Front. Mech. Eng. 2015, 10(1): 48–63 DOI 10.1007/s11465-015-0330-5
RESEARCH ARTICLE
Li LI, Yujin HU, Weiming DENG, Lei LÜ, Zhe DING
Dynamics of structural systems with various frequency-dependent damping models
© Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract The aim of this paper is to present the dynamic analyses of the system involving various damping models. The assumed frequency-dependent damping forces depend on the past history of motion via convolution integrals over some damping kernel functions. By choosing suitable damping kernel functions of frequency-dependent damping model, it may be derived from the familiar viscoelastic materials. A brief review of literature on the choice of available damping models is presented. Both the mode superposition method and Fourier transform method are developed for calculating the dynamic response of the structural systems with various damping models. It is shown that in the case of non-deficient systems with various damping models, the modal analysis with repeated eigenvalues are very similar to the traditional modal analysis used in undamped or viscously damped systems. Also, based on the pseudo-force approach, we transform the original equations of motion with nonzero initial conditions into an equivalent one with zero initial conditions and therefore present a Fourier transform method for the dynamics of structural systems with various damping models. Finally, some case studies are used to show the application and effectiveness of the derived formulas. Keywords damping, viscoelasticity, dynamic analysis, mode superposition method, Fourier transform method
1
Introduction
Problems involving vibration are encountered frequently in mechanical and aerospace engineering. In most of cases, vibration is not desirable and therefore we are usually Received January 15, 2015; accepted February 2, 2015
✉
Li LI, Yujin HU ( ), Weiming DENG, Lei LÜ, Zhe DING School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China E-mail:
[email protected]
forced on reducing it by energy dissipation. Damping plays a significant role in the dynamic behaviour of vibration systems since it is a characteristic parameter describing the energy dissipation mechanism from structural vibration systems. Understanding of damping mechanisms is quite difficult; hence these general damping phenomena of overall levels are seldom fully understood or defined precisely. As a result, the viscous damping is often considered and it has been a well known method to get rid of understanding complex damping mechanisms. The viscous damping can approximate the energy dissipation of overall levels or the local energy dissipation for grouping materials with similar characteristics of energy dissipation. A classical viscous damping model [1] assumes the damping matrix as a linear combination of inertia and stiffness matrices. This damping model is also known as “Rayleigh damping”, “classical damping” and “proportional damping”. With such a classical damping assumption, the real modal analysis, originally developed for undamped systems, can be extended to analyze structural damped systems in a similar manner. In spite of routinely assumed classical damping, the classical damping is still extensively used today due to its simplification. It was shown by Caughey and O’Kelly [2] that the viscously damped system must satisfy a rigorous mathematical condition to produce real modes. However, there is, of course, no reason why the mathematical condition must be satisfied with a physical system. In experimental modal testing, physical systems produce complex modes. In general, real modes mean that energy dissipation is almost uniformly distributed throughout the mechanical system. However, in practical, structural systems with two or more parts with significantly different levels of energy dissipation are encountered frequently in mechanical designs. With the development of viscoelastic damping techniques in modeling and analyzing mechanical systems (including nanocomposites and their applications in new generation of aircrafts, robotics and active structures, large wind turbines and so on, active control involving piezo-
Li LI et al. Dynamics of structural systems with various frequency-dependent damping models
electric and electromagnetic principles) [3,4], these demands cannot be solved without proper understanding of damping behaviour, and therefore the need to calculate the responses of damped systems in a more accurate manner is more urgent than ever before. As can be observed, the instantaneous velocity is assumed the only relevant state variable to determine the viscous damping force, which may work reasonably well for lightly damped systems lying in a small frequency range. Therefore, the viscous damping is a local damping model [5]. With the development of viscoelastic damping techniques, some nonviscous damping models [3,6–17] were developed considering the entire velocity history via convolution integrals over some kernel functions. The nonlocal character of time can be considered by this kind of damping models which are more likely to have a better match with experimental data. The nonviscous damping models are considered as the most generalized damping model within the scope of a linear mechanical analysis [18]. From the discussions, it emerges that some significant developments in the analysis of damped systems have taken place in the past decades. Although there are many detailed studies on the material damping and the damping or energy dissipation mechanisms in the joints, the major difficulty lies in representing all these tiny energy dissipation mechanisms in different materials or parts of the structural system in a unified manner. In practice, mechanical engineering systems with two or more parts with significantly different levels of energy dissipation are encountered frequently in dynamical designs, so these damping systems often involve multiple damping models. To the end, the aim of this paper is to present a dynamic analysis of the system involving various damping models. Due to the fact that various damping models are involved in a structural system, the dynamic analysis is often difficult or even impossible to be accurately carried out even for small-scaled problems. Firstly, we discuss the equations of motion of a linear structural system with various damping modes. Then a brief review of literature on the choice of available damping models for analyzing damped dynamic systems is presented. In this study, both the mode superposition method and Fourier transform method are developed for calculating the dynamic response of structural systems with various damping models. Finally, some case studies are used to show the application and effectiveness of the derived formulas.
