**5.1. Configuration and modeling of MR valve**

366 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

objective function. Thus, the direction vector is calculated by

calculated according to Polak-Ribiere recursion formula as follows:

and gradient computations.

FORTRAN, C languages is required.

where *OBJ0* is the reference objective function value that is selected from the current group of design sets, *q* is the response surface parameter which controls constraint satisfaction. *Px* is the exterior penalty function applied to the design variable *x*. *Pg* is extended-interior penalty function applied to state variable (the constraint) *g*. For the initial iteration (*j = 0*), the search direction of DVs is assumed to be the negative of the gradient of the unconstrained

( 1) ( ) ( ) *j jj*

where the line search parameter *sj* is calculated by using a combination of a golden-section algorithm and a local quadratic fitting technique. The analysis file is then executed with the new values of DVs and the convergence of the objective function, *OBJ,* is checked. If the convergence occurs, the values of DVs at the *jth* iteration are the optimum. If not, the subsequent iterations will be performed. In the subsequent iterations, the procedures are similar to those of the initial iteration with the exception of the direction vectors which are

> ( ) ( ) ( 1) <sup>1</sup> ( ,) *jj j k j d Qx q r d* <sup>−</sup>

*Qx q Qx q Qx q <sup>r</sup>*

Thus, each iteration is composed of a number of sub-iterations that include search direction

It is noted that ANSYS software supports optimal design problems by integrating an optimization tool. Therefore, in most cases the optimal solution of the MRF based devices can be solved directly by the ANSYS software without interfacing with any programming software. In order to use the ANSYS optimization tool, it is necessary to set up optimization parameters. To do this, firstly the analysis file should be manually executed once to load all parameters in the analysis file into software buffer memory. After that, from the *Design Opt* menu, we specify the analysis file which will be used during optimization process, the DVs with their limits and tolerances, the state variables (if there are any) with limits and tolerances, the objective function with a convergence criteria, the method for solving the optimal solution, and the optimal output control option if necessary. In some cases, it is expected to employ some advanced optimization algorithms such as Genetic Algorithms, Neural Network, or user defined algorithms, the interfacing between the ANSYS and other software to perform the optimization such as Matlab,

∇ −∇ <sup>∇</sup> <sup>=</sup> ∇

<sup>1</sup> <sup>2</sup> ( 1)

*<sup>j</sup> <sup>j</sup>*

<sup>−</sup> <sup>−</sup>

( ) ( 1) ( )

*jj j T*

−

( ,)

[ ( , ) ( , )] ( , )

*Qx q*

The values of DVs in next iteration (*j+1*) is obtained from the following equation,

(0) (0) *d Qx* = −∇ ( ,1) (32)

*<sup>j</sup> x x sd* <sup>+</sup> = + (33)

<sup>−</sup> = −∇ + (34)

(35)

Figure 10 shows the structural configurations of the two typical types of MR valves: the annular MR relief valve (Figure 10a) and MR valve with both annular and radial flow paths (Figure 10b). The valve in Figure 10a consists of valve coil, cores and covers. The MRF flows through annular ducts between core A and core B. When the power of the coil is turned on, a magnetic field is exerted on the MRF, which causes the MRF flowing through the ducts to change its state into semi-liquid or solid and stop the flow. Only when the supply pressure gets high enough to offset the yield stress, the fluid can flow through the valve again. The valve in Figure 10b consists of the valve core, magnetic disk and valve housing form a magnetic circuit of the valve. A non-magnetic washer is used to warrant the required thickness of the radial duct. When the magnetic disk is placed coaxially with the valve housing using the cone-shape cap, the annular and radial ducts are formed between the disk and the valve housing, and the disk and the valve core, respectively. MRF flows from the inlet through the first annular and radial duct, then flow along the hole at the center of the core and after that follows the second radial and annular duct to the outlet.

**Figure 10.** Schematic diagrams of MR valves.

Figure 11a shows a simplified structure and significant dimensions of a single-coil annular MR valve. The valve geometry is featured by the overall effective length *L,* the outside radius *R*, the valve housing thickness *th,* the MR duct gap *tg*, the core flange (pole) thickness *tf*, and the coil width *wc*.

