**3. Tensometric wheel pairs**

248 Advanced Topics in Measurements

The aforementioned five parameters are varied in the range from 250 to 640 mm. The design of experiment for 5 factors and 112 trial points is calculated with MSE criterion value 0.5394. In the next step responses of these models are calculated by SW Simulation using shell elements with global size 7.5 mm and total number of DoF ~108000. Then these responses are used for approximation by EDAOpt. Using obtained locally weighted polynomial approximations by global search procedure, the optimal cutout shape is obtained (see Figure 6) for the different aforementioned techniques. In table 2 the results are summarized and compared with the volume obtained in work (Papadrakakis et al., 1998). Variants

Fig. 6. The obtained plate shapes. Plate shapes obtained (a) in (Papadrakakis et al., 1998); and by current approach using different techniques: (b) the points that are connected with straight

> *vp mm3*

a - - - 280 - 4.38 b 8 48.73 0.00 255.93 255.84 4.38 5.22 c 12.6 47.48 2.27 251.76 251.38 4.38 4.35 d 7.3 31.94 1.17 251.00 250.69 4.38 4.34

Table 2. Quantitative indices of the shape optimization of cutout for the plate stretching

is chosen to minimize relative leave-one-out crossvalidation error *σerr* of approximations of appropriate responses, i.e. maximal equivalent stress *σmax* and volume *v* of plate. *vp* is predicted volume calculated using approximations and *va* is actual volume calculated using full FEM and geometrical models. Again the best results are achieved with the technique using the control points of NURBS polygon. This allows reducing volume of the plate by

*va mm3*

of the local quadratic polynomial approximation

*σmax p MPa*  *σmax a MPa* 

lines; (c) with the NURBS knot points; (d) with the control points of NURBS polygon.

*σerr <sup>v</sup> %* 

correspond to shapes shown in figure 6.

*Variant* 

problem.

The value of Gaussian kernel parameter

*σerr <sup>σ</sup>*

*%* 

At the present time special tensometric wheel pairs are used for the wheel - rail system monitoring. For each type of rolling stock these wheel pairs must fit the vehicle's wheel wearing condition, diameter and bearing box connection type. Using and delivering the tensometric wheel pairs is expensive and takes a lot of time for preparing strength – dynamics tests. In this work removable equipment for monitoring is proposed for mounting on the ordinary wheel pairs. The monitoring wireless system (Grigorov, 2004; Hart, 1986) for 80 tons wagon (freight car) is taken for prototype. The movable part of the equipment (Fig. 8) consists of a removable disk, two transmitters and a transmitting antenna as well as strain gauges bonded to the wheel at defined places. The removable disk is fixedly attached to the wheel pair's axis. A circular transmitting antenna and two transmitters are mounted on the outside of the disk.

Removable equipment must be lightweight to minimize distortions of measurements and at the same time it must be with appropriate durability. During testing dynamical loads caused by rail joints, railroad switches and other irregularities as well as due to defects of wheel geometry are transmitted from wheel pairs to the removable disk which is rigidly mounted on the wheelset axis. Therefore shape optimization of the mounting disk that is the main heavy-weight part of the equipment is very important to reduce its total weight.

Shape Optimization of Mechanical Components for Measurement Systems 251

The geometrical model of the disk is created using SW. It takes into account the shape, size and material of the disk and transmitters (dimensions 20х55х80 mm, mass 0.1 kg). The transmitting antenna (Fig. 8) is removed from the calculation model because it has small dimensions and lightweight material. The calculation model also doesn't consider fastening holes and fastening elements. The transmitting antenna works stable if its displacement in axial direction is less than 2 mm (Hart, 1986). This constraint is taken into account in the next optimization. The strength calculations are performed using SW Simulation. First, the shape of the removable disk (radius R = 300 mm and thickness b = 10 mm) is changed to an ellipse with semi-major axis length 300 mm and semi-minor axis E1 (Fig. 9 a) by simple size optimization of E1. This shape is convenient for the equipment mounting purposes and is taken for initial

