**3. Thermal model of the actuator**

In this section we present the procedure adopted to build the thermal model for the AMRK to carry out a transient temperature simulation across a long walk over the ground. We set up a model for each element that comprises the actuator: EC motor, harmonic drive, MR clutch, MR brake, bearings and housing. The detailed method used is described in the following subsections.

#### **3.1 Motor-reducer model**

According to Maxon's Key Information for DC and EC motors [42], the losses of efficiency in the motor are divided into losses due to friction, *P*mec, and due to the Joule effect, *P*J, of the winding, which has resistance *R*a. The energetic balance can be treated as:

$$P\_{cl} = P\_{mc} + P\_f \tag{5}$$

where *P*el is the electrical power of the motor. The dissipation of power due Joule effect is given by:

$$P\_f = R\_a \cdot I\_a^{\;\;\;\;\;\prime} \tag{6}$$

where *I*a is the input current in the motor armature.

The thermal properties of the motor are given by the supplier catalog. We considered the EC 60 flat model 411,678 for the analysis. The temperature of the motor housing can be predicted when the free surfaces are submitted to air as in [43]:

$$P\_f = \frac{T\_S - T\_\circ}{R\_{th2}} \tag{7}$$

where *T*S is the surface temperature, *T*∞ is the environment temperature (25°C), and *R*th2 is the thermal resistance between the surface and the environment, which is given by the supplier catalog, and can also be described by the following equation [43]:

$$R\_{th2} = \frac{1}{h\_{coub}A\_{\text{S}}} \tag{8}$$

**379**

**Figure 3.**

**Figure 2.**

28.86°C was obtained after 2 hours.

*Input current in the motor armature during the gait cycle.*

*Transient Thermal Analysis of a Magnetorheological Knee for Prostheses and Exoskeletons…*

We carried out a simulation considering the input current in the motor for the intermittent operating condition due to the gai cycle. A constant temperature of

*Simulation of the motor's external temperature in nominal operating conditions. The highlighted color map* 

Regarding the harmonic drive, the heat generation (*P*HD) occurs basically due

*P N HD f* =τ ω

(9)

friction between its movable parts and it can be calculated as follows:

*DOI: http://dx.doi.org/10.5772/intechopen.95372*

*represents the steady state temperature of the motor's surface.*

where *h*comb is the combined heat transfer coefficient for convection and radiation and *A*S is the heat transfer surface area. By rearranging Eq. (8), and considering the motor assembled in a plastic plate to reproduce the standard cooling conditions, as described in [44], the coefficient *h*comb = 1.72x10E-5 W/(mm2 °C) is obtained.

In order to validate the EC 60 motor thermal model to be used in the actuator, a transient simulation of the motor's temperature is shown in **Figure 2**. The external surface of the motor reaches a maximum temperature of 51.12°C in about 7200 s, which is very close to the temperature obtained by the Eq. (7) (51.15°C), thereby validating the used method.

As previously described, the motor is used just when positive work of the knee joint is required. For intermittent work, the supplier recommends using the average input current, IRMS, in the motor armature during the cycle operation. The input current variation for the proposed working modes (**Figure 1(d)**) is presented in **Figure 3**.

*Transient Thermal Analysis of a Magnetorheological Knee for Prostheses and Exoskeletons… DOI: http://dx.doi.org/10.5772/intechopen.95372*

**Figure 2.**

*Heat Transfer - Design, Experimentation and Applications*

method used is described in the following subsections.

where *I*a is the input current in the motor armature.

In this section we present the procedure adopted to build the thermal model for the AMRK to carry out a transient temperature simulation across a long walk over the ground. We set up a model for each element that comprises the actuator: EC motor, harmonic drive, MR clutch, MR brake, bearings and housing. The detailed

According to Maxon's Key Information for DC and EC motors [42], the losses of efficiency in the motor are divided into losses due to friction, *P*mec, and due to the Joule effect, *P*J, of the winding, which has resistance *R*a. The energetic balance can

where *P*el is the electrical power of the motor. The dissipation of power due Joule

The thermal properties of the motor are given by the supplier catalog. We considered the EC 60 flat model 411,678 for the analysis. The temperature of the motor housing can be predicted when the free surfaces are submitted to air as in [43]:

*S*

*th T T <sup>P</sup> R* <sup>−</sup> <sup>∞</sup> <sup>=</sup> 2

where *T*S is the surface temperature, *T*∞ is the environment temperature (25°C),

