1)INMA Bucharest / Romania; 2)P.U. Bucharest / Romania. Summary ... 14. 2.5.1. Elementary model of the axle as constant section bar, M1 14. 2.5.2.
TECHNICAL ASPECTS REGARDING AXLES CALCULATION OF ROAD TRANSPORT MEANS Sfîru Raluca1), VlăduŃ Valentin1), Cârdei Petru1), Ciupercă Radu1), Matache M.1), Ştefan Vasilica1), Ungureanu Nicoleta2) 1)
2)
INMA Bucharest / Romania; P.U. Bucharest / Romania
Summary Mathematical models of mechanical structures, structural models are characterized by three main components: geometry, border conditions (bearing or ties with the outside environment for dynamic problems and initial conditions) and the load (external actions on the mechanical system which is studied). The first component (geometry) is an idealized component of the reality – the shape of the body or the shape of the studied assemblies of bodies. This can be improved up to the level desired by the analysts. The other two components (border conditions and load) are engineering assumptions. Thus, these two components of the model must be either theoretically deduced and experimentally verified, either deduced experimentally directly by various measurements. Generally, when studying separate components of a whole assembly, these assumptions are difficult to formulate and thus, the overestimates lead to the design of some massive structures, which generate high consumptions and reduced maneuverability. As easily, the under sizing may occur, leading to more serious effects, and malfunctions resulting in failures or major accidents. This article aims to tackle the issue of conditions (structure bearing) at the border of the structural model, suggesting solutions for the situation in which the components of a whole assembly are separately analyzed.
1
TABLE OF CONTENTS
1. INTRODUCTION
3
2. MATERIAL AND METHOD
4
2.1. Model with rigid bearing - MR
4
2.2. ME - elastic bearing model
7
2.3. Model with rigid supporting on the lower supporting half-ring - MRS
10
2.4. Model of elastically bearing on semi-lower bearing ring – MES*
12
2.5. 1D Reference Model
14
2.5.1. Elementary model of the axle as constant section bar, M1
14
2.5.2. The one-dimensional model with finite elements, M2
15
2.6. Complex models including components connected to the axles
20
2.6.1. The structural model of the chassis, M4
20
2.6.2. The structural model of the chassis with axle and wheels, M5
24
3. RESULTS
25
3.1. Experiments for the collection of loading data
28
3.2. Effects of bearing on the fatigue state of axle material
32
4. CONCLUSIONS
42
5. REFERENCES
45
2
1. INTRODUCTION Shafts and axles calculation usually constitute a separate chapter in treaties of machine parts, and also, in some treaties for materials strength. The axles are defined, for example after [2], as fixed machine parts with rotating or oscillatory movements, designed to support other organs, which usually have rotating movement, without transmitting twisting moments, their main loading being the bending. The axles are framed within the broader category of shafts (improper called axes, after [3]), which is defined in [3] as being revolution solid bodies, having small maximum radius compared to their length. The shaft, after [3], is stressed primarily to bending and secondly to torsion, stretching and compression. Compared to most of the literature, [3] offers vague references to the axles bearing, which are classified by the number of bearings (statically determined, when they have two bearings and statically undetermined when they have three or more bearings. Also, the axles are classified by their mobility (fixed and rotary), or by the spindles position (bearing in console), etc. In [2] is given the classification of axes in static determinants and static un-determinants, without giving any definition of bearing. [3] is very laconic in defining the term bearing, as the external link that prevents some linear or rotating movements of a body. It defines the kinematical coupling through the mechanical system composed of two solid bodies in permanent and direct contact that allow the mutual relative motion and transmits mechanical actions (forces and torques) between them. In literature, the structural modeling of axles has been often discussed, but often with insufficient explanations, regarding the issue of border conditions (ties or supports), [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].
3
2. MATERIAL AND METHOD The study uses as application material an agricultural trailer axle of 7.5 tons, designed by INMA Bucharest. The axle has a central section by the filled square section with the side of 80 mm, and the length of 1590 mm, soon afterwards, following two short sections of 27 mm at the ends, of circular transverse section with the diameter of 75 mm. The remaining short sections at the ends do not interest the calculation. The material of which the axle is build is 1023 carbon steel (from the program materials library [4]), having the closest characteristics of the steel OLC 45: E= 2.05·1011 N/m2, ν= 0.29, G=8·1010 N/m2, ρ= 7858 kg/m3, σr=425 MPa, σc= 282.685 MPa. The shape of the axle is presented in Fig. 1.
Fig. 1 - Drawing provided by the designer of the agricultural trailer axle of 7.5 tons
2.1. Model with rigid bearing - MR The first model is apparently normal for a beginner, namely the model of the axle bearing on the recessed rings in rigidly, i.e. by canceling all six freedom degrees at each node, which means restraint (away from the real model). 4
According to [4], the bearing applied through fixing the geometry of the cylindrical surfaces from the contact area of the axle with bearings, corresponds to perfect restraint. These areas do not leave their positions corresponding to the undeformed configuration of the body. According to the used program of structural analysis, [4], this bearing or connection is made by fixing the geometry from the cylindrical surface of the contact areas between axle and bearings. The state of the relative displacement resulted in the structure is represented in Fig. 2, a and b, in which the values are given in mm.
a)
b) Fig. 2 - Deformed shape of the axle for the given conditions, a, and the state of the relative resultant displacement (deformation) in the axle model, in mm, b 5
The equivalent strain state in the axle model is represented by the map shown in Fig. 3, a, and the equivalent stress state in the built model of the axle is represented as a map on the deformed shape of the structure in Fig. 3, b. The values of the equivalent tension are given in Pa (N/m2).
a)
b) Fig. 3 - The state of total specific deformation in the axle model, a. The state of equivalent tension in the axle model în Pa (N/m2), b
The first two proper vibration frequencies of the axle and the deformed shapes at maximum elongation are given in Fig. 4. The two modes of vibration are nearly identical, differing only by the direction of deformation.