FE mesh (Fig. 9 b) is generated with second order tetrahedral solid elements and is generated to get high accuracy results. It consists of about 51000 elements with 86000 nodes

Removable disk's displacement is restrained on its cylindrical face (Fig. 9 c). The material of the disk is aluminum alloy (1060 H12) with elastic modulus *E* = 69000 MPa, Poisson's ratio

 (a) (b) (c) Fig. 9. (a) Ellipsoidal disk; (b) Computational finite element mesh of the model; (c) Scheme

The material's ultimate fatigue resistance is calculated as (State Railway Research Institute

<sup>1</sup> 0.4 

safety FOS = 2.75 is assumed to be sure that the disk will be durable in any worst-case

situation (SRRI, 1998). The acceptable stress in the disk material is reduced to max

Von Mises yield stress criterion is used for all strength calculations:

*<sup>y</sup>* = 27.5742 MPa.

*<sup>y</sup>* (5)

4 MPa.

11.0297 MPa. Additionally the value of the factor of

= 2700 kg/m3 and yield strength

**3.2 Disk model for strength calculation** 

design.

(258000 DOF).

= 0.33, mass density

of disk fastening.

So stresses must be less than 1

[SRRI], 1998):

Fig. 8. (a) Removable disc with elements of measurement system and (b) its mounting place.

#### **3.1 Loads acting on wheel pairs**

The main loads are acting in vertical direction and are caused by railroad irregularities and wheel defects in the wheel – railroad contact. The removable disk sustains all loads from the wheel pair because it is rigidly fastened. Strength of the removable disk is calculated for maximal possible loading. For example, in the case when the wheel pair has 2 mm flat of wheel (Fig.8 b), the loaded and empty wagon wheelsets undergo different loads in vertical direction at different velocities (see Table. 3) (Sladkowsky & Pogorelov, 2008). For an empty wagon the maximal load is at velocity 5 m/s, but for the loaded wagon at 10 m/s. Strength of the removable disk will be analyzed with maximal vertical load – 620.6 kN for two cases of orientation of the transmitters - horizontal and vertical when the wheel pair acceleration can reach 12 *g* (the gravitational acceleration *g* = 9.81 m/s2 is taken into account).


Table 3. Load versus velocity of wagon (Sladkowsky & Pogorelov, 2008).

Strength of the disk under centrifugal load will also be analyzed at maximum vehicle velocity 200 km/h (wheel angular velocity = 116.98 rad/s). In this case the disk is considered as new without wear on riding circle.

Besides, frequency analysis was made to find natural frequencies of the wheel pair and evaluate possible resonance in the case of flat of wheel. Obtained results show that excitation frequencies at velocities of operating conditions are significantly smaller than fundamental frequency.

#### **3.2 Disk model for strength calculation**

250 Advanced Topics in Measurements

Fig. 8. (a) Removable disc with elements of measurement system and (b) its mounting place.

The main loads are acting in vertical direction and are caused by railroad irregularities and wheel defects in the wheel – railroad contact. The removable disk sustains all loads from the wheel pair because it is rigidly fastened. Strength of the removable disk is calculated for maximal possible loading. For example, in the case when the wheel pair has 2 mm flat of wheel (Fig.8 b), the loaded and empty wagon wheelsets undergo different loads in vertical direction at different velocities (see Table. 3) (Sladkowsky & Pogorelov, 2008). For an empty wagon the maximal load is at velocity 5 m/s, but for the loaded wagon at 10 m/s. Strength of the removable disk will be analyzed with maximal vertical load – 620.6 kN for two cases of orientation of the transmitters - horizontal and vertical when the wheel pair acceleration

can reach 12 *g* (the gravitational acceleration *g* = 9.81 m/s2 is taken into account).

Table 3. Load versus velocity of wagon (Sladkowsky & Pogorelov, 2008).

considered as new without wear on riding circle.

fundamental frequency.

