represented as follows M~ x€ + C~ x_ + K~ x=~ P ðt Þ

ð1Þ

whose initial conditions are ~ xð0Þ =~ x0 ,

~ x_ð0Þ = ~ x_

ð2Þ

where M, C, and K are the constant mass, damping, and stiffness matrices, respectively, and all of them are of order N 3 N. Concurrently, M, C, and K are all real symmetrical matrices, in which M is a positive definite matrix while K is a positive semi-definite matrix. What ! is more, P (t) is the excitation column matrix while ! x is the response column matrix. Equation (1) may be transformed into the state equation for the system as below ! y_ + B! y = Q ðt Þ ð3Þ A~       0 0 M ~_ ! , A= , and where ~ y = x , Q ðt Þ = ! P ðt Þ M C ~ x  M 0 . It is easy to verify that AT = A and B= 0 K BT = B. Thereby both A and B are real symmetrical matrices. ! Letting ! x = f elt yields ! ! y = c elt

ð4Þ

" # ! ! ! lf where c = ! . Obviously, f is a column matrix f ! of order N 3 1 while c is a column matrix of order 2N 3 1. ! Letting Q (t) = 0 and substituting equation (4) into equation (3) leads to ! ! B c =lA c

ð5Þ

Because both A and B are not positive definite commonly, their Cholesky decompositions cannot be implemented.14 Therefore, in general, the generalized characteristic value problem (5) cannot be transformed into a characteristic value problem in regard to a real symmetrical matrix like the case of the real mode. By taking advantage of the determinant computing approach for the partitioned matrix,15 the determinant value of the matrix A can be obtained as follows

Zhao and Zhang  M jAj = ð1ÞN  C

3    0  = ð1ÞN jMjM  CM1 0  M

= ð1ÞN jMj2 6¼ 0

ð6Þ

Thereby the matrix A is reversible and the generalized proper value problem (5) can be transformed into a proper value problem in regard to a real square matrix as ! ! Dc = lc

ð7Þ

where D =A1 B and generally the real square matrix D is not a symmetrical matrix. Apparently the characteristic polynomial of the real square matrix D always has real coefficients. As a consequence, the 2N eigenvalues, l1 , l2 , ., and l2N , belonging to the proper value problem (7), are usually pair-wise conjugate in complex number field.16 This of course gives rise to that the corresponding 2N eigenvec! ! ! tors, c 1 , c 2 , ., and c 2N , are pair-wise conjugate as well. Parenthetically, the above eigenvalue lj may be called the complex eigenvalue of the system while the ! corresponding eigenvector c j may be called the state complex mode vector. The state complex mode vectors can compose the state complex vibration shape matrix, which can be written as h

C= ! c1  =

FL F



" ! i l1 f !    c 2N = ! 1 f1

! #    l2N f 2N !  f 2N ð8Þ

where F is the complex vibration shape matrix and ! F=½! f 1!    f 2N , and L = diag( l1 ,    , l2N ). Herein, f j (j = 1, 2, . . . , 2N) may be called the complex mode vector. Obviously, the matrix C is a complex square matrix of order 2N 3 2N and the matrix F is a complex matrix of order N 3 2N. Provided the system in question does not have repeated complex proper value, it is easy to build the orthogonal relation between two different state complex mode vectors from equation (5), which can be expressed as !T ! !T ! c s A c k = 0, c s B c k = 0, s, k = 1, 2, . . . , 2N , s 6¼ k

ð9Þ

Under the condition of no repeated complex proper ! ! ! value, c 1 , c 2 , ., and c 2N are the proper vector belonging to different proper value so that they are linearly independent. Consequently, the state complex vibration shape matrix, C, made up by them is invertible.

With the aid of equation (9), both the matrices A and B can be diagonalized and the results obtained can be expressed as follows CT AC = diagð a1 , BC = diagð b1 ,

..., ...,

a2N Þ = Ap , CT b2N Þ = Bp

ð10Þ

!T ! !T ! !T ! !T where aj = c j A c j = 2lj f j Mf j + f j Cf j , bj = c j !T ! !T ! ! B c j =lj f j Mf j + f j Kf j , j = 1, 2, . . . , 2N . By right of equations (5) and (9), the relationship among aj , bj , and lj can be attained as lj =bj =aj = mj + inj , j = 1, 2, . . . , 2N

ð11Þ

In order to ensure the stability of the system vibration, the real part of lj should be negative, that is, mj \0. Owing to the conjugacy of the 2N complex characteristic values, the natural circular frequency of the system can be defined as vr = jlr j =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2r + n2r , r = 1, 2, . . . , N

ð12Þ

where l1 , l2 , ., and lN denote the complex characteristic values extracted from the 2N complex characteristic values and they are not conjugate with each other. Furthermore, corresponding to the natural frequency vr , the modal damping ratio in the complex mode theory can be defined as follows jr =

mr mr mr = = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð0, 1Þ, r = 1, 2, . . . , N l v j rj m2r + n2r r ð13Þ

Equation (13) shows that, under the condition of the generalized viscous damping, the oscillation of the system can always pertain to the under-damped case in general. On the basis of jr and vr , the complex characteristic value can as well be represented as ð14Þ lr =jr vr + inr , r = 1, 2, . . . , N qffiffiffiffiffiffiffiffiffiffiffiffiffi where nr = vr 1  j2r . ! The complex mode vector f j (j = 1, 2, . . . , 2N ) can be written in the form of real and imaginary parts as !ðRÞ !ðI Þ ! j = 1, 2, . . . , 2N ð15Þ f j = f j + if j pffiffiffiffiffiffiffi where i = 1. By employing equation (15), it is possible to have !T ! ðRRÞ ðII Þ ðRI Þ !T ! ðRRÞ f j Mf j = mj  mj + 2imj , f j Cf j = cj ðII Þ ðRI Þ cj + 2icj , j = 1, 2, . . . , 2N ð16Þ

4

Advances in Mechanical Engineering

!(R) !(R) !(I) T !(I) m(RR) = (f j )T Mf j , m(II) j j = (f j ) Mf j , !(R) !(I) !(R) !(R) m(RI) = m(IR) = (f j )T Mf j ; c(RR) = (f j )T Cf j , j j j !(I) T !(I) !(R) !(I) (RI) c(II) = c(IR) = (f j )T Cf j . j = (f j ) Cf j , and cj j After substituting equation (16) into the expression of aj in equation (10), it is easy to acquire the expression for aj in the form of real and imaginary parts as follows where

