The analytical beam element model: novel approach for fast calculation of vibrations in electric machines

The stator of an electric machine can be described by a multi-mass oscillator. This is a multi-degree-of-freedom (DOF) system without gyroscopic effects that can be generally described by where \(\mathbf{M}\), \(\mathbf{D}\) and \(\mathbf{K}\) are the mass, damping and stiffness matrices, respectively. \(\vec{F}\) is the vector of the applied forces and \(\vec{x}\) is the vector of displacements for each DOF.

The system can be decomposed and diagonalized in terms of its eigenvectors \(\vec{\Psi_{r}}\), also known as its eigenmodes, and the corresponding eigenfrequencies \(f_{0,r}\). The eigenfrequencies \(\omega_{0,r}=2\pi f_{0,r}\) are calculated by solving which will provide \(q\) solutions for a system with \(q\) DOFs. \(r\) is the order of the eigenmodes. In case of a cylindrical structure such as the stator of an electric machine, the order \(r\) represents the number of maxima or minima around the circumference.

The eigenvectors \(\vec{\Psi_{r}}\) are determined by replacing each of the \(q\) obtained eigenfrequencies in (1), rewritten in the frequency domain with no damping and no external forces. The base transformation or modal matrix \(\boldmath{\Psi}\) is then defined as the matrix formed by the \(q\) eigenvectors placed in its columns

The relationship between the vectors \(\vec{x}\) (in the original base) and \(\vec{X}\) (in the transformed base) is given by

Since the eigenvectors provide only the directions of the new vector base, they can be arbitrarily scaled. Typically, the mass matrix is used for the scaling, giving the relations where \(r=1...q\). In (7), it is assumed that the damping matrix can be approximated as a linear combination of the mass and stiffness matrices, and \(d_{r}\) is the modal damping factor for eigenmode \(r\). As a result, the decoupled system equations in the transformed base become

Using this equation, the vibration response \(\vec{x}\) to a force excitation \(\vec{F}\) of the system can be calculated. This procedure forms the basis of the vibration and acoustic analysis. The purpose of the ABM is to describe the mass and stiffness matrices and to calculate the eigenfrequencies and eigenmodes.

In the basic approach introduced in [18], the Euler–Bernoulli beam element, as represented in Fig. 1a, is used to discretize the stator of an electric machine and to generate the mass and stiffness matrices.

The stiffness matrix \(\mathbf{K}_{\mathrm{e}}\) for such a beam element with three DOFs per node is \(E\) is the material’s Young’s modulus, \(l\) is the beam element length, \(A\) is the cross-sectional area and \(I\) is the moment of inertia of the beam [19]. Shear strains are neglected in this formulation. Similarly, the element’s mass matrix \(\mathbf{M}_{\mathrm{e}}\) is defined as where \(\rho\) is the beam material’s mass density [19]. The mass and stiffness matrices are defined by considering the node DOFs shown in Fig. 1a as a vector written in the sequence \([u_{\mathrm{A}},v_{\mathrm{A}},{\phi}_{\mathrm{A}},u_{B},v_{B},\phi_{B}]^{T}\).

The cross-sectional area of the elements in the yoke corresponds to the stator’s axial length \(l_{\mathrm{s}}\) multiplied by the yoke height \(h_{\mathrm{y}}\). The cross-section of the elements which represent the teeth is assumed to be constant and is defined by the stator’s axial length and the average width of the teeth. In case the teeth have tips with a larger width, the tip’s additional mass is accounted for by increasing the mass density of the last element of each tooth.

Initially, the elements are not connected to each other. This way, the free element equations, using their own local coordinate systems, can be written by replicating the single-element mass and stiffness matrices on the diagonal of a global matrix. This procedure produces the stiffness matrix \(\mathbf{K}_{\mathrm{raw}}\) and the mass matrix \(\mathbf{M}_{\mathrm{raw}}\) [18].

The next step is to connect the beam elements to each other. This is achieved by means of defined master nodes named P₁₁… \(P_{Tn}\) and Q₁₁…\(Q_{Tm}\), which use a cylindrical coordinate system and correlate with the beam nodes A, B, C, and D, as illustrated in Fig. 1c. The relation between an arbitrary element node EN in its local coordinate system and a corresponding master node MN in cylindrical coordinates is

\(\theta\) is the angle between the \(r\) and \(y\) axes of the cylindrical and local coordinate systems. Based on the trigonometric relations shown in Fig. 1c, the transformation matrices are calculated as for nodes A, B, C, and D, respectively. The angle \(\alpha\) is the angle between adjacent elements in the stator yoke.