6
This difference is caused by the plates which are not symmetrical to the median plane of the axle parallel with the xOz plane.
a)
b) Fig. 4 - The relative resultant displacement in the deformed structure of the axle when this vibrates in the fundamental vibration mode (204.66 Hz), a, respectively, in the vibration mode corresponding to the second proper frequency (205.85 Hz), b
2.2. ME - elastic bearing model This model corresponds to an elastic bearing in the areas where the axle is supported on the bearings, the elastic constant of the elastic bearings being 1080000000 (N/m) / m2. This constant was chosen in order to achieve, by simulation, a maximum relative resultant displacement and a maximum equivalent tension, close to the ones of the axle elementary theoretical model, [1]. 7
This type of bearing acknowledges that the axle is supported on the bearings and they are supported on the rim, which in turn are supported on the tire, which is an element with a higher elasticity than that of the metal. As a result, there is a redistribution of the stress in the weaker elements and simultaneously reduces the effort on the rigid elements.
a)
b) Fig. 5 - The state of relative resultant displacement in the axle model, in [mm] a, respectively, the equivalent tension state in the axle model in [Pa (N/m2)] b
The state of the relative resultant displacement in the axle, in case of elastic bearing and the loads used in case of rigid bearing are given as color map on the deformed shape of the structure in Fig. 5, a. In Fig. 5, b is graphically represented, as a color map, the equivalent stress state in the structure on the deformed shape. 8
a)
b) Fig. 6 - The relative resultant displacement in the axle deformed structure when it vibrates in the vibration fundamental mode (69,544 Hz), a, respectively, the relative movement resulting in the axle deformed structure when this vibrates in the vibration second mode (69,893 Hz), b
The deformed shape of the structure at vibration in the first and the second fundamental mode is shown in Fig. 6, a, respectively, b. On the deformed shapes are given the color maps of the relative displacements resulting in the structure, in mm.
9
2.3. Model with rigid supporting on the lower supporting half-ring - MRS The model with supporting on the lower half-ring comply more precisely with the real supporting surface in working regime with constant speed, because the contact between the supporting area of the axle and the bearing is estimated to be achieved on the inferior area on an arc less than 180°. The model tries to predict the distribution of stress and the displacement in this case. Obviously, the rigid propping, as we appreciated is an exaggeration which leads to stress increasing and obviously to an overestimation by its calculation. The areas of loading and bearing of the structure are shown in Fig. 7 and Fig. 8. There is a loading of 37500 N on each bearing plate of the chassis on the axle. The main results obtained following the analysis are given in Fig. 9, the map of distribution of the relative resultant displacement (a), respectively the map of the equivalent stress distribution (b), both on the boundary of the structure.
Fig. 7 - 3D structural model of the axle, resting on the lower semi-ring
Fig. 8 - Axle load is made on the loaded areas with strengths of each 37500 N on each plate It should be observed that the maximum relative displacement has a lower value, of 1.1 mm, being located in the middle area of the axle, with the 10
displacement oriented on the Oy axis (Fig. 1). The maximum stress occurs in the boundary supporting zone toward the central body of the axle, on the side where is applied the abutment (Fig. 9) and has a high value for the steel specified above, steel of which the structure is made at this time, amounting to 442.39 MPa. At this value of stress, even on an area of material greatly reduced as volume, there are quite probable the development of cracks and premature fatigue failure, as in [17]. Therefore, accidents with serious consequences can occur in case of severe stiffeners and use of inappropriate material, [18]. But the fatigue of materials is another subject that cannot be simulated without considering the supporting structures [19], [20], [21]. Such crashes are exemplified e.g. in [22]. The supporting mode of the MRS model makes that its own fundamental frequency gets from about 205 Hz (model MR) at 137 Hz.
a)
b) Fig. 9 - Distribution maps of the main state sizes of axle on the boundary: a) the relative resultant displacement (mm); b) equivalent stress (Pa) 11
Fig. 10 - Images of axles which have broken (by [22])
2.4. Model of elastically bearing on semi-lower bearing ring –MES* Loading of this model is similar to all the other models, being applied uniformly on the bearing plates of the chassis with the value of 37500 N. Bearing is only made on the lower semi-ring (as at the MRS model), Fig. 7, but the abutment is of elastic nature with the same qualities as the ME model).
a)
12
b) Fig. 11 - Maps of the distribution of main state sizes of the axle on boundary: a) relative resultant displacement (mm); b) equivalent stress (Pa)
a)
b) Fig. 12 - Comparative maps of the axle rigidly propped (a) and elastically propped (b), both on the lower abutment semi-ring 13
The main results of analysis on this model show that the location of the maximum relative resultant displacement remains at the middle of axle, but has a value appreciably higher than for rigid abutment, 11.84 mm (Fig. 11). The greater freedom of movement achieved by elastic abutment leads to this increase of the resulting relative displacement (however, within acceptable limits), in return decreases to less than half the maximum equivalent stress obtained in the case of the appropriate model beared through constraining, 204,772 MPa. Also, the location of the value of the maximum equivalent stress in the MES model is in the immediate vicinity of the chassis plate support (Fig. 12 b). The difference in behavior between MRS and MES models, different only through their abutment mode, in terms of the equivalent stress can be observed comparatively in Fig. 12. The abutment mode of MES model makes its fundamental frequency to be of only about 31 Hz, versus the elastically model beared over the entire surface of the abutment annular surface, ME, in which case its own fundamental frequency has a value of approximately 70 Hz.
