New approach based on operational strain modal analysis to identify dynamical properties of the high-speed reciprocating operation mechanism

Abstract

A high-speed reciprocating operating mechanism with fast running and frequent stopping produces a certain vibration due to inertia. The inherent vibration of this mechanism becomes a key factor affecting positioning accuracy and time. However, the existing vibration identification method is suitable only for static or low-speed structure operations. Therefore, a method of identification for operational modal parameters based on strain response is proposed. This method combines the strain mode with the operational modal analysis method, both of which are verified under the static and operational states, respectively. First, the theoretical background of the identification method is derived. Next, the proposed approach is used to identify the displacement mode shapes and the strain mode shapes of the beam structure with modal parameters that are obtained by experimental modal analysis. These results show that the beam structure presents different displacement mode shapes and strain mode shapes under different constraints. Finally, the displacement mode shapes of the sorting arm are obtained by the method under operation. The recognition results are compared with the computational modal analysis, and the two mode shapes are found to be highly consistent, which verifies that the method is equally reliable during operation. This method expands the application scope of the classic experimental modal analysis and reduces the error because of the additional mass of the actual sensor.

Introduction

A high-speed reciprocating operating mechanism with characteristics of fast running, frequent stopping and short moving distances presents a certain vibration due to inertia. The working vibration is a major factor hindering production efficiency. At present, it is difficult to evaluate the dynamics and identify the modal parameters of these mechanisms. Barre et al. ¹ pointed out that the inertial vibration caused by the repeated stop positioning of high-speed motion has a direct influence on the positioning accuracy of the end of the mechanism. The influence is negligible at low speeds. However, the influence cannot be ignored at higher speeds, which is a key factor of the positioning accuracy.

Certain pieces of equipment with pickup and placement functions present a time-varying boundary condition because the relative motion between the flexible components leads to a change in the boundary conditions of the joint. This property makes the natural frequency and mode shapes of the mechanical structure closely related to the working state. This nonlinear phenomenon inevitably affects the stability and performance of the mechanical system.^2–5 In the gain control of a variable structure and high-speed machine tool, Symens et al. ⁶ pointed out that the high acceleration will stimulate the new frequencies of the structure. The natural frequencies of these structures are not invariable but depend on the working position. Next, he proposed a gain scheduling control method related to a position for the mechanical and electrical system. Pritschow ⁷ pointed out that the speed gain coefficient of the position controller determines the dynamic accuracy of the machine tool. However, these methods cannot be applied to the high-speed movement because the speeds are tens of times or even hundreds of times higher than the speed of the conventional machinery.

Many studies simply consider the structural dynamics to be consistent under operation and static state such as for machine tools, bridges and buildings. However, the high-frequency reciprocating mechanism is composed of multiple components (movable components and immovable components) that have different stiffness, and its dynamics change with the movement of the movable components.^8–10 The boundary condition between the movable and immovable components during the operation will change with the working state, and the positioning accuracy will be affected by the inertial impact and structural self-vibration. Therefore, the structural dynamics under operation are different from the structural dynamics under the static state. A method must be found to evaluate modal parameters of such mechanisms at high speed. The evaluation methods for the dynamics of mechanical structures can be divided into two categories: the finite element method (FEM) and the experimental modal analysis (EMA) method.

The FEM has been widely applied in industrial applications and research. ¹¹ Many studies have used FEM to evaluate the dynamics of mechanical structures. Altintas et al., ¹² who used the finite element model of the virtual machine tool to analyse the structure of the machine tool and simulated the cutting process, explain the application of FEM in structural dynamics in detail. Law et al.^13,14 developed a dynamic finite element model for obtaining the dynamics of a virtual machine tool. Zaeh and Siedl ¹⁵ simulated the working process of a gear shaper to accurately predict the processing results, combining FEM with a multi-body dynamics method. Kolar et al. ¹⁶ developed a coupled finite element model for machine tools, spindle and frame, and the dynamics of the machine tool were obtained through the model. The FEM is used in structural dynamics for structural parameters and is used in the structural design stage for structural optimization. The FEM can achieve satisfactory results for simple structures but is not accurate for complex structures. Because the parameters (such as stiffness and damping) of the component joints of complex structures are still not fully identified, obtaining structural dynamics under operation using FEM is still inaccurate.

Compared to FEM, EMA is an effective instrument for describing, understanding and modelling the dynamic behaviour of a structure. EMA can be carried out both to determine the natural frequencies and mode shapes of a structure as well as to verify the accuracy and calibrate a finite element model. Additionally, the EMA can be used to troubleshoot vibration problems and calculating the frequency response function (FRF) from measurements of both input excitations and corresponding responses at the measured point when the machine is resting. ¹⁷ EMA can significantly reduce the iterative process and computation time, and the results can represent boundary conditions. Kono et al. ¹⁸ used EMA to obtain the natural frequency and mode shapes of the machine tool and found that the main deformation part of the machine tool changes under different modes. Kushnir ¹⁹ obtained the modes of the machine tool and the FRF of each component and found that the mode shapes are determined by certain specific components. These results depend on the comparison of the mode shape and FRF of each component. However, the measured FRF of the component is under the influence of other components; therefore, the comparison of the mode shapes can lead to errors in the results that are not accurate. ²⁰ Zaghbani and Songmene ²¹ reported that the modal parameters of the machine tool are very different under the static state and the machining state. The results show that the difference in the natural frequency is 2–8%, and the difference of the damping ratio is 2–10 times. For the high-frequency reciprocating mechanism, the traditional EMA cannot be applied. Therefore, for the high-frequency reciprocating mechanism, multiple components interact and cooperate with each other, and their flexible features are obvious. The multidimensional impact movement leads to greater complexity, which makes the dynamic model difficult to establish. Therefore, the dynamics of the high-frequency reciprocating mechanism under the operating state and the static state are different.

