Dynamic analysis of temperature-dependent vibration in electric drive gear transmission systems with magnetorheological fluid coupling

2.1. Torque model of MRF coupling unit

A three-dimensional model of MRF coupling is shown in Fig. 2. The MRF is distributed in the working clearance formed between the driving and driven cylinders, where the former is fixed to the driving motor input shaft and the latter to the driving gear. Initially, when the coil is unenergized, the magnetic particles in the working clearance are free in the carrier fluid. At this point, the torque generated by the MRF zero-field viscosity cannot rotate the driven cylinder. Upon energization of the coil, the MRF converts to a solid-like state, with its magnetic particles aligned into a chain-like structure along the magnetic lines of force. This structure enhances the shear yield stress, which transfers torque and enables the driven cylinder to rotate. After the coil is de-energized, the MRF reverts to a fluid state within a few millimeters, and the MRF coupling returns to its initial state. The stiffness, damping, and torque of the MRF coupling can be adjusted by varying the coil current, thereby regulating the mechanical performances of the MRF coupling and reducing the vibration of the EDGTS.

Fig. 2Three-dimensional model of MRF coupling

The torque model of MRF coupling is shown in Fig. 3, where the outer radius of the driving cylinder and the inner radius of the driven cylinder are $R_1$ and $R_2$, respectively, the input and output rotational speeds are ${\omega }_1$ and ${\omega }_2$, respectively, $L$ is the actual axial length of the MRF between the cylinders, and $L_e$ is the equivalent axial length of the MRF. The torque transmitted by the MRF is [21]:

where ${\eta }_0$ is the MRF zero-field viscosity, $\tau \left(H,T\right)$ is the shear yield stress of magnetic particles, which is dependent on variations in the applied magnetic field and temperature.

Fig. 3Torque model of MRF coupling

2.2. Thermodynamical model of MRF coupling unit

When the coil is energized, there is a rotational speed difference between the driving and driven cylinders, resulting in MRF coupling slip operation. By combining Eq. (1), the MRF slip speed can be expressed as:

where $\delta$ is the MRF working clearance width.

Considering the influence of the magnetic flux density, particle size, and coating thickness, the friction between the magnetic particles, cylinder wall, and carrier fluid in the MRF generates slip thermal power, as shown in Eq. (3):

where ${\mu }_0$ is the vacuum permeability, ${\chi }_0$ is the initial magnetic susceptibility, $M_s$ is the saturation magnetization, $H$ is the magnetic-field intensity, ${\phi }_a$ is the MRF volume fraction, $\gamma$ is the shear strain, ${\vartheta }_p$ is the friction coefficient, $N$ is the number of single-chain magnetic particles, $\beta$ is the material correlation constant, $h_c$ is the clearance between particles, $r_1$ and $r_2$ are the magnetic-particle radius and coating thickness, respectively, and $h$, $\zeta$, and ${\vartheta }_w$ are the computational intermediate variables, as shown in Eq. (4):

The slip thermal power of MRF coupling is shown in Fig. 4.

Fig. 4Slip thermal power of MRF coupling

Under slip conditions, the heat of MRF coupling primarily arises from the slip differential heat in the MRF working clearance and the energy loss of the coil. The heat source intensity per unit volume in the MRF working clearance can be expressed as:

where $V_m$ is the volume of the MRF working clearance. The energy loss $P_c$ of the coil is related to the coil current $I_c$ and coil resistance $R_c$ and can be expressed as:

The heat source intensity of the coil can be expressed as:

where the coil inner and outer radii are $R_2$ and $R_3$, respectively, and the coil width is $L_c$.

Thermal convection of MRF coupling mainly consists of convective heat transfer between the outer surface of the driven cylinder, the outer surface of the shell, and the coil end faces and the air [27, 28]. Newton’s law of cooling is used to describe the heat transfer equation for the above thermal convection:

where ${\alpha }_i$ and $A_i$ are the heat transfer coefficient and surface heat transfer area of each component of MRF coupling, respectively.

