Eigenvalue analysis and estimation on dynamic quality of mechanical assembly

2.1. Dynamic transfer-functions of mechanical assembly

Fig. 1FRFs on two-level of substructures before a) and after b) assembling

a) Component level

b) System level

Dynamic system of mechanical assembly can be modeled as a two-level of substructures as shown in Fig. 1. Where, two substructural components involved in assembling process are indicated by A and B, and dynamic transfer-function ($H$) from excitation-force ($F$) to displacement response ($X$) before assembling (in uncoupled state) and after assembling (in coupled state) are called “FRF at component level” (without subscript ‘$s$’) and “FRF at system level” (with subscript ‘$s$’), respectively. It represents dynamic transferring characteristics expressed specifically by spectral-based FRFs. The assembly-coupling dynamic stiffness is denoted by $K_s$ for assembly connection after assembling, which is different from the stiffness $K_{s0}$ for real assembly connector before assembling. Lowercase $a$, $b$ and $c$ denote substructures A, B and their coupling interface, and $i$ and $o$ denote input force and output displacement, respectively. For examples, $H_{oaca}$ and $H_{s,oaib}$ indicate FRF at component level from excitation on interface of side-A to output response also on A and FRF at system level from input excitation on side-B to output response on A.

Based on substructuring analysis, the relations between FRFs at component level ($H$) and FRFs at system level ($H_s$) of an assembling process can be expressed as follow [11]:

where:

The symbols $\alpha =$+1, −1 for $x=a$ and $b$, and $\beta =$+1, −1 for $x=b$ and $a$, respectively. Matrix $\left[C\right]$ is called “assembly-coupling matrix”, in which matrix $\left[K_s\right]$ is so-called assembly-coupling dynamic stiffness and every of its entry is generally simplified by:

The real part $k_s$ represents static stiffness that indicates capacity of dynamic energy conservation, and $c_s$ in imaginary part represents damping factor indicating capacity of dynamic energy dissipation of an assembly, respectively. Symbol $\omega =2\pi f$ and $f$ is analytical frequency in Hertz, and $j=\sqrt{-1}$.

Eqs. (1) and (2) show that, FRFs at system level (after assembling) can be explicitly expressed by that before assembling, which belongs to the ‘positive issue’ of substructuring analysis. Oppositely, FRFs at component level can also be inversely expressed by that after assembling which is so-called inverse substructuring analysis.

For convenience, the nine FRFs to be used in next discussions are simply denoted as:

In essence, FRFs $H_A$, $H_B$ and $H_s$ indicate the dynamic transferring characteristics of substructural components A and B before assembling, and the assembly system after assembling, respectively.

2.2. Changes of FRFs on substructural components due to assembling process

For the two-level substructural model of Fig. 1, the effects of assembling process on dynamic transferring characteristics of substructural components A and B involved in the process are discussed next.

The nine representative FRFs expressed in Eq. (4) are associated with a mechanical assembly, For the two substructural components, their FRFs at system level (after assembling) can be expressed individually by corresponding FRFs at component level (before assembling). From Eq. (1), the functional relations can be formulated as follow:

where, [$I$] is unit square matrix with the same size of matrix [$C$]. In order to measure the relative changes of dynamic transferring characteristics, ratios of the four representative FRF-matrices before assembling to those after assembly are formulated from Eqs. (5) and (6). A FRF-matrix-ratio compares the two matrices relative to each other. If the numbers of $oa$, $ca$, $cb$ and $ib$ are set to be equal, then the four FRF-matrix-ratios can be derived as:

Eqs. (7) and (8) demonstrate that, $R_A=R_{Ac}$ and $R_B=R_{Bc}$, that means an assembling process has identical effect on the same component. Therefore, $R_A$ and $R_B$ would be selected as two representative FRF-matrix-ratios for further study next. On the other hand, the effects are described specifically by mathematical product of $\left[C\right]$ and $\left[H_{Xc}\right]$$\text{(}X=A$ or $B\text{)}$. Whereas assembly-coupling dynamic stiffness [$K_s$] is the unique factor affected by assembling process, thus [$K_s$] (for assembly connection after assembling) $\ne$[$K_{s0}$] (for real assembly connector before assembling), see Fig. 1.

3.1. Lumped-parameter model

To investigate in detail the changes of FRFs on two substructural components before assembling due to assembling process, a lumped-parameter model is employed for theoretical analysis. It is shown in Fig. 2 with parameters listed in Table 1. For this model system, it is assumed that no real assembly connector before assembling the two substructural components, i.e., [$K_{s0}$]$=$0, and after assembling, the coupling dynamic stiffness of assembly connection turns out to be [$K_s$].

Fig. 2Lumped-parameter model of assembly system

a) Before assembling

b) After assembling

All associated FRFs in the model system can be calculated by following governing equation of either substructural components or their assembly system:

where, $p=$$A$, $B$ and $s$ corresponding to substructural components A and B, and assembly system, respectively. The analytical frequency $f$ is chosen to be 0-30 Hz for this model system. In addition, there are only two diagonal entries in assembly-coupling dynamic stiffness matrix, i.e., $\left[K_s\right]=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left[K_{s11},K_{s22}\right]$. They are given by Eq. (3) as follow:

Thus, a pair of dynamic force excitation input and displacement response output are set on both substructural components A and B (see Fig. 2), so that all FRF matrices from Eq. (7) are square with same size of ${\left[K_s\right]}_{2\times 2}$.