There are also limitations in the sensors for obtaining the vibration signal. For example, the accelerometers commonly used in the EMA test cannot be used due to their own mass and the light weight of the measured object. Sensors cannot adhere to the measured object and cannot obtain the experimental data accurately because of the strong centrifugal force caused by the high speed. The non-contact sensors can only measure the displacement of some fixed points and cannot get the real-time experimental results under operation. The use of strain gauges can avoid the above problems. Kranjc et al. ²² proposed a method for acquisition strain mode shapes (SMSs) by symmetrically arranging the accelerometers and strain gauges. However, this method can only obtain the structural mode shapes at the static state, and this method cannot eliminate the mass of the sensors on the experimental results.

Considering the drawbacks of the two methods mentioned above, an identification method for obtaining operational modal parameters based on strain response is proposed in this paper, expanding the application scope of the classic EMA and reducing errors because of the additional mass of the sensors. The main goals of this research are (1) to study the corresponding relationship between the strain and displacement mode shapes (DMSs) of the beam structure, (2) to develop an approach to identify the modal parameters of the sorting arm based on the strain response and (3) to obtain the dynamics of the sorting arm under operation.

The paper is organized as follows. The next section presents the theoretical background of the strain mode. In the case of a known excitation, the strain transducer is used to identify the displacement modal parameters. In the case of an unknown excitation, the identification method for the displacement modal parameters based on the strain response is deduced and explained in detail. ‘EMA experimental verification based on strain response’ section presents the experimental verification of the EMA method. We describe the method for identifying the displacement modal parameters of the beam based on the strain response in the case of known excitation. The strain test is carried out under three different states (cantilever, free–free and simply supported states) of the beam, and the displacement modal parameters are identified, after which the results are compared with the traditional modal test. Furthermore, the relationship between the SMSs and DMSs of the beam is determined. ‘OMA experimental verification based on strain response’ section presents the experimental verification of the operational modal analysis (OMA) method. We describe the method for identifying operational modal parameters of the structure (using the example of the rotating arm of the high-speed chip-sorting machine) based on the strain response in the case of an unknown excitation. The results are compared with the calculated modal analysis (FEA) results. In the ‘Conclusion’ section, the study is summarized, and future work is presented.

Theoretical background

Strain modal analysis theory

According to an analysis of the displacement modal theory, the mode shapes are the inherent equilibrium state of the deformation energy of the structure. Therefore, the displacement modes are independent of one another, and the structural displacement response can be expressed by the sum of the structural mode contributions $$\left\{\mathit{x}\right\}\text{=}{\displaystyle \sum _{\text{r=}1}^{\text{m}}{\mathit{q}}_r}\left\{{\Phi }_r\right\}$$ (1)

Similarly, the structural strain and displacement have a corresponding relationship. Therefore, the strain response can also be expressed as $$\left\{\epsilon \right\}={\displaystyle \sum _{\text{r=}1}^{\text{m}}{{\mathit{q}}^{\prime }}_r}\left\{{\Psi }_{\text{r}}^{\epsilon }\right\}$$ (2)

In the above expressions, $\left\{\mathit{x}\right\}$ is the structural displacement response, $\left\{{\Phi }_r\right\}$ is the structural displacement mode, $q_r$ is the displacement modal coordinate (the proportion of the displacement mode in the displacement response); the corresponding $\left\{\epsilon \right\}$ is the structural strain response, $\left\{{\Psi }_{\text{r}}^{\epsilon }\right\}$ represents the structural strain mode and ${{\mathit{q}}^{\prime }}_r$ is the strain modal coordinate (the proportion of the strain pattern in the strain response).

We can assume that there is a corresponding strain pattern for each order displacement mode. These modes represent two different implementations of the structure at the same time-energy balance. Therefore, the proportion of $\left\{{\mathit{\varphi }}_r\right\}$ in $\left\{\mathit{x}\right\}$ and $\left\{{\mathit{\varphi }}_r{\hspace{0.17em}}^{\epsilon }\right\}$ in the proportion of $\left\{\epsilon \right\}$ should be the same, that is $${{\mathit{q}}^{\prime }}_r={\mathit{q}}_r=\frac{{\left\{{\mathit{\varphi }}_r\right\}}^T\left\{\mathit{F}\right\}{e}^{j\omega t}}{{\mathit{k}}_r-{\omega }^2{\mathit{m}}_r+j\omega {\mathit{c}}_r}$$ (3)where ${\mathit{m}}_r$ , ${\mathit{k}}_r$ and ${\mathit{c}}_r$ are the modal mass, modal stiffness and modal damping matrix, respectively, and are diagonal matrices. The corresponding structural strain response can be expressed as follows $$\left\{\epsilon \right\}={\displaystyle \sum _{\text{r=}1}^{\text{m}}{{\mathit{q}}^{\prime }}_r}\left\{{\psi }_{\text{r}}^{\epsilon }\right\}={\displaystyle \sum _{\text{r=}1}^{\text{m}}\frac{\left\{{\psi }_{\text{r}}^{\epsilon }\right\}{\left\{{\mathit{\varphi }}_r\right\}}^T}{{\mathit{k}}_r-{\omega }^2{\mathit{m}}_r+j\omega {\mathit{c}}_r}}\left\{\mathit{F}\right\}{e}^{j\omega t}$$ (4)

Therefore, the strain pattern has the following derivation process. In the displacement of the structure represented by $\mu ,\nu ,\omega$ , the displacement response is the superposition of the modes by their contribution $$\mu ={\displaystyle \sum _{\text{r=}1}^{\text{m}}{q}_r}{\Phi }_r\left(x\right)$$ (5) $${\epsilon }_x=\frac{\partial \mu }{\partial x}=\frac{\partial }{\partial x}{\displaystyle \sum _{\text{r=}1}^{\text{m}}{\mathit{q}}_r}{\Phi }_r\left(x\right)={\displaystyle \sum _{\text{r=}1}^{\text{m}}{\mathit{q}}_r}\frac{\partial {\Phi }_r\left(x\right)}{\partial x}={\displaystyle \sum _{\text{r=}1}^{\text{m}}{\mathit{q}}_r{\psi }_{\text{r}}^{\epsilon }\left(x\right)}$$ (6)