*comb S*

where *h*comb is the combined heat transfer coefficient for convection and radiation and *A*S is the heat transfer surface area. By rearranging Eq. (8), and considering the motor assembled in a plastic plate to reproduce the standard cooling conditions, as

In order to validate the EC 60 motor thermal model to be used in the actuator, a transient simulation of the motor's temperature is shown in **Figure 2**. The external surface of the motor reaches a maximum temperature of 51.12°C in about 7200 s, which is very close to the temperature obtained by the Eq. (7) (51.15°C), thereby

As previously described, the motor is used just when positive work of the knee joint is required. For intermittent work, the supplier recommends using the average input current, IRMS, in the motor armature during the cycle operation. The input current variation for the proposed working modes (**Figure 1(d)**) is presented in

*J*

and *R*th2 is the thermal resistance between the surface and the environment, which is given by the supplier catalog, and can also be described by the following

*th*

*R*

described in [44], the coefficient *h*comb = 1.72x10E-5 W/(mm2

*PP P el mec J* = + (5)

*P RI J aa* = <sup>2</sup> · (6)

*h A* <sup>2</sup> <sup>=</sup> <sup>1</sup> (8)

°C) is obtained.

(7)

**3. Thermal model of the actuator**

**3.1 Motor-reducer model**

be treated as:

effect is given by:

equation [43]:

validating the used method.

**378**

**Figure 3**.

*Simulation of the motor's external temperature in nominal operating conditions. The highlighted color map represents the steady state temperature of the motor's surface.*

#### **Figure 3.**

*Input current in the motor armature during the gait cycle.*

We carried out a simulation considering the input current in the motor for the intermittent operating condition due to the gai cycle. A constant temperature of 28.86°C was obtained after 2 hours.

Regarding the harmonic drive, the heat generation (*P*HD) occurs basically due friction between its movable parts and it can be calculated as follows:

$$P\_{\rm HD} = \tau\_f a o \mathcal{N} \tag{9}$$

where *τ*f is the friction torque that can be gathered from the supplier datasheet (Harmonic Drive AG, Germany), and *ω* is the angular velocity of the knee joint, as shown in **Figure 1(d)**, and *N* is the HD reduction ratio (*N* = 100). The presented intermittent working conditions for the CSG-14-100-2a, results in a RMS heat generation of 1.04 W.

### **3.2 MR clutch and MR brake**

The proposed operating modes for the AMRK during over-ground walking consider that the MR clutch works in non-slip condition when the magnetic field is activated and in full-slip condition when no transmitting torque is required. Since the power is dissipated as heat in the MR fluid region and it is a function of the torque and the angular velocity between disks, the MR clutch does not present heat generation in the fluid area. On the other hand, the MR brake is always subjected to the full-slip operation and generates heat in the MR fluid region. However, the brake just works when braking torque or joint stabilization is required, thereby minimizing the heat generation.

The fluid used in the prototype is the MRF-140 CG, which has a recommended working temperature between −40°C and 130°C, according to the manufacturer. However, Chen et al. [33] reported that there is a reduction in the yield stress of the MR fluid at temperatures around 100°C, due to the deterioration of some additives. For this reason, it is safe to limit the working temperature of the MR fluid to 100°C. Heat generation due to the sliding condition is given by [35]:

$$
\dot{\Phi}\_d = \frac{T\phi}{V\_f} \tag{10}
$$

where Φ*<sup>d</sup>* is the volumetric generation of heat in the fluid, *T* and *ω* are the torque and angular velocity of the knee, respectively, and *V*f is the volume of the fluid.

The loss of electrical power due to the Joule effect on the coil can be written as [35].

$$
\dot{\Phi}\_{\text{C}} = \frac{I^2 R\_{\text{C}}}{V\_{\text{C}}} \tag{11}
$$

where Φ*<sup>C</sup>* is the volumetric generation of heat in each coil, *I* is the electric current in the coil, *R*C is the resistance of the coil, and *V*c is the volume of the coil. The electrical current variation in the MR clutch/brake coils during the gait cycle for the proposed working modes is shown in **Figure 4**.