Velocity of wagon m/s Maximal load in moment of shock, kN

Static load 22.8 104.5 1 136.1 251.3 2 170.2 316.4 5 297.8 367.2 10 271.3 620.6 20 276.6 604.9

Strength of the disk under centrifugal load will also be analyzed at maximum vehicle velocity 200 km/h (wheel angular velocity = 116.98 rad/s). In this case the disk is

Besides, frequency analysis was made to find natural frequencies of the wheel pair and evaluate possible resonance in the case of flat of wheel. Obtained results show that excitation frequencies at velocities of operating conditions are significantly smaller than

Empty wagon Loaded wagon

(a) (b)

**3.1 Loads acting on wheel pairs** 

The geometrical model of the disk is created using SW. It takes into account the shape, size and material of the disk and transmitters (dimensions 20х55х80 mm, mass 0.1 kg). The transmitting antenna (Fig. 8) is removed from the calculation model because it has small dimensions and lightweight material. The calculation model also doesn't consider fastening holes and fastening elements. The transmitting antenna works stable if its displacement in axial direction is less than 2 mm (Hart, 1986). This constraint is taken into account in the next optimization.

The strength calculations are performed using SW Simulation. First, the shape of the removable disk (radius R = 300 mm and thickness b = 10 mm) is changed to an ellipse with semi-major axis length 300 mm and semi-minor axis E1 (Fig. 9 a) by simple size optimization of E1. This shape is convenient for the equipment mounting purposes and is taken for initial design.

FE mesh (Fig. 9 b) is generated with second order tetrahedral solid elements and is generated to get high accuracy results. It consists of about 51000 elements with 86000 nodes (258000 DOF).

Removable disk's displacement is restrained on its cylindrical face (Fig. 9 c). The material of the disk is aluminum alloy (1060 H12) with elastic modulus *E* = 69000 MPa, Poisson's ratio = 0.33, mass density = 2700 kg/m3 and yield strength *<sup>y</sup>* = 27.5742 MPa.

Fig. 9. (a) Ellipsoidal disk; (b) Computational finite element mesh of the model; (c) Scheme of disk fastening.

The material's ultimate fatigue resistance is calculated as (State Railway Research Institute [SRRI], 1998):

$$
\sigma\_{-1} = 0.4 \cdot \sigma\_y \tag{5}
$$

So stresses must be less than 1 11.0297 MPa. Additionally the value of the factor of safety FOS = 2.75 is assumed to be sure that the disk will be durable in any worst-case situation (SRRI, 1998). The acceptable stress in the disk material is reduced to max 4 MPa. Von Mises yield stress criterion is used for all strength calculations:

Shape Optimization of Mechanical Components for Measurement Systems 253

The results for the stressed state of initial design of the disk from centrifugal load are shown in Fig. 11. Maximum von Mises stress in the disk from centrifugal load is at least 12 times

The shape of cross section of the disk is optimized, taking into account only centrifugal load. The constructive restrictions allow changing the disk cross section shape only at one side and in radial direction at range 150 mm to 300 mm from ellipse center. The section of the disk at radial distance 0-150 mm has constant thickness b = 10 mm. Three methods are used to define the cross section shape (Fig. 12): a) with NURBS knot points, b) with NURBS polygon points and c) with points that are connected with straight lines. Four parameters are stated to define the shape. Parameters are varied in the following ranges: 4 ≤ X 1 ≤ 10; 4 ≤ X 2 ≤ 10; 5 ≤ X 3 ≤ 12; 3 ≤ X 4 ≤ 5 for variants "a", "c" and 3 ≤ X 1 ≤ 10; 0.5 ≤ X 2 ≤ 20; 5 ≤ X 3 ≤ 25; 2 ≤ X 4 ≤ 5 for "b". The design of experiments is calculated with MSE criterion for 4 factors and 70 trial points. This design of experiment is also available on the web:

So the 70 strength studies are calculated for each considered method. SW Simulation results (volume, maximal von Mises stress, axial displacement of the disk etc.) are entered into

Some optimization and approximation characteristics are shown in Table 4. Results of variants "a" and "b" are obtained with second order local polynomial approximation. Third order local polynomial approximation is used for variant "c". Gaussian kernel coefficient