Let ${\psi }_{\text{r}}^{\epsilon }\left(x\right)=\frac{\partial }{\partial x}{\mathbf{\Phi }}_r\left(x\right)$ , ${\psi }_{\text{r}}^{\epsilon }$ be the strain modality, and discretizing the above expression, we obtain the following $$\left\{{\epsilon }_x\right\}={\displaystyle \sum _{\text{r=}1}^{\text{m}}{\mathit{q}}_r}\left\{{\psi }_{\text{r}}^{\epsilon }\right\}={\displaystyle \sum _{\text{r=}1}^{\text{m}}\frac{\left\{{\psi }_{\text{r}}^{\epsilon }\right\}{\left\{{\mathit{\varphi }}_r\right\}}^T}{-{\omega }^2{\mathit{m}}_r+j\omega {\mathit{c}}_r+{\mathit{k}}_r}}\left\{\mathit{F}\right\}{e}^{j\omega t}$$ (7)

The same method can be applied for equations $\left\{{\epsilon }_y\right\}$ and $\left\{{\epsilon }_z\right\}$ . In the 3D geometric space, the displacement response can be expressed as $\left\{x\right\}={\left[\begin{array}{ccc}\mu & \upsilon & \eta \end{array}\right]}^T$ , where the same displacement mode is $$\left\{{\mathit{\varphi }}_r\right\}={\left[\begin{array}{ccc}\left\{{\mathit{\varphi }}_r^{\mu }\right\}& \left\{{\mathit{\varphi }}_r^{\upsilon }\right\}& \left\{{\mathit{\varphi }}_r^{\eta }\right\}\end{array}\right]}^T$$ (8)

Therefore, the displacement response in the 3D space can be expressed as follows $$\left\{\begin{array}{c}\left\{\mathit{\mu }\right\}\\ \left\{\mathit{\upsilon }\right\}\\ \left\{\eta \right\}\end{array}\right\}={\displaystyle \sum _{\text{r=}1}^{\text{m}}{\mathit{q}}_r}\left\{\begin{array}{c}\left\{{\mathit{\varphi }}_r^{\mu }\right\}\\ \left\{{\mathit{\varphi }}_r^{\upsilon }\right\}\\ \left\{{\mathit{\varphi }}_r^{\eta }\right\}\end{array}\right\}$$ (9)

The relationship between the strain obtained from the elasticity knowledge and response displacement is $$\left\{\begin{array}{ccc}{\epsilon }_x& {\alpha }_{yx}& {\alpha }_{zx}\\ {\alpha }_{xy}& {\epsilon }_y& {\alpha }_{zy}\\ {\alpha }_{xz}& {\alpha }_{yz}& {\epsilon }_z\end{array}\right\}={\displaystyle \sum _{\text{r=}1}^{\text{m}}{\mathit{q}}_r}\left\{\begin{array}{ccc}\frac{\partial \mu }{\partial x}& \frac{\partial \upsilon }{\partial x}& \frac{\partial \eta }{\partial x}\\ \frac{\partial \mu }{\partial y}& \frac{\partial \upsilon }{\partial y}& \frac{\partial \eta }{\partial y}\\ \frac{\partial \mu }{\partial z}& \frac{\partial \upsilon }{\partial z}& \frac{\partial \eta }{\partial z}\end{array}\right\}$$ (10)

In the above expression, ${\epsilon }_x,{\epsilon }_y,{\epsilon }_z$ is the response to the normal strain along axis, while the shear strain is $${\gamma }_{xy}={\alpha }_{yx}+{\alpha }_{xy},{\gamma }_{yz}={\alpha }_{zy}+{\alpha }_{yz},{\gamma }_{zx}={\alpha }_{zx}+{\alpha }_{xz}$$ (11)

Using the modal superposition principle, the strain response expression is $$\left\{\begin{array}{ccc}{\epsilon }_x& {\alpha }_{yx}& {\alpha }_{zx}\\ {\alpha }_{xy}& {\epsilon }_y& {\alpha }_{zy}\\ {\alpha }_{xz}& {\alpha }_{yz}& {\epsilon }_z\end{array}\right\}={\displaystyle \sum _{\text{r=}1}^{\text{m}}{\mathit{q}}_r}\left\{\begin{array}{ccc}\frac{\partial }{\partial x}\left\{{\mathit{\varphi }}_r^{\mu }\right\}& \frac{\partial }{\partial x}\left\{{\mathit{\varphi }}_r^{\upsilon }\right\}& \frac{\partial }{\partial x}\left\{{\mathit{\varphi }}_r^{\eta }\right\}\\ \frac{\partial }{\partial y}\left\{{\mathit{\varphi }}_r^{\mu }\right\}& \frac{\partial }{\partial y}\left\{{\mathit{\varphi }}_r^{\upsilon }\right\}& \frac{\partial }{\partial y}\left\{{\mathit{\varphi }}_r^{\eta }\right\}\\ \frac{\partial }{\partial z}\left\{{\mathit{\varphi }}_r^{\mu }\right\}& \frac{\partial }{\partial z}\left\{{\mathit{\varphi }}_r^{\upsilon }\right\}& \frac{\partial }{\partial z}\left\{{\mathit{\varphi }}_r^{\eta }\right\}\end{array}\right\}$$ (12)

The strain gauge can measure only the normal strain, without measuring the shear strain $$\left\{\begin{array}{c}\left\{{\epsilon }_x\right\}\\ \left\{{\epsilon }_y\right\}\\ \left\{{\epsilon }_z\right\}\end{array}\right\}={\displaystyle \sum _{\text{r=}1}^{\text{m}}{\mathit{q}}_r}\left\{\begin{array}{c}\frac{\partial }{\partial x}\left\{{\mathit{\varphi }}_r^{\mu }\right\}\\ \frac{\partial }{\partial y}\left\{{\mathit{\varphi }}_r^{\upsilon }\right\}\\ \frac{\partial }{\partial z}\left\{{\mathit{\varphi }}_r^{\eta }\right\}\end{array}\right\}={\displaystyle \sum _{r=1}^m{\mathit{q}}_r\left[\begin{array}{l}\left\{{\psi }_r^{{\epsilon }_x}\right\}\\ \left\{{\psi }_r^{{\epsilon }_y}\right\}\\ \left\{{\psi }_r^{{\epsilon }_z}\right\}\end{array}\right]}$$ (13)

The above is based on the displacement modal theory for proving the response modal. The orthogonality among strain modes can also be demonstrated.