Assuming that the temperature rise time is $t$ and the temperature rise is $d{T}_i$ in $dt$ time, the heat representation of the energy conservation equation can be obtained by combining Eqs. (5-8):

where, $P_i$, $c_i$, and $m_i$ are the heat dissipation power (kW), specific heat capacity, and mass of each component of MRF coupling, respectively.

2.3. Dynamical model of MRF coupling unit

The stiffness and damping of the MRF coupling vary with the solidification degree of the MRF at different magnetic-field intensities, which, in turn, is related to the shear complex modulus. The shear complex modulus of the MRF can be expressed as [29]:

where the real and imaginary parts of the shear complex modulus are the shear energy storage modulus $G^{\text{'}}$ and the shear loss modulus $G^{\text{'}\text{'}}$, respectively. The ratio of the two moduli is the loss factor ${\delta }^{\text{'}}$ and is generally calculated using the following empirical formula [30].

The relationship between the shear storage modulus, magnetic flux density $B$, and temperature $T$ is approximated by fitting [31]:

where $B_1$ = 0.2 Tesla, $B_2$ = 0.6 Tesla. When the zero-field shear energy storage modulus $G_0^{\text{'}}$ = 0.15 MPa, the variation laws of the shear energy storage and shear loss moduli with magnetic flux density and temperature can be obtained by combining Eqs. (10) and (11), and the results are shown in Fig. 5.

Fig. 5Modulus versus temperature at different magnetic flux densities

a) Storage modulus

b) Loss modulus

The shear energy storage and shear loss moduli decrease nonlinearly with increasing temperature at different magnetic flux densities. Figs. 5(a) and (b) show that the maximum shear energy storage and shear loss moduli (under a 0.8-Tesla field intensity condition) decrease from 0.45 and 0.09 MPa to 0.20 and 0.04 MPa, respectively, both with a change rate of 55.56 %.

The three main parameters of MRF coupling coupled to the EDGTS are the equivalent torsional stiffness coefficient $k_h$, equivalent damping coefficient $c_h$ and equivalent moment of inertia $J_h$, respectively, which can be expressed as [32]:

where $m_{ch}$ and $R_h$ are the mass and equivalent rotary radius of the driven cylinder, respectively. Here, ${\zeta }_h$ is the damping ratio of the MRF, and $C_h$ is the critical damping coefficient [33].

The torsional stiffness and damping matrices for the MRF coupling unit are $K_H$ and $C_H$, respectively, with $C_H$ and $K_H$ having the same form. In addition, $M_H$ and $X_H$ are the mass and displacement matrices, respectively, which can be expressed as:

where $m_h$ is the mass of the MRF coupling. The dynamical equation for the MRF coupling unit can be expressed as:

3.1. Dynamical model of shaft segment unit

Fig. 6 shows the double nodes of the shaft segment unit are subjected to the bending moments, torsional moments, and axial loads, and its equivalent dynamical model can be established by the Timoshenko beam unit.

The displacement column vector of the double nodes of the shaft segment in the generalized coordinates $xyz$ can be expressed as:

where $x_i$, $y_i$, and $z_i$and $x_{i+1}$, $y_{i+1}$, and $z_{i+1}$ are the displacements of nodes $i$ and $i+1$ along the local coordinate direction, respectively, and ${\theta }_{x_i}$, ${\theta }_{y_i}$, and ${\theta }_{z_i}$ and ${{\theta }_x}_{i+1}$, ${{\theta }_y}_{i+1}$, and ${{\theta }_z}_{i+1}$ are the angles of the section around the three coordinate axes at nodes $i$ and $i+1$, respectively. The stiffness matrix for the double nodes of the shaft segment unit can be expressed as:

where specific calculations for $a_i$, $c_i$, $e_i$, and $f_i$ ($i=x,y,z$) are available in reference [35].

The mass matrix $M_j$ for the shaft segment unit can be expressed as:

where $\rho$ is the shaft segment density, $A_e$ is the unit cross-sectional area, and $l$ is the unit length, respectively.

