Complex mode superposition method of nonproportionally damped linear systems with hysteretic damping

Abstract

The vibration response of a damped linear system is always calculated based on the mode superposition method. However, the construction of the damping matrix is difficult for the conventional mode superposition methods based on the viscous damping model, and this problem is much more serious for nonproportionally damped linear systems. The damping matrix based on the hysteretic damping model is easy to construct and unique, which is determined only by the structural stiffness and material loss factor. The time-domain motion equation of a multi-degree-of-freedom nonproportionally damped linear system is easily constructed based on the hysteretic damping model. According to the characteristics of external excitation, the general solution of the corresponding homogeneous equation and special solution of the corresponding nonhomogeneous equation can be solved. By the aid of the easiness of the damping matrix, a complex mode superposition method based on the hysteretic damping model is proposed for the nonproportionally damped linear system. Based on the proposed method, a user subroutine in ANSYS and MATLAB is developed to calculate vibration responses in time-domain dynamic analyses. A shaking table test of a cantilever plate composed of host and damping layers is conducted to validate the proposed method. The method proposed in this article is unconditionally convergent, and its convergence is independent of the time step of time-domain analyses. Compared with the common complex mode superposition method based on the viscous damping model, the simulation results of the proposed method are closer to the test results, and its accuracy and efficiency are higher. In addition, the calculation results of the proposed method are unique, which is irrelevant to the choice of vibration modes.

Keywords

Hysteretic damping model nonproportionally complex mode superposition method vibration response shaking table test

Highlights

A complex mode superposition method for hysteretic nonproportionally damped linear systems is proposed.

The proposed complex mode superposition method is unique and has high computation efficiency.

The proposed complex mode superposition method is irrelevant to the choice of vibration modes.

Different complex mode superposition methods of nonproportionally damped linear systems are analyzed.

The user subroutines in ANSYS and MATLAB are developed in time-domain dynamic analyses.

A shaking table test is conducted to validate the proposed method.

1. Introduction

To improve performance against strong wind or earthquake ground motion, many building structures can be equipped with viscoelastic dampers (Lee et al., 2002) or different damping materials (Huang et al., 2015). These hybrid structures have one feature in common. The damping is inherently nonproportional. The damping matrices are not proportional to the mass and/or stiffness matrices in hybrid structures with multiple material damping characteristics. This type of damping is always classified as nonproportional damping (Jones, 2001; Nasif et al., 1985). The methods dealing with the problem can be classified into two categories: the real mode superposition method of approximately proportionally damped linear systems and the complex mode superposition method of nonproportionally damped linear systems (Villaverde, 1997).

For the first case, the nonproportionally damped linear systems can be approximately regarded as proportionally damped linear systems. The commonly used calculation method is based on the Caughey damping model (Caughey and O’Kelly, 1965), which is a generalized Rayleigh damping model (Strutt, 1945). The Caughey damping coefficients are obtained by finite element modes (Pant et al., 2013; Ryan and Polanco, 2008). The real mode superposition method can be directly applied in the Caughey damping model (Bilbao et al., 2006; Cruz and Miranda, 2017). However, the method has two shortcomings. The first is that the method depends on the choice of appropriate modes; specifically, the Caughey damping matrix is not unique. The second shortcoming is that the damping ratios are underestimated or overestimated in the frequency range of interest, and the calculated dynamic responses are overestimated or underestimated. To solve the question about the nonuniqueness of the real mode superposition method based on the Rayleigh damping model, a real mode superposition method based on the equivalent modal damping ratio was proposed (Chopra, 2001). The equivalent modal damping ratio was determined approximately through a structural energy-weighted function (Feriani and Perotti, 1996; Shen et al., 1995; Wang, 2009). However, this method is a forced decoupling method and cannot be applied to large damping ratio or large-scale systems because the calculation error is difficult to control.

In theory, mode vectors of nonproportionally damped linear systems are complex mode vectors rather than real mode vectors (Liang and Lee, 1991; Villaverde, 1988). Thus, the viscous damping motion equations of these systems can be transformed to first-order differential equations by state-space modeling. The corresponding complex mode superposition method is obtained as a result of complex modal orthogonality (De Domenico and Ricciardi, 2019; Foss, 1958; Igusa et al., 1984; Neugebauer et al., 2011; Papageorgiou and Gantes, 2010). However, the construction of a reasonable damping matrix is very difficult (Clough and Penzien, 2003). The viscous damping model exhibits the unreasonable physical phenomenon that the energy dissipation in one cycle of harmonic vibration is proportional to the structural frequency (Bert, 1973; Crandall, 1970). This shortcoming leads to the difficulty in the construction of a damping matrix because it is related to the selection of reasonable structural frequency. In addition, the dimension of the matrix is doubled, and the computation cost is large.