2 Structural systems with various damping models
_ þ M€x ðtÞ þ C 0 xðtÞ
n X
Ck
k¼1
49
dτ !– 1gk ðt – τÞ ∂xðτÞ ∂τ t
þKxðtÞ ¼ f ðtÞ
(1)
where M 2 ℝN N and K 2 ℝN N are, respectively, the mass and stiffness matrices; C 0 2 ℝN N is the frequencyindependent damping matrix; C k 2 ℝN N are the coefficient matrices of frequency-dependent damping; gk ðtÞ is the kth damping kernel function (in the literature under different names in the context of different areas, it is also known as the constitutive time varying characteristic relaxation function, heredity function, retardation function, characteristic relaxation function or after-effect function); f ðtÞ is the forcing vector; and €x ðtÞ denotes the secondorder time derivative of the displacement vector xðtÞ. The lower limit of the integration in Eq. (1) is taken from – 1, which means that the integration starts before the beginning of the motion. In general, the motion starts at _ ¼ 0 for t < 0. Here a step time t = 0, that is, xðtÞ ¼ xðtÞ discontinuity should be noted by considering the following initial conditions: _ ¼ 0Þ ¼ x_ 0 xðt ¼ 0Þ ¼ x0 , xðt
(2)
Since the discontinuous histories with a step discontinuity are considered, Eq. (1) can be then rewritten as _ þ M€x ðtÞ þ C 0 xðtÞ
n X k¼1
n X þ Ck k¼1
Ck
0þ
δðτÞdτ !0 gk ðt – τÞ∂xðτÞ ∂τ
dτ þ KxðtÞ ¼ f ðtÞ !0 gk ðt – τÞ∂xðτÞ ∂τ t
þ
(3)
which can be simplified as _ þ M€x ðtÞ þ C 0 xðtÞ
n X
gk ðtÞC k x0
k¼1 n X þ Ck k¼1
where the term
dτ þ KxðtÞ ¼ f ðtÞ !0 gk ðt – τÞ∂xðτÞ ∂τ t
þ
n X
(4)
gk ðt ÞC k x0 gives the effect of the initial
k¼1
conditions and arises from allowing an initial disturbance of xðtÞ at t = 0. Taking the Laplace transform of the previous equation, one obtains ! ! n X 2 s M þ s C0 þ Gk ðsÞC k þ K X ðsÞ k¼1
The equations of motion of an N degree-of-freedom (DOF) linear structural system with various damping models can be expressed as
¼ FðsÞ þ M x_ 0 þ sMx0 þ C 0 x0 or
(5)
50
Front. Mech. Eng. 2015, 10(1): 48–63
DðsÞUðsÞ ¼ QðsÞ with DðsÞ ¼ s M þ s C 0 þ 2
n X
^b 1k !
Gk ðsÞC k
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ^ k ^ k – ω ^ k ^k – 1, ^b 2k ¼ ω ^ k ^ k þ ω ^ k ^k – 1: ¼ω
And the ADF models express the damping kernel function as
þ K,
k¼1
QðsÞ ¼ FðsÞ þ M x_ 0 þ sMx0 þ C 0 x0
(6)
Gk ðsÞ ¼
m X k¼1
where X ðsÞ ¼ ‘½xðtÞ, Gk ðsÞ ¼ ‘½gk ðtÞ, FðsÞ ¼ ‘½f ðtÞ and ‘½ denotes the Laplace transform; DðsÞ is so-called the dynamic stiffness matrix. Inpthe ffiffiffiffiffiffiffi context of structural dynamics, s = iω, where i = – 1 and ω denotes the frequency in rad/s.