ð RÞ

aj = aj

ðI Þ

+ iaj ,

j = 1, 2, . . . , 2N

(RI) and a(I)  4jj vj m(RI) + 2nj (m(RR)  m(II) j = 2cj j j j ). On the other hand, from equation (15), it is easy to procure

Accurate solution to dynamic response Equation (19) is a linear non-homogeneous ordinary differential equation of first order with regard to zj , whose solution is

With the help of equation (10), letting ! y = C! z and submitting it into equation (3) gives rise to j = 1, 2, . . . , 2N

ð19Þ

As already stated, the state complex vibration shape matrix, C, is reversible and its inverse matrix can be T attained from equation (10) as C1 = A1 p C A. Consequently, it is likely to have   ~ x_ T ! y = A1 C A z = C1 ! p ~ x

lj t

ð20Þ

In view of equations (2) and (20), the initial condition in respect of equation (19) can be established as below

ðt 1 !T ! + f j P ðt Þelj ðttÞ dt, aj

j = 1, 2, . . . , 2N

0

ð22Þ After the coordinate transformation, the accurate solution of equation (1) can be acquired from equation (22) and the result is ! x=

2N 2N X X ! ! zj f j = zj ð0Þelj t f j j=1

j=1

 ðt 2N X 1 ! !T ! + fj fj P ðtÞelj ðttÞ dt a j=1 j

ð23Þ

0

From equation (23), making use of equation (18) and taking into account the conjugacy, the total response of the system in time domain can be represented as ! x ðt Þ = ! x in ðtÞ + 2 

Treatment of initial condition

1 !T ! f P ðtÞ, aj j

(R) (R) (R) ! (R) ! where h j = ½(nj a(I) j  jj vj aj )M + aj C x 0 + aj (I) (R) (I) ! (I) M~ x_0 and ! h j = ½(jj vj a(I) j + n j aj )M  aj C x 0  aj M~ x_0 .

ð18Þ

!(R) !(R) T !(I) !(I) T where F(1) and F(2) j = f j (f j )  f j (f j ) j = !(R) !(I) T !(I) !(R) T f j (f j ) + f j (f j ) . It is not difficult to understand that the intrinsic dynamical characteristics of the system, including the natural frequency and the complex mode, are all the functions regarding the mass, stiffness, and damping in accordance with the improved complex mode theory established here. Nonetheless, the situation in the real mode theory is far from this. There the intrinsic dynamical characteristics solely are functions regarding the mass and stiffness because the damping matrix always is dealt with to be the linear combination of the mass and stiffness matrices in terms of the Rayleigh assumption. From such a perspective, the improved complex mode theory proposed in this study is more impeccable and rigorous.

z_ j  lj zj =

ð21Þ

zj = zj ð0Þe

j = 1, 2, . . . , 2N

j = 1, 2, . . . , 2N

ð17Þ

(RR) (RI)  c(II)  2jj vj (m(RR)  m(II) where a(R) j = cj j j j )  4n j mj

! !T ð 1Þ ð 2Þ f j f j = Fj + iFj ,

 1 !T  ðRÞ ðI Þ zj ð0Þ =  2 f j ! h j + i! hj , aj 

ðt N X 1 r = 1 jar j

2

ejr vr ðttÞ

0  !ð AÞ !ðBÞ Q r ðtÞ cos nr ðt  tÞ  Q r ðtÞ sin nr ðt  tÞ dt ð24Þ

!(B) !(A) (1) (I) (2) ! where Q r (t) = (a(R) r Fr + ar Fr ) P (t), Q r (t) = (2) (I) (1) ! and ! x in ðtÞ denotes the (a(R) r F r  ar F r ) P (t); response gotten rise to by the initial condition. By virtue of equations (21) and (23), ! x in ðtÞ can be achieved as ! x in ðtÞ = 2

N i X ejr vr t h ð AÞ ð BÞ ~ ~ cos n t  r sin n t ð25Þ r r r r r 2 r = 1 jar j

(1) ! (2) ! where ~ r(A) r = Fr h r  F r h r (I) (R) ! h + F(2) ! h . (R)

r

r

(I)

and

(1) ~ r (B) r = Fr

r

As shown by equation (24), the total dynamic response of the system consists of two parts. One is,

Zhao and Zhang

5

! x in ðtÞ, which is the response caused by the initial condition and usually corresponds to the general solution. The other is, ! x e ðtÞ, the response caused by the external excitation and is generally analogous to the particular solution. Distinctly, the response caused by the initial conditions will decay in time domain following an exponential rule as indicated by equation (25). Therefore, the steady-state dynamic response is mainly determined by the response led to by the external excitation. This point is the same with the status in the real mode. After transformation, the integral in equation (24) may have the roughly same form with the Duhamel integral, that is, the convolution integral,17 in the real mode theory. Consequently, these integrals in equation (24) can be called the generalized Duhamel integral, which can be expressed as ~ Ir ðt Þ =

ðt

!ð AÞ Q r ðtÞejr vr ðttÞ cos nr ðt  tÞdt,

~ Jr ðtÞ

0

ðt =

!ðBÞ Q r ðt Þejr vr ðttÞ sin nr ðt  tÞdt

ð26Þ

0

!(A) !(B) In such integrals, Q r (t) and Q r (t) are associated with the excitation and they can be called the complex mode excitation. Resorting to the preceding concept of the generalized Duhamel integral, the representation of the total dynamic response can further be reduced to ! x ðt Þ = ! x in ðtÞ + 2

N i X 1 h~ ~ ð t Þ  J ð t Þ I r r 2 r = 1 jar j

ð27Þ

Generally speaking, it is simpler to obtain the timedomain response for a symmetrical system directly from equation (27) because it need not be with the aid of the integral transformation.18 Concurrently, the number of the equation essentially is not increased, nor is the quadratic eigenvalue problem required to be solved.