2.5. 1D Reference Model 2.5.1. Elementary model of the axle as constant section bar, M1 In accordance line with the provisions of the designer, the simplest model is taken from specialized books of resistance in which the axle is represented as a right beam (Bernoulli-Euler model, [23] and [24]) of constant section with opening equal to the distance between the centers of areas of the axle supporting on bearings and loaded in the centers of the two zones of symmetrical load (model further noted with M1). Such a model is shown in many treaties of material resistance, such as [2], [25], [26], [27], [28], [29], [16] or of machine parts, [30]. The scheme and the model formulas are given in Fig. 13. The bar is articulated in the two points of supporting points. 14
M max = Fa
umax =
(
Fa 3l 2 − 4a 2
,
)
24 EI
(1)
(2)
Fig. 13 - The geometry, supporting and axle loading modeled within the model M1
In formulas (1) and (2), Mmax is the maximum moment, F is half the total loading force acting on the axle, a and l are geometrical characteristics, E is the modulus of elasticity of the material of which the axle is made, I is the geometrical moment of inertia of the cross section of the bar which models the axle, and umax is the maximum relative displacement along the bar on its axis. For the model of the axle shown in Fig. 2 the cross-section of the bar is constant, square filled with the side of 80 mm, the material of which it is constructed being OLC 45 (E=2.1·1011 Pa, Poisson coefficient ν= 0.3, mass density ρ= 7850 kg/m3).
2.5.2. The one-dimensional model with finite elements, M2 The closest mathematical model of the preceding with finite elements model which was solved purely analytical (with simple, elementary formulas), but which also respects the real geometrical shape of the supporting zones, with support in a single point in each bearing area, is given in Fig. 14. The geometrical model of the axle is an elementary one: straight bar (reduced to 15
the axis of symmetry) with section full square on the central area (square with side of 80 mm) and with circular cross section in the bearing zones (diameter of 75 mm). The areas of the ends including the two finite elements each, bounded by three nodes, are the supporting zones of the axle on bearings. The discretization is done with one-dimensional elements type BEAM3D, located in the library of finite elements of the used program of structural analysis, [31]. Bearing is done by canceling the translational movements (the relative movement's ux, uy, uz) and the rotating movement around its own axis of the axle (the relative rotation, rx). The revolutions around the axis Oy and Oz remain free. The supporting described above is done in a single node, considering that practically the supporting area is very restricted and the catch bearing system rests on elastic elements (wheels). The similar reinforcement in the other two nodes in each of the two supporting zones leads to the increase of the stress in the axle, which does not correspond to the reality. This kind of supporting leads to results in agreement with the specialty literature on materials resistance, [2]. The model with finite elements given in Fig. 14 and described above, it is further noted with M2.
Fig. 14 - Elementary model with finite elements of the axle (geometry, supporting and loading, M2)
Fig. 15 - Resultant relative displacement map in the model M2 of the axle, under the conditions given above, given by vectors, on the deformed shape of the organ (the values are given in m) 16
Fig. 16 - The map of the equivalent stress in the model M2 of the axle, under the conditions given above [the values are given in Pa (N/m2)]
The distribution of relative resultant displacement and of the equivalent stress of the axle M2 in the model is given graphically by map in Fig. 15 and Fig. 16. The fundamental frequency of the axle is 71.76 Hz and the corresponding deformation is one of bending. The own frequencies are the following: 287.06, 645.818, 794.115, 1148.01, 1794.06, 2381.24, 2584.35, 3517.98, 3965.05 Hz. Reaction force in each supporting points has the value of 37500 N, half of the total load applied to the axle. The problem of supporting the axle generates a large number of further possible models. The way in which the axle sits on bearing, the shape and size of the contact surfaces is unknown and changes over time, at the beginning faster and according to loading. The change of the supporting conditions at the mathematical model with finite elements M2, leads to three different models which give results that differ substantially from those of the models M1 and M2. The geometry, discretization and loading of variants of the model M2 will be marked as M21, M22 and M23. For each of these models, the supporting area is discretized into two elements and three nodes that edge them. It will be named external nodes the nodes at the ends of the two bars bordering the bar and the supporting area. It will be called internal nodes of the supporting area, the nodes of the two zones located toward the center of the axle and which is situated at the border of section change of the axle. The nodes located on the center of the supporting areas will be called
17
central nodes. With these assignations can be defined for differentiation the models derived from M2: -
the model M2 has the supporting only in the central nodes, ux=0, only to the left side, uy=uz=0;
-
the model M21 maintains the supporting of the model M2 and in addition in the extreme nodes: uy=uz=0;
-
the model M22 maintains the supporting of the model M2 and in addition in the inner nodes: uy=uz=0;
-
the model M23 maintains the supporting of the model M2 and in addition in the extreme and in the inner nodes: uy=uz=0. The main results of the models M2 and of its derivatives are given in
Table 1. As a result of the multiple possible bearing hypotheses, there can be imagined a lot of mathematical models with finite elements, which have a higher level of complexity than the simple one-dimensional bar ends propped. One of these is the one whose geometry, loading and supporting are shown in Fig. 17. The model will be called M3. The rectangles on which the axle sits directly (Fig. 17, a), are shaping the rims, and the rectangles located in their extension are shaping the tires representing the elastic bearing of the axle. These geometric entities are discretized with two-dimensional finite elements type SHELL3 from the finite element library of the program [31]. Since in this model it only interests us the behavior of the axle, the rim and tire model are modeled with high fidelity towards reality. In this model, the rims and tires only play the role of elastic supports. Under these circumstances, the results must be well interpreted. The resultant relative displacement of the axle is calculated as the difference between its value on the entire structure at the middle of the axle and the value of the same field in the central node of the bearing. The value of this difference is about 4.98 mm. The maximum equivalent stress is localized in the axle in the central side in the area between the nodes where the loading is applied. Its value is of 134.7 MPa. 18
The first ten own frequency of the model M3 are: 0.0026, 12.181, 18.916, 19.432, 28.392, 79.147, 115.744, 116.681, 321.564, 335.12 Hz.