(a) single-coil annular MR valve (b) annular-radial flow MR valve

#### **Figure 11.** Simplified MR valve configurations

By using Bingham plastic model, the pressure drop of the valve is calculated by [80, 81]

$$
\Delta P\_A = \Delta P\_{A,\eta} + \Delta P\_{A,\tau} = \frac{6\eta L}{\pi t\_\chi^3 R\_d} \mathcal{Q} + 2c \frac{t\_f}{t\_\chi} \tau\_y \tag{36}
$$

where Δ*PA,*τ and Δ*PA,*η are the field-dependent and viscous pressure drop of the single annular MR valve, respectively, *Q* is the flow rate through the MR valve, *Rd* is the average radius of annular duct given by *Rd=R-th-0.5tg*, *c* is the coefficient which depends on flow velocity profile and has a value range from a minimum value of 2.0 (for Δ*PA,*τ /Δ*PA,*η greater than 100) to a maximum value of 3.0 (for Δ*PA,*τ /Δ*PA,*ηless than 1).

The multi-coil MR valve, which was first employed by Spencer et al. to make a high damping force MR damper used in seismic protection system [85], is now widely used in many applications. For MR valve with two coils, the pressure drop is calculated by

$$
\Delta P\_{2A} = \Delta P\_{2A, \eta} + \Delta P\_{2A, \tau} = \frac{6\eta L}{\pi t\_{\text{g}}^3 R\_d} Q + 2c\_1 \frac{t\_{f1}}{t\_{\text{g}}} \tau\_{y1} + c\_2 \frac{t\_{f2}}{t\_{\text{g}}} \tau\_{y2} \tag{37}
$$

where *∆P2A,*τ and *∆P2A,*η are the field-dependent and viscous pressure drop of the two- coil annular MR valve respectively, τ*y1* and τ*y2* are the yield stresses of the MRF in the end ducts and the middle duct, respectively. *c1* and *c2* are coefficient which depends on flow velocity profile of MR flow in the end ducts and the middle duct, respectively.

Similarly, for the three-coil annular MR valve, the pressure drop is calculated by

Optimal Design Methodology of Magnetorheological Fluid Based Mechanisms 369

$$
\Delta P\_{3A} = \Delta P\_{3A,\eta} + \Delta P\_{3A,\tau} = \frac{6\eta L}{\pi t\_{\mathcal{g}}^3 R\_d} \mathcal{Q} + 2(c\_1 \frac{t\_{f1}}{t\_{\mathcal{g}}} \tau\_{y1} + c\_2 \frac{t\_{f2}}{t\_{\mathcal{g}}} \tau\_{y2}) \tag{38}
$$

For the MR valve with both annular and radial flow paths shown in Figure 11b, the pressure drop can be calculated by

$$
\Delta P\_{AR} = \Delta P\_{AR,\eta} + \Delta P\_{AR,\mathfrak{r}} \tag{39}
$$

where *∆PAR,*τ and *∆PAR,*ηare determined by

368 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

**Figure 11.** Simplified MR valve configurations

than 100) to a maximum value of 3.0 (for

where

Δ*PA,*τ and Δ*PA,*η

where *∆P2A,*

τ

and *∆P2A,*

annular MR valve respectively,

η

By using Bingham plastic model, the pressure drop of the valve is calculated by [80, 81]

(a) single-coil annular MR valve (b) annular-radial flow MR valve

, , 3

 τ

*P P P Qc*

Δ =Δ +Δ = +

Δ*PA,*τ /Δ*PA,*η

many applications. For MR valve with two coils, the pressure drop is calculated by

*P P P Qc c*

π

Δ =Δ +Δ = + +

 τ

Similarly, for the three-coil annular MR valve, the pressure drop is calculated by

η

velocity profile and has a value range from a minimum value of 2.0 (for

η

τ*y1* and τ

profile of MR flow in the end ducts and the middle duct, respectively.