EMA based on strain response

From equation (7), the strain frequency response function (SFRF) is given by $${\mathit{H}}_{ij}^{\epsilon }\left(\omega \right)=\frac{{\epsilon }_i\left(\omega \right)}{f_i\left(\omega \right)}={\displaystyle \sum _{\text{r=}1}^{\text{n}}\frac{\left\{{\psi }_{ir}^{\epsilon }\right\}{\left\{{\mathit{\varphi }}_{jr}\right\}}^T}{-{\omega }^2{\mathit{m}}_r+j\omega {\mathit{c}}_r+{\mathit{k}}_r}}={\displaystyle \sum _{\text{r=}1}^n\frac{{\psi }_{ir}^{\epsilon }}{{\mathit{m}}_r\left[\left({\omega }_r^2-{\omega }^2\right)+2j{\mathit{\xi }}_r{\omega }_r\omega \right]}{\mathit{\varphi }}_{jr}^T}$$ (14)where ${\mathit{k}}_r={\mathit{m}}_r{\omega }_r^2$ , ${\mathit{\xi }}_r={\frac{{\mathit{c}}_r}{2{\mathit{m}}_r\omega }}_r=\frac{{\mathit{c}}_r}{2\sqrt{{\mathit{m}}_r{\mathit{k}}_r}}$ .

We apply the excitation force at point j and obtain the response at point i and the SFRF matrix $$\left[{\mathit{H}}_{ij}^{\epsilon }\right]={\displaystyle \sum _{\text{r=}1}^{\text{n}}\frac{\left\{{\psi }_{ir}^{\epsilon }\right\}{\left\{{\mathit{\varphi }}_{jr}\right\}}^T}{-{\omega }^2{\mathit{m}}_r+j\omega {\mathit{c}}_r+{\mathit{k}}_r}}={\displaystyle \sum _{\text{r=}1}^n\frac{\left[\begin{array}{cccc}\left\{{\psi }_{\text{r}}^{\epsilon }\right\}{\mathit{\varphi }}_{1r}& \left\{{\psi }_{\text{r}}^{\epsilon }\right\}{\mathit{\varphi }}_{2r}& \cdot \cdot \cdot & \left\{{\psi }_{\text{r}}^{\epsilon }\right\}{\mathit{\varphi }}_{nr}\end{array}\right]}{{\mathit{m}}_r\left[\left({\omega }_r^2-{\omega }^2\right)+2j{\mathit{\xi }}_r{\omega }_r\omega \right]}}$$ (15)

In the strain experiment, the N measuring points are arranged on the structure, and the strain value of a fixed-point P is tested. The following formula is obtained $$\left\{\begin{array}{cccc}{\mathit{H}}_{i1}^{\epsilon }& {\mathit{H}}_{i2}^{\epsilon }& \cdot \cdot \cdot & {\mathit{H}}_{in}^{\epsilon }\end{array}\right\}={\displaystyle \sum _{\text{r=}1}^n\frac{{\psi }_{ir}^{\epsilon }}{{\mathit{m}}_r\left[\left({\omega }_r^2-{\omega }^2\right)+2j{\xi }_r{\omega }_r\omega \right]}}\left\{\begin{array}{cccc}{\mathit{\varphi }}_{1r}& {\mathit{\varphi }}_{2r}& \cdot \cdot \cdot & {\mathit{\varphi }}_{nr}\end{array}\right\}$$ (16)

For the same order strain mode, the modal mass ${\mathit{m}}_r$ , modal stiffness ${\mathit{k}}_r$ , modal damping ${\mathit{c}}_r$ and SMSs $\left\{{\psi }_{\text{r}}^{\epsilon }\right\}$ are constant. Therefore, the definition of the strain-DMS constant coefficient C₁ is as follows $$C_1=\frac{{\psi }_{ir}^{\epsilon }}{{\mathit{m}}_r\left[\left({\omega }_r^2-{\omega }^2\right)+2j{\xi }_r{\omega }_r\omega \right]}$$ (17)

Then, equation (18) can be simplified to $$\left\{\begin{array}{cccc}{\mathit{H}}_{i1}^{\epsilon }& {\mathit{H}}_{i2}^{\epsilon }& \cdot \cdot \cdot & {\mathit{H}}_{in}^{\epsilon }\end{array}\right\}=C_1\left\{\begin{array}{cccc}{\mathit{\varphi }}_{1r}& {\mathit{\varphi }}_{2r}& \cdot \cdot \cdot & {\mathit{\varphi }}_{nr}\end{array}\right\}$$ (18)

The modal shape is represented by the ratio of the two amplitudes between the measuring points and selected reference points on the measured structure, which is independent of the vibration of each measuring point. Equation (18) shows that the DMS is related only to the strain transfer function amplitude of each measuring point but not to the mode coefficient C. In the strain mode experiment, only the multi-point excitation, with a single point for obtaining the strain response, can obtain the strain response function amplitude, to determine the DMS.

The maximum value of the strain amplitude of the corresponding measuring point is used as the normalization factor, while the positive and negative modal mode values are determined by the phase angle. Because the phase angle determines only the positive and negative values of the mode shapes, the value does not have a substantial effect. Therefore, we need not be concerned about the error in the phase measurement values during the measurement process. According to equation (18), the value of each measuring point is compared to the normalized factor value, and we can obtain the DMSs.