a)
b) Fig. 17 - The geometry (a), discretization, bearing and loading (b) of the model M3
Fig. 18 - The distribution of the resultant relative displacement field in the model M3 (the values are given in m)
19
Fig. 19 - The distribution of the equivalent stress in the model M3 (the values are given in Pa)
The relative resultant displacement field distribution and the equivalent stress in the model from Fig. 17 are given in Fig. 18, respectively in Fig. 19.
2.6. Complex models including components connected to the axles 2.6.1. The structural model of the chassis, M4 The gradual complication of the model continues with the structural model of the chassis which is now supported on the rear side on axle and in the front side on the coupling point to the tractor. The elementary structural model of the chassis, which will further be noted with M4, is constructed exclusively with finite elements of type BEAM3D, located in the library of finite elements of the structural analysis program with which we used. In Fig. 20 is given the structural model of the chassis used for the analysis in this calculation.
20
Fig. 20 - The elementary model (M4) with finite elements of the trailer chassis – the geometry, mashing, bearing and loading
For bearing were used the cancellation conditions of the three relative translation movements at the front side of the chassis (the one that binds to the tractor, ux=uy=uz=0) and cancellation of translations in the vertical and laterally, in the bearing points on axle of the chassis (uy=uz=0). Loading was made using a total force of 7500 daN, in six points, as indicated by the designer, equally loaded, therefore 1250 daN in every loading point (the red arrows in Fig. 20). According to the indications of the designer, the two longerons were made using U profile 180x80x5 mm, as the traverse from the back of the trailer, and the penultimate traverse of profile U 120x80x5 mm. The first two traverses were considered made of steel with profile square pipe 100x6 mm. The material is steel for longerons L42 (the flow limit 250 MPa), traverse back and central, S275 JR for the drawbar ear, OLT 35 the traverse from the front (the flow limit 230 MPa). It was considered for all the steels E= 2.1 ·1011 Pa, ν= 0.3, respectively the density of 7850 kg/m3. The analysis was made by the finite element method obtaining the relative resultant displacement field and the field of equivalent stress in structure, which are graphically represented in Fig. 21 and Fig. 22.
21
Fig. 21 - The field distribution of relative resultant displacement in the structure, in the bearing conditions and the loading specified above. The distribution is given on the deformed shape of the structure (the values are given in m)
Fig. 22 - The equivalent stress field in the trailer chassis structure, on the undeformed shape [the values are given in Pa (N/m2)]
It can be observed that the arrow in the structure has the maximum value of about 2 mm (1.9794 mm), in the specified conditions of bearing and loading, very small value compared with the structure span. Maximum stress in the structure components constructed of steel L42 is 94.31 MPa, while in structure components made of OLT 35 is about 42 MPa.
22
The fundamental own frequency of the chassis so propped has the value of 50.98 Hz, and the deformed shape of the structure is graphically represented in Fig. 23. In Fig. 24 are provided the reactions in the supporting points of the chassis, each value being of 26465.7 N on each axle bearing points, respectively 11053.9 N on each of the bearing points from the front of the trailer hitch. Their sum exceeds the value of loading with a force of 38 kN, which is due to the errors of calculation of the discretization, but this error only represents 0.05% of the total load.
Fig. 23 - The deformed shape of the structure when it is vibrating in the fundamental mode of vibration
Fig. 24 - The reactions in the supporting points of 26465.7 N on each of the supporting points on axle and of 11053.9 N on each of the two supporting points from the trailer hitch 23
The mass of the analyzed chassis model is of 148.6 kg. The center of mass in the coordinate system given in Fig. 26 has the following coordinates: xG=1.376 m, yG=-0.003545 m, zG= 0.00 m. All the results obtained above were calculated leaving aside the gravitational load generated by the own weight of the structure. By taking into account the gravitational load, the maximum relative displacement becomes 1.9987 mm, and the maximum equivalent stress reaches 95,703 MPa, i.e. minimal changes, which are negligible. Due to the simplification of the model, a series of joint elements and of reinforcing (plates, ears etc.) located on the chassis are not shown in the structural model, which is why the weight value of it is shorter than the value obtained by weighting for the real model. 2.6.2. The structural model of the chassis with axle and wheels, M5
Fig. 25 - The structural model of the chassis with axle, M4
Fig. 26 - Distribution of the relative resultant displacement in the submodel consisting of chassis and axle, as well as in the connecting elements between them 24
Fig. 27 - The distribution of the equivalent stress in the submodel formed from the chassis and axle, as well as in the jointing elements between these This model is obtained by coupling the patterns M3 and M4. The structural model of the chassis with axle and wheels is shown in Fig. 25. Bearing is done at the bottom of the wheels and on the front ears on the chassis. The loading is the same as in the case of the M4 model (each 1250 daN) in each loading point marked by red arrows in Fig. 25. The responses of the model at the given loadings and bearings are given in Fig. 26 and Fig. 27. The first ten fundamental frequencies of the model are as follows: 12.739, 13.459, 29.107, 45.837, 55.267, 66.358, 79.003, 86.442, 129.129, 139.729 Hz. Model M5 highlights complex issues, characteristic to large structures. These results show the effects of considering the natural applying or as close to the reality of loadings. It changes not only the maximum values of the relative displacement and equivalent stress fields, but also the location of the peaks.