Compared with the viscous damping model, the damping matrix of the hysteretic damping model can be more easily determined. Theodorsen and Garrick (1940) proposed a hysteretic damping model, also called a complex damping model. This model guarantees that the energy losses per cycle are independent of structural frequencies as observed in material experiments and building observations (Bert, 1973; Crandall, 1970). Therefore, the construction of the damping matrix based on the hysteretic damping model is determined only by material loss factors and the stiffness of the system. Regarding time-domain analysis, Inaudi and Kelly (1995) proposed a time-domain calculation method based on the Hilbert transform method. However, the method is only applied in a hysteretic proportionally damped linear system. In addition, the method is a time-domain step-by-step numerical method, which cannot consider the influences of structural natural frequency and vibration frequency of external excitation.

In this article, according to the characteristics of external excitation, a complex mode superposition method based on the hysteretic damping model is proposed and can be applied to nonproportionally damped linear systems. A user subroutine in ANSYS and MATLAB based on the proposed method is developed to calculate vibration responses. The proposed method is unconditionally stable and has high computational efficiency (Section 2). The results of the shaking table test are discussed to verify the proposed method in Section 3, and the differences between conventional mode superposition methods and the proposed method have been identified.

2. Complex mode superposition method for a nonproportionally damped system

2.1. Complex mode superposition method based on the Rayleigh damping model

The damping matrix of a nonproportionally damped system is not orthogonal, so the traditional real mode superposition method is no longer applicable. The viscous damping model is one of the most commonly used damping models. Several researchers have suggested that combined with the state-space method, a complex mode superposition method based on the viscous damping model can be realized, which can be applied to nonproportionally damped systems (De Domenico and Ricciardi, 2019; Foss, 1958; Igusa et al., 1984; Neugebauer et al., 2011; Papageorgiou and Gantes, 2010).

Based on the viscous damping model, the time-domain motion equation of the multidegree-of-freedom (MDOF) system is given by

M \frac{d^{2} x (t)}{d t^{2}} + K x (t) + C \frac{d x (t)}{d t} = - M G g (t)

(1)

where

M

is the mass matrix.

K

is the stiffness matrix.

C

is the damping matrix.

x (t) = {[\begin{matrix} x_{1} (t) & x_{2} (t) & \dots & x_{N} (t) \end{matrix}]}^{T}

is the displacement vector.

G = {[\begin{matrix} 1 & 1 & \dots & 1 \end{matrix}]}^{T}

g (t)

is the acceleration of external excitation.

Rayleigh damping is the most common method based on the viscous damping model. Based on the Rayleigh damping, the damping matrix of a nonproportionally damped system is expressed as (Papageorgiou and Gantes, 2010, 2011)

C = \sum_{j = 1}^{S} α_{j} M_{j} + β_{j} K_{j}

(2)

{\begin{cases} α_{j} = \frac{2 ξ_{j} ω_{m} ω_{n}}{ω_{m} + ω_{n}} \\ β_{j} = \frac{2 ξ_{j}}{ω_{m} + ω_{n}} \end{cases}

(3)

where

M_{j}

is the mass matrix of the j-th component.

K_{j}

is the stiffness matrix of the j-th component. S is the number of components.

ξ_{j}

is the damping ratio of the j-th component.

α_{j}

and

β_{j}

are Rayleigh coefficients of the j-th component.

ω_{m}

is the natural frequency of the m-th mode, and

ω_{n}

is the natural frequency of the n-th mode.

The damping ratio is half of the material loss factor (Lin and Zhu, 2009), and it is expressed as

ξ_{j} = 0.5 η_{j}

(4)

where

η_{j}

is the material loss factor of the j-th component.

By means of state-space modeling (Foss, 1958), the auxiliary equation is introduced as

M \frac{d x (t)}{d t} - M \frac{d x (t)}{d t} = 0

(5)

Equations (1) and (5) are combined as

C_{v} \frac{d u (t)}{d t} + K_{v} u (t) = F

(6)

where

C_{v} = [\begin{matrix} C & M \\ M & 0 \end{matrix}]

K_{v} = [\begin{matrix} K & 0 \\ 0 & - M \end{matrix}]

u (t) = [\begin{matrix} x (t) \\ d x (t) / d t \end{matrix}]

, and

F = [\begin{matrix} - M G g (t) \\ 0 \end{matrix}]

Equation (6) can be directly solved by the complex mode superposition method. Thus, a complex mode superposition method based on the Rayleigh damping model (CR) can be realized.

2.2. Complex mode superposition method based on the hysteretic damping model

CR is realized in the state space, rather than in the physical space, and the amount of calculation will be very large. Besides, equations (2) and (3) show that the determination of the damping matrix depends on the choice of modes. To overcome these shortcomings, this article is devoted to propose a complex superposition method based on the hysteretic damping model.

The frequency-domain motion equation of the MDOF system based on the hysteretic damping model is given by (Inaudi and Kelly, 1995)

- ϖ^{2} M x (i ϖ) + K x (i ϖ) + i sgn (ϖ) K_{η} x (i ϖ) = - M G g (i ϖ)

(7)

where

ϖ

is the vibration frequency of the response. i is the imaginary unit.

