Pierre Lemerle

Pierre Mistrot

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Parametrical study of the tire properties to optimise the vibratory behaviour of a forklift truck. Etude paramétrique des caractéristiques de pneumatiques pour l'optimisation du comportement vibratoire d

Keywords:

un chariot élévateur.

Introduction

Within the frame of the study called « Modelling of the dynamical behaviour of the forklift truck tires», INRS designed and validated a new numerical model of forklift truck tires [START_REF] Lemerle | A New Tire Model to Predict Vibration Emission of Counterbalance Trucks[END_REF].

From the quasi-static measurement of the physical parameters characterizing the tires (solid as well as pneumatic), the results have shown that it was possible to predict their dynamical behaviour. This model was implemented in the whole model of forklift truck including the chassis, the forks and the cab. It was able to predict the vibration emission at the driving position.

The interest of such a model lies in the ability of simulating the effects of design modifications on the vibration emission. The aim of this study was to run this model to analyse the influence of each of the physical parameters of the tires (stiffness, damping...) on the dynamical behaviour of the forklift truck and to optimise their performance in reducing vibration at the driving place.

More precisely, the aim was to evaluate the influence of theoretical physical properties (load/deflection law, damping parameters) on the attenuation properties of the tires. The aim was also to write guidelines, which may be useful to manufacture new tire prototypes. This objective is realistic from a technical point of view. This report is divided in five chapters :

The 1 st chapter is concerned with the forklift truck characterization. It is shown how to measure the mechanical properties of the forklift truck (position of the center of mass, mass and inertia). These data were used to set up the whole model of forklift truck equipped with the four tires.

The 2 nd chapter is dedicated to the characterization of the tires. The tests carried out to measure the static properties (stiffness and damping) were realized with a specific test bench designed by INRS. Three sets of tires were tested (front and rear tires) : one set of pneumatic tires, one set of solid tires and one set of mixed tires (pneumatic tires with an inner elastomer ring).

In the 3 rd chapter, the procedure used to test forklift trucks is detailed. It follows the recommendations of the corresponding European test code [START_REF]Safety of Industrial Trucks -Test Methods for Measuring Vibration[END_REF]. The protocol used to measure the vertical acceleration at the driving position for each configuration of tires is also explained.

The numerical model of the whole forklift truck equipped with the tires is shown in the 4 th chapter. The model was used to simulate the tests presented in the previous chapter. A good correlation between calculation and measurements is noted.

In the 5 th chapter, the model was used to test the influence of physical parameter changes : shape of the load/deflection law (degree of non-linearity of the stiffness), slope of the load/deflection law (stiffness), damping.

The model was able to predict the influence of these changes on the vertical acceleration at the driving position, according to the method recommended in the European test code.

Conclusion

This study aimed to analyse the effects of the tire properties on the vibratory behaviour of the forklift truck. It has been shown how to set up the model from static tests knowing the mechanical properties of the tires and of the forklift truck. This model was validated under the testing conditions prescribed by the European test code [START_REF]Safety of Industrial Trucks -Test Methods for Measuring Vibration[END_REF].

The parametrical study consisted in changing the tire characteristics of stiffness and damping and in calculating the effect of such changes on the transmitted vibration at the driving position, when the truck was running over obstacles (according to the European test code).

The numerous results obtained allow us to draw guidelines that may be used by tire manufacturers to take the vibration reduction into account in the process of design :

•

Concerning the stiffness, the calculation confirmed that the lower the stiffness, the higher the vibration attenuation. For a given stiffness value, the response of the forklift truck strongly depends on its velocity. The calculations and the experiments showed that there is a critical speed at which the vibration is the highest. Decreasing the stiffness leads to decrease the level of these vibration peaks and also to decrease the corresponding critical speed.

•

The calculations have shown that the use of different stiffness for front and rear wheels is not efficient to reduce vibration emission.

• Concerning the stiffness non-linearity, the calculations have shown that the vibration attenuation is the best for « softening tires», this meaning tires with a decreasing stiffness law in function of the deflection. The « softening » concept does not seem to be realistic. Moreover this property could have a bad influence on the forklift truck stability. Then, it may be considered that the aim to be reached corresponds to the linear case when the stiffness does not depend on the deflection.

These guidelines do not take stability or handling problems into consideration. These problems are also linked to the tire properties.

• Concerning the damping, the calculations have shown that it is always beneficial to increase it. Nevertheless it is very difficult to quantify the expected gains of attenuation, because the damping model used was too simple and the damping parameters are not directly connected to the material composition.

1-2-1 longitudinal position of the center of mass

The position was determined with a static balance procedure. This was an iterative method : the aim was to find the position where the forklift truck was stable when placed on the edge of a L section. When the equilibrium point was reached, the center of mass was just up on the section (cf. Figure I-3).

Characterizing the KOMATSU FD20 forklift truck I-l Measurement of the weight of the forklift truck KOMATSU FD20

The forklift truck was hung with ropes fixed to a force sensor. By so doing, the measured weight was 3420 kg.

Center of mass

I i

1-2-2 Verification of the longitudinal position of the center of mass

From the momentum law it was also possible to deduce the longitudinal position of the center of mass: the rear side was hung with ropes (cf. Figure I-4). When the forklift truck was horizontal the force in the ropes was 1435 kg. From the momentum law the distance between the projection of the center of mass on the floor and the center of the front wheel was deduced. This distance was 85 cm (87=1435*207/3420). So the position of the center of mass was made sure.