3. RESULTS The main results of the two structural models of the axle are given in the preceding paragraph. In this paragraph is presented only a summary of them. 25
Table 1 Synthesis of the results of the structural models considered in this study Resultant relative displacement Model
Maximum value [mm]
Location
Resultant Equivalent tension Maximum value [MPa]
The extremities
The middle MR
0.370
zone of the
Location
Own fundamental frequency [Hz]
225.090
axle
of the square section area of
204,860
the axle The vicinity of
ME
4.983
The middle part of the axle
the limitrophe 152.551 areas on which
69.544
is applied the load The
The middle MRS
1.100
zone of the
extremities of 442.390
axle
the square
136.800
section area of the axle The vicinity of
The middle MES
11.840
zone of the
the limitrophe 204.772 areas on which
axle
30.848
is applied the load
Reference model
The middle 5.141
zone of the
The middle 135.571
axle
zone of the axle
26
70.863
For the model M1 having the data: a= 0.3085 m, l= 1.617 m, F=37500 N, are obtained the results: Mmax= 11568.75 Nm, umax= 0.0050 m, respectively the maximum stress in the axle, σ= 135.57 MPa, the last one being obtained using the Navier's formula [2], page 317, or [1] page 122, for example, maximum stress obtained on the outer fibers of the axle. For the same above data, according to [25] for example, the fundamental frequency of the axle is 71.759 Hz. The model M1 also gives the reactions in the bearing points, equal each with F. The finite elements model M2, with the same distance between the supports and the same distance between the points of application of force, model whose characteristics are given in Fig. 3, has a bearing (conditions at the border) selected so as to obtain results as close to those given by the model M1, selected so as to obtain results as close to those given by the model M1, [1], [2]. Axle bent (the maximum resultant relative displacement) is localized at the middle of the axle, having the value of about 5 mm (Fig. 4). This value coincides with that calculated by the classical formula given in [1] and is small compared to the length of the axle between the bearings, 1617 mm. The resulting relative displacement practically coincides with the absolute value of the vertical displacement (Oy axis) because the loading is static. Maximum equivalent stress is localized in the central area, with the value of 135.571 MPa, the classical calculation after [1] giving practically the same value. The fundamental frequency of vibration is practically the same as that given by the model M1, 71.76 Hz. The model M2 gives in each of the bearing points a reaction force equal to half of the total load applied to the axle. Reactions can be used to calculate the crushing effort in the bearing. Main results of the model M2 and of the models derived of it are presented in Table 1. The structural model M3 is the separated model of the trailer chassis.
27
Table 2 Main results relative only to axle of the models M1 - M5 Model
Fundamental own frequency
umax, mm
σmax, MPa
M1
5.005
135.570
71.759
M2
5.016
135.571
71.762
M21
8.790
219.091
153.544
M22
7.290
214.869
164.251
M23
7.230
215.199
164.731
M3
4.980
134.700
115 – 335
M4
-
-
-
M5
0.700
102.000
143 – 250
[Hz]
3.1. Experiments for the collection of loading data In the first part of this article we discussed extensively the consequences of certain types of bearing of the structures, as well as those of the analysis of some isolated substructures and by subassemblies. In this chapter we will briefly illustrate how are selected the loadings for the structural models and we will give some guidance on their choosing. If the loading data for a starting model are useful at the design phase of a trailer or transportation equipment, can be theoretically chosen based on predicted loadings, the data for the loading of a specific model in transportation (data with dynamic characteristic), is recommended to be selected according to experiences that register deformations, velocities, accelerations and stresses in structures already constructed or at least similar to those which are to be built (similar in overall size especially). For the sensitive issues, such as resonance, lateral and longitudinal stability, fatigue, the minimum loading on the steered axles, the stability in the overtaking maneuvers, etc. The loading data are either arising from 28
experiences or are defined by standards. The selection of the critical stresses for transport equipment is a difficult task because it has to scan a wide range of speeds and a wide enough range of runways. The raceways are either roads from the public network or specially designed tracks for experiments.
Fig. 28 - Placement of measuring instruments on various component parts of an equipment of transportation
An example of placing the sensors on components of transportation equipment is given in Fig. 28. Other sequences during performing of tests on the runway for various tractors - agricultural trailer aggregates are given in Fig. 29 and Fig. 30.