K_{η}

is the damping matrix.

x (i ϖ)

is the Fourier transform of

x (t)

, and

g (i ϖ)

is the Fourier transform of

g (t)

Here, $sgn (ϖ)$ is a symbolic function as follows

sgn (ϖ) = {\begin{matrix} 1 & ϖ > 0 \\ 0 & ϖ = 0 \\ - 1 & ϖ < 0 \end{matrix}

(8)

The construction of a damping matrix based on the hysteretic damping model is determined only by the structural stiffness and material loss factor. The damping matrix is expressed as

K_{η} = \sum_{j = 1}^{S} η_{j} K_{j}

(9)

By the Fourier transform method, equation (7) is rewritten as

M \frac{d^{2} x (t)}{d t^{2}} + K x (t) - K_{η} x^{H} (t) = - M G g (t)

(10)

where

x^{H} (t) = {[\begin{matrix} x_{1}^{H} (t) & x_{2}^{H} (t) & \dots & x_{N}^{H} (t) \end{matrix}]}^{T}

x^{H} (t)

is the Hilbert transform of

x (t)

. It is expressed as (Hahn, 1996)

x_{n}^{H} (t) = \frac{1}{π} \int_{- \infty}^{+ \infty} \frac{x_{n} (τ)}{t - τ} d τ (n = 1,2, \dots, N)

(11)

g (t)

can be decomposed by signal recognition methods of external excitation, such as the fast Fourier transform method (Brigham, 1974), short-time Fourier transform method (Cohen, 1995), and Hilbert–Huang transform method (Huang et al., 1998). It is obtained that

g (t) = r (t) + s (t)

(12)

where

r (t)

is the nonharmonic part and the vibration frequency of

r (t)

is zero.

s (t)

is the harmonic part, and the vibration frequency of

s (t)

is not zero.

Equation (12) is substituted into (10), which is rewritten as

M \frac{d^{2} x (t)}{d t^{2}} + K x (t) - K_{η} x^{H} (t) = - M G [r (t) + s (t)]

(13)

The general solution of equation (13) is expressed as

x (t) = x_{c} (t) + x_{r} (t) + x_{s} (t)

(14)

Equation (14) is substituted into (13), which is decomposed into

M \frac{d^{2} x_{c} (t)}{d t^{2}} + K x_{c} (t) - K_{η} x_{c}^{H} (t) = 0

(15)

M \frac{d^{2} x_{r} (t)}{d t^{2}} + K x_{r} (t) - K_{η} x_{r}^{H} (t) = - M G r (t)

(16)

M \frac{d^{2} x_{s} (t)}{d t^{2}} + K x_{s} (t) - K_{η} x_{s}^{H} (t) = - M G s (t)

(17)

where

x_{c} (t)

is the general solution of equation (15).

x_{r} (t)

is a special solution of equation (16), and

x_{s} (t)

is a special solution of equation (17).

By the Hilbert transform method, equation (15) can be rewritten as

{\begin{cases} M \frac{d^{2} x_{c} (t)}{d t^{2}} + K x_{c} (t) - K_{η} x_{c}^{H} (t) = 0 \\ M \frac{d^{2} x_{c}^{H} (t)}{d t^{2}} + K x_{c}^{H} (t) + K_{η} x_{c} (t) = 0 \end{cases}

(18)

It is assumed that

y_{c} (t) = x_{c} (t) + i x_{c}^{H} (t)

(19)

Equation (18) is equivalent to

M \frac{d^{2} y_{c} (t)}{d t^{2}} + K y_{c} (t) + i K_{η} y_{c} (t) = 0

(20)

From equation (20), the corresponding complex eigenvectors are given by

Ψ = [\begin{matrix} ψ_{1} & ψ_{2} & \dots & ψ_{N} \end{matrix}]

(21)

$y (t)$ is linearly represented by these complex eigenvectors. Thus, $y_{c} (t)$ is given by

y_{c} (t) = Ψ z_{c} (t)

(22)

where

z_{c} (t) = {[\begin{matrix} z_{c, 1} (t) + i z_{c, 1}^{H} (t) & z_{c, 2} (t) + i z_{c, 2}^{H} (t) & \dots & z_{c, N} (t) + i z_{c, N}^{H} (t) \end{matrix}]}^{T}

Equation (22) is substituted into (20), which can be rewritten as

{\begin{cases} \frac{d^{2} z_{c, n} (t)}{d t^{2}} + k_{n} z_{c, n} (t) - c_{n} z_{c, n}^{H} (t) = 0 \\ \frac{d^{2} z_{c, n}^{H} (t)}{d t^{2}} + k_{n} z_{c, n}^{H} (t) + c_{n} z_{c, n} (t) = 0 \end{cases} (n = 1,2, \dots, N)

(23)

where

\frac{ψ_{n}^{T} (K + i K_{η}) ψ_{n}}{ψ_{n}^{T} M ψ_{n}} = k_{n} + i c_{n} (n = 1,2, \dots, N)

(24)