The damping matrix $C_I$ for the shaft segment unit is commonly derived using Rayleigh damping:

where $a_1$ and $a_2$ are weight coefficients of the mass and stiffness matrices, respectively, and $f_i$ and ${\xi }_i$ ($i=\mathrm{1,2}$) are the first two-order natural frequencies and damping coefficients of the shaft segment unit, respectively.

Combining Eqs. (15-18), the dynamical equation for the shaft segment unit can be derived as:

Fig. 6Dynamical model of shaft segment unit

3.2. Dynamical model of connecting unit

The connecting unit equivalent dynamical model consists of nodes $j$ and $j+1$ of shaft segment units 1 and 2, as shown in Fig. 7.

Fig. 7Dynamical model of connecting unit

Here, $K_J$ and $C_J$ are the coupled stiffness and damping matrix between nodes $j$ and $j+1$ on shaft segment units 1 and 2, respectively, which can be expressed as:

where $k_x$, $k_y$, and $c_x$, $c_y$ are the radial coupled stiffness and damping, respectively, $k_z$ and $c_z$ are the axial coupled stiffness and damping, respectively, and $k_{\theta i}$ and $c_{\theta i}$ ($i=x,y,z$) are the torsional coupled stiffness and damping in three directions around the generalized coordinates $xyz$, respectively.

The mass matrix $M_J$ for the connecting unit can be expressed as:

where $M_j$ and $M_{j+1}$ are the mass matrices of counterpart nodes $j$ and $j+1$, respectively, $m_j$ and $m_{j+1}$ are the masses of the shaft segment units of counterpart nodes $j$ and $j+1$, respectively, and $J_{ij}$ and $J_{ij+1}$ ($i=x,y,z$) are the rotational inertia values of the shaft segment units of the counterpart nodes $j$ and $j+1$, respectively.

The displacement column vector $X_J$ of the connecting unit has the same structural form as $X_I$ in Eq. (15), and the dynamical equation for the connecting unit can be expressed as:

3.3. Dynamical model of bearing-foundation unit

The bearing-foundation unit dynamical model was adopted to reflect the coupling characteristics and vibration transmission law of the bearing support structure accurately, as shown in Fig. 8.

Fig. 8Dynamical model of bearing-foundation unit

The support stiffness matrix for the bearing-foundation unit can be expressed as:

where $k_{xx}$, $k_{yy}$, and $k_{zz}$ are the radial and axial support stiffness values, respectively, and $k_{{\theta }_x{\theta }_x}$ and $k_{{\theta }_y{\theta }_y}$ are the torsional stiffnesses in the direction around the $x$- and $y$-axes of the generalized coordinates, respectively.

The structural form of the bearing support damping matrix $C_K$ remains the same as that of $K_K$, and the dynamical equation for the bearing-foundation unit is [36]:

where $X_K$ is the displacement column vector of node $k$ ofhe shaft segment in six directions.

3.4. Dynamical model of gear pair engagement unit

The dynamical model of the gear pair engagement unit is shown in Fig. 9, and its main parameters are described in Reference [21].

Fig. 9Dynamical model of gear pair engagement unit

As Fig. 9 shows, the displacement column vector of the gear pair engagement unit node is $X_L={\left(x_l,y_l,z_l,{\theta }_{x_l},{\theta }_{y_l},{\theta }_{z_l},x_{l+1},y_{l+1},z_{l+1},{{\theta }_x}_{l+1},{{\theta }_y}_{l+1},{{\theta }_z}_{l+1}\right)}^T$. By projecting the microdisplacements in each direction to the engagement line, the engagement vector can be obtained:

where $r_l$ and $r_{l+1}$ are the base circle radii of the gear pair, respectively, and ${\beta }_l$ is the base circle helix angle (positive for dextrorotation and negative for levorotation). Here, ${\alpha }_l$ is the working pressure angle of the gear pair, and ${\phi }_l$ is the angle between the line $o_l{o}_{l+1}$ connecting the centers of the gear pair and the $x$-axis.