Fig. 29 - Typical agricultural transportation equipment at testing on track
29
Fig. 30 - Checking the stability at the maximum tipping angle
Sequences of running on the testing track of the transportation equipment are given in Fig. 31. In Fig. 31 are graphically represented the signals from three accelerometers (one on the rear axle of the tractor, one on the frontal axle of the trailer and the third on the trailer chassis) disposed on a tractor - trailer unit of 5 t capacity. In Fig. 32 are given the frequency spectra of the three recordings, whose graphical representation over time is given in Fig. 31. On such spectrograms are selected the accelerations which correspond to some excitations originating from the contact between the undercarriage and the wheels of the aggregate. On these accelerations is being loaded the structural models for the mathematical models of this equipment in the stages of design, redesign, improvement or optimization. The structural models outlined above are used as simulators to increase the safety for prediction of structures behavior over time (fatigue, reliableness, etc.). Also, on these simulators are studied the problems of occurrence of resonant transportation regimes, regimes that can reduce the adherence to the rolling track and can lead to some serious events in traffic. Such events are particularly amplified now, when vehicles have a high or very high overall size (frequently trailers with a capacity exceeding 10 t) and the transportation speeds with tractor, have already reached 50 km/h. The tests mentioned above are both sources of information for the construction of structural 30
models (especially loadings but also bearings), and means of validation and correction for the complex simulators which are obtained.
Fig. 31 - Typical signalizing sequences obtained on trailers in various operating modes
31
Fig. 32 - The spectra of the three records shown in Fig. 31
3.2. Effects of bearing on the fatigue state of the axle material We considered the structural model whose geometry is presented in Fig. 33. We tried to estimate the effects of a bearing difference on the fatigue state of the axle material. Before proceeding to the evaluation of fatigue state, it is performed a static analysis that also defines the maximum value of the considered single loading cycle on the axle, in dynamic regime. The 7 tons load is evenly distributed on the two bearing plates of the carried weight. The difference between the two cases is that in the first case, the axle is embedded throughout the entire retainer ring in the bearing (360o), while in the second case the embedding is done only on the inferior semi-ring (180o). Main results of static analysis for the two cases are presented below. In Fig. 33 is drawn the color map of the distribution of the resultant relative displacement on the boundary of the structure supported by embedding on the whole bearing ring.
32
Fig. 33 - Field distribution of the relative resultant displacement (strain) on the boundary of axle structural model – the case of bearing on the whole bearing ring
Fig. 34 shows the distribution of the field of specific strains on the boundary of the structural model, in the same case of bearing, and Fig. 35 presents the distribution of the stress field.
33
Fig. 34 – Field distribution of the resultant strain on the boundary of axle structural model – the case of bearing on the whole bearing ring
Fig. 35 – Field distribution of equivalent stress (Von Mises) on the boundary of the axle structural model – the case of bearing on the whole bearing ring
The same characteristic state fields generated in the structural model, when the embedding is done only on the inferior semi-ring are given, in the same order, in Fig. 36, 37 and 38. 34
Fig. 36 – The distribution of the relative resultant displacement on the boundary of axle structural model – the case of bearing on the inferior semiring
Fig. 37 – The distribution of the resultant strain on the boundary of axle structural model – the case of bearing on the inferior bearing semi-ring
35
It can be observed that the embedding on a smaller area leads to higher values of the resultant relative displacement, and also to somewhat higher values of equivalent stress in the stress concentrators in comparison with the embedding throughout the whole bearing ring.
Fig. 38 – The distribution of the equivalent stress (Von Mises) on the boundary of axle structural model – the case of bearing on the whole inferior semi-ring
For fatigue analysis, it is considered a symmetrically oscillating event at the stress intensity calculated in static analysis, the applied number of cycles being 1000. The endurance curve for axle material (carbon steel) is chosen from the database of SolidWorks program and is drawn in Fig. 39. The curve is adopted by ASME for carbon steels, [4]. The main results of the fatigue analysis are the percentage damage and the number of life cycles supported by the structure at this intensity. For the structure embedded on the whole bearing ring, were graphically drawn the distribution of percentage damage, respectively the number of cycles of the
36
type of the tested event on boundary of the structure, both globally and in detail, in Fig. 40, 41, 44 and 45. For the structure embedded only on the inferior bearing semi-ring, the same representations are given in Fig. 42, 43, 46 and 47.