*AA A y*

annular MR valve, respectively, *Q* is the flow rate through the MR valve, *Rd* is the average radius of annular duct given by *Rd=R-th-0.5tg*, *c* is the coefficient which depends on flow

The multi-coil MR valve, which was first employed by Spencer et al. to make a high damping force MR damper used in seismic protection system [85], is now widely used in

2 2, 2, 3 1 12 2

and the middle duct, respectively. *c1* and *c2* are coefficient which depends on flow velocity

*AA A y y*

η π

η

<sup>6</sup> <sup>2</sup> *<sup>f</sup>*

*t R t*

*g d g L t*

are the field-dependent and viscous pressure drop of the single

less than 1).

<sup>6</sup> <sup>2</sup> *f f*

*t R t t*

*g d g g L t t*

are the field-dependent and viscous pressure drop of the two- coil

τ

τ

1 2

*y2* are the yield stresses of the MRF in the end ducts

 τ

(36)

(37)

Δ*PA,*τ /Δ*PA,*ηgreater

$$
\Delta P\_{AR,\tau} = \mathcal{Q}(c\_a \frac{t\_f}{t\_g} \boldsymbol{\sigma}\_{ya} + c\_r \frac{R\_2 - R\_0}{t\_g} \boldsymbol{\sigma}\_{yr}) \tag{40}
$$

$$\Delta P\_{AR,\eta} = 2\left[\frac{6\eta t\_f}{\pi t\_\mathcal{g}^3 R\_d} \mathcal{Q} + \frac{6\eta \mathcal{Q}}{\pi t\_\mathcal{g}^3} \ln(\frac{R\_d}{R\_o})\right] + \frac{8\eta (L - 2t\_f)\mathcal{Q}}{\pi R\_0^4} \tag{41}$$

In the above, τ*ya* and τ*yr* are the induced yield stresses of the MRF in the annular duct and the radial duct, respectively. *Ro* is the radius of the hole at the center of the valve core and *R2* is the outer radius of the radial duct. Here, *ca* and *cr* are coefficients that depend on the velocity profile of MRF flowing through the annular and radial ducts, respectively.

#### **5.2. Optimization of MR valves considering pressure drop and dynamic range**

The optimal objective is to minimize the valve ratio defined by the ratio of the viscous pressure drop to the field-dependent pressure drop of the MR valve. This ratio has great effect on the characteristics of the MR valve. It is desirable that the valve ratio takes a small value. The valves are constrained in a cylinder of the radius *R*=30*mm* and the height *H*=50*mm*. Magnetic properties of valve components are given in Table 1. The post-yield viscosity of the MRF is assumed to be constant, η=0.092*Pa.s* and the flow rate of the MR valves is *Q*=3E-4 *m3/s*. The commercial MR fluid (MRF132-DG) from Lord Corporation is used. The induced yield stress of the MR fluid as a function of the applied magnetic field intensity (*Hmr*) can be approximately expressed by

$$\sigma\_y = p(H\_{mr}) = \mathbf{C}\_0 + \mathbf{C}\_1 H\_{mr} + \mathbf{C}\_2 H\_{mr}^2 + \mathbf{C}\_3 H\_{mr}^3 \tag{42}$$

In Eq. (42), the unit of the yield stress is *kPa* while that of the magnetic field intensity is *kA/m*. The coefficients *C0, C1, C2*, and *C3*, determined from experimental results by applying the least square curve fitting method, are respectively identified as 0.3, 0.42, -0.00116 and 1.05E-6.

