The structural dynamics can be described by the continuous impulse response function (IRF,

$h\left(t\right)\phantom{\rule{0.166667em}{0ex}}(\mathrm{m}/(\mathrm{N}\phantom{\rule{0.166667em}{0ex}}\mathrm{s})$), derived from experimental FRFs (

$H\left(\omega \right)\phantom{\rule{0.166667em}{0ex}}(\mathrm{m}/\mathrm{N})$) by means of the inverse Fourier transform as

Accordingly, the integral form of the dynamical system can be represented as

where

$G(\theta ,\vartheta )=h(\theta +\vartheta )$ is the Green function defined over the interval

$\theta \in [0,T]$. The finite length

T of the measured IRF is directly related to the frequency resolution

$\delta f$ of the FRF, since

$T=1/\delta f$. The first term of (

8) actually corresponds to the homogeneous dynamics as a result of initial forcing via the Green function

G. In this manner, as an alternative to the usually used initial conditions (IC’s), the system reaches its initial state at time

t by the (initial) past force pattern described by

${F}_{x}^{-}$. This hypothetical force evolution pushes the system to the opening state at

t, afterwards it gradually behaves as a forced (excited) dynamic system described by the second term of (

8). The present (excitation) force

${F}_{x}^{+}$ operates via the IRF, which results in the time periodic stationary forced vibration combined with the transient response of the system to zero IC, without considering the state dependency, e.g., in (

3). In milling, (

8) has a periodic (stationary) solution,

$\overline{x}\left(t\right)=\overline{x}(t+{T}_{Z})$, due to (

4) that can be determined, e.g., by harmonic balance or considering the long term behavior of the second term of (

8). It can be defined for a time-period

${T}_{Z}$ after the vanishing transient response at

${T}_{\mathrm{t}}$ (

${T}_{\mathrm{t}}:=M\phantom{\rule{0.166667em}{0ex}}{T}_{Z}$, where

$M\in {\mathbb{N}}^{+}$ to keep the phase) as

satisfying the causality condition

$h(\theta \le 0)=0$. Around the stationary solution

$\overline{x}\left(t\right)$, a perturbation

$u\left(t\right)$ is considered in the form

$x:=\overline{x}+u$. Consequently, the perturbed motion is described by the following so-called variational system that is given as

The measured FRF is given with a certain frequency resolution

$\delta f$ due to the digital sampling, which determines the length of the IRF as

$T=1/\delta f$. Therefore, the solution is defined over the time mesh

${t}_{j}=j\delta t\phantom{\rule{1.em}{0ex}}(\delta t=\delta \theta =1/{f}_{\mathrm{max}}$,

$j=0,1,\dots ,N-1$,

${f}_{\mathrm{max}}=(N/2-1)\delta f$) as

where

$\Delta {\mathbf{F}}_{-T}\left(t\right)={\mathrm{col}}_{k}\Delta {F}_{x}^{-}(t-(k-1)\delta \theta )$,

$\mathbf{C}\left(t\right)={\mathrm{diag}}_{k}C(t+(k-1)\delta \theta )$, while

${\mathbf{u}}_{T}\left(t\right)={\mathrm{col}}_{k}u(t+(k-1)\delta \theta )$. Similarly, the IRF is also considered in its sampled form as

${h}_{k}=h\left((k-1)\delta \theta \right)$. The Toeplitz and Hankel representations of the forced IRF and the Green function are given as

respectively. Note that

$\mathbf{G}$ is theoretically a full matrix, but

${h}_{k}=0$ if

$k>N-1$ due to practical reasons.

#### 2.2.1. Impulse Dynamic Subspace (IDS) Transformation

Instead of the usual modal parameter-based description of the system, the homogeneous dynamics is constructed over its singular impulse response characteristics [

24], defined by SVD on the Hankel representation of the Green function (

12):

It is known that the SVD gives the dominant principle directions with respect to their “stored energy” or, based on receptance IRF, with respect to their compliance. The singular values are given in

$\Sigma ={\mathrm{diag}}_{k}{\sigma}_{k}$, such that

${\sigma}_{1}>{\sigma}_{2}>\cdots >{\sigma}_{k}>\cdots >{\sigma}_{N}$. Many of the

${\sigma}_{k}$ singular values almost vanish, which means that these vanishing directions have little influence in (

13), thus truncation can be performed. Using

$n<N$ dimensional truncation, the new reduced set of singular IRFs can be defined as

${\mathbf{V}}_{n}=[{\mathbf{v}}_{1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{v}}_{2}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{v}}_{3}\phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}}{\mathbf{v}}_{n}]$. Knowing that

${\mathbf{V}}_{n}$ is an orthonormal set due to its very definition in (

13), a new abstract state

$\mathbf{p}\left(t\right)={\mathrm{col}}_{k}{\mathbf{p}}_{k}\left(t\right)$ can be defined as

By using the continuous description, like in (

8), the solution and its derivative can be deduced similarly to (

14):

where

$\dot{a}=\mathrm{d}a/\mathrm{d}t$ and

${a}^{\prime}=\mathrm{d}a/\mathrm{d}\theta $. Thus, the system matrix can be derived using the product of the singular IRFs with its derivatives as

The derivative of the singular IRFs,

${\mathbf{V}}_{n}^{\prime}$, can be determined using the sampled representation

$\mathbf{G}\prime $ of

${G}^{\prime}(\theta ,\vartheta )$, and consequently

Therefore, the solution for the initial forcing (IF) is

based on (

16) from a given initial time

${t}_{l}$.