Based on this, a complete nodal transformation from ABCD (element nodes) to PQ (master nodes) is performed by

\(\mathbf{A}\) is the assembly matrix obtained by combining the previously defined transformation matrices \(\mathbf{H}_{\mathrm{A}}\), \(\mathbf{H}_{\mathrm{B}}\), \(\mathbf{H}_{\mathrm{C}}\) and \(\mathbf{H}_{\mathrm{D}}\), surrounded by zeros. The final system stiffness matrix \(\mathbf{K}_{\mathrm{A}}\) and mass matrix \(\mathbf{M}_{\mathrm{A}}\) of the connected beams become and These two matrices are used for calculating the stator eigenmodes and eigenfrequencies according to the procedure described in Sect. 2.1.

The first approach involves to replace the Euler-Bernoulli beam element used in [18] with the Timoshenko beam element. The Euler-Bernoulli beam element assumes that the beam cross-section remains perpendicular to the beam’s intermediate axis when deformed. This assumption neglects the shear deformations and is typically suitable for problems where the beam thickness is relatively small compared to its length. The Timoshenko formulation allows the beam cross-section to rotate with respect to the intermediate axis and, therefore, takes into account possible shear deformations. This allows the method to account for the natural tendency of the cross-section to rotate back to its initial position, relieving the internal stresses and making the Timoshenko formulation suitable for moderately thick beams. Fig. 2 illustrates the assumptions made using the two formulations discussed.

According to the Timoshenko formulation, the mass matrix can be written as the sum of the translational mass \(\mathbf{M}_{\mathrm{trans}}\) and the rotational mass \(\mathbf{M}_{\mathrm{rot}}\). Together with the corresponding stiffness mass \(\mathbf{K}_{\mathrm{e}}\) they are [19] and

\(C\) and \(\gamma\) are defined by \(\kappa\) is the shear correction factor used to consider the non-uniform shear distributions. It is typically assumed to be \(5/6\) [19]. \(G\) is the shear modulus. Under the assumption that the material is isotropic, this can be calculated as [20] \(\mu\) is the Poisson’s ratio. When considering an isotropic material, the material is defined if two of the three quantities \(E\), \(G\) and \(\mu\) are known. It is worth noting that (9) and (10) from the investigation of the Euler-Bernoulli beam correspond to a particular case of the Timoshenko beam in which the shear modulus \(G\) tends to infinity.

Another possible approach to considering the stator yoke’s thickness accurately is to discretize the yoke with multiple Euler-Bernoulli beam element layers, as represented in Fig. 3. The layers are connected to each other by vertical beam elements with a cross-section dependent on the number of elements around the circumference and a length dependent on the number of layers. It is worth noting that the vertical elements’ masses would overlap with that of the yoke elements, since they occupy the same volume. Therefore, the mass density of all the vertical elements in the yoke is set to zero, so that they act solely as an ideal elastic connection between the layers.

This approach also requires an extension of the assembly matrix \(\mathbf{A}\) defined in (13) to consider the connection of the newly introduced beams. The extensions are made following the same principle as presented in Sect. 2.2.

Since the beam elements representing the stator yoke correspond to the yoke’s intermediate surface, a direct connection between the tooth base and the yoke nodes would produce a partial overlap of the tooth and yoke beam elements. Taking the yoke height as \(h_{\mathrm{y}}\), the overlapping height is \(\frac{h_{\mathrm{y}}}{2}\) as illustrated in Fig. 4. Evidently, this fact becomes more relevant for the case of a thick stator yoke.