It is noted that, a small change in the valve gap *tg* would drastically alter the performance of the MR valve. Therefore, in MR valve design, a fixed gap is chosen according to each application. In this study, the valve gap is chosen as 1*mm*. From Eqs. (36)-(41), the valve ratios of the single-coil, two-coil and radial-annular MR valve are respectively calculated by

$$\mathcal{A}\_A = \frac{\Delta P\_{A,\eta}}{\Delta P\_{A,\pi}} = \frac{3\eta H Q}{\pi t\_g \,^2 R\_d c t \sigma\_y} \tag{43}$$

$$\beta\_{2A} = \frac{\Delta P\_{2A,\eta}}{\Delta P\_{2A,\tau}} = \frac{3\eta H Q}{\pi t\_{\text{\textdegree\textquotedblleft}}^2 R\_d c (t\tau\_{yt} + 0.5a\tau\_{yu})}\tag{44}$$

$$\lambda\_{AR} = \frac{\frac{2\eta Q}{\pi} \text{(I}\frac{3t}{t\_{\text{g}}} + \frac{3}{d^2}\ln(\frac{R\_d}{R\_o})\text{I} + \frac{2t\_{\text{g}}(H - 2t)}{R\_0^4}\text{)}}{c(t\tau\_{ya} + (R\_2 - R\_0)\tau\_{yr})}\tag{45}$$

The ANSYS APDL program is the analysis ANSYS APDL code used in optimal design of the annular single MR valve. The analysis ANSYS APDL code for other types of MR valve can be prepared in the same maner.


**Table 1.** Magnetic properties of the valve components

**Figure 12.** Magnetic properties of silicon steel and MR fluid


*A*

η

π

λ

*AR*

λ

be prepared in the same maner.

Valve

Components

Nonmagnetic Cap/Bobbin

*A*

2 , 2 2 2 ,

*A*

τ

<sup>=</sup> + −

τ

Valve Core Silicon Steel B-H curve (Fig. 12a) 1.5 Tesla Valve Housing Silicon Steel B-H curve (Fig. 12a) 1.5 Tesla

MR Fluid MRF132-DG B-H curve (Fig. 12b) 1.6 Tesla

Coil Copper 1 x

Nonmagnetic

Steel

**Figure 12.** Magnetic properties of silicon steel and MR fluid

**Table 1.** Magnetic properties of the valve components

λ

<sup>2</sup> , *<sup>A</sup>* 3

<sup>Δ</sup> = = <sup>Δ</sup> η

τ

*P HQ P t Rct a* <sup>Δ</sup> = = <sup>Δ</sup> <sup>+</sup> η

*g d o*

*ct R R*

The ANSYS APDL program is the analysis ANSYS APDL code used in optimal design of the annular single MR valve. The analysis ANSYS APDL code for other types of MR valve can

*P HQ P t R ct*

π

*A g d y*

3

η

πττ

*A g d yt ya*

2 2 4

2 33 2 ( 2) {[ ln( )] )

*Q t R tH t tR d R R*

( ( ))

*ya yr*

2 0

η

> τ

( 0.5 )

*g d*

 τ

Material Relative Permeability Saturation Flux

1 x

<sup>−</sup> + +

0

(43)

(44)

Density

(45)

,



**Table 2.** ANSYS APDL program

372 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

LSEL, , , ,P51X CM,\_Y1,LINE CMSEL,,\_Y

FLST,5,8,4,ORDE,7

FITEM,5,1 FITEM,5,-2 FITEM,5,10 FITEM,5,13 FITEM,5,18 FITEM,5,-20 FITEM,5,22 CM,\_Y,LINE LSEL, , , ,P51X CM,\_Y1,LINE CMSEL,,\_Y

! Meshing MSHAPE,0,2D MSHKEY,0

AMESH,all

!\*

!\* FINISH

/SOL

FITEM,2,1 FITEM,2,-2 FITEM,2,7 FITEM,2,17 FITEM,2,20 FITEM,2,-27 DL,P51X, ,ASYM

!\*

LESIZE,\_Y1, , ,msize\*2, , , , ,1

LESIZE,\_Y1, , ,msize, , , , ,1

! Solving magnetic circuit

! Apply current density to the coil area

! Boundary condition FLST,2,12,4,ORDE,6

FLST,2,1,5,ORDE,1

BFA,P51X,JS, , ,J,0

NCNV,0,0,0,0,0,

FITEM,2,1

! Solving

SOLVE

MPTEMP,1,0

pi=3.1416

Rw=R-th-d

Ac=pi\*dc\*\*2/4 rrc=res/Ac

Resistance

!\*

!\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

!Geometric definition H=0.05 ! Height R=0.03 ! Outer Radius w=0.010 ! Coil width t=0.017 ! Pole length

