The adequate modelling of damping is a major issue in vibrational analysis. Only some works deal with the investigation of damping in the context of
Mikota’s vibration chain. Based on [
12], the influence of a single absolute damper and a single relative damper on the vibration behaviour of
Mikota’s vibration chain was investigated in [
17] for some
n DOF. In [
20],
Mikota’s vibration chain with additional dampers was investigated concerning some issues of model order reduction. However, a systematic investigation is still missing, yet. Thus, some general properties shall be discussed in what follows.
The eigenvalue problem of Eq. (
2.5) reads
$$\begin{aligned} \left( \mathbf{M} \lambda _l + \mathbf{K} \right) \mathbf{u} _l = \mathbf{0} \end{aligned}$$
(4.1)
with
\(\mathbf{M} \),
\(\mathbf{K} \) according to Eqs. (
2.2)–(
2.5) and
$$\begin{aligned} \lambda _l = -\varOmega _l^2 = - (l)^2 . \end{aligned}$$
(4.2)
In the present case, for the spectral matrix
$$\begin{aligned} {\varvec{\varOmega }}^2 = \text {diag} \left( \varOmega _l^2 \right) = \text {diag} \left( 1, 4, \ldots , n^2 \right) \end{aligned}$$
(4.3)
holds. Using the modal matrix as introduced with Eq. (
3.7), the following relations hold
$$\begin{aligned} \mathbf{U }^\mathrm{T} \mathbf{M }\mathbf{U }&= \text {diag} \left( m_1, m_2, \ldots , m_n \right) \end{aligned}$$
(4.4)
$$\begin{aligned} \mathbf{U }^\mathrm{T} \mathbf{K }\mathbf{U }&= \text {diag} \left( \kappa _1, \kappa _2 , \ldots , \kappa _n \right) , \quad \text {where} \quad \frac{\kappa _i}{m_i} = \varOmega _i^2 = i^2 \end{aligned}$$
(4.5)
$$\begin{aligned} {\varvec{\varOmega }}^2&= \left( \mathbf{U }^\mathrm{T} \mathbf{M }\mathbf{U }\right) ^{-1} \mathbf{U }^\mathrm{T} \mathbf{K }\mathbf{U }= \mathbf{U }^{-1} \mathbf{M }^{-1} \mathbf{K }\mathbf{U }. \end{aligned}$$
(4.6)
In the preceding sections,
\(\mathbf{M }\) and
\(\mathbf{K }\) denote number matrices. Thus, for describing modal damping, a number matrix
\(\mathbf{D }\) is introduced, too:
$$\begin{aligned} d \mathbf{D }\quad \text {with} \quad \mathbf{D }\mathbf{M }^{-1} \mathbf{K }= \mathbf{K }\mathbf{M }^{-1} \mathbf{D }. \end{aligned}$$
(4.7)
where condition (
4.7) has been given by [
1].
This number matrix
\(\mathbf{D }\) shall be determined in what follows. Starting point is the equation of motion describing free oscillations
$$\begin{aligned} \left( \mathbf{M }\mu _l^2 + \mathbf{D }\mu _l + \mathbf{K }\right) \mathbf{u }_l = \mathbf{0 }. \end{aligned}$$
(4.8)
Right-multiplication of
\(\mathbf{u }_l\) with Eq. (
4.7) yields
$$\begin{aligned} \mathbf{D }\underset{\varOmega _l^2 \mathbf{u }_l }{\underbrace{\mathbf{M }^{-1} \mathbf{K }\mathbf{u }_l}} = \mathbf{K }\mathbf{M }^{-1} \mathbf{D }\mathbf{u }_l \end{aligned}$$
(4.9)
and consequently
$$\begin{aligned} \left( \varOmega _l^2 \mathbf{I }- \mathbf{K }\mathbf{M }^{-1} \right) \mathbf{D }\mathbf{u }_l = \mathbf{0 }, \end{aligned}$$
(4.10)
where
\(\mathbf{I }\) denotes the unit matrix. This can be rewritten
$$\begin{aligned} \left( \varOmega _l^2 \mathbf{M }- \mathbf{K }\right) \mathbf{M }^{-1} \mathbf{D }\mathbf{u }_l = \mathbf{0 }, \end{aligned}$$
(4.11)
showing the eigenvector
$$\begin{aligned} \mathbf{M }^{-1} \mathbf{D }\mathbf{u }_l = \alpha _l \mathbf{u }_l , \quad \alpha _l \ne 0 \end{aligned}$$
(4.12)
with an arbitrary scaling factor
\(\alpha _l\). Then,
$$\begin{aligned} \mathbf{D }\mathbf{u }_l&= \alpha _l \mathbf{M }\mathbf{u }_l \end{aligned}$$
(4.13)
$$\begin{aligned} \mathbf{D }\mathbf{U }&= \mathbf{M }\mathbf{U }\varvec{\alpha } , \quad \varvec{\alpha } = \mathrm {diag} \left( \alpha _l \right) \end{aligned}$$
(4.14)
holds. The eigenvalue problem for the damped
Mikota’s vibration chain can now be formulated as follows:
$$\begin{aligned} \left( \mu _l^2 + \alpha _l \mu _l + \varOmega _l^2 \right) \mathbf{M }\mathbf{u }_l = \mathbf{0 }. \end{aligned}$$
(4.15)
As
\(\mathbf{M }\mathbf{u }_l \ne \mathbf{0 }\),
$$\begin{aligned} \mu _l = -\frac{\alpha _l}{2} \pm \sqrt{ \frac{\alpha _l^2}{4} - \varOmega _l^2 } . \end{aligned}$$
(4.16)
The
\(\alpha _l\) can be determined e.g. by setting the
Lehr’s damping measure
\(D_l\) to
$$\begin{aligned} D_l := 1 = \frac{\alpha _l}{2 \varOmega _l} , \end{aligned}$$
(4.17)
cf. amongst others [
8]. Then,
$$\begin{aligned} \alpha _l&= 2l \quad \text {and with Eq.~(4.16)} \quad \mu _l = -l , \end{aligned}$$
(4.18)
which is a twofold eigenvalue. Equations (
4.13), (
4.14) then read
$$\begin{aligned} \mathbf{D }\mathbf{u }_l = 2l \mathbf{M }\mathbf{u }_l \quad \Leftrightarrow \quad \mathbf{D }\mathbf{U }= 2 \mathbf{M }\mathbf{U }{\varvec{\varOmega }}\end{aligned}$$
(4.19)
and thus the sought damping matrix is
$$\begin{aligned} \mathbf{D }= 2 \mathbf{M }\mathbf{U }{\varvec{\varOmega }}\mathbf{U }^{-1} . \end{aligned}$$
(4.20)
Considering
\(\mathbf{U }^\mathrm{T} \mathbf{D }\mathbf{U }\) which is a diagonal matrix it is obvious that
\(\mathbf{D }\) is a symmetric positive definite matrix. As can be directly seen, the modal matrix
\(\mathbf{U }\) and thus all mode shapes
\(\mathbf{u }_l\) are needed to calculate the damping matrix
\(\mathbf{D }\). The question arises of how the damping matrix can be determined without knowing the mode shapes in advance. Using the relation
$$\begin{aligned} \left( \mathbf{U }{\varvec{\varOmega }}\mathbf{U }^{-1} \right) ^2&= \mathbf{U }{\varvec{\varOmega }}^2 \mathbf{U }^{-1} = \mathbf{U }\mathbf{U }^{-1} \mathbf{M }^{-1} \mathbf{K }\mathbf{U }\mathbf{U }^{-1} \end{aligned}$$
(4.21)
$$\begin{aligned}&= \mathbf{M }^{-1} \mathbf{K }, \end{aligned}$$
(4.22)
where Eq. (
4.6) was applied, gives
$$\begin{aligned} \mathbf{U }{\varvec{\varOmega }}\mathbf{U }^{-1}&= \sqrt{ \mathbf{M }^{-1} \mathbf{K }} . \end{aligned}$$
(4.23)
Thus, the damping matrix can be determined without knowing the mode shapes by means of the following equation
$$\begin{aligned} \mathbf{D }= 2 \mathbf{M }\sqrt{ \mathbf{M }^{-1} \mathbf{K }} . \end{aligned}$$
(4.24)
However, finding the roots necessitates the eigenvalues of the matrix root to be positive. This preliminary is always fulfilled for the vibration chain dealt with here. But still the roots of matrices in general are ambiguous.
Setting
$$\begin{aligned} \alpha _l = \varOmega _l \end{aligned}$$
(4.25)
yields
$$\begin{aligned} \mu _l = - \frac{\varOmega _l}{2} \pm \iota \varOmega _l \frac{\sqrt{3}}{2} = l \left( -\frac{1}{2} \pm \iota \frac{\sqrt{3}}{2} \right) \end{aligned}$$
(4.26)
and consequently
$$\begin{aligned} \mathbf{D }\mathbf{u }_l = \varOmega _l \mathbf{M }\mathbf{u }_l \end{aligned}$$
(4.27)
according to Eq. (
4.13). For this case, the damping matrix reads as
$$\begin{aligned} \mathbf{D }= \mathbf{M }\mathbf{U }{\varvec{\varOmega }}\mathbf{U }^{-1} = \mathbf{M }\sqrt{\mathbf{M }^{-1} \mathbf{K }} . \end{aligned}$$
(4.28)
As can readily be seen, the damping matrix in the present case is half of the damping matrix as given with Eq. (
4.24). The eigenvalues of the present case are
$$\begin{aligned} \mathfrak {R}(\mu _l) = -\frac{\varOmega _l}{2} = -\frac{l}{2} , \quad \mathfrak {I}(\mu _l) = \pm \sqrt{\frac{3}{4}} l \end{aligned}$$
(4.29)
and
Lehr’s damping measure follows to
$$\begin{aligned} D_l = \frac{\varOmega _l}{2 \varOmega _l} = \frac{1}{2} . \end{aligned}$$
(4.30)
It should be noted that finding the matrix root
\(\sqrt{\mathbf{M }^{-1} \mathbf{K }}\) again is the major issue for determining the damping matrix
\(\mathbf{D }\) without using the mode shapes.