OMA based on strain response

In response to the fact that the structure can be measured, while the motion excitation (input) cannot be measured, we can assume that the structure of a response reference point is a motion excitation. The response of the other measuring points exhibits a certain linear correlation with the reference point response. It is possible to establish a transfer function between the response and reference points for system identification. In the structure, we take a fixed reference point g. The transmissibility function is $$T_i\left(\omega \right)=\frac{{\epsilon }_i\left(\omega \right)}{{\epsilon }_g\left(\omega \right)}$$ (19)

The strain response at point b can be expressed by the excitation force of point j and the $H_{ij}^{\epsilon }\left(\omega \right)$ $${\epsilon }_i\left(\omega \right)={\mathit{H}}_{ij}^{\epsilon }\left(\omega \right)\cdot f_j\left(\omega \right)$$ (20)

Assuming the excitation force signal applied to the structure is a flat spectrum signal, its power spectral density (PSD) function is approximately uniformly distributed within the frequency range of the covering structure. Therefore, the excitation of the various points of the structure to meet is expressed as follows $$f_j\left(\omega \right)=C_2$$ (21)where C₂ is a constant. Equation (20) can be simplified as $${\epsilon }_i\left(\omega \right)=C_2\cdot {\mathit{H}}_{ij}^{\epsilon }\left(\omega \right)$$ (22)

From equation (22), the response spectrum of the structure is equivalent to the FRF (real mode) of the structural system. Therefore, the natural frequency of the structure can be obtained directly from the response spectrum. Substituting equation (22) into equation (19), we can formulate equation (23), following the rearrangement of equation (22) $$T_i\left(\omega \right)=\frac{{\epsilon }_i\left(\omega \right)}{{\epsilon }_g\left(\omega \right)}=\frac{f_j\left(\omega \right)\cdot {\mathit{H}}_{{\hspace{0.17em}}_{ij}}^{\epsilon }\left(\omega \right)}{f_j\left(\omega \right)\cdot {\mathit{H}}_{{\hspace{0.17em}}_{\text{g}j}}^{\epsilon }\left(\omega \right)}=\frac{{\mathit{H}}_{{\hspace{0.17em}}_{ij}}^{\epsilon }\left(\omega \right)}{{\mathit{H}}_{{\hspace{0.17em}}_{\text{g}j}}^{\epsilon }\left(\omega \right)}$$ (23)

From equation (23), $T_i\left(\omega \right)$ is directly related to the modal parameters of the structure.

We assume that the solid modes of the structure can be separated from one another effectively, and they are neither coupled nor small relative to one another. Therefore, the system response at the natural frequency is dominated by the r-order mode, and the contribution of the other order modes is negligible at that frequency. As explained in ‘EMA based on strain response’ section, the N measuring points are the excitation points, and the strain value of a fixed-point P is tested. If i and g are the excitation points and j is the measuring point, the transmissibility function is as follows $$T_i\left(\omega \right)=\frac{{\epsilon }_{ji}\left(\omega \right)}{{\epsilon }_{jg}\left(\omega \right)}=\frac{f_i\left(\omega \right)\cdot {\mathit{H}}_{ji}\left(\omega \right)}{f_g\left(\omega \right)\cdot {\mathit{H}}_{jg}\left(\omega \right)}=C_3\frac{{\mathit{H}}_{ji}\left(\omega \right)}{{\mathit{H}}_{jg}\left(\omega \right)}$$ (24)where C₃ is a constant, $\frac{f_i\left(\omega \right)}{f_g\left(\omega \right)}=C_3$ .

Substituting equation (14) into equation (24), we obtain the corresponding r-order transmissibility function $$T_i\left(\omega \right)=C_3\frac{{\mathit{H}}_{ji}\left(\omega \right)}{{\mathit{H}}_{jg}\left(\omega \right)}=C_3\frac{\frac{\left\{{\psi }_{jr}^{\epsilon }\right\}}{{\mathit{m}}_r\left[\left({\omega }_r^2-{\omega }^2\right)+2j{\xi }_r{\omega }_r\omega \right]}{\left\{{\mathit{\varphi }}_{ir}\right\}}^T}{\frac{\left\{{\psi }_{jr}^{\epsilon }\right\}}{{\mathit{m}}_r\left[\left({\omega }_r^2-{\omega }^2\right)+2j{\xi }_r{\omega }_r\omega \right]}{\left\{{\mathit{\varphi }}_{gr}\right\}}^T}=C_3\frac{{\left\{{\mathit{\varphi }}_{ir}\right\}}^T}{{\left\{{\mathit{\varphi }}_{gr}\right\}}^T}$$ (25)where point g as a reference point is fixed, so it corresponds to a certain natural frequency, and ${\left\{{\phi }_{gr}\right\}}^T$ is the definite value corresponding to a certain natural frequency. Equation (24) can be rewritten as follows $$T_i\left(\omega \right)=\frac{{\left\{{\mathit{\varphi }}_{ir}\right\}}^T}{{\left\{{\mathit{\varphi }}_{gr}\right\}}^T}=C_3{\left\{{\mathit{\varphi }}_{ir}\right\}}^T$$ (26)where C₃ is a constant.

From equation (25), the amplitude and phase of $T_i\left(\omega \right)$ at ${\omega }_r$ can be obtained directly. The maximum strain amplitude value is the normalization factor. The positive and negative of the mode shapes are determined by the phase angle. Next, the operational deflection shape of the structure with a frequency of ${\omega }_r$ can be obtained, which is like the r-order mode structure.

From the theoretical knowledge, the PSD estimation method is as follows $$P_{ii}\left(\omega \right)\propto {\left|{\epsilon }_i\left(\omega \right)\right|}^2\propto {\left|{\mathit{H}}_i(\omega )\right|}^2$$ (27) $$P_{ij}\left(\omega \right)\propto {\epsilon }_i\left(\omega \right)\cdot conj\left({\epsilon }_j\left(\omega \right)\right)$$ (28)where $P_{ii}\left(\omega \right)$ is the auto-power spectrum of the i response, and $P_{ij}\left(\omega \right)$ is the cross-power spectrum of the i and j responses.

From equation (27), $P_{ii}\left(\omega \right)$ is proportional to ${\left|{\mathit{H}}_i(\omega )\right|}^2$ , and they have the same pole. Therefore, the amplitude of ${\mathit{H}}_i(\omega )$ can be replaced by $P_{ii}\left(\omega \right)$ for the response point. Then, the natural frequency and damping ratio can be obtained by identifying the mode stabilization diagram of the auto-power spectrum.