1-2-3 Height of the center of mass (measurement by rotating around the pitch axis)

This experiment consists in rotating the forklift (ruck by pulling ropes fixed on its rear side and measuring the angle of rotation (or equivalently measuring the height of the rear wheel) in function of the static force in the ropes.

a=:207.cm'< ' ' ' .

. » The formula linking the height Z G of the center of mass to the rotation angle of the forklift truck a and the tensile force F is :

Z G = X G cotga- f F cos f Arctg ) 4a 1 +b 2 + H Z G = X G cotga- Mg sin a I J + H
A proof of the formula (1) is given in appendix [START_REF] Lemerle | A New Tire Model to Predict Vibration Emission of Counterbalance Trucks[END_REF].

1-2-4 Height of the center of mass (measurement by rotating around the roll axis)

The same principle was applied by rotating the forklift truck around the roll axis. In the same manner, the tensile force was measured in function of the rotation angle of the forklift truck. The forklift truck was disposed on the edge of a L section, longitudinally placed on the floor (cf. Figure I-8). The distance between the fixing point of the ropes and the floor was called Z. The wheels of the forklift truck laid on wedges. When the forklift truck was horizontal, Z was 50 cm. To calculate Z G , the same method as previously was used, with the analytical formula (1) after replacing X G with (52.75-Y G ). In this case, a was 25 cm, b was 115.5 cm and H was 15.5 cm.

The evolution of the Z G value was calculated (with the analytical formula (1)) for each value of the rotation angle of the forklift truck (cf. Figure I-10). The results showed that the calculated value Z G did not depend of the rotation angle, for significant values of the angle. It was also verified that the calculated value of Z G was the same as the value previously calculated : Z G was 64 cm.

1-3-1 Ixx Inertia measurements :

The principle of these measurements consisted in hanging the forklift truck on springs in parallel (when the forklift truck was horizontal and laid on the edge of a L section, cf. Figure I-13) and in measuring the frequency of the free oscillations. The springs had a well-known stiffness (see their characteristics in appendix 2). With the knowledge of the free oscillation time period, the analytical formula (2) made it possible to calculate the inertia corresponding to the roll axis. For the experiment held with the 8 springs Ixx=1480 kg.m 2 For the experiment held with the 7 springs Ixx=1410 kg.m 2 This gave the mean value of 1450 kg.m 2

1-3-2 Iyy Inertia measurements :

The same principle was used for the measurements around the pitch axis (cf. The measurement was repeated with different numbers of springs, this meaning with different suspension stiffness. Different locations of the forklift truck support were also tested .

For each tested configuration, the frequency of the free oscillations was measured. The values of the I yy inertia calculated from the natural frequencies are reported in The cases 3 and 4 are covered with errors because the rotation was around the front wheel axle : for this reason, the inertia of the wheels was not taken into account. Moreover, it was observed that the wheels were slightly moving during the experiment, so that the axis of rotation was not always coincident with the front wheel axle. Then, the mean value of I yy was calculated from the cases 1,2,5,6,7, this giving :

I yy = 3100 kg.m 2
Taking into account all the symmetries, it was assumed that the axes (xx') and (yy') were "closed" from the principal axes of inertia, or, in other terms, the off-diagonal terms of the matrix of inertia were neglected.

Forklift truck model

Characterizing the tires

The mechanical properties of 3 sets of tires were measured :

• Pneumatic tires (inflated at 10 bars).

Front wheel tire called Gl (0=68 cm) Rear wheel tire called G2 (0=54 cm)

• Solid tires.

Front wheel tire called PI (0=68 cm) Rear wheel tire called P2 (0=54 cm)

• Mixed tires composed of pneumatic tires filled with an inner elastomer ring.

Front wheel tire called HI (0=68 cm) Rear wheel tire called H2 (0=54 cm)

According to the modelling method designed by INRS [START_REF] Lemerle | A New Tire Model to Predict Vibration Emission of Counterbalance Trucks[END_REF], the quasi-static properties characterizing the tires may be measured with two types of tests. One test at constant height (height of the wheel axle) gives the « shape factor » which is the polynomial characterizing the evolution of the contact forces when the tire is running over an obstacle of a given thickness. One other quasi-static test is used to get the load/deflection law of each tire.

II-1 Measurement of the shape factors

This test consists in measuring the changes of the contact force between the tire and the ground and/or the obstacle, in function of the horizontal position of the wheel and for a given constant height of the wheel axle.

The shape factor for the upper part gives the evolution of the contact force between the tire and the obstacle and the shape factor of the lower part is related to the changes of the contact force between the tire and the ground.

The shear of the tire on the edge of the obstacle is characterized by a parameter called « shock advance ». This parameter was also deduced from this quasi-static test F tire/ ground ~~ Pi j ground V ^0

J ^tire / obstacle P h ( jobst \ r obst \ L 0 J •F(z)

(3)

F (z + Kbst) (4)
With:

^tire/ground = contact force between the tire and the ground.

Ftire/obstacie = contact force between the tire and the obstacle. Pi: shape factor for the lower part, polynomial of 5 th order. P n : shape factor for the upper part, polynomial of 5 th order.

jjwund

. t h eoret i ca i t j r e p r j nt on trie ground ( see appendix 4).

jjround

. trieoret j ca j t j r e p r j nt on an infinite plane surface (see appendix 4).

L obst : theoretical tire print on the obstacle (see appendix 4).