From equation (23), the results are obtained as

{\begin{cases} z_{c, n} (t) = e^{- λ_{n} t} (A_{n} \sin ϖ_{n} t + B_{n} \cos ϖ_{n} t) \\ z_{c, n}^{H} (t) = e^{- λ_{n} t} (- A_{n} \cos ϖ_{n} t + B_{n} \sin ϖ_{n} t) \end{cases}

(25)

where

A_{n}

and

B_{n}

are undetermined coefficients

{\begin{cases} λ_{n} = \frac{\sqrt{2} c_{n}}{2 \sqrt{k_{n} + \sqrt{k_{n}^{2} + c_{n}^{2}}}} \\ ϖ_{n} = \sqrt{\frac{1}{2} k_{n} + \frac{1}{2} \sqrt{k_{n}^{2} + c_{n}^{2}}} \end{cases}

(26)

z_{c, n} (t)

and

z_{c, n}^{H} (t)

are obtained from equation (25).

x_{c} (t)

is obtained by the complex mode superposition method.

The vibration frequency of $r (t)$ is zero. Thus, the results are obtained as

{\begin{cases} \frac{d^{2} x_{r} (t)}{d^{2} t} = 0 \\ x_{r}^{H} (t) = 0 \end{cases}

(27)

Then, equation (16) is rewritten as

K x_{r} (t) = - M G r (t)

(28)

$x_{r} (t)$ is obtained from equation (28) and is linearly expressed as

x_{r} (t) = \sum_{n = 1}^{N} ψ_{n} [z_{r, re, n} (t) + i z_{r, im, n} (t)]

(29)

Both sides of equation (29) are multiplied by $ψ_{n}^{T} M$ . The results are obtained as

ψ_{n}^{T} M x_{r} (t) = ψ_{n}^{T} M {\sum_{n = 1}^{N} ψ_{n} [z_{r, re, n} (t) + i z_{r, im, n} (t)]}

(30)

By the orthogonality of the complex eigenvectors, equation (30) is rewritten as

z_{r, re, n} (t) + i z_{r, im, n} (t) = \frac{ψ_{n}^{T} M x_{r} (t)}{ψ_{n}^{T} M ψ_{n}}

(31)

x_{r} (t)

is also obtained by the complex mode superposition method from equation (29).

By the Hilbert transform method, equation (17) can be rewritten as

{\begin{cases} M \frac{d^{2} x_{s} (t)}{d t^{2}} + K x_{s} (t) - K_{η} x_{s}^{H} (t) = - M G s (t) \\ M \frac{d^{2} x_{s}^{H} (t)}{d t^{2}} + K x_{s}^{H} (t) + K_{η} x_{s} (t) = - M G s^{H} (t) \end{cases}

(32)

It is assumed that

y_{s} (t) = x_{s} (t) + i x_{s}^{H} (t)

(33)

Equation (32) is equivalent to

M \frac{d^{2} y_{s} (t)}{d t^{2}} + K y_{s} (t) + i K_{η} y_{s} (t) = - M G [s (t) + i s^{H} (t)]

(34)

$y_{s} (t)$ is given by

y_{s} (t) = Ψ z_{s} (t)

(35)

where

z_{s} (t) = {[\begin{matrix} z_{s, 1} (t) + i z_{s, 1}^{H} (t) & z_{s, 2} (t) + i z_{s, 2}^{H} (t) & \dots & z_{s, N} (t) + i z_{s, N}^{H} (t) \end{matrix}]}^{T}

Equation (35) is substituted into equation (34), which can be rewritten as

{\begin{cases} \frac{d^{2} z_{s, n} (t)}{d t^{2}} + k_{n} z_{s, n} (t) - c_{n} z_{s, n}^{H} (t) = - γ_{re, n} s (t) + γ_{im, n} s^{H} (t) \\ \frac{d^{2} z_{s, n}^{H} (t)}{d t^{2}} + k_{n} z_{s, n}^{H} (t) + c_{n} z_{s, n} (t) = - γ_{re, n} s^{H} (t) - γ_{im, n} s (t) \end{cases} (n = 1,2, \dots, N)

(36)

where

\frac{ψ_{n}^{T} M G}{ψ_{n}^{T} M ψ_{n}} = γ_{re, n} + i γ_{im, n} (n = 1,2, \dots, N)

(37)

z_{s, n} (t)

and

z_{s, n}^{H} (t)

are obtained from equation (36).

x_{s} (t)

is also obtained by the complex mode superposition method.