Truncation of complex mode superposition and related acceleration method Truncation of complex modes superposition At heart, the approach to obtain the response from equation (27) should pertain to the superposition of the complex modes in time domain. By employing such an approach, the accurate response can be worked out. However, if the number of the degree of freedom is relatively large, the preceding approach to search for accurate solution may be difficult or inconvenient to be

implemented at least to some degree. To handle this challenge, one way is to truncate the summation of the complex modes in equation (27) and the outcome truncated can be represented as   ! x ðt Þ = ! x in ðtÞ + 2

S i X 1 h~ ~ ð t Þ  J ð t Þ , I r r 2 r = 1 jar j

0\S\N ð28Þ

S P  ! x in ðtÞ = 2

where

r=1

ejr vr t jar j2

ð AÞ  ~ rðrBÞ sin nr t . r r cos nr t  ~

Herein, S symbolizes the number of the terms kept in the superposition summation after the truncation. In fact, equation (28) can be viewed as the truncation of the complex mode superposition, which can merely present an approximate solution to the dynamic response. However, the relevant computing workload is sure to be notably slighter. Given this, the investigation on how to truncate the complex mode superposition rationally is of great significance. The truncation of the real mode superposition, covering the corresponding accelerating method, has been discussed in detail in the literature.9,19 Nevertheless, the truncation of the complex mode superposition and the related accelerating method have never been reported until nowadays as far as the author has known.

Acceleration of complex mode superposition after truncation So as to enhance the accuracy of the truncated approximate solution, the accelerating method of the complex mode superposition after truncation needs to be discovered. For this, the complex mode expansion of the flexibility matrix of the system ought to be established in advance. From equation (10), the inverse matrix of the matrix B can be acquired and the result is "

# 0 T = CB1 B p C K1 2 1 3 " ! 0 ! # b1 l1 f 1    l2N f 2N 6 7 .. = ! 4 5 ! . f1  f 2N 1 0 b2N 2 2N 3 2N P l2j ! !T P lj ! !T " ! # T f f f f ! j j j j 7 6 b bj l1 f 1    l2N f 2N j=1 6 j=1 j 7 = 6 2N 7 ! ! 2N T T P P ! ! ! ! 4 5 lj f1  f 2N 1 f f f f j j j j bj bj 1

M1 = 0

j=1

j=1

ð29Þ By comparing with the counterparts of the partitioned matrices on both sides of equation (29), the following equivalence relations can be constructed as

6

M

Advances in Mechanical Engineering

1

2N 2N X X lj ! !T 1 ! !T = f j f j , K1 = F = f f , a al j j j=1 j j=1 j j 2N X 1 ! !T fj fj = 0 ð30Þ a j=1 j

where F is the flexibility matrix of the system, which is the inverse matrix of the stiffness matrix K. The second equality in equation (30) is the so-called complex mode expansion of the flexibility matrix of the system. It is assumed that the initial condition is zero and the external excitation is simple harmonic, that is, ! ! ! P (t) = P 0 sin vt, in which P 0 is a constant column vector. Under these conditions, the steady-state dynamic response of the system can roughly be estimated from equation (22), whose consequence is

’

aj l2j

lj sin vt =

ð31Þ On the basis of equation (8), the relationship between the state vector, ! y , and the vector, ! z , can be built as ð32Þ

such that the relationship between ! z and ! x is ! ! x =F z . If both the complex vibration shape matrix F and the vector ! z are split into the higher and lower order parts, respectively, the above relation can be partitioned as !  z ! ! z L + FH ! zH x = F z = ½ FL FH  ! L = F L ! zH 2S 2N X X ! ! = zj f j + zj f j ð33Þ j=1

2N X

j=1

! 2N 2S X 1 ! !T X 1 ! !T ! P ðt Þ f f  f f al j j al j j j=1 j=1 j j j=1 j j ! 2S 2S X X 1 ! !T ! ! P ðt Þ zj f j + F + f f = al j j j=1 j=1 j j

2S X ! zj f j  =

ð34Þ From equations (17) and (18), it is possible to have 1 ! !T 1 f j f j =  2 2 aj lj vj aj  h  i ð1Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ hj Fj  hj Fj + i hj Fj + hj Fj , ð35Þ

(R) (I) (2) (R) (I) where h(1) j = jj vj aj + n j aj and hj = nj aj  jj vj aj . In terms of the conjugacy, from equation (35), it is easy to obtain

1 !T ! f P ðtÞ, j = 1, 2, . . . , 2N aj lj j

    FL ! ~_ ! y = x = C! z = z F ~ x

! 1 ! !T ! P ðt Þ f f al j j j = 2S + 1 j j

2S X ! zj f j 

j = 1, 2, . . . , 2N

!T ! ðt fj P 0 elj t !T ! lj t  zj = f j P 0 sin vte dt =  aj 2 + l2 a v j j 0 lj t

ve  v cos vt  lj sin vt !T ! fj P 0

  lj sin vt + v cos vt ’ 2 2 aj v + lj !T ! fj P 0

! x ðt Þ ’

j = 2S + 1

Substituting equation (31) into equation (33) yields an approximate solution to the response under the zero initial condition. By right of the complex mode expansion of the flexibility matrix in equation (30), the result acquired can be simplified to

" # 2S 2S X X 1 ! !T 1 ! !T f f = 2Re f f = al j j al j j j=1 j j j=1 j j S  X 1  ð1Þ ð1Þ hr Fr  hðr2Þ Fðr2Þ 2 2 2 r = 1 vr jar j

ð36Þ

Substituting equation (36) into equation (34) and making utilization of equation (28), an accelerated approximate solution to the response after the truncation during the complex mode superposition can be attained as ~ xin ðtÞ + 2 xa ðtÞ =~

S i X 1 h~ ! ~ ð t Þ  J ð t Þ + DF P ðtÞ I r r 2 r = 1 jar j

ð37Þ where DF may be called the surplus flexibility matrix and DF = F  2

S X

1

2 2 r = 1 vr jar j



hðr1Þ Fðr1Þ  hðr2Þ Fðr2Þ



ð38Þ

Equation (37) is the computing formula for the approximate dynamic response by adopting the accelerating method of the complex mode superposition after ! truncation. In comparison to equation (28), DF P ðtÞ in equation (37) is the accelerating term, which is capable of compensating the truncation error at least in some measure. Hence, ! x a ðtÞ usually comes to be a better  approximate solution to ! x ðtÞ than ! x ðtÞ, which is attained only by truncating the superimposed summation of the complex modes.

Zhao and Zhang

7

6c

5c 3m

4c

3c

2m

k

m

2k

2m

3k

x1

2c 3m

4k

x2

x3

c

5k x4

6k x5

Figure 1. Physical structure of the system in question. Table 1. Natural frequencies of the system.