The stiffness matrix $K_L$ and damping matrix $C_L$ for the gear pair engagement unit are:

where $k_l$ is the normal composite engagement stiffness, and $c_l$ is the gear pair engagement damping.

The equivalent displacement and velocity column vectors for the composite engagement error after decomposition in each degree of freedom are:

where $e_l$ is the normal composite error of the gear pair. The mass matrix $M_L$ for the gear pair engagement unit is:

where $m_l$ and $m_{l+1}$ are the masses of the gear pair, respectively, and $J_{xi}$, $J_{yi}$, and $J_{zi}$ ($i=l,l+1$) are the rotational inertia of the gear pair around the $i$-axis ($i=x,y,z$), respectively. Combining Eqs. (25-28) yields the following dynamical equation for the gear pair engagement unit:

The engagement impact force column vector $F_L$ is the product of the gear normal impact force $f_l$ and the engagement transpose vector $V_L^T$ [34].

3.5. Mechanical-electromagnetic-thermal coupled dynamical model

As shown in Fig. 10, each unit of the EDGTS is respectively discretized into separate nodes, which are positioned and numbered sequentially according to the order of each unit. When both axes are divided into 12 units, the MRF coupling is positioned at node 6, the driving and driven gears are positioned at nodes 10 and 17, respectively, and the driving and load torques ($T_{in}$ and $T_{out}$) are imposed at nodes 4 and 23, respectively.

Fig. 10Schematic diagram of the discrete nodes of EDGTS

Fig. 11Schematic diagram of the total assembling rules of EDGTS

Based on the idea of the classical structural mechanics analysis method, such an infinite degree-of-freedom continuously distributed mechanical vibration system as the EDGTS is translated into a finite degree-of-freedom discrete system. Subsequently, according to the assembly relationship between each unit node and the total nodes of the system, the submatrices of each unit are sequentially filled into the coupled position of the total matrix of the system to assemble the total stiffness matrix $K_N$ for the EDGTS, as shown in Fig. 11. The total mass matrix $M_N$ and damping matrix $C_N$ for the EDGTS are assembled in the same way as described above.

By assembling each basic unit according to the rules shown in Fig. 11, the overall dynamical equation for the EDGTS can be obtained:

where $X\left(t\right)$ and $F\left(t\right)$ are the overall displacement column vector and external excitation force (including the driving and load torques) of the EDGTS, respectively. The equivalent displacement column vectors $e\left(t\right)$ and velocity column vectors $\dot{e}\left(t\right)$ of the composite engagement error only have values for the gear pair engagement unit.

4.1. Magnetic-field FEA of MRF coupling

A two-dimensional axisymmetrical model was established to conduct the finite-element analysis (FEA) to characterize the transmission characteristics of MRF couplings more precisely at different currents. As shown in Fig. 12, the materials for each component of the MRF coupling are represented by different colors, and its dimensional design parameters are listed in Table 1.

Fig. 12Two-dimensional axisymmetrical model of the MRF coupling

The material of both the driving and driven cylinders is steel 1010 and the material of the shell is aluminum. Considering the magnetisation characteristic of the materials used in the MRF coupling to obtain more accurate magnetic flux and magnetic line of force distributions are shown in Fig. 13.

Fig. 13Three-dimensional cloud maps of magnetic flux distribution at different currents

a)$I=$0.5 A

b)$I=$1.0 A

c)$I=$1.5 A

d)$I=$2.0 A

At different currents excitation, the direction of the magnetic lines of force passing through is vertical to the MRF shear flow direction, indicating that the MRF coupling is well-designed. When $I$ = 0.5 A, the mean value of magnetic flux density in the working clearance is 0.24 Tesla. As the current increases incrementally to 1.0, 1.5, and 2.0 A, the corresponding mean values of magnetic flux density in the working clearance rise to 0.48, 0.66, and 0.77 Tesla, respectively.