Fig. 39 – Stress-number of limit cycles curve, for the carbon steel of which the axle is made
Fig. 40 – Distribution of the damage on the boundary of axle structural model - the case of bearing on the whole bearing ring 37
Fig. 41 – Distribution of the number of life cycles on the boundary of axle structural model – the case of bearing on the whole bearing ring
Fig. 42 – Distribution of the damage on the boundary of axle structural model – the case of bearing on the inferior semi-ring 38
Fig. 43 – Distribution of the number of life cycles in the boundary of axle structural model – the case of bearing on the inferior semi-ring
Fig. 44 – Distribution of the damage on the boundary of axle structural model – the case of bearing on the whole ring, detail 39
Fig. 45 – Distribution of the number of life cycles on the boundary of axle structural model – the case of bearing on the whole ring
Fig. 46 – Distribution of the damage on the boundary of axle structural model – the case of bearing on the inferior semi-ring, detail
40
Fig. 47 – Distribution of the number of life cycles on the boundary of axle structural model – the case of bearing on the inferior semi-ring, detail
Table 3
Model
Maximum relative Maximum equivalent resultant displacement, stress, MPa mm
Life cycle (cycles of the Maximum type of the damage, % considered event)
Embedding on the whole
0.85
317
20.91
47810
1.05
404
20.91
47810
bearing ring Embedding on the bearing semi-ring
41
4. CONCLUSIONS The reference model with which the results should be compared should be the experimental model. The ability of the structural model to reflect the reality would then be estimated by the difference between the theoretical and experimental results. In this manner might be chosen the most faithful model of the real structure. In the absence of some experiments, for example for a structure which is in the design stage or maybe in the conceptual stage, the reference model used by the authors of this paper is the analytical theoretical model proposed in [1], [2] or [16], the articulated bar at the ends and loaded symmetrically towards the middle of it. The results of the reference model appear in Table 1 were calculated simply by analytical formulas and Saint Venant's formula. The comparison terms are the maximum resultant relative displacement (and its localization), the maximum equivalent stress (and its localization) and, where possible, the first few own frequencies. A first conclusion of the study is that the abutment mode of the structure influences decisively the effects produced by a same loading on the structure: displacement, specific deformation, the stress, the spectrum of own frequencies. A second conclusion shows that the stress in structure increases with its degree of rigidity, while maintaining the same load. At the same time, the displacement in structure decreases with increasing the stiffening degree. The degree of stiffening represents the ratio between the number of freedom degrees canceled in the given case and the number of freedom degrees canceled in the reference situation for the same model and the same discretization. A precise measurement of the stiffening degree of a structure is not standardized. The next conclusion refers to the fact that the elastical bearing gets closer more to the reference model (model validated through the good results achieved with its help over a long period of time). 42
The elastic bearing used has adjusted characteristics so that the model elastically beared to have a maximum resultant relative displacement as close to the reference model. For the improvement of the model, the characteristics of this elastic bearing must be deducted from the elastic characteristics of a natural suspension of the axle: rim and tire. The superior model will also include and the submodels of supporting structures. Similarly, a better sizing will be obtained and by respecting as accurate as possible of the application mode of the loads on structure (plates for forces and even the full representation of the chassis). It will not only get accurate information, but even more. As a result of differences in the results provided by the models are two important conclusions. The first conclusion is that physical experiences are the only ones that can validate or select the most accurately model of a collection of models proposed. In absence of experimental data, the reference results for the modeling of the axle are those of the model used for a long time in designing, M1. Such models are used successfully for over a century and gave good results in major European and world industries. The majority of the proposed models give the same deformation of the axle, located in the middle of it, ranging between 5 and 9 mm. However, all these models give close values of the deformation in axle, are loaded with independent forces. A value less than 10 times of the axle deformation indicates the most complex model, M5. This result is due to the special charge compared to the other models, loading with a numerically equivalent force, but by means of a rigid frame. Regarding the maximum equivalent stress in the structure of the main objective of the study – the axle – the reference model gives a value that overestimates by approximately 25% the maximum value of it. In addition, if the models M1 - M3 locate the maximum equivalent stress in the middle of
43
the axle, while the complex model M5 locates it and on the axle in the vicinity of the fastening points of the chassis, but also in the chassis frame. It is observed that the model results [2], although much different from those of the complex model M5, are covering, which explains the proper functioning of the axles in a very long operating time. The reality seems to validate this complex model, [32], [33], [34], [35], [36], [37]. If the bearing mode has an appreciable influence on the main fields of structure state (structural model of the axle), Table 3, in terms of accumulation of fatigue or lifetime, the difference of stress fields in the two cases shows no differences in the resolution limits which are required in practice. This situation is due to the fact that, on the Woher curve of the material structural model (Fig. 39), the maximum stress in Table 1, appropriate to the two cases of bearing, corresponds to very similar number of cycles, which differ by values of orders of tens of cycles or even less, being practically insignificant. Moreover, the shape and location of areas affected by fatigue for the two cases (Fig. 44, 45, 46, 47), are very similar. In both cases, fatigue is triggered and obviously will lead to the emergence and propagation of cracks. When the fatigued area will have a large enough surface, it will significantly shrink the useful section of the axle, making possible the occurrence of the sudden failure due to achieving the limit breaking stress on a vital area as stretch. Firmer
conclusions
regarding
the
recommendation
to
use
of
sophisticated models that would lead to structures more flexible, lighter, and more efficient; we will be able to issue once we have enough experimental data. However, the structural models are a source of increasing of safety for prediction in important issues such as fatigue or reliability of agricultural transportation equipment and we recommend them in the testing activity of them.