It is assumed that

ψ_{n} = ψ_{re, n} + i ψ_{im, n} (n = 1,2, \dots, N)

(38)

The results are obtained as

x_{c} (t) = \sum_{n = 1}^{N} [ψ_{re, n} z_{c, n} (t) - ψ_{im, n} z_{c, n}^{H} (t)]

(39)

x_{r} (t) = \sum_{n = 1}^{N} [ψ_{re, n} z_{r, re, n} (t) - ψ_{im, n} z_{r, im, n} (t)]

(40)

x_{s} (t) = \sum_{n = 1}^{N} [ψ_{re, n} z_{s, n} (t) - ψ_{im, n} z_{s, n}^{H} (t)]

(41)

From equations (39)–(41), the solution of (10) is rewritten as

x (t) = \sum_{n = 1}^{N} [ψ_{re, n} z_{re, n} (t) - ψ_{im, n} z_{im, n} (t)]

(42)

{\begin{cases} z_{re, n} (t) = z_{c, n} (t) + z_{r, re, n} (t) + z_{s, n} (t) \\ z_{im, n} (t) = z_{c, n}^{H} (t) + z_{r, im, n} (t) + z_{s, n}^{H} (t) \end{cases}

(43)

Hence, a complex mode superposition method based on the hysteretic damping model (CH) is realized, which can be directly applied to a nonproportionally damped system. The procedure of this method is presented in Figure 1. The problem of determining the initial conditions of MDOF systems should be solved first.

From equation (43), the results are obtained as

{\begin{cases} z_{re, n} (t_{0}) = B_{n} + z_{r, re, n} (t_{0}) + z_{s, n} (t_{0}) \\ z_{im, n} (t_{0}) = - A_{n} + z_{r, im, n} (t_{0}) + z_{s, n}^{H} (t_{0}) \end{cases}

(44)

{\begin{cases} \frac{d z_{re, n} (t_{0})}{d t} = ϖ_{n} A_{n} - λ_{n} B_{n} + \frac{d z_{s, n} (t_{0})}{d t} \\ \frac{d z_{im, n} (t_{0})}{d t} = λ_{n} A_{n} + ϖ_{n} B_{n} + \frac{d z_{s, n}^{H} (t_{0})}{d t} \end{cases}

(45)

Equations (44) and (45) are substituted into (42). The results are obtained as

{\begin{cases} x (t_{0}) = \sum_{n = 1}^{N} {ψ_{re, n} [B_{n} + z_{r, re, n} (t_{0}) + z_{s, n} (t_{0})] - ψ_{im, n} [- A_{n} + z_{r, im, n} (t_{0}) + z_{s, n}^{H} (t_{0})]} \\ \frac{d x (t_{0})}{d t} = \sum_{n = 1}^{N} {ψ_{re, n} [ϖ_{n} A_{n} - λ_{n} B_{n} + \frac{d z_{s, n} (t_{0})}{d t}] - ψ_{im, n} [λ_{n} A_{n} + ϖ_{n} B_{n} + \frac{d z_{s, n}^{H} (t_{0})}{d t}]} \end{cases}

(46)

x (t_{0})

and

d x (t_{0}) / d t

are known.

A_{n}

and

B_{n}

can be obtained from equation (46). The initial conditions of MDOF systems can be accommodated.

Combined with finite element modeling (Naghinejad and Ovesy, 2019; Wang and Wang, 2016), the numerical simulation of CH can be realized. The user subroutines in ANSYS and MATLAB are developed to calculate vibration responses in time-domain dynamic analyses. The algorithm of the user subroutines in ANSYS and MATLAB is presented in Figure 2.

CH is a semianalytical method, which is unconditionally stable. Here, some common time-domain numerical calculation methods are compared, such as Newmark-β (Chopra, 2001), Wilson-θ (Keierleber and Rosson, 2005), finite differences (Civalek, 2006), discrete singular convolution (Civalek and Acar, 2007), and harmonic differential quadrature (Civalek, 2004). These methods are usually realized based on the viscous damping model, and the corresponding calculation is implemented in state space. However, CH is directly realized in the physical space, and its computational efficiency is higher. The comparisons are shown in Table 1.

Table 1.

Comparisons of different methods.

	CH	CR	Newmark-β (β = 0.25 and γ = 0.50)	Wilson-θ $(θ \geq 1.37)$	FD	DSC	HDQ
Unconditional stability	Yes	Yes	Yes	Yes	No	Yes	No
Computational efficiency	High	Low	Medium	Medium	Medium	Medium	Medium

Note: CH: complex mode superposition method based on the hysteretic damping model; CR: complex mode superposition method based on the Rayleigh damping model; FD: finite difference; DSC: discrete singular convolution; HDQ: harmonic differential quadrature.

3. Experiment examples

The nonproportionally damped model is used with a cantilever plate composed of host and damping layers. The host layer is made of steel plate, and the damping layer is made of high-molecular polymer. There is a thin layer of damping rubber between the host layer and damping layer, which can make the host and damping layers work together very effectively. Numerical simulation and experiment analysis are performed on the cantilever plate. The purpose behind these experiment example studies is to verify the accuracy of CH. Two identical specimens (specimen CP-1 and specimen CP-2) are used in the test. The structural dimension of the specimens is shown in Figure 3. The bottom of the specimen is fixed and subjected to axial (X-axis) dynamic excitation. The basic physical information is listed in Table 2.

Table 2.

Model properties of the cantilever plate.

