Frequency/Laplace domain methods for computing transient responses of fractional oscillators

Abstract

Although computing the transient response of fractional oscillators, characterized by second-order differential equations with fractional derivatives for the damping term, to external loadings has been studied, most existing methodologies have dealt with cases with either restricted fractional orders or simple external loadings. In this paper, considering complicated irregular, but deterministic, loadings acting on oscillators with any fractional order between 0 and 1, efficient frequency/Laplace domain methods for getting transient responses are developed. The proposed methods are based on pole-residue operations. In the frequency domain approach, “artificial” poles located along the imaginary axis of complex plane are designated. In the Laplace domain approach, the “true” poles are extracted through two phases: (1) a discrete impulse response function (IRF) is produced by taking the inverse Fourier transform of the corresponding frequency response function (FRF) that is readily obtained from the exact transfer function (TF), and (2) a complex exponential signal decomposition method, i.e., the Prony-SS method, is invoked to extract the poles and residues. Once the poles/residues of the system are known, those of the response can be determined by simple pole-residue operations. Sequentially, the response time history is readily obtained. Two single-degree-of-freedom (SDOF) fractional oscillators with rational and irrational derivatives, respectively, subjected to sinusoidal and complicated earthquake loading are presented to illustrate the procedure and verify the correctness of the proposed method. The verification is conducted by comparing the results from both the Laplace and the frequency domain approaches with those from the numerical Duhamel integral method. Furthermore, the proposed method applied to a 3-DOF system with fractional damping is also demonstrated.

Appendix I: Summary of the Prony-SS method

Given the IRF signal h(t) that are sampled with interval $\Delta t$, $h(t_k)=h(k\Delta t)$, $k=0,1,\cdots ,N-1$, using the Prony-SS method is to approximate h(t) as $\sum _{n=1}^{N_n} \beta _n\, \exp (\mu _n t)$. Summarized here is the three sequential steps of the Prony-SS method to determine $N_n$, $\mu _n$ and $\beta _n$, respectively.

Step 1 for determining $N_n$: Construct the Hankel matrix ${\mathbf {H}}(0)$ $\in {\mathbb {R}}^{\xi \times \eta }$ from the sampled signal $h(t_k)$:

$$\begin{aligned} {\mathbf {H}}(0) = \left[ \begin{array}{cccc} h(t_0) &{} h(t_1) &{} \cdots &{} h(t_{\eta -1}) \\ h(t_1) &{} h(t_2) &{} \cdots &{} h(t_{\eta }) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ h(t_{\xi -1}) &{} h(t_{\xi }) &{} \cdots &{} h(t_{\xi +\eta -2}) \end{array}\right] \end{aligned}$$

(53)

where $\xi +\eta =N$, and a better choice is both $\xi$ and $\eta$ are close to N/2. Applying the singular value decomposition of ${\mathbf {H}}(0)$ gives

$$\begin{aligned} {{\mathbf {H}}}(0)=[\begin{array}{cc} {\mathbf {U}}_{1}&{\mathbf {U}}_{2} \end{array}] \left[ \begin{array}{cc} {\mathbf {S}}_{1} &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\mathbf {0}} \end{array}\right] \left[ \begin{array}{c} {\mathbf {V}}_{1}^{T} \\ {\mathbf {V}}_{2}^{T} \end{array} \right] ={\mathbf {U}}_{1} {\mathbf {S}}_{1} {\mathbf {V}}_{1}^{T} \end{aligned}$$

(54)

Where ${\mathbf {U}}_{1} \in {\mathbb {R}}^{\xi \times N_n}$, ${\mathbf {S}}_{1} \in {\mathbb {R}}^{N_n \times N_n}$ and ${\mathbf {V}}_{1} \in {\mathbb {R}}^{\eta \times N_n}$; theoretically, $N_n$ is the rank of ${\mathbf {H}}(0)$, i.e., the number of nonzero singular values in Eq. 54. Mathematically, the singular values go to zero when the rank of the matrix is exceeded; however, for data involving random errors or inconsistencies, some singular values will be very small, but not exactly zero. In this situation, $N_n$ could be determined based on the magnitudes of the singular values having been ordered sequentially from the largest to the smallest; a conventional way to determine $N_n$ is based on a significant drop of the normalized singular values.

Step 2 for determining $\mu _n$: Construct the Hankel matrix ${\mathbf {H}}(1)$ $\in {\mathbb {R}}^{\xi \times \eta }$ from the sampled signal $h(t_k)$:

$$\begin{aligned} {\mathbf {H}}(1) = \left[ \begin{array}{cccc} h(t_1) &{} h(t_2)&{} \cdots &{} h(t_{\eta }) \\ h(t_2) &{} h(t_3) &{} \cdots &{} h(t_{1+\eta }) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ h(t_{\xi }) &{} h(t_{1+\xi }) &{} \cdots &{} h(t_{\xi +\eta -1}) \end{array}\right] \end{aligned}$$

(55)

Using ${\mathbf {U}}_{1}$, ${\mathbf {S}}_{1}$, ${\mathbf {V}}_{1}$ and ${{\mathbf {H}}}(1)$, one computes

$$\begin{aligned} {\mathbf {A}}={\mathbf {S}}^{-1/2}_{1}{\mathbf {U}}^{T}_{1} {{\mathbf {H}}}(1) {\mathbf {V}}_{1}{\mathbf {S}}^{-1/2}_{1} \end{aligned}$$

(56)

After computing the eigenvalues $z_n$, $n=1,\cdots ,N_n$, of ${\mathbf {A}}$, one can get $\mu _n=\ln (z_n)/\Delta t$.

Step 3 for determining $\beta _n$: Compute the complex coefficients $\beta _n$ by solving a set of linear equations using a least-square procedure, based on the obtained $z_n$ and the sampled $h(t_k)$:

$$\begin{aligned} \left[ \begin{array}{cccc} z_{1}^{0} &{} z_{2}^{0} &{} \cdots &{} z_{N_n}^{0} \\ z_{1}^{1} &{} z_{2}^{1} &{} \cdots &{} z_{N_n}^{1} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ z_{1}^{N-1} &{} z_{2}^{N-1} &{} \cdots &{} z_{N_n}^{N-1}\\ \end{array} \right] \left\{ \begin{array}{c} \beta _{1} \\ \beta _{2} \\ \vdots \\ \beta _{N_n} \\ \end{array} \right\} =\left\{ \begin{array}{c} h(t_0) \\ h(t_1) \\ \vdots \\ h(t_{N-1}) \\ \end{array} \right\} \nonumber \\ \end{aligned}$$

(57)