L°Q St

: theoretical tire print on an infinite plane surface at the obstacle height (see appendix 4). F : load/deflection law on an infinite plane surface.

II-2 Measurement of the load/deflection law

Cyclic tests were carried out with the hydraulic actuator to obtain the load/deflection curve. The load/deflection curves were fitted with a 3 rd order polynomial of z (deflection expressed in cm).

II-3 Measurement of the damping coefficient

The damping was assumed linear and of viscous type. This is an important simplification because the damping depends strongly on the magnitude of the oscillations and on their frequency. Nevertheless previous calculations run with tires of different sizes (see ref. [START_REF] Lemerle | A New Tire Model to Predict Vibration Emission of Counterbalance Trucks[END_REF]) have shown that the viscous model is good enough to predict the vibratory behaviour of the forklift truck. To measure the viscous damping constant, the tire, loaded with 2 tonnes, was lifted up and released to generate free oscillations. The logarithmic decrement method was then used to identify the damping constant from the time histories of the free oscillating deflection. The logarithmic decrement method consists in measuring two different extrema Xi and X i+ i and the two corresponding times tiand ti+i of the time history. The following formula gives then the damping :

S - 2co 0 M -Imi 2n Í ( 5
)
Where 8 is the critical damping, which is a function of the natural pulsation of the structure co 0 . 0) 0 = the linear case. To measure the viscous damping constant under dynamical conditions, the tires were tested with the laboratory tire test bench (see the description of the test bench in appendix 5): the arm supporting the wheel was loaded with 1 tonne. A 1x15 cm PVC obstacle was stuck inside the rolling ring. The tires were due to run over the obstacle and their speed was controlled. At each turn of the ring, after running over the obstacle, the system could freely oscillate. The vertical acceleration of the wheel was measured. The viscous damping constant was deduced from these measurements using the logarithmic decrement method. Three different velocities were tested : 5 km/h, 10 km/h and 12 or 14 km/h depending on the tire. For each of these tests, 2 values were calculated, one coming from the maximum acceleration values and the other from the minimum acceleration values. Indeed it was observed that for solid and mixed tires the curve decrease was not symmetrical 1 Caution must be care of acceleration peaks of -9.81m.s~2. They do not have to be used because when the aceleration is -9,81m.s 2 , that means that the tire has left the ground and it falls down to the ground. It was observed that damping values strongly depend on the rolling velocity for the solid and mixed tires. Then the damping was modelled as a 3 rd order polynomial of the rolling velocity :

C = p x .v 3 + p 2 .v 2 + p 3 .v + p 4 ( 6
)
where v is the speed expressed in km/h and C the viscous damping constant (N/m/s).

Tire G1

Measurement of the vibration emission of the KOMATSU FD20 III-l Test protocol

The KOMATSU FD20 forklift truck was tested according to the procedure described in the European test code [START_REF]Safety of Industrial Trucks -Test Methods for Measuring Vibration[END_REF]. The principle of this test code consists in moving the forklift truck along a flat test-track on which two 1x15 cm obstacles were disposed and in measuring the vertical acceleration at the driving position. The space between the two obstacles was 10 m. Unlike the recommendation of the test code [START_REF]Safety of Industrial Trucks -Test Methods for Measuring Vibration[END_REF], the speed of the forklift truck was not fixed at 10 km/h : several values of velocity were tested, between 2 and 15 km/h with an increment of 1 km/h. Each test was carried out at a constant velocity. An optical encoder sensor was used in contact with the front right wheel to measure the velocity in real-time.

The driver could use this information displayed on a multimeter to adjust the forklift truck velocity to the specified value and to keep it constant as much as possible. The velocity and the 3 accelerations were acquired simultaneously. These 4 channels were transmitted in real-time to the laboratory, by means of a telemetry system (see Figure III-1). They were also acquired with the LMS system.

The same experiment was repeated the three types of tires : forklift truck successively equipped with solid, pneumatic and mixed tires. The aim was to compare the vibration emission for each type of tires tested.

Remark;

For the tests at higher speeds (> 3m/s), three wave trains were observed. These wave trains correspond respectively to the starting point (marked with a few mm thick plank), to the first obstacle of the test code (1x15 cm) and to the second obstacle (1x15 cm). For the lower speeds each of these wave trains is divided into two wave trains : the first wave train is due to the front wheels running over the obstacle and the second wave train is due to the rear wheels.

HI-3-Analvsis of the results

To quantify the measured acceleration responses, a criterion equivalent to a dose was calculated from each time history 1 . This criterion was expressed as the maximum of a mobile RMS value calculated over a period of 1 s, according to the formula (7). It comes from the ISO standard 2631-1 [START_REF]Mechanical Vibration and Shock -Evaluation of Human Exposure to Whole Body Vibration[END_REF]. It was assumed to be representative of the human perception of shocks. It takes into account both peak values and shock duration. This criterion was calculated for each obstacle crossing, so that two values were deduced from one run.