Un-damped natural frequency (rad/s) Natural frequency (rad/s)

v1

v2

v3

v4

v5

18.4269 18.6650

34.4389 34.9984

49.8757 49.1199

71.7138 71.6522

97.5224 96.2798

Numerical example investigation The main goal of this section is to research the characteristics of the truncation and acceleration of the complex mode superposition via the numerical examples. To this end, a 5-degree-of-freedom system, as shown in Figure 1, is taken into consideration. The mass, stiffness, and damping matrices of the system are M = diagð3m, 2m, m, 2m, 3mÞ 2

3k 2k 0 6 2k 5k 3k 6 K=6 3k 7k 6 0 4 0 0 4k 0 0 0 2 11c 5c 0 6 5c 9c 4c 6 C=6 4c 7c 6 0 4 0 3c 0 0 0 0

0 0 4k 9k 5k 0 0 3c 5c 2c

3 0 0 7 7 0 7 7, 5k 5 11k 3 0 0 7 7 0 7 7 2c 5 3c

ð39Þ

corresponding relation between the complex mode and the related inherent frequency. In addition, the complex modal damping ratios are tabulated in Table 2. Hereinafter two cases of the external excitation are studied: the simple harmonic excitation and the slope step excitation.

Simple harmonic external excitation In this case, the external excitation is ! ! P ðtÞ = P 0 cos vt

ð40Þ

in which m = 1kg, k = 1000 N=m, and c = 5 kg=s. After solving eigenvalue problem equation (7), the natural frequencies of the system can be obtained, which are listed in Table 1. For the sake of contrast, the natural frequencies without damping are listed in the same table as well, which are attained by means of the real mode method. Table 1 shows that, strictly speaking, the damping effect makes the lower order natural frequencies rising while makes the higher order natural frequencies decreasing. However, the influence of the damping on the natural frequency is rather limited. Because of the pair-wise conjugacy among the 10 complex modes, only 5 complex modes among them are given in Table 2. The subscripts reflect the

ð41Þ

! where P 0 = ½ 10, 000, 0, 0, 0, 0 T (N) without loss of generality. Substituting equation (41) into equation (27), the response of the system caused by the simple harmonic external excitation can be worked out as ! x e ðt Þ = 2

 5 X 1 r = 1 jar j

2

!ð AÞ !ðBÞ ðr Þ ðr Þ fA ðtÞQ r  fB ðtÞQ r

 ð42Þ

where in accordance with the formulae in equation (24), !(B) !(A) Q r and Q r can be figured out as h

i !ð AÞ !ð AÞ !ð AÞ !ð AÞ !ð AÞ = Q1 , Q2 , Q3 , Q4 , Q5 2 3 0:4147 0:0176 0:0317 0:0003 0:0000 6 0:1229 0:1061 0:0322 0:0019 0:0001 7 6 7 6 0:0229 0:0466 0:0329 0:0023 0:0005 7 6 7 4 0:0790 0:0134 0:0060 0:0032 0:0002 5 0:0611 0:0311 0:0139 0:0030 0:0000 ð43Þ

8

Advances in Mechanical Engineering

Table 2. Complex modes and complex modal damping ratios of the system. ! f1 j1 ! f2 j2 ! f3 j3 ! f4 j4 ! f5 j5

½ 0:0044  0:0309i, 0:0001  0:0312i, 0:0016  0:0239i, 0:0020  0:0163i, 0:0013  0:0081i T 0.1417 ½ 0:0018  0:0143i, 0:0078 + 0:0017i, 0:0058 + 0:0121i, 0:0024 + 0:0157i, 0:0000 + 0:0106i T 0.1487 ½ 0:0026 + 0:0070i, 0:0029  0:0144i, 0:0040  0:0055i0:0050 + 0:0047i, 0:0018 + 0:0069i T 0.2003 ½ 0:0008 + 0:0004i, 0:0014  0:0041i, 0:0030 + 0:0071i, 0:0018 + 0:0072i, 0:0008  0:0077i T 0.1088 ½ 0:0002  0:0002i, 0:0003 + 0:0022i, 0:0020  0:0090i, 0:0021 + 0:0034i, 0:0009  0:0006i T 0.2200

h

i !ðBÞ !ðBÞ !ðBÞ !ðBÞ !ðBÞ = Q1 , Q2 , Q3 , Q4 , Q5 2 3 2:0948 0:2081 0:0201 0:0003 0:0000 6 2:1311 0:0475 0:0668 0:0001 0:0000 7 6 7 6 1:6357 0:1884 0:0100 0:0024 0:0002 7 6 7 4 1:1198 0:2294 0:0342 0:0005 0:0001 5 0:5578 0:1502 0:0337 0:0017 0:0001

~ x0 = ½ 1,

h

0

  ðr Þ ejr vr t sin nr t  uA1 ffi = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 1  q2r + 4j2r q2r vr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

2ffi   q2r 1  2j2r  q2r + j2r 1 + q2r ðr Þ h i  sin vt  uA2

2 vr 1  q2r + 4j2r q2r

ð45Þ ðr Þ f B ðt Þ =

ðt

ejr vr ðttÞ sin nr ðt  tÞ cos vtdt =

0     ðr Þ ðr Þ vr ejr vr t cos nr t  uA1  nr cos vt  uB2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 

2 1  q2r + 4j2r q2r v2r

where

qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 = arctan (j (1 + q ))=((1  q ) 1  j2r ), u(r) r r r A1

2 2 2 u(r) A2 = arctan (jr (1 + qr ))=(qr (1  2j r  qr )),

arctan (2jr qr )=(1  q2r ), ðr Þ

ðrÞ

ð46Þ

qr = v=vr .

Hereinto

uA2 , uB2 2 ½0, p). The initial condition is presumed to be

u(r) B2 = ðrÞ

uA1 ,

0 T ,

~ x_0 = ½ 0,

1,

1,

1,

1 T

i ð AÞ ð AÞ ð AÞ ð AÞ ð AÞ ! = r1 , ! r2 , ! r3 , ! r4 , ! r5 2 0:0100 0:0020 0:0004 0:0000 6 0:0110 0:0000 0:0010 0:0000 6 6 0:0087 0:0016 0:0003 0:0001 6 4 0:0061 0:0022 0:0004 0:0000 0:0031 0:0015 0:0005 0:0001

3 0:0000 0:0000 7 7 0:0000 7 7 0:0000 5 0:0000 ð48Þ

h ejr vr ðttÞ cos nr ðt  t Þ cos vtdt

0,

Taking advantage of the expressions of ~ r (A) r (B) r and ~ r in equation (25), it is possible to achieve

On the other hand, the generalized Duhamel integrals can thus be reduced and the integral outcomes ðr Þ acquired in equation (42) are denoted by fA ðtÞ and ðr Þ fB ðtÞ. By employing equations (62) and (63) in Appendix 2, it is easy to have ðt