In this work, an additional translational transformation of the master nodes \(P\) is considered as means to connect the nodes of the tooth bases to the yoke. This translation corresponds to the lever arm effect of the yoke cross-section modeled as a rigid body when it rotates. For an arbitrary translation of (\(\mathrm{\Delta}x\), \(\mathrm{\Delta}y\)) applied to a point \(P^{\prime}\) with respect to the point \(P\), the transformation matrix \(\mathbf{T}(\mathrm{\Delta}x,\mathrm{\Delta}y)\) can be defined as

In combination with the rotation matrix presented in (11), a generalized coordinate transformation including translations and rotation effects can be defined as the multiplication \(\mathbf{H}(\theta)\cdot\mathbf{T}(\mathrm{\Delta}x,\mathrm{\Delta}y)\). For the connection of the nodes in the tooth base to the yoke, for instance, the transformation matrix is calculated using \(\theta=\frac{\pi}{2}\), \(\mathrm{\Delta}x=0\) and \(\mathrm{\Delta}y=\frac{-h_{\mathrm{y}}}{2}\). Such complete transformation can be considered when generating the assembly matrix \(\mathbf{A}\), as explained in (13), so that the lever arm effect of the tooth base with respect to the yoke’s intermediate surface can be taken into account.

Table 2 summarizes the results of the ABM with the Euler-Bernoulli (EB) beam, the Timoshenko (TS) beam and the multi-layer (ML) approach to modeling the stator yoke. 120 elements are considered around the circumference for all the ABM models and the multi-layer model is calculated with two layers. The results are compared to the FE simulation results and the deviations \(f_{\mathrm{\Delta},r,EB/TS/ML}=\frac{f_{0,r,EB/TS/ML}-f_{0,r,\text{FE}}}{f_{0,r,\text{FE}}}\) between the individual approaches (EB/TS/ML) and the FE model are determined.

It can be seen that the model with the Euler-Bernoulli beam element produces the highest deviations \(f_{\mathrm{\Delta},r}\) when compared to the FE model. This is due to the fact that the shear strains are neglected in this method, which is appropriate for slender beams but becomes less accurate when the thickness of the beam increases in relation to its length. The ratio of the yoke height \(h_{\mathrm{y}}\) to the mean stator diameter \(D_{\mathrm{o}}\) is 0.103 for the stator S1. In this case, the results calculated using the Timoshenko beam element and the multi-layer approach are more accurate. Although these two aforementioned methods present similar deviations for the case studied (\(f_{\mathrm{\Delta},r}=0.02\%{\ldots}4.88\%\)), the Timoshenko approach is the one which offers more advantages. One is that it requires less elements than the multi-layer approach, since only one layer is necessary. Another is that the effect of the shear strain is intrinsically considered in the constituent equations of the Timoshenko beam, whereas, in the multi-layer approach, this is accounted for by means of the virtual vertical beams connecting the layers. As a consequence, the stiffness for shear displacements between the layers becomes dependent on the number of elements in the system in such a way that an excessive number of elements (with consequently very thin vertical beams) would produce a model which underestimates the shear stiffness. Accordingly, the multi-layer approach leads to different results for different numbers of vertical beam elements.

In Table 3, the results for the stator with teeth are summarized and compared to measurements and results of the FE simulations. In this case, only the ABM with the Timoshenko beam element is considered since, as discussed in Sect. 4.1, this produces better results for the modeling of the stator yoke’s thickness. The ABMs are composed of 120 elements in the yoke and 5 elements per tooth. Two variants of the Timoshenko approach are compared: one including the effect of the lever arm discussed in Sect. 3.3 (TS,LA) and another one neglecting this effect (TS).

Furthermore, the results are compared to the FE model and to results of an experimental modal analysis. The experimental modal analysis is performed by means of an instrumented hammer and one accelerometer installed in the back of the yoke. Details concerning the test procedure and the results obtained can be found in the related papers [10, 21].

The deviations \(f_{\mathrm{\Delta},r}\) shown in Table 3 are calculated with respect to the measured values. An exception is made for the eigenmode \(r=5\), which could not be identified in the measurements due to its high frequency (see [10, 21]). In this case, the deviations marked with an asterisk (*) are calculated with respect to the FE model.

In general, it can be seen that the calculated eigenfrequencies show a good agreement with the measurements, with a maximum absolute deviation of 7.43%. Nevertheless, the deviations are generally lower when the lever arm effect discussed in Sect. 4.1 is taken into account. This becomes more evident for the mode \(r=4\), where large deflections of the teeth are present, as can be seen in Table 5. The absolute deviation decreases from 7.43% to 4.06% when the lever arm effect is considered.