RMS (J)

Figure III-14 shows with an example of measured acceleration the meaning of the criterion (7). 1 The signals were firstly filtered with a numerical low-pass filter whose cut frequency was 10 Hz. The same tendencies as previously with the KOMATSU FG15 [START_REF] Lemerle | A New Tire Model to Predict Vibration Emission of Counterbalance Trucks[END_REF] may be observed, e.g. the phenomenon of interference between the shock on the front axle and the shock on the rear. Depending on the speed, the second shock can attenuate or amplify the vibration generated by the first shock. So a "critical speed" value still exists. For this value, which is around 12 km/h, (in the case of the FG15 it was around 8.5 km/h, see [START_REF] Lemerle | A New Tire Model to Predict Vibration Emission of Counterbalance Trucks[END_REF]), the ratio between calculated y RM s values for solid and mixed tires is around 1.7. Considering the overall range of speeds, the best behaviour was achieved with mixed tires

Modelling the forklift truck -Description and validation

IV-l Description of the KOMATSU FD20 model

The whole model of forklift truck with tires was implemented with SDS [START_REF]SDS : kinematical and mechanical multi-body modelling software is designed and edited by the company Solid Dynamics[END_REF] (the program code is given in appendix 6).

The forklift truck (chassis, engine, cab, forks ...) was modelled as a mass with inertia. The tires were also modelled as masses on which were applied contact forces with the ground and the obstacles. These contact forces were analytically calculated using the INRS model [START_REF] Lemerle | A New Tire Model to Predict Vibration Emission of Counterbalance Trucks[END_REF]. The tires were linked to the chassis with spherical joints.

The velocity was imposed on the 2 front wheels (see Figure Remark : the SDS model was "skinned" to give it a realistic look. The shapes representing the chassis, the forks or the tires have only an esthetic function and they do not have any influence on the calculation. For example, the mass and the inertia were entered as numerical values (previously measured data) and they did not result from a geometrical calculation based on the geometries.

IV-2 Results

The model allowed us to simulate the tests carried out within the frame of the European test code [START_REF]Safety of Industrial Trucks -Test Methods for Measuring Vibration[END_REF] and to compare the calculations with the measurements. For each type of tires, 14 runs at different speeds between 2 and 15 km/h by steps of 1 km/h were computed. In the following figures, the calculated y RM s values are compared to the measured ones.

IV-2-1 Comparison between calculated and measured yRMS

It can be seen that there is a close agreement between calculations and measurements: the model made it possible to predict the evolution of the y RM s criterion in function of the forklift truck speed. In previous studies [START_REF] Lemerle | A New Tire Model to Predict Vibration Emission of Counterbalance Trucks[END_REF], it has been explained that the observed extrema (minima and maxima of the y RM s criterion) come from an interference phenomenon between the free response of the forklift truck after the shock on the front axle and the shock on the rear axle. Depending on the speed, the second shock may occur in phase or in phase opposition with the free response of the forklift truck, involving attenuation or amplification. The good correlation between the calculated and measured extrema shows that the natural frequency of the forklift truck was well estimated.

Parametrical study of the influence of tires on the vibration emission at the driving position

This study was realized with the Komatsu FD20 numerical model. The forklift truck equipped with pneumatic tires was considered to be the reference configuration. The reason is that pneumatic tires had almost linear stiffness (around 6000 N/cm) as well for front wheel tires as for rear wheel tires. The aim of this study was to calculate the vibration emission at the driving position [START_REF]Safety of Industrial Trucks -Test Methods for Measuring Vibration[END_REF], and to evaluate the effects of physical parameters of tires on the transmission of vibration.

The evaluation criterion was the maximum of the mobile RMS value calculated over a 1 s period (see chapter III). This criterion is called «MAX acc RMS/ls » or «YRMS» in the following figures. Like in the previous chapter, the calculations were made for various speeds between 2 and 15 km/h by steps of 1 km/h.

Firstly our interest was focused on the tire stiffness. The shape factors were always considered as constant and not depending on the stiffness. Indeed, the quasi-static measurements carried out with the three types of tires (cf. fig. II-3 to II-8) showed that the shape factors were very similar, though their stiffnesses were extremely different. So, we only dealt with the load/deflection curve. Four different cases were considered : Linear case. In this case the load/deflection curve was fully described with one parameter : the stiffness, e.g. the slope of the load/deflection line. A non-linearity was added by changing the convexity of the load/deflection curve. A first step was to modify the convexity without changing the static equilibrium conditions, this meaning without changing the deflection value of the tires under the forklift truck weight (see §V-l-3-l).

A second step was to modify, the convexity without changing the tire stiffness around the static equilibrium (the slope was kept constant for the force value corresponding to the weight of the truck, §V-l-3-2 ).

The effect of the tire damping was also evaluated. The damping model used was basic and not really realistic. So, the aim was not to predict gains of attenuation but to analyse the tendencies.

V-l Influence of the tire stiffness

V-l-1 Same linear stiffness for front wheels and rear wheels

Six values were successively used : 5000,5500,6000,6500,7000,7500 N/cm.

The results are shown in Figure V-1.

+ Fr ont sti ff ness =7500, Rear sti ff ness =7500 Fr ont st i f f ness =7000, Rear sti ff ness =7000 Fr ont st i f f ness =6500, Rear sti ff ness =6500 O Fr ont st i f f ness =6000, Rear sti ff ness =6000 V Fr ont sti ff ness =5500, Rear sti ff ness =5500 • Fr ont st i f f ness =5000, Rear st i f f ness =5000

0 • 0 Velocilvfkntfi) Figure V-1 :
Influence of the tire stiffness on the forklift truck vibration emission.

Linear stiffness -same values for front wheels and rear wheels

The increase of stiffness leads to higher vibration emission, especially within the range of speed where the vibratory behaviour of the forklift truck is governed by interference phenomena between the front and rear axle (speeds higher than 5 km/h). This interference phenomenon, detailed in reference [START_REF] Lemerle | A New Tire Model to Predict Vibration Emission of Counterbalance Trucks[END_REF], is linked to the natural frequency of the forklift truck. It can be seen that, the higher the stiffness, the higher the natural frequency and consequently the curve is shifted towards the higher speeds.