Fig. 14Variation of magnetic flux density with distance and current within the MRF working clearance

As shown in Fig. 14, the magnetic flux density within the MRF working clearance is uniformly distributed and increases nonlinearly with increasing current. According to the distribution law of magnetic flux density, the maximum flux density is 1.14 Tesla at axial distances $d$ = 0.035 and 0.065 m, and the minimum flux density is 0.003 Tesla at axial distances $d$ = 0.025 and 0.075 m.

4.2. Thermal-field FEA of MRF coupling

The temperature distribution of MRF coupling was analysed using a thermal-field FEA model, which is identical to the magnetic-field FEA model shown in Fig. 12. The material thermodynamic parameters of MRF coupling are listed in Table 2.

In this study, $Q_{MRF}$ and $Q_c$ are taken as the main heat sources of MRF coupling and are calculated using Eqs. (5) and (7), respectively. The heat transfer coefficients of each MRF coupling component are shown in Table 3.

As shown in Fig. 15, the average temperatures of the driving and driven cylinders are 80.53 °C and 69.04 °C, respectively, when $I$ = 2.0 A. The maximum MRF coupling temperature (91 °C) is located at the geometrical center of the MRF working clearance between the two coils because of the low thermal conductivity and high thermal resistance of this area. Meanwhile, the lowest MRF coupling temperature (39 °C) is at the end of the output shaft.

Fig. 15Temperature distribution cloud maps of the MRF coupling unit under varied currents

a)$I=$0.5 A

b)$I=$1.0 A

c)$I=$1.5 A

d)$I=$2.0 A

Fig. 16Effects of current on the MRF working clearance temperatures: a)-d) temperature distribution cloud maps of the MRF working clearances under varied currents, and e) relationship between operating temperature and time of MRF at different currents

a)$I=$0.5 A

b)$I=$1.0 A

c)$I=$1.5 A

d)$I=$2.0 A

e)

A temperature distribution cloud map of the MRF working clearances is shown in Figs. 16(a-d). Taking $I$ = 1.5 A as an example. The temperature is symmetrically distributed along the radial centerline of the MRF working clearance and gradually decreases along the axial centerline to both sides. When the temperature at the geometrical center of the MRF working clearance reaches 69 °C, the temperature on both sides of the MRF working clearance is 55.35 °C. As shown in Fig. 16(e), the maximum steady-state temperatures of the MRF working clearance at different currents are 36.10 °C, 60.20 °C, 77.50 °C, and 90.90 °C. The corresponding temperature rise times are 19.67 min, 23.47 min, 29.48 min, and 33.42 min, respectively.

Fig. 17MRF coupling torque versus temperature at different currents

The variation law of the MRF coupling torque with current and temperature is shown in Fig. 17. The torque generated by the MRF coupling increases with increasing current at any constant temperature. The shear yield stress of the MRF increases nonlinearly with the current. As a result, the current-torque curve also demonstrates a high degree of nonlinearity. When $T$ = 20 °C, the MRF coupling can generate a torque of 12.40 N·m at a current of 0.125 A. As the current increases to 1.625 A, the torque rapidly increases to 65.41 N·m. However, as the current continues to increase, the magnetic saturation characteristic of the MRF slows the growth rate of the shear yield stress, resulting in a plateau in the torque growth trend. When $I$ =2.0 A, the rheological properties of the MRF, including the shear yield stress and viscosity, decrease significantly as the temperature rises from 20 to 90 °C. The torque decreases from 67.44 to 50.21 N·m, representing a reduction rate of 25.55 %. Therefore, it is essential to research the impact of the operating temperature on the MRF transmission performance during system operation.