m X Δk or gk ðtÞ ¼ Δk exp ð – Ωk tÞ s þ Ωk k¼1
Some authors considered the rheologic theory to model the linear viscoelasticity models, such as the generalized Maxwell model [23]: Gk ðsÞ ¼
m X k¼1
3
In this section, a brief review of literature on the choice of available damping models for analyzing damped dynamic systems is presented. Any causal model may be considered as a candidate for a damping model if it can make the energy dissipation function nonnegative. By choosing different kernel functions, Eq. (1) can be reduced to the systems with different damping forces. In the special case, when gðτÞ ¼ δðτÞ where δðτÞ is the Dirac delta function, the system can be reduced to a familiar viscously damped system. Some suitable damping kernel functions may be derived from familiar viscoelastic materials, and several physically realistic mathematical forms of damping models are available in the literature [15,16]. The Biot model [12,13] allows one to incorporate a wide range of experimental data as the following formula: m X k¼1
ak s þ bk
or gk ðtÞ ¼ a0 δðtÞ þ
m X
ak expð – bk tÞ
(7)
k¼1
Although some other viscoelastic models, including the Golla-Hughes-McTavish (GHM) model [19,20], Anelastic Displacement Field (ADF) model [21,22] and generalized Maxwell model [23], are physically different, they can be mathematically represented by a pole and residue form in the frequency or Laplace domain similar to Eq. (7). The GHM damping model can be given by Gk ðsÞ ¼ G0
m X
αk
k¼1
or gk ðtÞ ¼
m X k¼1
where
αk
sGk ðsÞ ¼ K0 þ
(10)
k – b^2k t
(8)
m X Kk Ck sαk ð0 0
(61)
Thus, the velocity and acceleration responses can be, respectively, expressed as J X lj t lj ðt – τÞ T _ ¼ xðtÞ e ψ j fðτÞdτ þ elj t ψ Tj ðMx_ 0 þ C0 x0 0 j¼1 j
!
þlj Mx0 Þφj , 8t > 0 x€ðtÞ ¼
By using HðsÞ ¼ DðsÞ – 1 and in view of Eq. (38), the frequency response from Eq. (5) can be carried out in terms of the poles and residues (i.e., the complex mode superposition method): J X ψ Tj ½FðsÞ þ Mx_ 0 þ sMx0 þ C0 x0
i ðs – lj Þ
j¼1
φj
(63)
In the context of structural dynamics, we consider s = iω. The displacement in the time domain of damped systems subjected to any forcing function can be obtained by taking the inverse Laplace transform of Eq. (63) and given by J X 1 t lj ðt – τÞ T xðtÞ ¼ ‘ – 1 fXðsÞg ¼ e ψ j fðτÞdτþ 0 j¼1 j
!
elj t ψ Tj
Mx_ 0 þ C0 x0 þ lj þ δðtÞ Mx0 φj
(64)
When considering t > 0, the previous equation can be simplified as J X 1 t lj ðt – τÞ T e ψ j fðτÞdτ þ elj t ψ Tj ðMx_ 0 þ C0 x0 xðtÞ ¼ 0 j j¼1
!
j
(66)
!0el ðt – τÞψTj fðτÞdτ þ el t ψTj ðMx_ 0 þ C0x0 t
j
j
þlj Mx0 Þφj , 8t > 0
(67)
Furthermore, when considering the zero initial condition, we have xðtÞ ¼
J X 1 φj j¼1 j
!0el ðt – τÞψTj fðτÞdτ t
j
(68)
It is essential to consider the following form: xðtÞ ¼ hðtÞ fðtÞ ¼
Transient response analysis
XðsÞ ¼
J X l2j j¼1
Dynamic response analysis
4.4.1
(65)
!0hðt – τÞfðτÞdτ t
(69)
which means that the transient responses can be calculated by using a convolution integral over the impulse response function. 4.4.2
Harmonic response analysis
The steady-state response of structural systems is of interest in dynamic design problems subjected to harmonic loading that are caused by a reciprocating power train or such other rotating machine parts such as motors, forging hammers, fans and compressors. When considering the harmonic loading, as is well known that for any linear system, if the loading function is harmonized, the response is also harmonic, that is, the force and response can be respectively expressed as fðtÞ ¼ FðωÞeiωt , xðtÞ ¼ XðωÞeiωt
(70)
Substituting u(t) and f(t) into Eq. (1) and considering Eq. (38), we obtain XðωÞ ¼ HðωÞFðωÞ ¼¼
J X ψ Tj FðωÞ φ ðiω – lj Þi j j¼1
(71)
Note that, in the case of the steady-state response analysis, it makes little sense to consider the initial conditions. Under such case, the displacement, velocity and accelerated responses can be expressed as
Li LI et al. Dynamics of structural systems with various frequency-dependent damping models
xðtÞ ¼ eiωt
J X ψ Tj FðωÞ φ ðiω – lj Þ j j¼1 i
_ ¼ iωeiωt xðtÞ
J X ψ Tj FðωÞ φ ðiω – lj Þ j j¼1 i
x€ðtÞ ¼ – ω2 eiωt
J X ψ Tj FðωÞ φ ðiω – lj Þ j j¼1 i
(72)
(73)
(74)
When considering real system matrices, the real part of the steady-state responses of the system, that is, xðtÞ¼ Re½X ðωÞexp ðiωtÞ, is used in the context of structural dynamics. From the previous discussions, it can be noted that, in the case of non-deficient systems, the dynamics of structural systems with various damping models can be analyzed in a familiar manner used in undamped or viscously damped systems. It should be mentioned that the dynamic responses of structural systems with various damping models can be calculated even when the equations of motion cannot be decoupled, i.e., the normal modes do not exist or the system may not satisfy the mathematical condition [56] to process normal modes.