From equations (15) and (22) $${\epsilon }_i\left(\omega \right)=C_2\cdot {\mathit{H}}_{ij}^{\epsilon }\left(\omega \right)=C_2\frac{\left\{{\psi }_{ir}^{\epsilon }\right\}}{-{\omega }^2{\mathit{m}}_r+j\omega {\mathit{c}}_r+{\mathit{k}}_r}{\left\{{\mathit{\varphi }}_{jr}\right\}}^T$$ (29)

From equation (28), the pole value of ${\displaystyle {\sum }_{k=1}^n{\mathit{H}}_{ik}\left(\omega \right)}$ is independent of the response point position. We find that equations (27) and (28) are similar, so $P_{ii}\left(\omega \right)$ and $P_{ij}\left(\omega \right)$ have the same poles. Therefore, the amplitude of ${\mathit{H}}_i(\omega )$ can be replaced by $P_{ij}\left(\omega \right)$ between the response and reference points. Then, the natural frequency and damping ratio can be obtained by identifying the mode stabilization diagram of the cross-power spectrum.

EMA experimental verification based on strain response

Experimental setup and measurements

The size of the #45 carbon steel beam is 1500 mm × 30 mm × 8 mm, the elastic modulus is 210 GPa and the density is 7890 kg/m³. The beam is divided into 14 equal parts, arranged with 13 points, and each point number is shown in Figure 1. For each measuring point, two mutually perpendicular strain gauges are arranged in the x (length) and y (width) directions, where the x-direction is the measuring sheet and the y-direction is the compensation sheet. Furthermore, the accelerometer (PCB 356A15) is used at the 13 measuring points to carry out the experiment through comparison. The exciter produces a white noise signal. The strain data are collected by a dynamic strain signal acquisition and analysis instrument (DH 5929), while the acceleration response data are collected by an LMS signal acquisition and data analysis system (LMS SCADAS). The sampling frequency is set to 5000 Hz. Four groups of experiments were conducted, and Figure 2 illustrates the experimental setup.

Figure 1.

Tested beam and measuring point.

Figure 2.

(a) Experimental setup, (b) strain gauge placement and (c) accelerometer placement.Note: LMS is a software owned by Siemens PLM.

Three confinement state types (cantilever, free–free and simply supported states) were performed for the steel beams with three experiments conducted for each state. The recognition method proposed in this paper is aimed at the strain response in steel beams with different confinement conditions. The displacement mode obtained by the transformation is compared with the traditional modal test to verify the reliability of the method. Furthermore, the difference between the displacement and strain modes is investigated.

Test #1 Multi-point input and single-point output – strain: The hammer (PCB-086C04) is used to excite the structure at all points (along the y-direction), and the strain response measurement is taken at point 5 using the strain gauge (BF350). The strain response is measured by the DH5929 dynamic strain signal acquisition and analysis system. A row of the strain response function matrix is obtained from test #1. The strain–displacement mode shapes (S-DMSs) are obtained using the method described in ‘EMA based on strain response’ section.

Test #2 Single-point input and multi-point output (SIMO) – strain: The hammer is used to excite the structure at point 5 (along the y-direction), and strain response measurements are taken at all points using strain gauges. The strain response is measured by the DH5929 dynamic strain signal acquisition and analysis system. A column of the strain response function matrix is from test #2. Next, the SMSs are obtained using the method described in ‘EMA based on strain response’ section.

Test #3 SIMO – acceleration: The acceleration response at all points, as shown in Figure 2(c), is measured using accelerometers, and excitations are created by using a hammer at point 5 along the y-direction. The acceleration response of the measuring point is obtained by the LMS signal acquisition and analysis system. A column of the displacement FRF is obtained from test #3. Next, the DMSs can be obtained by the traditional modal test method.

Test #4 Single-point input and single-point output – acceleration: The contrast experiment was conducted with the aim of excluding the effects of accelerometers and the cables. The acceleration response at point 5, as shown in Figure 2(c), is measured using an accelerometer, and excitations are created by using a hammer at point 1 along the y-direction. The acceleration response of the measuring point is obtained by the LMS signal acquisition and analysis system. Next, the natural frequency and damping ratio can be obtained.

DMSs of the cantilever beam based on strain response

The modal parameters of the cantilever beam were estimated from the FRFs of all measurement points using the PolyMAX method. The bandwidth of modal analysis is 0–1000 Hz. Figure 3 presents the stability diagram of the cantilever beam, where the green solid curve represents the summation of the FRFs of all measurement points. The horizontal axis is a frequency and the vertical axis represents the amplitude. The letters ‘s’, ‘v’ and ‘o’ represent the relationship between the mode size and computation result. The ‘s’ represents stable, denoting that the frequency, damping, and vector of the pole are stable. The ‘v’ represents vector, denoting that the vector of the pole is stable. The ‘o’ represents pole, denoting that the pole is not stable. If the letters ‘s’ and ‘v’ appear at some frequency when a different modal size is chosen, this appearance indicates that this frequency is very likely to become a natural frequency of the cantilever beam.

Figure 3.

Mode stabilization diagram using PolyMAX.

During the test, we performed multiple sampling tests to ensure data reliability. Table 1 presents the natural frequency and damping ratio identified by the strain and displacement modal experiment. From Table 1, the natural frequencies identified by the strain and displacement modal methods are very close, and the maximum error rate is 2.6%, indicating that the classical modal method and strain modal method are equally effective. However, their damping ratio is high because the steel has a small damping ratio. The contrasting experiment (test #4) was conducted to determine the cause of the higher damping, which can be explained by three things. (i) The additional damping could come from the cables and sensors attached to the beams as these attachments interfere with the beam motion. (ii) The boundary conditions are changing in real-time online measurement. (iii) The damping fluctuates by using the algorithm. Mode shapes become more accurate with an increase in the number of sensors. However, the damping ratios are affected more by the additional sensors than by natural frequencies. The frequency and mode shapes are the focus of this article, not damping.

Table 1.

Modal parameters.