To get a global comparison of the different sets of tires, the mean value of the criterion was calculated over the 14 velocity values. Linear stiffness -different values for front wheels and rear wheels. The differences are not so high as previously. Then, the use of tires with different stiffness for front wheels and rear wheels is not efficient in terms of vibration reduction.

V-l-3 Same non~ linear stiffness for front wheels and rear wheels V-l-3-1 Static deflection unchanged

The principle of these calculations consisted in changing the stiffness of the tires by increasing the curvature of the load/deflection curve without changing the equilibrium point (static equilibrium defined by the Force F 0 = 8500 N and the deflection z 0 = 1.5 cm).

The chosen load/deflection curves were defined by the coefficients: Pi = -444 z 2 + 6333 z, p 2 = -111 z 2 + 5833 z, (decreasing stiffness or « softening » system), p 3 = 555 z 2 + 4833 z, p 4 = 888 z 2 + 4333 z, (increasing stiffness or « hardening » system). Non-linear stiffness -same equilibrium point.

The lowest acceleration responses were obtained with the polynomial pi and p 2 corresponding to decreasing stiffness. Like the linear case, the lower the stiffness, the lower the acceleration response and consequently the curve is horizontally shifted towards the lower speeds.

Remark:

It was also verified that the stiffness law tested for each tire did not involve non-physical behaviour like, for example, very large deflection. For each configuration and for each run, the maximal dynamical deflection of each wheel was computed. We called wheel deflection the difference between the height of the wheel center and the wheel radius. Between 2 km/h and 15 km/h, the maximum deflection values ranged between 1.6 and 3.2 cm for the front tires and between 1.7 and 3.2 cm for the rear tires ; the minimum values ranged between -0.4 and 1.2 cm for the front tires and between -1.2 and 1cm for the rear tires. All these values seem to be reasonable.

V-1-3-2 constant stiffness around the static equilibrium position

The constant parameters were the initial static force of 8500 N, and the stiffness around the static equilibrium position 6000 N/cm. Unlike the previous study, the static deflection was varied: 1.1; 1.3 ; 1.7 ; 1.9 cm.

The chosen load/deflection curves were defined by the coefficients:

(1,1 cm) ql = -1570 z 2 + 9454 z, (1,3 cm) q2 = -414 z 2 + 7077 z, (decreasing stiffness), (1,7 cm) q3 = 586 z 2 + 4000 z, (1,9 cm) q4 = 803 z 2 + 2947 z, (increasing stiffness). This parameter has a real effect on the vibration reduction for each value of the velocity. But it is not easy to set up material with such high levels of damping (> 2000 Ns/m). For example, the value of 5000 N/m/s does not seem to be realistic for conventional tires.

V-3 Influence of tire properties at 10 km/h

The European test code stipulates that the trucks shall be tested at 10 km/h. The criterion recommended by the European test code was also calculated [START_REF]Safety of Industrial Trucks -Test Methods for Measuring Vibration[END_REF] as a weighted RMS value (according to the ISO standard 2631 [START_REF]Mechanical Vibration and Shock -Evaluation of Human Exposure to Whole Body Vibration[END_REF]), calculated over the whole acceleration signal emitted by the forklift truck when running over the two obstacles. Calculation made with different characteristics for front and rear tires Although this criterion differs from the previous criterion (YRMS) it can be seen that it may vary from a factor 2, depending on the type of tire configuration used. The configuration giving the best performance with the YRMS criterion (configuration 5050) does not give the best result with the procedure of the European test code. The reason is that the European procedure does not take into account the shift of the curves in function of the stiffness changes V-l-1 because only one velocity is considered. View of the system used to measure the stiffness One of the 10 springs was tested. It was fixed to the ground on one of its extremities and pulled by a cranage on the other extremity. A system made of pulleys was used to reduce the travel of the spring by a constant factor and to measure it with a LVDT sensor. The results of the traction test are shown in the following figure. The characteristics given by the manufacturer were confirmed with this test : stiffness of 0.75 daN/mm instead of 0.73 daN/mm and a preload of 50 daN instead of 51 daN. this giving F do :

The equation of motion is now : The equation of motion ( 19) is then :

Figure I- 1 : 1 - 2

 112 Figure I-1: Weighing of the forklift truck

Figure I- 2 :

 2 Figure I-2 : Measurement of the longitudinal position of the center of mass

Figure I- 4 :

 4 Figure I-4 : measurement of the height of the center of mass

Figure I- 5 :

 5 Figure I-5 : View of the measurement of the height of the center of mass by rotating around the pitch axis Figure I-6 shows the evolution of the calculated value Z G (with the exact formula (1)) for each value of the rotation angle of the forklift truck. The results showed that the calculated value Z G did not depend of the rotation

Figure I- 8 :

 8 Figure I-8 : View of the measurement of the height of the center of mass by rotating around the roll axis

Figure I- 9 :

 9 Figure I-9 : Measurement of the height of the center of mass

Figure 1 - 3

 13 Figure I-10 : Calculation of the height of the center of mass in function of the rotation angle (roll)

Figure I- 12 :

 12 Figure I-12 : Measurement of roll inertia by free oscillations.

Figure I- 13 :

 13 Figure I-13 : Measurement of the I xx inertia

  Figure I-16).