Here F[ ] denotes the Fourier transform. The Fourier transform form can be established under the assumption that the complex input force can be interpolated by using trigonometric polynomials. The equation of motion can be then solved by inserting the sample frequency points into the equation of motion. The equation at each input sample frequency can be therefore solved in a similar manner of solving a static-force problem. Once the frequency responses are calculated, the responses in the time domain can be determined easily by using the inverse Fourier transform. The transient responses can be solved by using the inverse Fourier transform, that is xðtÞ ¼ F – 1 ½XðωÞ
The Fourier transform method may be the easiest approach to compute dynamic responses and in some cases it has advantages of high precision and efficiency [57]. One of the problems in using the Fourier transform method is how to deal with the nonzero initial conditions for structural systems with various damped models. Here by using the pseudo-force approach [31,58,59], we transform the original equations of motion with nonzero initial conditions into an equivalent one with zero initial conditions.
FT
(78) Here NFT is a number of sample discrete points; xn are the elements of discrete time displacements and Xk are the elements of frequency spectrums of the discrete time series. Note that the inverse Fourier transform procedure is defined using a positive sign in the exponential terms and may be efficiently calculated by considering the FFT and IFFT algorithms, which have been developed into a mature technology applied successfully to calculate the responses of engineering problems (for details on this aspect can be found in Refs. [60–62]). 5.2
5.1
(77)
In practice, a general analytical loading function cannot be easily obtained and the discrete Fourier transform and inverse discrete Fourier transform algorithms given below should be used. We usually consider the techniques of the fast Fourier transform (FFT) and the inverse fast Fourier transform (IFFT). 8 NX FT – 1 > 1 – 2πikn > > xn exp , 8k ¼ 0,1,2,:::,NFT – 1 > < Xk ¼ N NFT FT n¼0 NX FT – 1 > 2πikn > > > , 8n ¼ 0,1,2,:::,NFT – 1 Xk exp : xn ¼ N k¼0
5 Fourier transform method for structural systems with various damping models
57
Considering the nonzero initial conditions
Considering the zero initial conditions
When considering the zero initial conditions, the time domain equation of motion Eq. (1) can be cast into a frequency domain form by using the Fourier transform technique: ! ! n X – ω2 M þ iω C0 þ Gk ðωÞCk þ K XðωÞ k¼1
¼ FðωÞ
(75)
One of the problems on using the Fourier transform method is how to deal with the nonzero initial conditions for structural systems with various damped models. By considering the pseudo-force techniques [31,58,59], Eq. (1) with the nonzero initial conditions expressed by Eq. (2) can be equivalently written as the following form with zero initial conditions: € _ MxðtÞþC 0 xðtÞþ
k¼1
¼ fðtÞ þ f
where FðωÞ ¼ F½fðtÞ and XðωÞ ¼ F½xðtÞ
(76)
n X
where
pseudo ðtÞ x0
þf
Ck
dτþKxðtÞ ! – 1gk ðt – τÞ ∂xðτÞ ∂τ t
pseudo ðtÞ x_ 0
(79)
58
Front. Mech. Eng. 2015, 10(1): 48–63
_ ¼ 0Þ ¼ 0, xðtÞ ¼ xðtÞ – x0 , xðt ¼ 0Þ ¼ 0, xðt € ¼ x€ðtÞ: _ ¼ xðtÞ, _ xðtÞ xðtÞ ðtÞ stands for the The displacement pseudo-force f pseudo x0 contribution of the initial displacement x0 and can be given by [31,59]: f
pseudo ðtÞ x0
¼ – Kx0 uðtÞ
(80)
where uðtÞ is the unit step function and can be defined as ( 1, 8 t³0 uðtÞ ¼ : 0, 8 t