0,

ð47Þ

ð44Þ

ðrÞ f A ðt Þ =

0,

i

ð BÞ ð BÞ ð BÞ ð BÞ ð BÞ ! r2 , ! r3 , ! r4 , ! r5 = r1 , ! 2 0:0078 0:0006 0:0002 0:0000 6 0:0064 0:0012 0:0001 0:0000 6 6 0:0043 0:0012 0:0003 0:0000 6 4 0:0026 0:0008 0:0003 0:0000 0:0012 0:0003 0:0001 0:0000

3 0:0000 0:0000 7 7 0:0000 7 7 0:0000 5 0:0000 ð49Þ

Then, the response caused by the initial condition, ! x in ðtÞ, can be obtained by means of equation (25). Furthermore, the total response, ! x ðtÞ, can be derived from equation (27) on this basis. The image of ! x ðtÞ is shown in Figure 2. In order to validate the correctness of the theoretical solution of ! x ðtÞ, its numerical solution is also acquired by virtue of the Newmark-b method (d = 1=2, b = 1=4),20 which is plotted as in Figure 2 by the intermittent curves formed by the mark symbols. Figure 2 indicates that the theoretical and numerical solutions tally with each other quite well. From equations (45) and (46), it is easy to make clear that the full response contains three parts: the damped vibration aroused by the initial condition, the damped vibration triggered by the external excitation, and the simple harmonic vibration under the same frequency with the simple harmonic excitation. Evidently

Zhao and Zhang

9

Figure 2. Accurate response curves under simple harmonic excitation (v = (v1 + v2 )=2). Blue line: x1 ðtÞ, —*—: numerical solution of x1 ðtÞ; red line: x2 ðtÞ, — —: numerical solution of x2 ðtÞ; black line: x3 ðtÞ, —3—: numerical solution of ° x3 ðtÞ; green line: x4 ðtÞ, —+—: numerical solution of x4 ðtÞ; pink line: x5 ðtÞ, —h—: numerical solution of x5 ðtÞ.

Figure 3. Curves of x1 ðtÞ, x1 ðtÞ, and xa1 ðtÞ (v = (v1 + v2 )=2). Blue line: x1 ðtÞ, black line: x1 ðtÞ, and red line: xa1 ðtÞ.

the first two parts are both the attenuation vibration and the last one is the steady-state vibration caused by the external excitation. In consequence, only the third part vibration, namely, the steady-state simple harmonic vibration, can be reserved after an enough long time. Figure 2 reflects such characteristics definitely. As an example for the lower frequency and narrow band external excitation, the frequency of the simple harmonic external excitation is selected to be v = (v1 + v2 )=2. At this time, let the truncation take place at the second-order complex mode to observe the effect. This means that only the complex modes of the first and second orders are kept in the complex mode superposition, that is, letting S = 2 in equation (28). Thereupon the truncated response can be procured.

The truncated and accelerated responses, x1 ðtÞ and xa1 ðtÞ, are taken to be the examples to be investigated at this place since the external excitation locates at the mass 1 as shown by equation (41). Their images are plotted in Figure 3 by the black and red lines, respectively. With the purpose of contrast, the image of x1 ðtÞ, the precise response of the mass 1, is plotted together in Figure 3 by the blue line. As manifested in Figure 3, if the external excitation mostly covers the low-frequency ingredient, truncating the complex vibration shape superposition at the lower order mode may give rise to a good approximate solution. Moreover, the precision of the truncated approximate solution can be further improved with the aid of the accelerating method given in

10

Advances in Mechanical Engineering

Figure 4. Curves of x1 ðtÞ and x1 ðtÞ (v = 5v5 ).

Figure 5. Curves of x3 ðtÞ and x3 ðtÞ (v = 5v5 ).

equation (37). In fact, the remarkable difference between the precise response and the proximate response, including the truncated response and the accelerated response after truncation, may invariably take place at the peaks of the response curve as demonstrated in Figure 3. Now investigate another instance with the higher frequency and narrow band external excitation, in which v = 5v5 . Under such condition, the displacement response curves of the mass 1, 3, and 5 are plotted in Figures 4–6, respectively. In these figures, only the accurate and related truncated response curves are considered and they are expressed by the blue and red lines in several. Because the frequency of the external excitation is higher, the truncation takes place at the fourthorder complex mode and the complex modes on the first four orders are reserved in the superposition. In Figure 4, the red and blue lines are almost coincident with each other. In Figure 6, the two lines are well coincident in most of the time except a short period of time at the beginning. In Figure 5, the two lines

coincide slightly worse. Plainly, even under the circumstance of the high-frequency external excitation with narrow bandwidth, if the truncation is placed at the high-order mode, the precision of the truncated response may still be satisfactory. Meanwhile, if the truncation takes place at the higher order complex mode, the acceleration after the truncation may lose its own significance. From equations (45) and (46), it is uncomplicated to know ðr Þ ðr Þ lim f ðtÞ = lim fB ðtÞ = 0 t!‘ A t!‘ v!‘ v!‘

x ðtÞ = 0. such that lim ! t!‘ v!‘

Figures 4–6 reflect such a tendency well.

Slope step external excitation The expression of the slope step external excitation can be expressed as

Zhao and Zhang

11

Figure 6. Curves of x5 ðtÞ and x5 ðtÞ (v = 5v5 ).

where by right of equations (76) and (79) in Appendix 2, it is undemanding to have ðrÞ gA ðtÞ =

ðt

tejr vr ðttÞ cos nr ðt  tÞdt =

0

jr vr t + 1  2j2r ejr vr t  sinðnr t + ur Þ v2r v2r ðrÞ gB ðtÞ =

Figure 7. Graph of the slope step function pðtÞ.