The other components to be modeled are the bearings and the convection coefficient on the free surfaces of the actuator. According to the NTN bearing catalog [45], bearing friction becomes a cause of heat generation that must be considered. Eq. (12) describes the heat generation applied to the bearings.

$$
\dot{\Phi}\_{\text{B}} = \frac{1.05 \times 10^{-4} \,\mu P d \Delta \phi}{2V\_{\text{B}}} \tag{12}
$$

**381**

**Figure 5.**

*Transient Thermal Analysis of a Magnetorheological Knee for Prostheses and Exoskeletons…*

Natural convection and radiation are also considered for the components whose surfaces are exposed to the air. The heat transfer coefficient, *h*T, composed by radia-

The 3D modeling of the AMRK is shown in **Figure 5**. Since the actuator presents an axisymmetric configuration, a part corresponding to 1/360 of the actuator was

where *h*c is the natural convection coefficient and *h*r is the heat radiation

*h hh T cr* = + (13)

°C) [35].

*DOI: http://dx.doi.org/10.5772/intechopen.95372*

tion and natural convection, is given by:

**3.3 FEM analysis**

**Figure 4.**

coefficient. Here we considered *h*T = 9.7x10E-6 W/(mm<sup>2</sup>

*AMRK Discretized model reduced to 1/360 to improve time processing.*

*Input current on the MR Clutch and MR Brake coils during the gait cycle.*

where Φ*<sup>B</sup>* is the volumetric heat generation in the bearing, *μ* is the friction coefficient given by the manufacturer's datasheet, *P* is the load to which the bearing is subjected, *ω* is the knee joint angular velocity, *d* is the inner diameter of the bearing, and *V*B is the volume of the bearing.

*Transient Thermal Analysis of a Magnetorheological Knee for Prostheses and Exoskeletons… DOI: http://dx.doi.org/10.5772/intechopen.95372*

**Figure 4.** *Input current on the MR Clutch and MR Brake coils during the gait cycle.*

Natural convection and radiation are also considered for the components whose surfaces are exposed to the air. The heat transfer coefficient, *h*T, composed by radiation and natural convection, is given by:

$$h\_T = h\_c + h\_r \tag{13}$$

where *h*c is the natural convection coefficient and *h*r is the heat radiation coefficient. Here we considered *h*T = 9.7x10E-6 W/(mm<sup>2</sup> °C) [35].

#### **3.3 FEM analysis**

*Heat Transfer - Design, Experimentation and Applications*

generation of 1.04 W.

**3.2 MR clutch and MR brake**

minimizing the heat generation.

where *τ*f is the friction torque that can be gathered from the supplier datasheet (Harmonic Drive AG, Germany), and *ω* is the angular velocity of the knee joint, as shown in **Figure 1(d)**, and *N* is the HD reduction ratio (*N* = 100). The presented intermittent working conditions for the CSG-14-100-2a, results in a RMS heat

The proposed operating modes for the AMRK during over-ground walking consider that the MR clutch works in non-slip condition when the magnetic field is activated and in full-slip condition when no transmitting torque is required. Since the power is dissipated as heat in the MR fluid region and it is a function of the torque and the angular velocity between disks, the MR clutch does not present heat generation in the fluid area. On the other hand, the MR brake is always subjected to the full-slip operation and generates heat in the MR fluid region. However, the brake just works when braking torque or joint stabilization is required, thereby

The fluid used in the prototype is the MRF-140 CG, which has a recommended working temperature between −40°C and 130°C, according to the manufacturer. However, Chen et al. [33] reported that there is a reduction in the yield stress of the MR fluid at temperatures around 100°C, due to the deterioration of some additives. For this reason, it is safe to limit the working temperature of the MR fluid to 100°C.

*d*

where Φ*<sup>d</sup>* is the volumetric generation of heat in the fluid, *T* and *ω* are the torque and angular velocity of the knee, respectively, and *V*f is the volume of

*C*

where Φ*<sup>C</sup>* is the volumetric generation of heat in each coil, *I* is the electric current in the coil, *R*C is the resistance of the coil, and *V*c is the volume of the coil. The electrical current variation in the MR clutch/brake coils during the gait cycle

The other components to be modeled are the bearings and the convection coefficient on the free surfaces of the actuator. According to the NTN bearing catalog [45], bearing friction becomes a cause of heat generation that must be considered.