Fig. 18Dynamical parameters of the MRF coupling versus temperature at different currents

a) Stiffness curve of MRF

b) Damping curve of MRF

When $I=$2.0 A, the stiffness and damping decrease from 1.71×10⁵ and 4.52×10² N·m·s·rad⁻¹ to 0.76×10⁵ and 3.01×10² N·m·s·rad⁻¹, respectively, as the temperature rises from 20 to 90 °C. The reduction rates are 53.56 % and 33.41 %, respectively. The solidification characteristic of the MRF changes with the current, and its stiffness and damping increase exponentially with the current. For instance, at $T=$90 °C, the maximum stiffness and damping are 1.08×10² N·m·rad⁻¹ and 11.38 N·m·s·rad⁻¹, respectively, when the current is in the range of 0~0.875 A. Subsequently, when the current surpasses 0.875 A, the maximum stiffness and damping surge to 0.76×10⁵ N·m·rad⁻¹ and 3.01×10² N·m·s·rad⁻¹. This is because of the significant increase in the apparent viscosity of the MRF when subjected to an applied magnetic field. Consequently, the stiffness and damping increase several hundred times [21].

4.3. Coupled vibration characteristics of EDGTS

To analyze the influence law of MRF coupling on the coupled vibration characteristics of the EDGTS under multiple excitations, a step load excitation was applied. As Fig. 19 shows, the load torque of the system abruptly increased from 0 to 50 N·m at 2.0 s.

Fig. 19Step loads applied to the EDGTS

Numerical simulations were performed to analyze the dynamical behaviors of the EDGTS both with and without considering the operating temperature of the MRF coupling (cases 2 and 1, respectively). According to the results of the thermal-field analysis in Section 4.2, the operating temperature of the MRF coupling reached its maximum value under 2.0 A current excitation, resulting in the most significant attenuation of its vibration-damping performance. To ensure generality, 2.0 A current was used as the excitation power source for the MRF coupling in the following. The variation laws of the vibration displacement amplitudes of the motor input shaft node (node 4), MRF coupling (node 6), driving gear (node 10), and driven gear node (node 17) under the temperature influence were investigated.

Figs. 20 and 21 show the time-domain variation laws of the vibration displacement amplitudes of the four nodes of EDGTS in the x and y directions for the no-load (0-2.0 s) and rated-load operation (2.0-4.0 s) phases of the two cases, respectively. During 0-2.0 s phase, the system vibration increases after motor startup but stabilizes after 0.8 s. In case 1, the amplitude ranges of the four nodes in the x-direction are –0.06~0.01×10^-4 m, –0.06~0.01×10^-4 m, –0.06~0.01×10^-4 m, and –0.04~0.05×10^-4 m, respectively. In the y-direction, the amplitude ranges are –0.22~0.03×10^-5 m, –0.22~0.03×10^-5 m, –0.22~0.03×10^-5 m, and –0.13~0.19×10^-5 m, respectively. In case2, the amplitude of vibration displacements increased, and the amplitude ranges of the four nodes in the $x$-direction are –0.07~0.02×10^-4 m, –0.07~0.02×10^-4 m, –0.07~0.02×10^-4 m, and –0.05~0.08×10^-4 m, respectively. ln the $y$-direction, the amplitude ranges are –0.25~0.08×10^-5 m, –0.24~0.08×10^-5 m, –0.25~0.08×10^-5 m, and –0.17~0.28×10^-5 m, respectively. After 2.0 s, the amplitude ranges of the four nodes of case 1 in the $x$-direction are –2.42~0.01×10^-4 m, –2.43~0.01×10^-4 m, –2.44~0.01×10^-4 m, and –0.06~2.86×10^-4 m, respectively. The amplitude ranges of the four nodes in the $y$-direction are –8.82~0.01×10^-5 m, –8.85~0.01×10^-5 m, –8.88~0.01×10^-5 m, and –0.21~10.42×10^-5 m, respectively. As the operating temperature increases the mechanical properties of the MRF decrease, resulting in a significant increase in the range of vibration displacement amplitudes of the system. In case2, the amplitude ranges of the four nodes in the $x$-direction are –2.85~0.25×10^-4 m, –2.87~0.25×10^-4 m, –2.88~0.24×10^-4 m, and -0.36~2.93×10^-4 m, with growth rates of 27.57 %, 27.87 %, 27.35 %, and 12.67 %, respectively. The amplitude ranges in y-direction are –10.38~0.91×10^-5 m, –10.43~0.89×10^-5 m, –10.48~0.88×10^-5 m, and –1.29~10.67×10^-5 m, with growth rates of 27.86 %, 27.77 %, 27.78 %, and 12.51 %, respectively.