! ! P (t) = P 0 pðtÞ,

 p0 pðtÞ =

t0

ð50Þ

! where P 0 = ½ 0, 1, 0, 0, 0 T , p0 = 1000 N, and t0 = 20p=v1 . The graph of the slope step function, pðtÞ, is drawn in Figure 7. With the aid of equation (68) in Appendix 2, the amplitude spectrum of the slope step function, pðtÞ, can be deduced as   Fp ðvÞ = p0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1  cos vt0 Þ ½1 + pv2 t0 dðvÞ + p2 dðvÞ v4 t02

0 qffiffiffiffiffiffiffiffiffiffiffiffi ffi 1  j2r

t0  t\ + ‘

ðrÞ g^A ðtÞ =

5 2p0 X 1 !ð AÞ !ðBÞ ðr Þ ðr Þ ! x e ðtÞ = g ðtÞQ r  gB ðtÞ Q r , t0 r = 1 jar j2 A

ðt0

tejr vr ðttÞ cos nr ðt  t Þdt + t0

0

ðt

ðr Þ

ejr vr ðttÞ cos nr ðt  t Þdt = tgA1 ðtÞ  ðt  t0 Þ

t0 ðrÞ

ðrÞ

gA1 ðt  t0 Þ  gA2 ðtÞ + gA2 ðt  t0 Þ =



0  t\t0

sinðnr t + ur Þ +

ð52Þ

ð55Þ

where with the aid of equations (71) and (75) in Appendix 2, it is possible to attain

ðrÞ



ðvr t  2jr Þ +

  5 2p0 X 1 !ð AÞ !ðBÞ ðrÞ ðrÞ ! g^ ðtÞQ r  g^B ðtÞQ r , x e ðt Þ = t0 r = 1 jar j2 A

ð51Þ where dðÞ is the so-called Dirac function. Equation (51) demonstrates the slope step excitation is a wideband excitation. From equation (27), the response caused by the slope step external excitation over the time interval ½0, t0  can be acquired as

tejr vr ðttÞ sin nr ðt  t Þdt

ejr vr t cosðnr t + ur Þ ð54Þ v2r v2r qffiffiffiffiffiffiffiffiffiffiffiffiffi where ur = arctan ((1  2j2r )=(2jr 1  j2r )). Similarly, the response caused by the slope step external excitation over the time interval ½t0 , + ‘ can be acquired as =

t 0  t\t0 p 0 t  t0

ðt

ð53Þ

jr t0 ejr vr t  vr v2r

ejr vr ðtt0 Þ sin½nr ðt  t0 Þ + ur  v2r

ð56Þ

12

Advances in Mechanical Engineering

Figure 8. Response curves under slope step external excitation.

ðrÞ g^B ðtÞ =

ðt0

tejr vr ðttÞ sin nr ðt  tÞdt

0

ðt

+ t0 ejr vr ðttÞ sin nr ðt  t Þdt t0

t0 j vr t + 1 ðrÞ j vr ðt  t0 Þt + 1 ðrÞ =  r gA1 ðtÞ + r gA1 ðt  t0 Þ nr nr nr i j vr h ðrÞ ðr Þ gA2 ðtÞ  gA2 ðt  t0 Þ + r nr n r t0 ejr vr t = 2 + cosðnr t + ur Þ vr v2r 

ejr vr ðtt0 Þ cos½nr ðt  t0 Þ + ur  v2r ð57Þ

x_0 = 0, the total response, If presuming that ~ x0 = ~ ! x ðtÞ, can be obtained with the help of equations (52) and (55), whose image is drawn in Figure 8. In order to verify the correctness of the theoretical solution of ! x ðtÞ, its numerical solution is also attained by right of the Newmark-b method (d = 1=2, b = 1=4), which is together plotted as in Figure 8 by the intermittent curves formed by the mark symbols. Figure 8 illustrates that the theoretical and numerical solutions are in keeping with each other quite well. Equations (52) and (55) state clearly that the total dynamic response, ! x ðtÞ, is a piecewise function. Equations (53) and (54) depict that the response over the time interval ½0, t0  contains two parts: one is a linear motion and the other is a degenerative simple harmonic vibration. As depicted by equations (56) and (57), the response over the time interval ½t0 , + ‘ is a damped simple harmonic vibration around a new equilibrium position. Moreover, the simple harmonic sections in the response suffer from attenuation with the

passage of time following exponential law. Figure 8 reveals such properties very well. Here, the truncating and accelerating characteristics of the responses of the masses 1 and 3 are selected to be the instances for investigation. The images of x1 ðtÞ, x1 ðtÞ, and xa1 ðtÞ are drawn together in Figure 9, among which x1 ðtÞ and xa1 ðtÞ are obtained under the condition of S = 2. Such a figure indicates that the curves of x1 ðtÞ and xa1 ðtÞ are nearly of coincidence. In consequence, for the response of the mass 1, the acceleration can observably boost the precision of the truncating solution and can supply a favorable approximate solution. In Figure 10, the graphs of x3 ðtÞ, x3 ðtÞ, and xa3 ðtÞ almost coincide mutually, among which x3 ðtÞ and xa3 ðtÞ are attained in the case of S = 2 as well. It is thus evident that, under some certain conditions, merely the truncation without acceleration may lead up to a beneficial approximate solution to the response as well. Although the slope step excitation belongs to the broadband excitation, its leading energy chiefly distributes over the lower frequency ingredient as shown by the expression of its spectrum, that is, equation (51). Therefore, the truncation and acceleration can give rise to a nice proximate solution for the dynamical response like the case of the low-frequency excitation even if the truncation occurs at the lower order complex mode.

Conclusion For a symmetrical system with general linear viscous damping, a method to solve its accurate dynamic response is proposed. In essence, the method is an improvement of the traditional complex mode technique. In the process of employing such an approach, the number of the differential equations to be solved need not be increased actually although the concept of state equation is quoted. Meanwhile, solving quadratic characteristic

Zhao and Zhang

13

Figure 9. Graphs of x1 ðtÞ, x1 ðtÞ, and xa1 ðtÞ under slope step external excitation.

Figure 10. Graphs of x3 ðtÞ, x3 ðtÞ, and xa3 ðtÞ under slope step external excitation.

value problem is not required. In addition, by reason of taking advantage of the conjugacy of the complex characteristic value and vector, the complex operation is avoided completely. In comparison with the integration transformation method, the improved complex mode method suggested is more simple and convenient because it pertains to the time-domain method essentially and the integration transformation is not needed. The truncation approach of the complex mode superposition is proposed and investigated using the improved complex mode theory established. The complex mode expansion of the flexibility matrix of the system and some other important and basic relations are discovered. Based on these, an accelerating method of the complex mode superposition after truncation is put forward. For the low-frequency excitation with narrow band, only keeping the lower order modes during the superposition and accelerating the truncated solution can generally give a good approximate solution to the dynamic response. Under such circumstance, the accelerated solution after truncation is always more accurate than the pure truncated one.