th=0.0065 ! Housing thickness d=0.001 ! MRF duct gap

res=0.0172e-6 ! Wire Resistivity

J=I\*4/dc/dc/pi ! Current density PP=I\*\*2\*Rc ! Consumption Power

Nturn=w\*(H-2\*t)/Ac ! No of turns Rc=rrc\*Nturn\*pi\*2\*(Rw-0.5\*w) ! Wire

dc=0.00052 ! Wire Radius

I=2.5 ! Applied current

! geometric model RECTNG,0,R,0,H, RECTNG,Rw,R-Th,0,H, RECTNG,Rw-W,Rw,t,H-t, RECTNG,0,Rw,0.0,t, RECTNG,0,Rw,H-t,H, RECTNG,Rw,R,t,H-t,

FLST,2,6,5,ORDE,2

FLST,5,3,5,ORDE,3

FITEM,2,1 FITEM,2,-6 AOVLAP,P51X NUMCMP,LINE NUMCMP,AREA ! Material assignment

! Housing

FITEM,5,5 FITEM,5,9 FITEM,5,-10

MPDATA,MURX,4,,1 ! Bobbin

msize=12 !Basic No. of elements/line

Figure 13 shows the optimal solution of a single-coil annular MR valve constrained in the specific volume when a current of 2.5*A* is applied to the valve coil. Initial values of *t*, *wc* and *th* are 17*mm*, 10*mm* and 6.5*mm*, respectively. The valve ratio, pressure drop and power consumption of the valve at these initial values are λ*0*=0.08274, Δ*P0*=15*bar* and *N0*=38.83*W*, respectively. From the figure, it is observed that the solution is convergent after 13 iterations and the minimum value of valve ratio (objective function) is λ*opt* =0.033. The corresponding pressure drop isΔ*Popt*=37.32*bar*, which is also the maximum. At the optimum, the power consumption is *Nopt*=7.92*W* which is much smaller than at the initial. The DVs at the optimum are *topt*=7.23 *mm*, *wc,opt*=1.78 *mm*, *th,opt*=7.43*mm*.

**Figure 13.** Optimal solution of single-coil MR valves considering the valve ratio and pressure drop

Figure 14 shows the optimal solution of the two-coil annular MR valve. Initial values of *a*, *t*, *wc* and *th* are 10*mm*, 5*mm*, 10*mm* and 4*mm*, respectively. The valve ratio, pressure drop and power consumption at these initial values are λ*0*=0.0381, Δ*P0*=28.2*bar* and *N0*=83.2*W*, respectively. The solution is convergent after 11 iterations and the minimum value of valve ratio isλ*opt* =0.023. The corresponding pressure drop is Δ*Popt*=48.6*bar*, which is also the maximum. The optimal DVs are *aopt*=19.7 *mm, topt*=10.6 *mm*, *wc,opt*=6.38 *mm*, *th,opt*=5.33*mm*.