Modes	Strain mode	Displacement mode	Error rate	Contrast
1
Frequency (Hz)	10.254	9.986	2.6%	10.084
Damping ratio (%)	2.25	1.97	–	0.52
2
Frequency (Hz)	78.454	76.904	1.9%	78.174
Damping ratio (%)	0.48	2.02	–	1.18
3
Frequency (Hz)	216.483	213.623	1.3%	215.496
Damping ratio (%)	1.96	2.75	–	0.42
4
Frequency (Hz)	333.269	329.593	1.1%	332.358
Damping ratio (%)	2.71	2.42	–	0.81
5
Frequency (Hz)	469.154	458.984	2.2%	463.284
Damping ratio (%)	2.69	1.76	–	1.17
6
Frequency (Hz)	690.765	678.334	1.7%	684.795
Damping ratio (%)	2.83	3.54	–	0.84

Through the SFRF of the cantilever beam, we can obtain the magnitude and phase of the measuring point. Table 2 presents the amplitude (normalized) and phase of each measurement point. Because the value of the phase does not have a substantial effect, we only use positive or negative signs in the table.

Table 2.

Amplitude and phase of SFRF from first to sixth order.

Measuring Point	1st Amplitude	1st Phase	2nd Amplitude	2nd Phase	3rd Amplitude	3rd Phase	4th Amplitude	4th Phase	5th Amplitude	5th Phase	6th Amplitude	6th Phase
1	0.04	+	0.05	+	0.11	+	0.08	+	0.12	+	0.03	+
2	0.05	+	0.12	+	−0.33	−	−0.29	−	0.44	+	0.35	+
3	0.08	+	0.37	+	−0.69	−	−0.75	−	1	+	0.94	+
4	0.11	+	0.55	+	−0.86	−	−0.78	−	0.66	+	0.59	+
5	0.16	+	0.68	+	−1	−	−0.89	−	0.06	+	0.12	+
6	0.21	+	0.74	+	0.35	+	0.29	+	−0.86	−	−0.08	−
7	0.24	+	1	+	0.78	+	1	+	0.31	+	0.49	+
8	0.32	+	0.91	+	0.92	+	0.72	+	0.75	+	0.69	+
9	0.42	+	0.73	+	0.95	+	0.61	+	0.88	+	0.67	+
10	0.54	+	0.66	+	0.65	+	−0.22	−	−0.13	−	−0.16	−
11	0.67	+	0.21	+	0.21	+	−0.68	−	−0.87	−	−0.25	−
12	0.84	+	−0.16	−	−0.43	−	−0.39	−	0.12	+	1	+
13	1	+	−0.33	−	−0.77	−	0.59	+	0.94	+	−0.7	−

SFRF: strain frequency response function.

By using the method described in ‘EMA based on strain response’ section, the DMSs can be obtained from test #1. (To avoid confusion with the traditional modal test #3, the DMS obtained by the strain test is abbreviated as SDMS.) In the same manner, the SMSs can be obtained from test #2, while the DMSs can be obtained from test #3. The cantilever beam mode shapes from the first to sixth order are shown in Figure 4.

Figure 4.

Cantilever beam mode shapes. (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5 and (f) Mode 6. (Blue lines: SMSs, green lines: S-DMSs and red lines: DMSs.) DMS: displacement mode shape; SDMS: displacement mode shape obtained by the strain; SMS: strain mode shape.

DMSs of free–free beam based on strain response

We also carried out the same experiment on the free–free beam, and Figure 5 presents its mode shapes. The result indicates that the DMSs and SMSs of the beam may be opposite in the free–free state.

Figure 5.

Free beam mode shapes. (a)–(f): Mode 1–6. (Blue lines: SMSs, green lines: SDMSs and red lines: DMSs.) DMS: displacement mode shape; SDMS: ; SMS: strain mode shape.

DMSs of simply supported beam based on strain response

We also conducted the experiment using the simply supported beam, and the experimental process is shown in Figure 6.

Figure 6.

Simply supported beam mode shapes. (a)–(f): Mode 1–6. (Blue lines: SMSs, green lines: SDMSs and red lines: DMSs.) DMS: displacement mode shape; SDMS: ; SMS: strain mode shape.

Strain and traditional modal testing under three boundary conditions (cantilever, free–free and simply supported beams) were conducted for steel beams. By using the method described in ‘EMA based on strain response’ section, we can first obtain the SFRF through the strain gauges. Next, we can obtain the frequency and damping ratio. Finally, the SDMSs are obtained through transformation. In comparison with the traditional displacement modal test, it is found that the SDMSs and DMSs are highly consistent. Meanwhile, the validity and reliability of the method are demonstrated. We can use SDMSs instead of DMSs completely.

The mass of the accelerometer is much greater than the mass of the strain gauge, and the mass of the strain gauge can be ignored. The DMSs based on the strain test are more realistic than the DMSs of the traditional modal test.

Furthermore, we can obtain the corresponding relationship between the DMSs and SMSs of the structure under different constraints. In the cantilever state and the free–free state, the DMSs and SMSs may be opposite; in the simply supported state, the DMSs and SMSs may be the same. Therefore, it is possible to derive DMSs through SMSs for a simple structure.

OMA experimental verification based on strain response

High-speed LED sorting machine experimental setup

In the chip-sorting field, there is an urgent need for fast starting and stopping, short distance, high frequency and multi-degree of freedom equipment. For example, the high-speed LED motion precision of a sorting machine is at the micron level, and the movement acceleration of its rotary arm is up to 15 g. The reciprocating frequency of the arm is greater than 15 Hz, and the chip-picking time is 10 ms. In the double rotating arm system shown in Figure 7, the rotating arm exhibits 180° back-and-forth rotation. When arm 1 takes the chip, rotating arm 2 places the chip at the same time.

Figure 7.

Experimental setup: (a) LED chip-sorting machine, (b) rotating arm, (c) LMS vibration signal data acquisition instrument and (d) strain gauge placement.