With regard to the high-frequency excitation with narrow band, the higher order modes should usually be reserved in the superposition to guarantee the accuracy of the truncated solution. Under such a condition, the acceleration after truncation may not be necessary. As to the broadband excitation, if the superimposition only comprises the complex modes of the lower orders, a nice approximate solution to the dynamical response can also be obtained in general. Furthermore, the acceleration after the truncation can be utilized to enhance the precision of the truncated solution. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research work in this paper was fully supported

14

Advances in Mechanical Engineering

by National Natural Science Foundation of China under Grant No. 51475083, New Century Excellent Talents Project by Education Ministry of China under Grant No. NCET-130116, National Key Basic Research Development Plan of China (the 973 Program) under Grant No. 2014CB046303, and Excellent Talents Support Program in Institutions of Higher Learning in Liaoning Province China under Grant No. LJQ2013027.

21. Teaching and Researching Group for Mathematics of Southeast University China. Integral transformations. Beijing, China: Higher Education Press, 1994.

References

a, b, c A, B, D C F ~ IðtÞ, ~ JðtÞ K M ! ! P (t), Q (t) ! x, ! y, ! z ! h, ~ r

1. Foss KA. Co-ordinates which uncouple the equation of motion of damped linear dynamic systems. J Appl Mech 1958; 25: 361–364. 2. Schmitz PD. Normal mode solution to the equations of motion of a flexible airplane. J Aircraft 1973; 10: 318–319. 3. Fawzy I and Bishop RED. On the dynamics of linear non-conservative systems. P R Soc London A Mat 1976; 352: 25–40. 4. Simpson A. A generalization of Kron’s eigenvalue procedure. J Sound Vib 1973; 26: 129–139. 5. Li D. Some general concepts of complex mode theory. J Tsinghua Univ 1985; 25: 26–37. 6. Ji W, Fang T and Chen S. Mechanical vibration. Beijing, China: Science Press, 1985. 7. Qiu J, Xiang S and Zhang Z. Computational structural dynamics. Hefei, China: China University of Science and Technology Press, 2009. 8. Meirovitch L. Fundamentals of vibrations. Long Grove, IL: Waveland Press, 2010. 9. Ni Z. Vibration mechanics. Xi’an, China: Xi’an Jiaotong University Press, 1989. 10. Zhang X and Wang T. Computational dynamics. Beijing, China: Tsinghua University Press, 2007. 11. Craig RR Jr and Ni Z. Component mode synthesis for model order reduction of nonclassically damped systems. J Guid Control Dynam 1989; 12: 577–584. 12. Fu Z and Hua H. Modal analysis theory and application. Shanghai, China: Shanghai Jiao Tong University Press, 2000. 13. Li D and Lu Q. Experimental modal analysis and its application. Beijing, China: Science Press, 2001. 14. Pozrikidis C. Numerical computation in science and engineering. Oxford: Oxford University Press, 1998. 15. Cheng Y, Zhang L, Xu Z, et al. Matrix theory. Xi’an, China: Northwestern Polytechnical University Press, 2000. 16. Qiu W. Higher algebra (volumes 1 and 2). Beijing, China: Higher Education Press, 1996. 17. Thomson WT and Dahleh MD. Theory of vibration and application. 5th ed. Upper Saddle River, NJ: Prentice Hall. 18. Hu H. On the vibration of linear damped systems. Acta Mech Solida Sin 1980; 1: 30–37. 19. Editorial board of vibration and shock handbook. Vibration and shock handbook (vol. 1). Beijing, China: National Defence Industry Press, 1988. 20. Roy D and Rao GV. Elements of structural dynamics. Hoboken, NJ: John Wiley & Sons, 2012.

Appendix 1 Notation the components of matrix matrices damping matrix flexibility matrix column matrix stiffness matrix mass matrix excitation column matrix column matrix column matrix complex characteristic value real and imaginary parts modal damping ratio complex mode vectors natural circular frequency diagonal matrix complex vibration shape matrices

l m, n j ! ! f, c v L F, C

Appendix 2 ðrÞ

ðrÞ

Calculation of functions fA ðtÞ and fB ðtÞ In equation (45), letting t1 = t  t, t = t  t1 and dt =dt1 . Thereby ðrÞ

f A ðt Þ =

ðt

ejr vr ðttÞ cos nr ðt  t Þ

0

cos vtdt =

i 1 h ðrÞ ðr Þ fA1 ðtÞ + fA2 ðtÞ 2

ð58Þ

Ðt ðrÞ ðtÞ = 0 ejr vr t1 cos½(nr + v)t1  vtdt1 and where fA1 Ð t ðr Þ fA2 ðtÞ = 0 ejr vr t1 cos½(nr  v)t1 + vtdt1 . By right of the integration by parts, it is possible to have ðrÞ fA1 ðtÞ =

ðt

ejr vr t1 cos½ðnr + vÞt1  vtdt1

0

1 jr vr t j r vr e = sin nr t + sin vt  nr + v ðn r + v Þ2 2 2 j v t

jr vr ðr Þ e r r cos nr t  cos vt  f ðt Þ 2 A1 ðn r + v Þ Solving equation (59) yields

ð59Þ

Zhao and Zhang ðr Þ

fA1 ðtÞ =

15

ejr vr t ½ðnr + vÞ sin nr t  jr vr cos nr t ðn r + v Þ

2

ðnr + vÞ sin vt + jr vr cos vt

+

ðn r + v Þ

2

where dðÞ is the so-called Dirac function and uðtÞ is the unit step function. That is to say

+ j2r v2r

Displacing v with v in equation (60) leads to ðr Þ fA2 ðtÞ =

ðt



ð60Þ

+ j2r v2r

uðtÞ =

Making use of the translation property of the Fourier transformation leads to

ejr vr t1 cos½ðnr  vÞt1 + vtdt1

F ½uðt  t0 Þ = e

0

= 

ejr vr t ½ðnr  vÞ sin nr t  jr vr cos nr t ðnr  vÞ2 + j2r v2r ðnr  vÞ sin vt  jr vr cos vt

ð61Þ

ðnr  vÞ2 + j2r v2r

Substituting equations (60) and (61) into equation (58) yields





ejr vr t nr v2r  v2 sin nr t  jr vr v2r + v2 cos nr t ðrÞ h ih i fA ðt Þ = ðnr + vÞ2 + j2r v2r ðnr  vÞ2 + j2r v2r 2



v nr  j2r v2r  v2 sin vt  jr vr v2r + v2 cos vt h ih i  ðnr + vÞ2 + j2r v2r ðnr  vÞ2 + j2r v2r

e ðr Þ fB ðtÞ =

jr vr t

ðrÞ f B ðt Þ



 nr 2jr vr v sin vt + n2r  v2 + j2r v2r cos vt ih i + h ð63Þ ðnr  vÞ2 + j2r v2r ðnr + vÞ2 + j2r v2r