The methodology how the system matrix is introduced in (

16) can be a base of an alternative modal analysis technique. The eigenvalues of

${\mathbf{V}}_{n}^{\mathsf{H}}{\mathbf{V}}_{n}^{\prime}$ are actually the poles that directly define the natural frequency and damping factor shown in [

24].

#### 2.2.2. Stability Prediction by Semidiscretization Method

The delayed nature of the regenerative milling process [

29,

30] requires discretization on the past motion of the tool that directly leads to a possible semidiscretization description, where the stationary forcing plays an important role.

For this purpose, another new time sampling

$\Delta t=D\delta t$ (

$D\in {\mathbb{N}}^{+}$) is needed to define a mesh

${t}_{l}={t}_{j}+(l-1)\Delta t$, as the time step

$\delta t$ is likely too small to construct an efficient numerical method (SDM). Afterwards,

$r=\lceil \frac{\tau}{\Delta \theta}-\frac{1}{2}\rceil $ intervals (

$\Delta t=\Delta \theta $) are defined for simplicity with

$\tau :={T}_{Z}$ (regular cutter). The second term of (

11) describes the forced solution

${\mathbf{p}}_{\mathrm{F}0}\left(t\right)$ with its transient for zero ICs considering constant (zero-order hold) actual and delayed states measured to

${t}_{l}$. Taking into account the transformation in (

14), the forcing solution combined with the zero transient solution (F0) is given in the time interval

$t\in [{t}_{l},{t}_{l+1}]$ as

where

${\mathbf{p}}_{l}=\mathbf{p}\left({t}_{l}\right)$ and

${\mathbf{p}}_{l,\tau}\approx {\mathbf{p}}_{l-r}$. The solution in

$t\in [{t}_{l},{t}_{l+1}]$ in the form

${\mathbf{p}}_{l}\left(t\right)={\mathbf{p}}_{\mathrm{IF},l}\left(t\right)+{\mathbf{p}}_{\mathrm{F}0,l}\left(t\right)$ can be expressed as

Considering that only the initial forcing (IF) term adopts to nonzero ICs,

${\mathbf{p}}_{\mathrm{h},l}\left({t}_{l}\right)={\mathbf{p}}_{l}$, the next time step has the following form,

where

${\mathbf{H}}_{D}$ and

${\mathbf{V}}_{n,D}$ are truncated matrices related to the integer number ratio

D between

$\Delta t$ and

$\delta t$.

The exponential expression of the initial forcing can be obtained by shifting the IDS by

$\Delta t$ as shown in [

24]. This can be defined by a shift matrix

$\mathbf{S}$, resulting in

A discretized state,

${\mathbf{z}}_{l}={\mathrm{col}}_{k}({\mathbf{p}}_{l},{\mathbf{p}}_{l-1},{\mathbf{p}}_{l-2},\phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}},{\mathbf{p}}_{l-r})$, can be introduced by which both (

21) and (

22) can be transformed to a map as

The map in (

23) is reapplied successively over the time period

${T}_{Z}$ in order to have the so-called transition matrix

$\Phi $
where

${N}_{\mathrm{SDM}}:=r$ is the applied SDM scheme resolution.

The characteristic multipliers (eigenvalues) of the transition matrix directly represent the asymptotic behavior of the stationary solution $\overline{x}$. The less than one magnitudes for all characteristic multipliers corresponds to stable stationary solution. Otherwise, $\overline{x}$ is unstable or critically stable. Thus, SLDs can be constructed based on the magnitudes of the characteristic multipliers.

In summary, the IDS description can provide a methodology, in which the step of the parameter extraction can be avoided (see flowcharts in

Figure 2). This is an advantage since parameter extraction is a time consuming action, which typically needs expert interaction. The effectivity of parameter-based pure numerical algorithms, like SDM, are usually measured without considering the modal parameter extraction step, which can carry significant errors, too, depending on the complexity of the initial FRF. The proposed methodology is initiated from a corresponding IRF, which is usually originated from measured FRFs (

Figure 2b). By performing an SVD on the Green function, the entire dynamical system is described using a selected set of singular IRFs

${\mathbf{V}}_{n}$, by determining the system matrix

$\mathbf{A}={\mathbf{V}}_{n}^{\mathsf{H}}{\mathbf{V}}_{n}^{\prime}$ (

16) or the solution operator

${\mathbf{V}}_{\mathbf{n}}^{\mathsf{H}}\mathbf{S}\phantom{\rule{0.166667em}{0ex}}{\mathbf{V}}_{\mathbf{n}}$ (

22) for the core homogeneous (ordinary) dynamics. By choosing an appropriate time step, based on the parameters taken from the stability chart (

$n\left(\mathrm{rpm}\right),a\left(\mathrm{mm}\right)$), an interpolation is performed on the singular-IRF set and the corresponding forcing term can be added (

16). This map actually describes the step matrix (approximated solution operator)

${\mathbf{B}}_{j}$ (

$j=0,1,\cdots ,r-1$) of the original regenerative (delayed) system, which leads to the transition matrix

$\Phi $ in an iterative manner.