Model information	Host layer material	Damping layer material
Young’s modulus (Pa)	2.1 × 10¹¹	4.0 × 10⁷
Mass density (kg m⁻³)	7500	1600
Poisson’s ratio	0.30	0.49
Loss factor	0.003	0.59

The energy losses per cycle are independent of structural frequencies as observed in material experiments within a wide frequency range (Bert, 1973; Crandall, 1970) so that the material loss factor is approximately constant. With the aid of material damping tests, from 5 Hz to 200 Hz, the loss factor of the host layer material is approximately 0.003, and the loss factor of the damping layer material is approximately 0.59.

Figure 4(a) shows that the vibration test system includes a signal collecting system, laser Doppler vibration measurement system, shake table, and control system. FNV-R4D-VD1 of the HoloBright company in Singapore is adopted in the laser Doppler vibration measurement system. The velocity time–history of four vibration points in the specimen can be obtained at the same time. The locations of the vibration points are shown in Figure 4(b). Because these points are symmetric, for convenience, only the vibration responses of points A and B are analyzed.

3.1. Frequency response curve

The structural steady-state vibration response is an important aspect of the structural vibration response. To verify the proposed method, the frequency response curves of different methods are analyzed. CR requires the choice of modes. Complex mode superposition method of Rayleigh damping model based on the first and the second modes (CRS) is adopted, and the method based on the first and the third modes (CRT) is also used for comparison. The frequency response curves of specimens CP-1 and CP-2 are obtained by the sine sweeping-frequency vibration method in the test. The dynamic excitation is the X-direction relative wave. The frequency of excitation vibration increases from 80 Hz to 120 Hz. The rate of frequency increase is 0.5 Hz/s. The peak acceleration of the excitation vibration is 9.8 m/s².

Following the methods mentioned above, the amplitude–frequency characteristic curves and phase–frequency characteristic curves for the third mode are obtained, as shown in Figures 5 and 6. Because of the discreteness of material properties and manufacturing errors of the test specimens, there are some differences between the velocity frequency response curves of specimens CP-1 and CP-2. The velocity amplitude–frequency response results of the different methods are shown in Tables 3 and 4.

Table 3.

Velocity amplitude–frequency response results of point A by different methods.

	CH	CRS	CRT	CP-1-test	CP-2-test
Frequency of the peak velocity amplitude (Hz)	101.0	99.5	98.5	98.5	95.5
Peak velocity amplitude (mm/s)	39.5106	24.7888	35.3396	40.7991	41.3098
$δ_{1}$ (%)	3.16	39.24	13.38	—	—
$δ_{2}$ (%)	4.36	40.00	14.45	—	—

Note: $δ_{1}$ or $δ_{2}$ is the relative error of the peak velocity amplitude between the simulation results and the test results of specimen CP-1 or specimen CP-2. CH: complex mode superposition method based on the hysteretic damping model; CRS: complex mode superposition method of Rayleigh damping model based on the first and the second modes; CRT: complex mode superposition method of Rayleigh damping model based on the first and the third modes.

Table 4.

Velocity amplitude–frequency response results of point B by different methods.

	CH	CRS	CRT	CP-1-test	CP-2-test
Frequency of the peak velocity amplitude (Hz)	94.0	96.5	93.5	92	89.5
Peak velocity amplitude (mm/s)	48.3011	23.7710	42.8966	52.3101	51.8101
$δ_{1}$ (%)	7.66	54.56	18.00	—	—
$δ_{2}$ (%)	6.77	54.12	17.20	—	—

Compared with the test results, the maximum relative errors of peak velocity amplitudes based on CRS and CRT are 54.56% and 18.00%, respectively. The phase–frequency characteristic curve of CRS is obviously different from the test results. For CR, the simulation results of CRT are more similar to the test results than those of CRS. The reason is that the frequency of the third mode is between 80 Hz and 120 Hz. The choice of the first mode and the third mode is reasonable. For CR, the damping ratios are underestimated or overestimated in the frequency range of interest (Ryan and Polanco, 2008), and the corresponding calculation results will be overestimated or underestimated. Regarding the shortcoming of the Rayleigh damping model, compared with the result of other mode, the modal damping ratio of the third mode based on CRS is larger, and the vibration responses are smaller. Therefore, the results of CR do not exhibit uniqueness, and the choice of reasonable modes based on the Rayleigh damping model is very important.

The simulation results of CH and the test results have great consistency. The maximum relative error of peak velocity amplitudes based on CH is only 7.66%. The error of the phase–frequency characteristic curve of CH is the smallest compared with the test results. For CH, the structural loss factor is among material loss factors, and the damping ratios are not underestimated or overestimated in the interested frequency range. The computational accuracy of CH is higher than that of CR. In addition, the calculation results of CH are unique, and the problem of the choice of reasonable modes is avoided.