The eigenfrequencies of the eigenmodes \(r=2\), \(r=3\), \(r=0\), \(r=4\) and \(r=5\) of stator S1 are shown in Table 5. As discussed in Sect. 4, the deviations are lower than \(5\%\). The eigenmodes calculated using the ABM show a good accordance with the eigenmodes calculated using the FE model. Beside the eigenmode \(r=0\), the deviations \(f_{\mathrm{\Delta},r}\) between the ABM and the FE model are negative, indicating that the ABM is tendentially less stiff than the FE model in this case.

Due to the larger size of stator S2, its eigenfrequencies are reduced in comparison to stator S1. According to the conclusions of Jordan [1], the bending eigenmodes \(r=2\), \(r=3\), \(r=4\), \(r=5\) of the stator are reduced more sharply than the eigenmode \(r=0\) due to the increased size of the machine. The deviations \(f_{\mathrm{\Delta},r}\) between the calculated eigenfrequencies of the ABM and the FE models are lower than \(2\%\), except for eigenmode \(r=5\). For the eigenmode \(r=5\), the deviation is around \(6\%\). The calculated eigenmodes show a good accordance. For the stator S2, the deviations between the ABM and the FE model are both negative or positive.

The eigenmodes and eigenfrequencies of stator S3 show a good accordance between the ABM and the FE model. The deviations between the eigenfrequencies \(f_{\mathrm{\Delta},r}\) are lower than \(3\%\) (see Table 5). It is striking that the deviations are all negative. In comparison to stators S1 and S2, the eigenfrequencies are lower due to the larger size of the stator.

The calculated eigenmodes and eigenfrequencies of stator S4 confirm the results so far. The deviations between the ABM and FE model \(f_{\mathrm{\Delta},r}\) are lower than \(1.5\%\). Accordingly, it can be concluded that the ABM and the FE model show a very high accordance. As with the stator S3, the deviations \(f_{\mathrm{\Delta},r}\) are all negative (see Table 5).

The comparison of the four stators’ vibration behavior using the ABM and the FE model proves that the ABM is capable of correctly predicting the vibration behavior of electric machines with outer diameters from \(D_{\mathrm{o}}=160\,\text{mm}\) to \(D_{\mathrm{o}}=7\,\text{m}\). Despite the eigenfrequency \(f_{r=5}\) of stator S2, the deviation \(f_{\mathrm{\Delta},r}\) is lower than \(5\%\), indicating a good agreement between the ABM and the respective FE model.

The deviations between the ABM and the FE model are mainly negative. This indicates that the ABM tends to be less stiff than the FE method. Furthermore, it is noteworthy that the deviation \(f_{\mathrm{\Delta},r}\) decreases with increasing size of the machine. To analyze this, the mean deviation \(\overline{f}_{\mathrm{\Delta}}=\frac{\sum_{r_{\mathrm{min}}}^{r_{\mathrm{max}}}|f_{\mathrm{\Delta},r}|}{n_{r}}\) is calculated.

The ratio \(\frac{h_{\mathrm{y}}}{D_{\mathrm{o}}}\) between the yoke height and the outer diameter decreases with increasing size as well (see Table 4). Accordingly, the influence of the shear strain in the stator yoke is higher for machines with a high ratio of \(\frac{h_{\mathrm{y}}}{D_{\mathrm{o}}}\). The shear strain is only approximated by the Timoshenko beam element and is better described by the FE models. Thus, the deviations between the FEM and ABM decrease with increasing size of the stators S1 to S4. These observations can also be made when calculating the vibration behavior using the analytical model of Jordan [1]. This analytical model is not capable of correctly calculating the vibration behavior of small machines with a high ratio of \(\frac{h_{\mathrm{y}}}{D_{\mathrm{o}}}\). For machines with a small ratio of \(\frac{h_{\mathrm{y}}}{D_{\mathrm{o}}}\), the analytical model leads to better results. Accordingly, the correct description of the stator yoke’s vibration behavior is the major challenge in the calculation of the vibration behavior for machines with a high ratio of \(\frac{h_{\mathrm{y}}}{D_{\mathrm{o}}}\). It can be concluded that the approach presented, using the Timoshenko beam element model, is capable of calculating the vibration behavior of machines of different sizes, taking into account the vibration behavior of the stator yoke.