The deformation vibration of the arm directly affects the chip-picking accuracy during the operation, increasing the positioning time and reducing efficiency. Therefore, the arm vibration is the main problem to be addressed in improving the working performance of the operating system. At present, for vibration suppression, it is first necessary to obtain the mechanical dynamic characteristics during the operation, requiring knowledge of the modal parameters (frequency, damping and modal shapes) of the arm during working.

Modal parameter identification

Figure 8 illustrates the sum of the PSD of all signals within the mode stabilization diagram for the results of the strain test using the PolyMAX method. In Figure 8, the red solid curve represents the summation of the FRFs of all measurement points, whereas the green hidden curve represents the mode indicator function. The horizontal axis is a frequency axis, and the vertical axis represents the amplitude axis. The meaning of letters ‘s’, ‘v’ and ‘o’ is the same as in Figure 3. If the letters ‘s’ and ‘v’ appear at some frequency when a different modal size is chosen, then this appearance indicates that this frequency is very likely to become a natural frequency of the sorting arm.

Figure 8.

Mode stabilization diagram.

Multiple samples are used to ensure data reliability. Table 3 gives the natural frequency and damping ratios identified by the strain and displacement modal tests.

Table 3.

Comparison of modal parameters.

Modal parameters		FEA	SDMS	Error rate
1	Frequency (Hz)	193.153	190.505	1.5%
	Damping ratio (%)	2.36	1.81	–
2	Frequency (Hz)	456.267	439.171	3.7%
	Damping ratio (%)	1.74	1.98	–
3	Frequency (Hz)	630.746	596.194	5.4%
	Damping ratio (%)	3.56	2.41	–
4	Frequency (Hz)	860.356	823.591	4.3%
	Damping ratio (%)	1.32	2.85	–
5	Frequency (Hz)	1126.985	1069.459	5.1%
	Damping ratio (%)	2.85	3.69	–
6	Frequency (Hz)	1381.591	1333.693	3.5%
	Damping ratio (%)	3.48	3.12	–
7	Frequency (Hz)	1906.421	1836.789	3.7%
	Damping ratio (%)	1.35	2.33	–

FEA: finite element analysis; SDMS: .

Strain modal tests were carried out on the rotating arm during operation to obtain the strain response of the arm test point. The SDMSs can be obtained using the method described in ‘OMA based on strain response’ section. For the sake of comparison, the DMSs are also obtained by finite element analysis (FEA). The DMSs obtained by means of strain response and FEA are presented in Table 4.

Table 4.

Comparison between estimated and theoretical mode shapes.

Mode	DMSs (FEA)	DMSs (strain)
1
2
3
4
5
6
7

DMS: displacement mode shape; FEA: finite element analysis.

Furthermore, a powerful tool, namely the modal assurance criterion (MAC), was used to evaluate the quality of the modal shapes. The MAC is defined as the squared correlation coefficient between two mode shape vectors, which assesses the correlation between the two vectors $$MAC\left({\varphi }_x,{\varphi }_y\right)=\frac{{\left|{\varphi }_x^H{\varphi }_y\right|}^2}{\left({\varphi }_x^H{\varphi }_x\right)\left({\varphi }_y^H{\varphi }_y\right)}$$ (30)

If two vectors are estimates of the same physical mode shape, the MAC should approach unity (100%); otherwise, the MAC should be low. ²³ A high-quality mode set normally contains diagonal elements that are 100% (by definition), as well as off-diagonal elements that have a low value (close to 0%). Figure 9(a) indicates the MAC values for the FEA and strain test. From Figure 9(a), the MAC is of high quality because the diagonal elements are close to 100%, and the off-diagonal elements are small. ²⁴ The DMSs identified by the FEA and strain test are high. The accuracy of the method for obtaining the DMSs based on the strain response is verified.

Figure 9.

MAC values for different mode shapes using PolyMAX: (a) MAC between mode shapes of FEA and strain test and (b) auto-MAC of mode shapes of strain test. FEA: finite element analysis; MAC: modal assurance criterion.

The MAC of the mode shapes for the x-direction of the arm in the strain test is shown in Figure 9(b). From Figure 9, the maximum MAC value is less than 40% for Modes 1 and 6 (32.578%); Modes 1 and 6 can be considered as two independent modes. The remaining values are less than 30%, which can ensure that the rest of the various modes are independent. The SDMSs identified by the strain response test are independent modes indicating that the identification results in the x-direction are valid.

In operation, acquiring the modal parameters of the lightweight high-speed operating mechanism is a problem. Using the high-speed LED sorting machine as an example, the arm vibration directly affects the chip-picking accuracy. The modal parameters cannot be obtained by traditional modal testing. First, the strain response is obtained by the strain gauge. Second, the PSD function of each measuring point is obtained. Then, the amplitude and phase are extracted. Third, the SDMSs are obtained through normalization. When comparing the experimental results with the theoretical calculation based on FEA, we find that the displacement modes are highly consistent. The reliability of the method is verified, which constitutes the difficulty of obtaining the displacement modal parameters under operation.

Conclusions

In this paper, a new approach based on strain response is proposed to identify the modal parameters of a lightweight, high-speed structure under operation and to identify the DMSs and SMSs accurately. First, the theoretical background of the identification method is derived and verified under static and operational states, respectively. Next, the proposed approach is used to identify the DMSs and SMSs of the beam structure with modal parameters that are obtained by EMA, verifying that the method is reliable under the static state. The result shows that the beam structure presents different DMSs and SMSs at different constraints, as follows: (i) In the cantilever state, the DMSs and SMSs are the opposite. (ii) In the free–free state, the DMSs and SMSs are the opposite. (iii) In the simply supported state, the DMSs and SMSs are the similar. Therefore, it is possible to derive DMSs through SMSs for a simple structure. Finally, the DMSs of the sorting arm are obtained by the method under high-speed operation. When the experimental results are compared with the computational modal analysis, the two mode shapes are found to be highly consistent, thus verifying that the method is equally reliable during operation. This method expands the application scope of the classic EMA method and reduces the error because of the additional mass of the actual sensor. This method provides effective guidance for obtaining dynamics of lightweight, high-speed and thin-walled structures.