**Figure 14.** Optimal solution of two-coil MR valves considering the valve ratio and pressure drop

Figure 15 shows the optimization solution of the annular-radial MR valve. Initial values of *R0*, *t*, *wc* and *th* are 6*mm*, 10*mm*, 6*mm* and 8*mm*, respectively. The valve ratio, pressure drop and power consumption at these initial values are λ*<sup>0</sup>*=0.041, *∆P0*=47*bar* and *N0*=44.3*W*, respectively. The convergence occurs at 10 iteration, at which the minimum value of valve ratio is λ*opt*=0.023 and the optimal design parameters are *R0,opt*=14.41 *mm*, *topt*=6.47 *mm*, *wc,opt*=2.32 *mm*, *th,opt*=4.81*mm*. The corresponding pressure drop is *∆Popt* =37.2*bar*, which is **not** the maximum pressure drop. The reason for this is that the uncontrolled pressure drop (viscous pressure drop) of the valve significantly depends on the valve core radius. An increase of the valve core radius results in a decrease of the viscous pressure drop by which reduces the valve ratio. However, the increase of the valve core radius causes a decrease of the magnetic flux density, and by which reduces the pressure drop of the valve. In order to improve the valve performance, the valve core radius should be fixed at an appropriate. In case the valve core radius is fixed at 6*mm*, it was found that the optimal value of valve ratio is λ*opt*=0.0293 and the corresponding pressure drop is *∆Popt* =64.4*bar* , which is also the maximum. The optimal DVs are *topt*=8.6*mm*, *wc,opt*=3.1 *mm* and *th,opt*=6.36 *mm*. At these optimal DVs the power consumption is *N0*=29.1*W*.

374 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

N(W)

0

λ

wc

λ

*0*=0.0381,

Δ

Δ

t a

2 4 6 8 10

Iteration

*<sup>0</sup>*=0.041, *∆P0*=47*bar* and *N0*=44.3*W*,

5

10

Value of Parameters (mm)

**Figure 13.** Optimal solution of single-coil MR valves considering the valve ratio and pressure drop

Figure 14 shows the optimal solution of the two-coil annular MR valve. Initial values of *a*, *t*, *wc* and *th* are 10*mm*, 5*mm*, 10*mm* and 4*mm*, respectively. The valve ratio, pressure drop

respectively. The solution is convergent after 11 iterations and the minimum value of

the maximum. The optimal DVs are *aopt*=19.7 *mm, topt*=10.6 *mm*, *wc,opt*=6.38 *mm*,

5

**Figure 14.** Optimal solution of two-coil MR valves considering the valve ratio and pressure drop

Figure 15 shows the optimization solution of the annular-radial MR valve. Initial values of *R0*, *t*, *wc* and *th* are 6*mm*, 10*mm*, 6*mm* and 8*mm*, respectively. The valve ratio, pressure drop

respectively. The convergence occurs at 10 iteration, at which the minimum value of valve

*opt*=0.023 and the optimal design parameters are *R0,opt*=14.41 *mm*, *topt*=6.47 *mm*,

10

15

Value of Parameters[mm]

20

25

*opt* =0.023. The corresponding pressure drop is

15

20

2 4 6 8 10 12

wc

t t h

*P0*=28.2*bar* and *N0*=83.2*W*,

*Popt*=48.6*bar*, which is also

t h

Iteration

ΔP(bar)

2 4 6 8 10 12

Iteration

and power consumption at these initial values are

N(W)

ΔP(bar)

2 4 6 8 10

Iteration

and power consumption at these initial values are

0

valve ratio is

*th,opt*=5.33*mm*.

10

ratio is

λ

20

30

Value of Parameters

40

50

60

λ

 λx103 

10

20

30

Value of Parameters

40

50

60

λx10<sup>3</sup> 

**Figure 15.** Optimal solution of annular-radial MR valves considering the valve ratio and pressure drop

Table 3 summarizes the optimization results for MR valve design abovementioned. The results show that the geometry of MR valve has a great effect on the valve performance. By choosing an optimal geometry, the valve performance such as pressure drop can be much improved and the power consumption can be significantly reduced. Among the MR valves constrained in the same volume, the two-coil annular MR valve provides the best value of valve ratio while the annular-radial can provide the best pressure drop at the optimal design parameters. For MR valves with three coils or more, it was shown that the performance of these valves is not better than that of the two-coil MR valve at optimal design parameters.

It was also shown by Nguyen et al. [79] that the optimal solution is affected by the applied current. The higher value of the applied current is the better performance of the valve is. However, when the applied current increases to a certain value the optimal solutions tends to be saturation. Therefore, it is advised that the applied current should be set by it maximum allowable value in the optimization problem of the MR valve.


**Table 3.** Optimization results for MR valve design