44
5. REFERENCES [1]. Drobotă V., Strength of Materials, Didactic and Pedagogical Publishing House, Bucharest, 1982; [2]. Comănescu A., Suciu I., Weber Fr., Comănescu D., Mănescu T., Grecu B., Mikloş I., Mechanics, Strength of Materials and Machine Parts, Didactic and Pedagogical Publishing House, Bucharest, 1982; [3]. Iacob C., GheorghiŃă I., S., Soare M., Dragoş L., Dictionary of Mechanics, Scientific and Encyclopedic Publishing House, Bucharest, 1980; [4]. Documentation SolidWorks Simulation 2012; [5]. GafiŃanu M., Poteraşu V. F., Mihalache N., Finite and Boundary Elements with Applications of Machine Parts Computation, Technical Publishing House, Bucharest 1987; [6]. Sanjay Aloni, Sandip Khedkar, Comparative Evaluation of Tractor Trolley Axle by Using Finite Element Analysis Approach, International Journal of Engineering Science and Technology (IJEST), vol. 4 No .04 April 2012; [7]. Vinod Kumar Verma, Dinesh Redkar, Arun Mahajan, Weight Optimization of Front Axle Support By OptimStruct Technology Application, HTC 2012, Mahindra Rise; [8]. Lalit Bhardaj, Amit Yadav, Deepak Chhabra, Design & Static Analysis of rear
axle,
Academia.edu,
http://www.academia.edu/1055381/DESIGN
_and_STATIC_ANALYSIS_OF_ REAR_AXLE; [9]. GhiŃă E. Gilbert-Rainer Gilich, An Analysis about the State of Stress in a Railway Axle, Annals of University "Eftimie Murgu" Resita, Year XIV, no. 1, 2007; [10]. Meng Qinghua, Zheng Huifeng, Lv Fengjun, Fatigue Failure Fault Prediction Of Truck Rear Axle Housing Excited By Random Road Roughness, International Journal of the Physical Sciences, ol. 6(7), pp. 1563 – 1568, 4 April, 2011; [11]. Mahanty D., K., Manohar V., Khomane B. S., Nayak S., Analysis and eight Reduction of a Tractor’s Front Axle, 45
http://designspace.com/staticassets/ANSYS/staticassets/resourcelibrary/conf paper/2002-Int-ANSYS-Conf-106.PDF; [12]. Davids W., G., EverStressFE1.0 Software for 3D Finite-Element Analysis of Flexible Pavement Structures, The Washington State Department of transportation, June 2009; [13]. Leon N., Martinez O., Orta P. C., Adaya P., Reducing the Weight of a Frontal Truck Axle Beam Using Experimental Test Procedures to Fine Tune FEA, 2end Worldwide MSCAutomotive Conference Dearbom, Michigan, october 9-11, 2000; [14]. Jafari A., Khanali M., Mobli H., Rajabipour A., Stress Analysis of Front of JD 955 Combine Harvester Under Static Loading, Journal of Agricultural &Social Sciences, 2006; [15]. Khanali M., Jafari A., Mobli H., Rajabipour A., Analysis And Design Optimization Of A Frontal Combine Harvester Axle Using Finite Element And Experimental Methods, Journal of Food, Agriculture & Environmental vol. 8 (2):359-364, 2010; [16]. Buzdugan Gh., Strength of Materials, Technical Publishing House, Bucharest, 1980; [17]. Osman A., Fatigue Failure of Rear Axle Shaft of an Automobile, Engineering failure analysis 13 (2006), p. 1293-1302 [18]. Smith R.A., European Structural Integrity Society, TC24 – Structural Integrity of railway components, Seminar: 3-4 March 2011, Predicting real world axle failure and reliability [19]. Topac M. M., Gunai H., Kuralay N.S., Fatigue failure prediction of a rear axle housing prototype by using finite element analysis, Engineering Failure Analysis, vol. 18, issue 5, 2009, p. 1474-1482 [20]. Dikmen F., Bayraktar M., Guclu R., Raylway Axle Analysis: Fatigue Damage and Life Analysis of Rail Vehicle Axle, Journal of Mechanical Engineering, 58, 2012, 9, p. 545-552
46
[21]. Moore H.F., A Study Of Fatigue Cracks In Car Axles, Production Note University of Illinois at Urbana-Champaign Library Large-scale Digitization Project, 2007 [22].
https://www.google.ro/search?q=axle+fatigue+failures&newwindow=
1&sa=N&tbm=isch&tbo=u&source=univ&ei=Af7KUtm6NIO_ygOFm4KYAQ&v ed=0CDwQsAQ4Cg&biw=1284&bih=638#imgdii=_ [23]. Radeş M., Finite elements analysis, 2006; [24]. Marin C., Hadăr A., Popa F., I., Albu L., Finite element modeling of mechanical structures, Romanian Academy Publishing House, AGIR Publishing House, Bucharest, 2002; [25] .Buzdugan Gh., Fetcu L., Radeş M., Mechanical vibrations, Didactic and Pedagogical Publishing House, Bucharest, 1982; [26]. Sofonea G., Pascu A. M., Strength of Materials, University "Lucian Blaga", Sibiu, 2007; [27]. Dupen B., Applied Strength of Materials for Engineering Technology, National Center for Educational Statistics, U.S. Department of Education, 2012; [28]. Bansal R. K., A Textbook of Strength of Materials, revised fourth edition, Laxmi
Publications
(P)
LTD,
2010,
http://books.google.
ro/books?id=2IHEqp8dNWwC&printsec=frontcover&source=gbs_ge_summar y_r&cad=0#v=onepage&q&f=false; [29]. Morley A., Strength of Materials, fourth edition, Logmans, Greeen and CO. 39, London, 1916; [30]. Tudor A., Machine parts, Polytechnic University of Bucharest, 2004; [31]. Documentation program for finite element structural analysis COSMOS / M 2.8; [32]. Klinger C., Bettge D., Hacker R., Heckel T., Gohlke D., Klingbeil D., Failure Analysis on a Broken ICE3 Railway Axle, Bundesanstalt fur Materialforschung und–prufung, ESIS TC24 Railway Structures, 11.Oct.2010;
47
[33]. Kendall C.C., Halimunanda D., Failure Analysis of Introduction Hardened Automotive Axles, Journal of Failure Analysis and Prevention, 2008; [34]. Transportation Safety Board of Canada, Railway investigation report R07T0240, 2007; [35]. Asi O., Fatigue failure of a rear axle shaft of an automobile, Engeineering Failure Analysis 13 (2006) 1293-1302; [36]. Transportation Safety Board of Canada, Railway investigation report R04V0173, 2004; [37].
LVVTA
Tech
Team,
Magnum
Axle
Safety
Warning,
2012,
http://lvvta.proboards.com/index.cgi?board=general&action=display&thread= 274.
48