*B*

µ

*V* <sup>−</sup> × ∆ Φ = <sup>4</sup> 1.05 10 2

where Φ*<sup>B</sup>* is the volumetric heat generation in the bearing, *μ* is the friction coefficient given by the manufacturer's datasheet, *P* is the load to which the bearing is subjected, *ω* is the knee joint angular velocity, *d* is the inner diameter of the

*Pd*

 ω

Φ =

The loss of electrical power due to the Joule effect on the coil can be written as [35].

2

*C*

*C I R V*

*f T V* Φ =

ω

(10)

(11)

(12)

Heat generation due to the sliding condition is given by [35]:

for the proposed working modes is shown in **Figure 4**.

bearing, and *V*B is the volume of the bearing.

Eq. (12) describes the heat generation applied to the bearings.

*B*

**380**

the fluid.

The 3D modeling of the AMRK is shown in **Figure 5**. Since the actuator presents an axisymmetric configuration, a part corresponding to 1/360 of the actuator was

**Figure 5.** *AMRK Discretized model reduced to 1/360 to improve time processing.*


#### **Table 2.**

*Boundary conditions for the transient thermal analysis simulation.*

simulated, to reduce the processing time. Unstructured tetrahedral elements were used for the model's mesh. Considering the dimensional differences of the actuator components, after a heuristic analysis of the actuator's mesh, different mesh sizes were adopted to obtain an acceptable precision of the results with reduced processing time. Mesh size for the MR fluid regions was 0.25 mm and for the other components we used mesh size of 2 mm.

To perform the transient thermal analysis, the AMRK is subjected to the operating modes presented in **Figure 1(d)** for 10,000 seconds, which represents 10.0 km walk, to reach the steady state condition. In such a simulation when the MU is activated, operating mode II, the EC motor, HD, MR clutch coil and bearings are the heat sources. When the MR brake is activated, operating mode I and III, the MR fluid and the brake coil and the larger bearings are the heat sources. The values of the boundary conditions are given in **Table 2**.

## **4. Results and discussion**

After build the thermal model of the AMRK, we carried out a transient thermal analysis considering the actuator is subjected to over-ground walking for a long period of time. The initial temperature of the actuator and environment was 25°C. **Figure 6(a)** presents the temperature color map for the steady state condition and **Figure 6(b)** shows the temperature variation across time in each component of the actuator.

After 6000 s of simulation, which represents a 6.0 km walk, the AMRK reached the steady state temperature distribution, **Figure 6(b)**. The simulation ran up to 10,000 s, but no change in the temperature distribution was observed, as presented in **Figure 6(a)**. The MR fluid is considered the most sensible element of the actuator to temperature rise and must be carefully evaluated. Moreover, for safe operation, it is important that the temperature of the copper wire does not exceed 150°C and the actuator's housing surface temperature does not exceed 43°C, so it does not cause injury to the user [46].

The maximum temperature observed after the simulation time was 30.86°C in the MR fluid region of the MR brake. As previously mentioned, the MR brake works in full-slip condition and dissipate energy as heat in the MR fluid region. The fullslip operation is the most critical condition for heat generation in MR fluid-based

**383**

**Figure 6.**

*Transient Thermal Analysis of a Magnetorheological Knee for Prostheses and Exoskeletons…*

devices and can lead to critical temperatures [41]. However, unlike the common applications of shear transmissions, the proposed operating modes for the AMRK just activate the MR brake, which works in full-slip condition, when the knee is subjected to negative work, thereby reducing the heat generated. The MR clutch, as previously mentioned, works in non-slip condition and the heat is generated by Joule effect in the coil and not in the fluid area. Compared to previous applications of MR fluid-based transmissions, such as the shear transmission of Chen et al. [33] and the MR clutches proposed in Wang et al. [35, 41] and Leal-Junior et al. [36], where the temperature reached more than 120°C in full-slip condition, the temperature reached by the AMRK while working with the proposed operating modes, is not critical to the MR fluid integrity. On the other hand, from a braking torque control point of view, a more careful analysis should evaluate if this temperature increase can affect torque capacity of the MR brake. To improve the braking torque control in a rising temperature scenario a temperature-dependent shear stress, as

*Transient thermal analysis of the AMRK. (a) Temperature color map for steady state condition.* 

proposed in [35], should be taken in to account.