Figure I- 16 :

 16 Figure I-16 : Measurement of the pitch inertia by free oscillations.

Figure I- 21 :Figure I- 24 :

 2124 Figure I-19: Case 2 ; X=54 cm ; 6 springs ; natural frequency = 0.825 Hz

Figure I- 25 :

 25 Figure I-25 : Equivalent mechanical diagram for the KOMATSU FD20

Figure II- 1 :Figure II- 3 : 1 :Figure II- 4 :- 2 :Figure

 13142 Figure II-1: Contact forces between the tire and the ground and/or the obstacle Figure II-2 shows the test bench used for the measurement of the shape factor: a PVC obstacle (same height as the obstacle used for the vibratory tests carried out with the forklift truck : 1cm) was glued on a thinner PVC strip. The wheel is squeezed on the PVC strip by a hydraulic actuator. The PVC strip is placed on a support covered with TEFLON. The height of the wheel axle is kept constant by the actuator. The wheel is motorized. It can slowly rotate. With the rotation of the wheel, the PVC strip can slide on the support and force the relative displacement between the obstacle and the wheel.

Figure

  Figure II-7 : Shape factor of the mixed tire (0=68 cm)

Figure

  Figure II-8 : Shape factor of the mixed tire (0=54 cm)

FigureFigureFigureFigureFigure

  Figure II-9 : Load/deflection law for the pneumatic tire (0=68 cm) Stiffness law of the tire Gl : F(z) = -86 z 3 + 1024 z 2 + 5788 z

FigureFigureFigure

  Figure II-15 : Free oscillation of the pneumatic tire (0=68 cm)

Figure

  Figure 11-21 : Run over an obstacle Pneumatic tire (0=54 cm)

FigureFigureFigureFigureFigure 11 Figure

 11 Figure II-23 : Run over an obstacle Pneumatic tire (0=54 cm)

Figure II- 36 :Figure

 36 Figure II-36 : Run over an obstacle Solid tire (0= 68 cm)

FigureFigure II- 44 :

 44 Figure II-43 : Evolution of the viscous damping constant in function of the tire rolling velocity Mixed tire (0= 68 cm)

Figure III- 1 :

 1 Figure III-1 : View of the equipped KOMATSU FD20 and the test-trackThe acceleration was measured below the suspension seat. By so doing the seat attenuation was not taken into account (the accclerometer was placed in the middle of the seat base on the engine cover; coordinates of the accelerometer position in the center of mass axes : x = -13 cm; y = 32 cm; z = 3 cm) (see FigureIII-2).

Figure III- 2 :

 2 Figure III-2 : Position of the accelerometer to measure the vibration emission

Figure III- 4 :

 4 Figure III-4 : Speed measurement system

Figure III- 5 :III- 2 ResultsIII- 2 - 1 Figure III- 6 :Figure III- 7 :III- 2 - 2 Figure III- 8 :Figure III- 9 :FigureIII- 2 - 3 FigureFigureFigure

 522167228923 Figure III-5 : Acceleration measured for the forklift truck equipped with pneumatic tires

Figure

  Figure III-14 : Definition of the y RM s criterion. Illustration with an example of measured acceleration.

Figure IV- 1 :

 1 Figure IV-1 : Diagram of the KOMATSU FD20 model

Figure IV- 2 :

 2 Figure IV-2 : View of the SDS model

Figure IV- 3 :Figure IV- 4 :IV- 2 - 2 ForkliftFigure IV- 6 :FigureForkliftFigure IV- 8 :ForkliftFigure

 342268 Figure IV-3 : Comparison between calculated and measured vibration emission of the KOMATSU FD20equipped with mixed tires.

Figure V- 2 :

 2 Figure V-2 : Evolution of the mean value of the YRMS criterion versus the tire stiffness configurations Linear stiffness -same values for front wheels and rear wheels

Figure V- 3 :

 3 Figure V-3 : Influence of the tire stifftiess on the forklift truck vibration emission.Linear stifftiess -different values for front wheels and rear wheels.

Figure V- 4 :

 4 Figure V-4 : Evolution of the mean value of the YRMS criterion versus the tire stifftiess configurations.

Figure V- 5 :

 5 Figure V-5 : Load/deflection curves with the same equilibrium point (1.5 cm, 8500 N).

Figure V- 5

 5 Figure V-5 shows the shape of these polynomials.

Figure V- 6 :Figure V- 7 :

 67 Figure V-6 : Influence of the tire stiffness on the forklift truck vibration emission.Non-linear stiffness -same equilibrium point.

Figure V- 8 :Figure V- 9 :

 89 Figure V-8 : Case 1-stiffness defined by the polynomial pi : max and min dynamical deflection for each run of the forklift truck

Figure V- 12

 12 Figure V-12 shows the shape of these polynomials.

FFigure

  Figure V-12 : Load/deflection curves with constant stiffness for a given force of 8500 N.

Figure V- 13 Figure V- 14 :

 1314 Figure V-13 shows the calculation results. The values corresponding to the nominal case (initial deflection of 1.5 cm for all the tires) are also presented as a reference. In the case of 1.1 cm of initial deflection, the stiffness value is 0 at around 3 cm and becomes negative for higher deflection which is a non-sense from a mechanical point of view.

Figure V- 15 V- 2 Figure V- 16 :Figure V- 17 :

 1521617 Figure V-15 shows the max and min dynamical deflection for all the calculated runs. It was verified that in any case the wheel moved in the range of negative stiffness (deflection higher than 3 cm, see Figure V-12).