3.2. Effects on substructural components

Eigenvalues of FRF-matrix-ratios $R_A$ and $R_B$ characterize the principal vectors on which relative changes of dynamic transferring characteristics of substructural components A and B are presented, respectively. From Eqs. (7) and (8), the eigenvalue modules can be calculated by:

where $X=A$ or $B$ for $R_A$ and $R_B$, and ‘$\mathrm{e}\mathrm{i}\mathrm{g}$’ denotes ‘eigenvalue-finding’. If $M_{eg}>1\to \left|R_X\right|>1$ in principal vectors, then the FRF before assembling ($H_X$) would be greater than that after assembling ($H_{s,X}$).

Strengthening assembly connection of the model system results in five levels of [$K_s$], as listed inTable 2. Subsquently, the eigenvalue modules $M_{eg}\left(R_A\right)$ and $M_{eg}\left(R_B\right)$ are computed by Eq. (11) in full frequency range (0-30 Hz). The results are shown in Figs. 3 and 4.

It can be seen from Figs. 3 and 4 that, in full frequency range, (1) at most frequencies, $M_{eg}>1\to \left|\left[H_X\right]\right|>\left|\left[H_{s,X}\right]\right|$. It means that assembling process of the model system reduces the dynamic transfer-functions (FRFs). It can be explained that assembling process changes boundary conditions of the substructrual components, which constrain the dynamic responses; (2) at some frequencies, such as at about 6 Hz for 1st eigenvalue and 4 Hz and 13 Hz for 2nd eigenvalue, $M_{eg}<1\to \left|\left[H_X\right]\right|>\left|\left[H_{s,X}\right]\right|$. This is resulted from destructive modal couplings. In this case, assembling process increases the transfer-functions; (3) $M_{eg}$ increases with strengthening assembly connection that makes [$K_s$] increased, and thus $\left|\left[H_{s,X}\right]\right|$ tends to be much lower than $\left|\left[H_X\right]\right|$. Simultaneously, the peak frequencies move upto higher modes, at which the modal couplings get to be weak. The analytical results of (1) and (2) can also be seen clearly in Fig. 5, where the magnitudes of four entries of [$H_A$] and [$H_B$] are directly compared with those of [$H_{s,A}$] and [$H_{s,B}$].

Fig. 3Eigenvalue module MegRA

a) 1st eigenvalue

b) 2nd eigenvalue

Fig. 4Eigenvalue module MegRB at five [Ks]-levels

a) 1st eigenvalue

b) 2nd eigenvalue

3.3. Indication of the assembling effects

In order to synthetically estimate and thus simply indicate the assembling effects on substructural components of an assembly process, mean eigenvalue modules of FRF-matrix-ratios $R_A$ and $R_B$ are computed in full analytical frequency range for all eigenvalues as follow:

where $N_f$ and $N_{eg}$ are the numbers of discrete frequencies and eigenvalues respectively. The mean eigenvalue module computed by Eq. (12) can be used to indicate the overall effect of assembling process.

Fig. 5Comparisons of a) HA with Hs,A, and b) HB with Hs,B (on [Ks]-level 3)

a)

b)

For the lumped-parameter model system, $N_f=$100 within 0-30 Hz $\to$ analytical frequency interval $∆f=$0.3 Hz, and $N_{eg}=$2 due to all square FRF-matrices with size of 2 by 2. the mean-values of eigenvalue modules $M_{eg}\left(R_A\right)$ and $M_{eg}\left(R_B\right)$ are computed for the five [$K_s$]-levels. Table 3 lists the resultant data, which are also compared in Fig. 6. It can be seen that, (1) the overall effect of assembling process on dynamic transfer-functions are increased with strengthening the assembly connection; (2) the overall effect of an assembling process on substructural component A is very close to that on B; (3) both of the mean eigenvalue modules indicate that the assembling process of the model system reduce the FRFs [$H_A$] and [$H_B$] about 2-3.5 times with strengthening the assembly connection at the five [$K_s$]-levels.

Fig. 6Mean MegRA and MegRB at five [Ks]-levels

4.1. Formulating a quantitative measure

For a mechanical assembly system, dynamic quality inspection and estimation require a right quantitative measure and/or indicator in engineering application. According to the definition of “dynamic quality of mechanical assembly” explained in ‘Introduction’ of this paper, a relative change of systematic dynamic transferring characteristics due to assembling process is to be determined quantitatively to measure the dynamic quality of the whole assembly system.