Fp ðvÞ =

pðtÞe

‘

p0 e iv

+ ðt0 ð‘ p0 ivt dt = te dt + p0 eivt dt t0 0

ivt0

=

ivt

+

p0 ivt0 e  1 + p0 v2 t0

+ ð‘

 ð66Þ

F ½uðt  t0 Þ =

+ ð‘ ‘

uðt  t0 Þe 

ivt

+ ð‘

dt =

1 + pdðvÞ iv



eivt dt = eivt0

t0

ð67Þ

Substituting equation (67) into equation (A7), it is possible to achieve the Fourier transformation of pðtÞ as follows   Fp ðvÞ 1 = 2 + pdðvÞ cos vt0 v t0 p0   1 1  2  i 2 + pdðvÞ sin vt0 v t0 v t0 ðrÞ

ð68Þ

ðrÞ

Calculation of functions gA ðtÞ and gB ðtÞ In equation (53), letting t1 = t  t leads to t = t  t1 and dt =dt1 . As a result, it is trouble-free to have ðt

ðr Þ

ðrÞ

ðt  t1 Þejr vr t1 cos nr t1 dt1 = tgA1 ðtÞ  gA2 ðtÞ

0

By definition, the Fourier transformation of the slope step function, pðtÞ, in equation (50) can be obtained as + ð‘

1 + pdðvÞ iv

1 t  t0 . 0 t\t0 To begin with the definition the Fourier transformation, employing equation (65) yields

ðrÞ gA ðtÞ =

Frequency spectrum of slope step function



where uðt  t0 Þ =

in





jr vr v2r + v2 sin nr t + nr v2r  v2 cos nr t h ih i ðnr  vÞ2 + j2r v2r ðnr + vÞ2 + j2r v2r

ivt0



ð62Þ In a similar way, it is possible to procure equation (46), whose computing result is

1 t0 0 t\0

t0

eivt dt ð64Þ

ð69Þ Ðt ðr Þ ðr Þ where gA1 ðtÞ = 0 ejr vr t1 cos nr t1 dt1 and gA2 ðtÞ = Ð t j v t r r 1 cos n t dt . r 1 1 0 t1 e By virtue of the integration by parts, it is possible to have ðr Þ

gA1 ðtÞ =

t0

ðt

1 nr 1  ejr vr t cos nr t  ejr vr t1 sin nr t1 dt1 j r vr j r vr 0

The Fourier transformation of the unit step function can be expressed as21 F ½uðtÞ =

1 + pdðvÞ iv

ð65Þ

1 nr n 2 ðr Þ = 1  ejr vr t cos nr t + 2 2 ejr vr t sin nr t  2 r 2 gA1 ðtÞ j r vr j r vr j r vr

ð70Þ

16

Advances in Mechanical Engineering Substitution of equations (70) and (75) into equation (69) leads to

Solving (70) yields ðr Þ gA1 ðtÞ =

ðt

ejr vr t1 cos nr t1 dt1 =

0

jr vr ejr vr t + n2r + j2r v2r n2r + j2r v2r

ðnr sin nr t  jr vr cos nr tÞ

ð71Þ

Also by virtue of the integration by parts, it is possible to have ðrÞ gA2 ðtÞ =

ðt

t1 ejr vr t1 cos nr t1 dt1 =

0

ðrÞ gB ðtÞ =

= t ejr vr t1 sin nr t1 dt1 0

1 ðr Þ nr ðnr sin nr t  jr vr cos nr tÞ + g ðt Þ  2 2 jr vr A1 j r vr e 0

n2 ðrÞ sin nr t1 dt1  2 r 2 gA2 ðtÞ j r vr

ðt

 t1 ejr vr t1 sin nr t1 dt1 =

ðr Þ

gA2 ðtÞ =

0

ðt j r vr t ðr Þ  g ðtÞ  t1 ejr vr t1 sin nr t1 dt1 nr A1

ð72Þ

From equation (72), it is straightforward to have

j r vr ðr Þ g ðt Þ n2r + j2r v2r A1

ðt

t1 ejr vr t1 sin nr t1 dt1 =

ðt

nr ejr vr t1 sin nr t1 dt1 n2r + j2r v2r

t jr vr t e cos nr t nr

 2j nr ejr vr t 2 = r3 + nr  j2r v2r sin nr t  2jr vr nr cos nr t 4 vr vr 

 From equation (72), it is effortless to have ejr vr t1 sin nr t1 dt1 =

j vr ðrÞ 1 1  ejr vr t cos nr t  r gA1 ðtÞ nr nr

0

tejr vr t ðjr vr sin nr t + nr cos nr tÞ v2r ð78Þ Substituting equation (78) into equation (77) yields

ð74Þ Substituting equation (74) into equation (73) yields ðr Þ gA2 ðtÞ =

tejr vr t ðnr sin nr t  jr vr cos nr tÞ n2r + j2r v2r

n2r  j2r v2r

2 ð75Þ n2r + j2r v2r

 ejr vr t 2jr vr nr sin nr t + n2r  j2r v2r cos nr t +

2 n2r + j2r v2r 

1 ðr Þ j vr ðrÞ gA1 ðtÞ  r gA2 ðtÞ nr nr

0

ð73Þ

0

ðt

ð77Þ

0

tejr vr t t1 ejr vr t1 cos nr t1 dt1 = 2 nr + j2r v2r

ðnr sin nr t  jr vr cos nr tÞ + 

t 1  ejr vr t cos nr t nr

0

Solving (72) gives rise to ðt

tejr vr ðttÞ sin nr ðt  t Þdt

0

0

jr vr t1

ðt

ðt

jr vr t

1 ðrÞ nr te g ðt Þ  t1 ejr vr t1 sin nr t1 dt1 = 2 2 j r vr jr vr A1 jr vr

ðt

ð76Þ

In equation (54), letting t1 = t  t and resorting to equation (74), it is uncomplicated to have

t jr vr t e cos nr t jr vr

ðt

+

ejr vr t 1 3 2 2 2 j v t + n  j v  r r r r r v4r v4r 2

 2jr vr nr sin nr t + nr  j2r v2r cos nr t ðr Þ

gA ðtÞ =

ðrÞ gB ðtÞ =

ðt 0



tejr vr ðttÞ sin nr ðt  t Þdt =

nr ðvr t  2jr Þ v3r



ejr vr t 2 nr  j2r v2r sin nr t  2jr vr nr cos nr t 4 vr ð79Þ