Fig. 20Vibration displacement amplitude of the four nodes in the x-direction

a) Node 4

b) Node 6

c) Node 10

d) Node 17

Fig. 21Vibration displacement amplitude of the four nodes in the y-direction

a) Node 4

b) Node 6

c) Node 10

d) Node 17

Figs. 22 and 23 show the variation laws of the vibration velocity RMS values with temperature for the four nodes in the $x$ and $y$ directions, respectively. The vibration velocity RMS values of each node in the EDGTS increase with the rise of the operating temperature. When the operating temperature increases from 20 °C to 90 °C, the RMS values of the four nodes in the $x$-direction increase from 1.22×10^-1 m/s, 1.26×10^-1 m/s, 1.23×10^-1 m/s, and 1.08×10^-2 m/s to 1.44×10^-1 m/s, 1.45×10^-1 m/s, 1.42×10^-1 m/s, and 1.86×10^-2 m/s, with growth rates of 17.09 %, 15.28 %, 15.67 %, and 72 %, respectively. In the $y$-direction, the RMS values increase from 4.47×10^-2 m/s, 4.58×10^-2 m/s, 4.48×10^-2 m/s, and 3.93×10^-3 m/s to 5.24×10^-2 m/s, 5.28×10^-2 m/s, 5.18×10^-2 m/s, and 6.75×10^-3 m/s, with growth rates of 17.09 %, 15.28 %, 15.67 %, and 72 %, respectively. The variation laws of the vibration velocity RMS values in the $x$ and $y$ directions remain consistent, and the vibration of node 17 in these two directions increases significantly when the temperature rises to 90 °C, with a growth rate of 72 %. The above results show that the rise in operating temperature has an inhibitory effect on the vibration-damping performance of the MRF coupling, which in turn affects the operating efficiency and work stability of the EDGTS.

Fig. 22The relationship between vibration velocity RMS values of four nodes in the x-direction and temperature

a) Node 4

b) Node 6

c) Node 10

d) Node 17

Fig. 23The relationship between vibration velocity RMS values of four nodes in the y- direction and temperature

a) Node 4

b) Node 6

c) Node 10

d) Node 17

Figs. 24 and 25 show the variation laws of the vibration acceleration maximum values with temperature for the four nodes in the $x$ and $y$ directions, respectively. The vibration acceleration maximum values of each node in the EDGTS is positively correlated with the operating temperature.

Fig. 24The relationship between vibration acceleration maximum values of four nodes in the x-direction and temperature

a) Node 4

b) Node 6

c) Node 10

d) Node 17

Fig. 25The relationship between vibration acceleration maximum values of four nodes in the y-direction and temperature

a) Node 4

b) Node 6

c) Node 10

d) Node 17

As the operating temperature rises from 20 °C to 90 °C, the maximum values of the four nodes in the $x$-direction increase from 67.54 m/s², 71.40 m/s², 171.59 m/s², and 8.20 m/s² to 92.75 m/s², 94.91 m/s², 277.33 m/s², and 12.17 m/s², with growth rates of 37.33 %, 32.92 %, 61.63 %, and 48.45 %, respectively. In the y-direction, the maximum values increase from 23.77 m/s², 25.99 m/s², 62.45 m/s², and 2.98 m/s² to 33.76 m/s², 34.54 m/s², 100.94 m/s², and 4.43 m/s², with growth rates of 42.02 %, 32.93 %, 61.62 %, and 48.45 %, respectively. As the temperature of the MRF rises, the base fluid then evaporates and irreversible thickening phenomenon occurs. This leads to a reduction in the vibration-damping performance of the MRF coupling, and deterioration in stability and working accuracy. The vibration amplitude of the EDGTS is exacerbated, affecting the service life of the system components. The most significant increase in the vibration acceleration maximum value of node 10 in the $x$-direction is observed when the temperature increases to 90 °C, with a value of 61.63 %.