3.2. Vibration responses under random excitation

In the sine sweeping-frequency vibration method, the structural vibration responses are mainly steady-state vibration responses, and the proportion of structural transient state responses is small. The frequency response curves cannot reflect the general situation of structural vibration. Therefore, the structural vibration responses under random excitation are analyzed. The vibration frequency of the random excitation is set from 40 Hz to 150 Hz. The random excitation of specimen CP-1 (random excitation (1)) and that of specimen CP-2 (random excitation (2)) can be obtained by a signal collection system. These random excitations and corresponding Fourier transform spectrum are shown in Figure 7. Then, the simulated velocity time-history responses of CH and CR under random excitation (1) or random excitation (2) can be calculated. The structural velocity time-history responses have been obtained by a laser Doppler vibration measurement system as described above.

The results shown in Figures 8 and 9 indicate that the characteristics of the structural vibration responses under random excitation are similar to those of structural steady-state vibration responses. The problem of the choice of reasonable modes is avoided based on CH. In Tables 5 and 6, compared with CR, the relative errors of peak velocity responses of CH are smaller. Figures 8 and 9 indicate that the simulation velocity time-history responses of CH are closer to the test results than those of CR. Therefore, the calculation accuracy of CH is higher. In addition, CH does not need the aid of state-space modeling method, and the computational efficiency of CH is relatively high.

Table 5.

Velocity responses of different methods under random excitation (1).

	CH	CRS	CRT	CP-1-test
Peak velocity time of point A (s)	37.5	39.9	37.5	37.5
Peak velocity of point A (mm/s)	58.2414	48.5796	55.1387	63.4593
Error (%)	8.22	23.45	13.11	—
Peak velocity time of point B (s)	37.5	37.5	37.5	37.9
Peak velocity of point B (mm/s)	68.1306	39.4871	60.7179	75.2177
Error (%)	9.42	47.50	19.28	—

Note: CH: complex mode superposition method based on the hysteretic damping model; CRS: complex mode superposition method of Rayleigh damping model based on the first and the second modes; CRT: complex mode superposition method of Rayleigh damping model based on the first and the third modes.

Table 6.

Velocity responses of different methods under random excitation (2).

	CH	CRS	CRT	CP-2-test
Peak velocity time of point A (s)	39.4	39.0	39.4	39.0
Peak velocity of point A (mm/s)	60.2475	49.4900	53.3541	62.8332
Error (%)	4.12	21.24	15.09	—
Peak velocity time of point B (s)	38.3	38.3	38.3	38.9
Peak velocity of point B (mm/s)	54.3841	35.4270	53.9069	59.9342
Error (%)	9.26	40.89	10.06	—

4. Conclusions

The objective of this study is to develop and calibrate a complex mode superposition method for nonproportionally damped linear systems, which is based on the hysteretic damping model. This method is the basis for the mode decomposition response spectrum method and can be used to calculate the seismic design forces of linear building structures with nonproportional damping.

The construction of a damping matrix based on the Rayleigh damping model does not exhibit uniqueness, which is significantly related to the choice of reasonable modes. Compared with the Rayleigh damping model, the damping matrix of hysteretic damping model is only related to the structural stiffness and the damping properties of materials and is irrelevant to the choice of modes.

The general solution of the corresponding homogeneous equation and special solution of the corresponding nonhomogeneous equation can be solved and superposed linearly. Then, a complex mode superposition method of the MDOF nonproportionally damped linear system based on a hysteretic damping model is realized. Compared with some common methods which are usually established based on the viscous damping model, the computational efficiency of the proposed method is higher. Besides, the proposed method is realized based on the assumption of relationship of external excitations, which is unconditionally stable.

Based on the proposed method, a user subroutine in ANSYS and MATLAB is developed to calculate vibration responses in time-domain dynamic analyses. This subroutine was verified by the comparisons of shaking table test results. The frequency response curves and vibration responses under random excitation show that the accuracy and efficiency of the proposed computational method are higher than those of the complex mode superposition method based on the Rayleigh damping model.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the National Natural Science Foundation of China (Grant No. 51878100) and partially supported by the Graduate Research and Innovation Foundation of Chongqing, China (Grant No. CYB18036).

ORCID iDs

Panxu Sun

Hong Yang

Yunjun Deng

References

Bert

(1973) Material damping: an introductory review of mathematic measures and experimental technique. Journal of Sound and Vibration 29(2): 129–153.

Bilbao

Avilés

Agirrebeitia

, et al. (2006) Proportional damping approximation for structures with added viscoelastic dampers. Finite Elements in Analysis and Design 42(6): 492–502.

Brigham

(1974) The Fast Fourier Transform. New Jersey: Prentice Hall.

Caughey

O’Kelly

MEJ

(1965) Classical normal modes in damped linear dynamic systems. Journal of Applied Mechanics 27(3): 269.

Chopra

(2001) Dynamics of Structures: Theory and Applications to Earthquake Engineering. New Jersey: Pearson Education, Inc.

Civalek

(2004) Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Engineering Structures 26(2): 171–186.

Civalek

(2006) Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation. Journal of Sound and Vibration 294(4–5): 966–980.