> Volume *v* [mm3]

a 6 20.56 0.06 1003944 1003891 0.005 3.9999 3.833816 4.33 b 3 40.28 2.03 923421 921740 0.018 3.9999 4.200354 4.77 c 4 10.93 0.00 946180 946173 0.001 3.9998 4.125750 3.05 d - - - - 1394900 - - 3.3 -

Table 4. Quantitative data of approximation and shape optimization of the ellipsoidal disk

The obtained metamodels are used for optimization of factors. The ellipsoidal disk volume is minimized by taking into account the specified constraints on displacement and stress level. The obtained shapes are presented in Fig. 12. As shown in table 4, the best results are obtained for variant "b" (Fig. 13), where the volume is lower by 8.2 % in comparison to variant "a" and on 2.6 % - to "c". All 3 variants give significant advantage in volume (28.1 – 33.9 %), comparing to variant "d"- the initial shape design with constant 10 mm thickness.

*vonMises <sup>v</sup>*Predicted Real Error

[%]

Maximal von Mises stress

*vonMises* [MPa]

[%] Predicted Real Error

greater than in the case of loading from flat of wheel.

http://www.mmd.rtu.lv.

Variant

cross section.

**3.4 Shape optimization of cross section of the ellipsoidal disk** 

EDAOpt for approximation and subsequent global search.

was varied for least value of crossvalidation error (4).

Approximation's *err* [%]

$$
\sigma\_{\text{ronMises}} = \sqrt{\frac{(\sigma\_1 - \sigma\_2)^2 + (\sigma\_2 - \sigma\_3)^2 + (\sigma\_1 - \sigma\_3)^2}{2}},\tag{6}
$$

where 123 , , are principal stresses.

Thereby the von Mises stress at any point of the disk should be less than acceptable stress:

$$
\sigma\_{\text{conMises}} < \sigma\_{\text{max}} \tag{7}
$$

#### **3.3 Stresses in initial design disk**

Three variants of stressed state of the disk are analyzed, i.e., from loads due to flat of wheel in two cases of the disk orientation: when the major axis of ellipse is vertical and horizontal as well as from centrifugal loads.

We consider a loaded wagon with maximal loading in moment of shock that occurs at velocity 10 m/s (Table 3). Maximal stresses in moment of shock (acceleration *a* = 119.3 m/s2) are shown on Fig. 10. As we can see, values of maximal stress levels for both orientations of the disk are very similar.

Fig. 10. Von Mises stresses distribution in initial design disk for (a) horizontal and (b) vertical orientation.

Fig. 11. (a) Von Mises stresses distribution in initial design disk from centrifugal loads; (b) Disk's displacements in axial direction.

 

Thereby the von Mises stress at any point of the disk should be less than acceptable stress:

Three variants of stressed state of the disk are analyzed, i.e., from loads due to flat of wheel in two cases of the disk orientation: when the major axis of ellipse is vertical and horizontal

We consider a loaded wagon with maximal loading in moment of shock that occurs at velocity 10 m/s (Table 3). Maximal stresses in moment of shock (acceleration *a* = 119.3 m/s2) are shown on Fig. 10. As we can see, values of maximal stress levels for both orientations of

(a) (b)

Fig. 10. Von Mises stresses distribution in initial design disk for (a) horizontal and

(a) (b)

Fig. 11. (a) Von Mises stresses distribution in initial design disk from centrifugal loads;

2 *vonMises* 

> *vonMises*

**3.3 Stresses in initial design disk** 

as well as from centrifugal loads.

the disk are very similar.

(b) vertical orientation.

(b) Disk's displacements in axial direction.

are principal stresses.

where 123 , , 

222

 

, (6)

max (7)

12 23 13 ( )( )( )

The results for the stressed state of initial design of the disk from centrifugal load are shown in Fig. 11. Maximum von Mises stress in the disk from centrifugal load is at least 12 times greater than in the case of loading from flat of wheel.