Figure V- 3

 3 Figure V-3 and Figure V-12 show that at this velocity, the YRMS criterion (as defined in the introduction of chapter 5) may vary from a factor 2, depending on the type of tires used.

  Our model was used again to predict the values for different configurations of tires. The calculation results are shown in Figure V-18 and Figure V-19.

Figure V -Figure V- 18 :Figure V- 19 :

 V1819 Figure V-18 corresponds to the same stiffness for front and rear tires, (Index 7575 means that the stiffness is 7500 N/cm both for front and rear tires) ;

  Orthogonal projection of S on (Ox) :Ol = OS cos(OI, OS) = OS cos\Pythagoras's theorem : of yhcosa from the equation (7) in equation (8), it gives the expression of z G in function of y G , Fda and a:

1 [sin ( 2 «_Mg yi +fcy 2 z 2 Equation 2 (

 1222 00, = ON-O.N = OPcosa-MP = OP cos a -GP sin a -yx cosa-z^ sin yi AND a SMALLCalculation of FH" :Assuming small rotations, the strain of the springs is due to the change of height of S : motion becomes now:J *LJL = -Mg (y} cos a -zx sin a) + (Fda0 -k(z2 cos a + y2 sin a)\y2 cos a -z 2 sin or) (6) Assuming small rotations, we have : fcos Û: « In this case, equation (6) becomes : constant coefficients are 0. So can we determine F da0 : 10) the oscillation period T of the forklift truck may be deduced. It can be expressed with equation (11) in function of I xx : can be expressed in function of T : The elongation of the springs is characterized by the difference between the lengths: COS a -1) + y2 sin a)) 2 + (y2 (l -cos a) + z 2 sin a) 2 -L (16) Assuming small rotations, AL becomes then : AL « ^(L-y 2 a) 2 + (z 2 a) 2 -

  (l ~ cos A ) + Z 2 si n a ) (L + Z 2 ) (L + Z 2 (lcos A) -Y 2 sin A) \L-Y 2 A) L

3AI

  ^7T = ~MS(YI ~f a « o -KY 2 CC{Y 2 -Z 2 A^L + ZL ^ = fo«oY2 ~ MGY, ) + [MGZ X -KY\ -F DA0 Z 2 fe±£î) V L \AAt equilibrium the constant coefficients are 0. So can we determine F<j a0 :Then the free oscillation period of the forklift truck is deduced: YI J I L J -M OG 2 )

Table I -

 I

1 : Measured values of the tensile force in function of the rotation angle of the forklift truck

  Table I-3. In theory, all the natural frequencies measured must lead to the same value of I vv .

	X(cm)			L(m)	Number of		Natural	Pitch inertia
					springs used		frequency	Iyy (kg.m 2 )
							(Hz)
	Case 1	54		44	8		0.95	3138
	Case 2	54		49	6		0.825	2981
	Case 3	0		54.5	8		1.025	3823
	Oscillations aroui
	the front wheel axle
	Case 4	0		62.5	6		0.9	3561
	Oscillations around
	the front wheel axle
	Case 5 Case 6 1 Case 7 |	64 64 __	43 " 48 53	6 -3	_	Ô78~ 0.602 0.5	3042 3170 illlllllll	1
	Table I-3 : Different tested configurations of free oscillations
					c o
					05
					j-0 0) o o < y /1 1 w 0 1 2 3 4 5 6 1^	7
		Time	&				Frequency (Hz)
	Figure I-18 : Case 1 ; X=54 cm ; 8 springs ; natural frequency = 0.95 Hz

Table II -

 II

		3.524886
		-8.613286
	Coefficient [L OBST Coefficient (L OHS '11% s ') 2 1C) 3 Coefficient {lf bst 1 l^s t ) Constant	6.510233 -0.183172 -0.000091
	Polynomial interpolating the shape factor (lower part)
	Coefficient [f*™* 1 UT^Y Coefficient (l'^/IT"*)*	-1.446549 9.090475
	Coefficient (L^/Lr"" 4 ) 3	-18.260364
	Coefficient ( L '™" D / T* ) R °'"" I Y	13.736148
	Coefficient (L 9 "^ / L^)	-2.117782
	Constant	-0.007583
	Shock advance (m)
	A	0.045000

4 : Shape factor parameters of the solid tire (0=54 cm)

Table II -

 II

		4.717954
	-10.064918 Coefficient [L obst 1 tf") 2 7.231672 Coefficient {L obst 1 I$ st f -0.296328 Coefficient {lS bst 1 Constant -0.000439
	Polynomial interpolating the shape factor (lower part)
	Coefficient (i'^/I*"""') 5	3.915364
	Coefficient [fT"* Coefficient {jJ romd / L g 1 VT^Y 0 r ° Y -4.054526 -5.162269 Coefficient {[T"* 1 VT^Y 7.093846
	Coefficient \p™ d 11}™*) Constant	-0.792396 0.000099
	Shock advance (m)
	A	0.055000

5 : Shape factor parameters of the mixed tire (0=68 cm)