*(b) Temperature variation of actuator's components over time.*

*DOI: http://dx.doi.org/10.5772/intechopen.95372*

*Transient Thermal Analysis of a Magnetorheological Knee for Prostheses and Exoskeletons… DOI: http://dx.doi.org/10.5772/intechopen.95372*

#### **Figure 6.**

*Heat Transfer - Design, Experimentation and Applications*

**Variable Value** Motor heat generation 7.83E-6 W/mm3 Harmonic drive heat generation 3.83E-5 W/mm3 MR Brake coil heat generation 1.81E-4 W/mm3 MR Clutch coil heat generation 2.22E-5 W/mm3 Large bearings heat generation 3.98E-7 W/mm3 Small bearing heat generation 4.51E-7 W/mm3 MR fluid heat generation (MR brake) 1.03E-3 W/mm3 Motor heat transfer coefficient1 1.72E-5 W/mm2

Heat transfer coef. remaining components1 9.70E-6 W/mm2

nents we used mesh size of 2 mm.

**4. Results and discussion**

cause injury to the user [46].

the boundary conditions are given in **Table 2**.

*Boundary conditions for the transient thermal analysis simulation.*

simulated, to reduce the processing time. Unstructured tetrahedral elements were used for the model's mesh. Considering the dimensional differences of the actuator components, after a heuristic analysis of the actuator's mesh, different mesh sizes were adopted to obtain an acceptable precision of the results with reduced processing time. Mesh size for the MR fluid regions was 0.25 mm and for the other compo-

°C

°C

To perform the transient thermal analysis, the AMRK is subjected to the operating modes presented in **Figure 1(d)** for 10,000 seconds, which represents 10.0 km walk, to reach the steady state condition. In such a simulation when the MU is activated, operating mode II, the EC motor, HD, MR clutch coil and bearings are the heat sources. When the MR brake is activated, operating mode I and III, the MR fluid and the brake coil and the larger bearings are the heat sources. The values of

After build the thermal model of the AMRK, we carried out a transient thermal analysis considering the actuator is subjected to over-ground walking for a long period of time. The initial temperature of the actuator and environment was 25°C. **Figure 6(a)** presents the temperature color map for the steady state condition and **Figure 6(b)** shows the temperature variation across time in each component of the

After 6000 s of simulation, which represents a 6.0 km walk, the AMRK reached the steady state temperature distribution, **Figure 6(b)**. The simulation ran up to 10,000 s, but no change in the temperature distribution was observed, as presented in **Figure 6(a)**. The MR fluid is considered the most sensible element of the actuator to temperature rise and must be carefully evaluated. Moreover, for safe operation, it is important that the temperature of the copper wire does not exceed 150°C and the actuator's housing surface temperature does not exceed 43°C, so it does not

The maximum temperature observed after the simulation time was 30.86°C in the MR fluid region of the MR brake. As previously mentioned, the MR brake works in full-slip condition and dissipate energy as heat in the MR fluid region. The fullslip operation is the most critical condition for heat generation in MR fluid-based

**382**

actuator.

*1*

**Table 2.**

*Room temperature = 25°C.*

*Transient thermal analysis of the AMRK. (a) Temperature color map for steady state condition. (b) Temperature variation of actuator's components over time.*

devices and can lead to critical temperatures [41]. However, unlike the common applications of shear transmissions, the proposed operating modes for the AMRK just activate the MR brake, which works in full-slip condition, when the knee is subjected to negative work, thereby reducing the heat generated. The MR clutch, as previously mentioned, works in non-slip condition and the heat is generated by Joule effect in the coil and not in the fluid area. Compared to previous applications of MR fluid-based transmissions, such as the shear transmission of Chen et al. [33] and the MR clutches proposed in Wang et al. [35, 41] and Leal-Junior et al. [36], where the temperature reached more than 120°C in full-slip condition, the temperature reached by the AMRK while working with the proposed operating modes, is not critical to the MR fluid integrity. On the other hand, from a braking torque control point of view, a more careful analysis should evaluate if this temperature increase can affect torque capacity of the MR brake. To improve the braking torque control in a rising temperature scenario a temperature-dependent shear stress, as proposed in [35], should be taken in to account.

The maximum temperature reached in the motor and harmonic drive was 29.94°C and 29.92°C, respectively, which are within the working limit temperature recommended by the supplier's catalog, up to 125°C and 120°C, respectively. The maximum temperature reached on the surface of the actuator was 30.68°C, which is not harmful to human skin [46]. These results validate the purpose usage of the AMRK as an actuator for knee replacement/assistance for over-ground walking.