Similarly to discussion in Section 2.2, relative change of systematic dynamic transferring characteristics from the FRFs $H_A$ and $H_B$ of both substructural components, together with the dynamic stiffness [$K_{s0}$] of real assembly connector, before assembling, to the systematic FRF $H_s$ of their assembly system can also be described by a FRF-matrix-ratio. On the other hand, Eqs. (1) and (2) show that assembly-coupling matrix [$C$] just combines the transfer-functions from the FRF [$H_{Bc}$] on side B of connecting-interface, through assembly connection [$K_s$], to the FRF [$H_{Ac}$] on another side A of the interface. Thereby it describes completely the overall dynamic transferring characteristics of an assembling process. Thus, matrix [$C$] does dominate the overall assembling effect on the systematic dynamic transferring characteristics of whole assembly system. Thereby, matrix [$C$] would be the right choice for finding out the FRF-matrix-ratio to estimate the relative change.

Two entries (1, 3) and (3, 1) of FRF-matrix on both sides of Eq. (1) provide such equalities that relate the systematic transfer-functions after assembling to that before assembling. Here choosing entry (1, 3) leads to:

Before assembling, the two substructural components are disconnected, Eq. (13) would become:

Specially, if there is no any real connector or joint for an assembling process, i.e., $\left[K_{s0}\right]=$0, as it is assumed in the lumped-parameter model, then Eq. (14) turns out to be:

If set same numbers of $oa$, $ca\left(cb\right)$ and $ib$, and combine Eqs. (14) and (15) for [$H_0$], then the FRF-matrix-ratio to be found can be formulated by $\left[H_0\right]{\left[H_s\right]}^{-1}$, denoted specially by $Q_C$, as follow:

In consideration of the matrix-ratio is adopted essentially for comparing [$H_0$] with [$H_s$] and thus generating the relative change $Q_C$, Eq. (16) can be equivalently rewritten in normalized form as:

where:

The FRF-matrix-ratio $Q_C$ expressed in Eqs. (16) and/or (17) describes definitely the overall change of systematic transfer-function $\left[H_s\right]$ after assembling, relative to the ‘systematic transfer-function’ [$H_0$] before assembling (as reference base), of whole assembly system due to assembling process. Therefore, it is supposed to be applied as the right quantitative measure on overall effect of the process of a mechanical assembly. Consequently, it is renamed as “dynamic quality matrix of mechanical assembly” here by this study.

4.2. Eigenvalue analysis on dynamic quality matrix

Again, the lumped-parameter model of Fig. 2 is applied for eigenvalue analysis on the dynamic quality matrix $Q_C$. In the model system, as it is supposed $\left[K_{s0}\right]=$0, thus Eq. (17b) is adopted to do the eigenvalue analysis on matrix $Q_C$ as done in Section 3.2 and 3.3.

Fig. 7 illustrates the eigenvalue module $M_{eg}\left(Q_C\right)$ computed by Eq. (17b) at the five $\left[K_s\right]$-levels. It shows that, (1) at most frequencies, $M_{eg}>1\to \left|\left[H_s\right]\right|<\left|\left[H_0\right]\right|$ for both eigenvalues of $Q_C$, even up to 20-30 Hz near peak frequencies. Additionally, modal couplings still exist so that $M_{eg}<1\to \left|\left[H_s\right]\right|>\left|\left[H_0\right]\right|$ at some frequencies, such as 4 Hz and 14 Hz; (2) strengthening the assembly connection ([$K_s$] goes up to higher levels) lead $M_{eg}\left(Q_C\right)$ getting down, that is, $\left|\left[H_s\right]\right|$ goes higher relative to $\left|\left[H_0\right]\right|$. This can be considered as that the dynamic quality of whole assembly system of the model turns out to be poor.

Fig. 7Eigenvalue module MegQC at five [Ks]-levels

a) 1st eigenvalue

b) 2nd eigenvalue

Fig. 8 compares directly $\left|\left[H_s\right]\right|$ to $\left|\left[H_0\right]\right|$. Again, it shows that the ‘overall systematic dynamic transfer-function’ [$H_0$] of whole model system before assembling (in free or uncoupling state) is generally greater than that after assembling (in coupled or constrained state). This proves the analytical results above from Fig. 7 and results in $M_{eg}\left(Q_C\right)>$1 generally. On the whole, dynamic quality matrix [$Q_C$] is proved to be the expected measure for estimating the dynamic quality of a mechanical assembly system.

To indicate the dynamic quality quantitatively, the mean eigenvalue modules of $Q_C$ over full frequency range is computed by Eq. (12) as well here at the five [$K_s$]-levels. The computed data are listed in Table 4 and compared in Fig. 9. It shows that strengthening assembly connection makes mean-value of $M_{eg}\left(Q_C\right)$ decreased, and thus the systematicdynamic transfer-function $\left|\left[H_s\right]\right|$ increased relative to $\left|\left[H_0\right]\right|$. This indicates the dynamicquality of the assembly system getting poor. Summarily, dynamic quality matrix $Q_C$ and its mean eigenvalue module can be considered as the right quantitative measure and indicator for dynamic quality inspection and estimation of mechanical assembly.

Fig. 8Comparing Hs with H0

Fig. 9Mean MegQc at five [Ks]-levels