/ L°0 bsl ) 5

					-1.628748
	Coefficient (L ohil	/	Lf 5 ') 4		6.749230
	Coefficient (r*"/Z; te ) 3		-10.740428
	Coefficient^"*'"/If') 2		6.591950
	Coefficient (l/'*'*' /L'q*')		0.031157
	Constant				-0.003541
	Polynomial interpolating the shape factor (lower part)
	Coefficient ([found f ^		y	7.507269
	Coefficient (L sro "" d /Lr Hnd ) 4	-15.656030
	Coefficient (l/""""' / U™ und	) 3	6.376905
	Coefficient	{p^/TJ^y	3.268253
	Coefficient (L gnmd /1^ ) Constant	-0.501015 -0.017393
			Shock advance (m)
	A				0.045000
	Table II-6 : Shape factor parameters of the mixed tire (0=54 cm)

Verification of the spring characteristics •y

	2
	Spring characteristics
	1 manufacturer data :
	STRAIN CALCULATION GIVING THE MINIMAL DIAMETER FOR A GIVEN SUPPORTING EFFORT
	Known data
	P MAX daN	300
	0 mean mm	52
	Max ratio daN/mm 70
	Strain calculation
	D wire mm	8.28
	TRACTION CALCULATION FOR A LOAD PROPORTIONAL TO THE DEFLECTION
	G Shear Modulus daN/mm 2
	G steel=8000
		KNOWN DATA
	N coils		40
	G shear modulus		8000
	Deflection mm
	Load daN under deflection
	0 mean mm		52
	Initial tension daN	51
	CALCULATION OF N IF UNKNOWN
	if d>8 step = Dm*0.35 (modified formula)
	L0 body mm	328
	n	40
	n chosen	40
	d calculated
	d chosen	8
	FINAL CHARACTERISTICS OF THE TRACTION SPRING
	material		steel
	D fil mm		8
	0 ext mm		60
	L0 body mm		328
	N coils		40
	step mm		8.20
	stiffness daN/mm		0.73
	ratio 5<Dm/d<12		6.5
	Verif step <0.35*Dm	0.16
	L mm if A english	432
	L max mm		832
	max length mm		400
	Initial tension daN	51

Velocity (km/h) Velocity (km/h)

Abstract

The aim of this study was to analyse by means of numerical simulation the influence of the tire properties on the vibratory behaviour of a forklift truck. It was shown how to build a

Calculation of the oscillation period of the suspended forklift truck

Theorem of the angular momentum applied in O :

We called J the angular momentum of the forklift truck around the longitudinal axis in the point 0. : length of the theoretical tire print calculated with an infinite plane surface at height h obst (considering the tire print as a line segment defined by the intersection between the circle representing the tire and the line representing the plane surface). ^ground . j en g t j 1 Q ^ t j le theoretical tire print on the ground (considering the tire print as a line segment defined by the intersection between the circle representing the tire and the line representing the ground). ground = 2^R 2 -Z 2 z height of the wheel axle, R wheel radius A : shock advance, shift defining the position of the virtual edge of the obstacle. This shift is in fact a geometrical model of the shear of the tire on the edge of the obstacle (in the area near the edge, the tire is lifted up).

Tire test bench

A tire test bench was designed by INRS to measure the input data for the tire model, e.g. the load/deflection curves and the shape factors. This test bench was also designed to test the tires in dynamical conditions especially with obstacles. Figures below show a view and a diagram of the tire test bench. The test bench is composed of 4 parts : 1. a support (1) with ball-bearing rollers to guide the ring as it is rotating. The bottom of the support includes the force measurement system (a platform with four load cells) and is firmly attached to the floor. 2. a ring (2) inside of which the wheel being tested can rotate. The ring is 2 cm thick, 28 cm wide and has an inside diameter of 146 cm. The linear speed of the hoop, limited to 20 km/h, is measured by means of an optical encoder. If necessary, obstacles made of PVC material may be stuck inside the ring 3.

a body (3) which guides the motorized wheel inside the hoop. This articulated framework includes a 6 kW variable speed motor and can receive additional masses (of up to 2.3 t) in order to load the wheel properly. The height of the joint with respect to the support may be set to ensure that the part of the body supporting the additional masses is horizontal. Both the displacement between the wheel axis and the ring and the acceleration of the wheel axis are measured. 4.

an electro-hydraulic actuator (4) which has a 250 mm stroke and allows a force of up to 25 kN to be generated. For the quasi-static tests, like load/deflection measurements and shape factor identification, this was connected to the body and secured to the floor by means of two spherical joints. Safety systems limit the maximum values of acceleration and force transmitted to the body.

For the dynamic tests including rotating the wheel over the obstacle stuck inside the ring, the electro-hydraulic actuator was disconnected from the body. By so doing, the wheel could behave as an unconstrained system of one degree of freedom.

The limitations imposed by this test bench are due to the curvature of the hoop. For example, for load/deflection measurements, the forces induced by the contact on a curved surface are higher than those on a flat surface. Consequently, when the wheel is moving over an obstacle stuck inside the ring it does not behave as if it were moving over a flat obstacle placed on the ground. In order to measure the characteristics of the tire on a flat surface, an interface system was developed. It is placed on the support above the ring and is illustrated next page.

The system comprised a rigid plate covered with a "Teflon" sheet on to which a thin, sliding PVC band was fitted. Obstacles of the same material could be stuck on the band. The motorized wheel induced the movement of the band which was guided on the rigid plate.

APPENDIX 6

KOMATSU FD20 model : SDS program source

] [1] = "GONFLES " tt[2][2] = 1 tt[3][1] = "PLEINS " tt[3][2] = 1 tt[4][1] = "HYBRIDES " tt[4][2] = 1 tt[5][1] = "type