Civalek

Acar

(2007) Discrete singular convolution method for the analysis of Mindlin plates on elastic foundations. International Journal of Pressure Vessels and Piping 84(9): 527–535.

Clough

Penzien

(2003) Dynamics of Structures. 3rd edition. California: Computers and Structures Inc.

10.

Cohen

(1995) Time-Frequency Analysis. New Jersey: Prentice Hall.

11.

Crandall

(1970) The role of damping in vibration theory. Journal of Sound and Vibration 11(1): 3–IN1.

12.

Cruz

Miranda

(2017) Evaluation of the Rayleigh damping model for buildings. Engineering Structures 138: 324–336.

13.

De Domenico

Ricciardi

(2019) Dynamic response of non-classically damped structures via reduced-order complex modal analysis: two novel truncation measures. Journal of Sound and Vibration 452: 169–190.

14.

Feriani

Perotti

(1996) The formation of viscous damping matrices for the dynamic analysis of MDOF systems. Earthquake Engineering & Structural Dynamics 25(7): 689–709.

15.

Foss

(1958) Coordinates which uncouple the equations of motion of damped linear dynamic systems. Journal of Applied Mechanics 25: 361–364.

16.

Hahn

(1996) Hilbert Transforms in Signal Processing. Boston: Artech House.

17.

Huang

Shen

Long

, et al. (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454(1971): 903–995.

18.

Huang

Qian

Zhou

, et al. (2015) An approach to equivalent damping ratio of vertically mixed structures based on response error minimization. Soil Dynamics and Earthquake Engineering 72: 119–128.

19.

Igusa

Kiureghian

Sackman

(1984) Modal decomposition method for stationary response of non-classically damped systems. Earthquake Engineering & Structural Dynamics 12(1): 121–136.

20.

Inaudi

Kelly

(1995) Linear hysteretic damping and the Hilbert transform. Journal of Engineering Mechanics 121(5): 626–632.

21.

Jones

(2001) Handbook of Viscoelastic Vibration Damping. New York: Wiley.

22.

Keierleber

Rosson

(2005) Higher-order implicit dynamic time integration method. Journal of Structural Engineering 131(8): 1267–1276.

23.

Lee

D-G

Hong

Kim

(2002) Efficient siesmic analysis of building structures with added viscoelastic dampers. Engineering Structures 24(9): 1217–1227.

24.

Liang

Lee

(1991) Representation of damping matrix. Journal of Engineering Mechanics 117(5): 1005–1019.

25.

Lin

Zhu

(2009) On the relationship between viscous and hysteretic damping models and the importance of correct interpretation for system identification. Journal of Sound and Vibration 325(1–2): 14–33.

26.

Naghinejad

Ovesy

(2019) Viscoelastic free vibration behavior of nano-scaled beams via finite element nonlocal integral elasticity approach. Journal of Vibration and Control 25(2): 445–459.

27.

Nasif

Jones

Henderson

(1985) Vibration Damping. New York: Wiley.

28.

Neugebauer

Scheffler

Wabner

, et al. (2011) State space modeling of non-proportional passive damping in machine tools. The International Journal of Advanced Manufacturing Technology 53(9–12): 945–952.

29.

Pant

Wijeyewickrema

ElGawady

(2013) Appropriate viscous damping for nonlinear time-history analysis of base-isolated reinforced concrete buildings. Earthquake Engineering & Structural Dynamics 42(15): 2321–2339.

30.

Papageorgiou

Gantes

(2010) Equivalent modal damping ratios for concrete/steel mixed structures. Computers and Structures 88(19–20): 1124–1136.

31.

Papageorgiou

Gantes

(2011) Equivalent uniform damping ratios for linear irregularly damped concrete/steel mixed structures. Soil Dynamics and Earthquake Engineering 31(3): 418–430.

32.

Ryan

Polanco

(2008) Problems with Rayleigh damping in base-isolated buildings. Journal of Structural Engineering 134(11): 1780–1784.

33.

Shen

Soong

Chang

, et al. (1995) Seismic behaviour of reinforced concrete frame with added viscoelastic dampers. Engineering Structures 17(5): 372–380.

34.

Strutt

(1945) Theory of Sound. 2nd edition. Mineola: Dover Publications.

35.

Theodorsen

Garrick

(1940) Mechanism of Flutter: A Theoretical and Experimental Investigation of the Flutter Problem. Washington, DC: National Advisory Committee for Aeronautics.

36.

Villaverde

(1997) Seismic design of secondary structures: state of the art. Journal of Structural Engineering 123(8): 1011–1019.

37.

Villaverde

(1988) Rosenblueth’s modal combination rule for systems with non-classical damping. Earthquake Engineering & Structural Dynamics 16(3): 315–328.

38.

Wang

(2009) Intrinsic damping: modeling techniques for engineering systems. Journal of Structural Engineering 135(3): 282–291.

39.

Wang

(2016) Vibration modeling of carbon-nanotube-based biosensors incorporating thermal and nonlocal effects. Journal of Vibration and Control 22(5): 1405–1414.