Finger-blade mower

Finger-blade mower

Introduction

The double movement finger-blade mower is an agricultural implement that can be rear-mounted on an agricultural tractor equipped with lift and universal three-point hitch. It operates by means of a cardan shaft applied to the pto (power take off) of the tractor. The mow of grass is made possible by a tooth-blade motion that allow high speed, sharp and clean cut. A multibody model of the kinematic chain that allows the cut of the grass will be created and simulated, observing how the belts modelling affects the torque applied to the pto.

Objectives

After creating the model of the mower with a 3D CAD software, it was studied with the multibody software Adams View to explore the follow main topics:

  • First of all, a kinematic analysis was performed with the main objective of doing a first check of the kinematic chain, evaluating in particular the torque that the motor has to supply to keep the speed of the power take off constant. Furthermore, in this phase, a simple but realistic way to apply the resistant force on the blades caused by the mowing grass was researched.
  • In the previous analysis a coupler joint constrains the rotation of the driving pulley to the rotation of the driven pulley giving a constant transmission ratio. In the mower, this constraint was obtained by 3 belts, so the coupler joint used in the kinematic analysis was substituted by belts modelled with the tool available in the simulation suite Adams Machinery integrated in Adams View. Before introducing the belts in the model of the mower, another model with only the 3 belts is created in order to understand how the software creates the belt systems from the parameters set during the creation wizard and the influence of their value on the results of the simulation. Than on the model created in this phase was performed a dynamic analysis to understand what is the maximum torque that the motor applied to the power take off can applied to the belt system avoiding the belts sliding.
  • The 3 belts are introduced to the full model of the mower and a dynamic analysis was performed in order to observe the influence of the flexibility introduced in the system on the simulation results. In particular, the motor torque at the power take off was evaluated and compared with the torque obtained in the kinematic simulations, especially to understand what is the value of the resistant force that produces the sliding of the belts and if this value is the same that was assumed in the kinematic simulation.
  • Finally, an analysis in the domain of frequency of the motor torque was performed to understand what is the influence of the flexibility of the system on this parameter.

The modelling problem

Before starting any kind of analysis with the multibody software, it is necessary to create a model of the object that must be studied. In this case, the model was created in a 3D CAD software and then imported into the multibody software because of the complex geometry of this tractor tool. The parts of the mower, necessary to study the aspects briefly described in the previous paragraph, were identified from the use and maintenance manual [1]; the sizes of these parts that are not available from the manual, were found mainly from a spare parts website. However, some sizes were not found from these sources, thus they were estimated looking at the available photos so that each part matches with the others. A final control on the geometry was done looking at another spare parts catalogue [2] in which there are the masses of some parts of the mower.

In the following figure, there is the model of the full mower created using Solidworks; in this model, some parts that don’t play an important role in the analysis that will be performed, like sheet metal protections, chains, bolts, were omitted.

Full model renderIn the next figure, a second model is represented after it was imported into Adams view: this one was extracted from the full model and contains the only parts necessary for the analysis that will be performed.

Model bodies with labelsIn the multibody model, the PTO support, the hinge bearing arm, the bar and hinge are fixed to the ground; however, this 3 bodies are helpful to define the position of the other body that are part of the kinematic chain. So, if all the bodies are considered rigid bodies and connected each other with rigid constraints, the system has only 1 DOF: after defining the motion of the power take off, the motion of all other bodies is defined. In the follow table, the joints used in the model are summarized.

Joints tableAt this point, the number of DOFs is still 4: it remains to connect the two blades to the respective rocker arm and the two pulleys. In the real machine, the connection between the blades and the rocker arms is done using a joint that leave completely free only the rotation about the ground Z axis between the two bodies. But if a revolute joint is applied, the system doesn’t move and some redundant constraints are introduced. Maybe clearances among blades and their upper and lower guides allows the motion of the system. Since the angular displacements of the two rocker arms are small during mower working, also vertical motion of the two blades and related accelerations are small if compared with the horizontal movements. So, as in the table, it was decided to connect the blades to the bar with translation joints and to the rocker arms using two general constraints (GCON). The displacement along the ground X axis between two markers, one fixed to the rocker arm and one to the blade, was put equal to zero; this operation was repeated for each blade.

One of the two remaining DOF could be removed coupling the motion of the two pulleys. At the beginning for the kinematic analysis a coupler joint was used to enforce a transmission ratio of 2.37 obtained assuming a pitch diameter of 280mm for the driving pulley and of 118mm for the driven pulley. Then the coupler joint was substituted by 3 belts modelled using the tool in Adams Machinery. Finally, to complete the model, a motor was applied to the revolute joint of the power take off with a constant angular velocity of 540rpm, as reported in the use and maintenance manual.

Before talking about how the belts were modelled in Adams, a few words are spent about the resistant forces. When the mower is working, the cutting blade and the teeth blade move always in opposite directions. So, every half turn of the eccentric shaft, teeth-cut section pairs pinch the grass near the ground cutting it quickly. The cutting process is quite simple to understand, but the estimation of the real cutting forces that act on the cutting blade and the teeth blade is not easy because the process is not stationary and depends on many factors as the type and the quantity of grass that the mower has to cut. However, to test the mower in operating condition with the multibody software it is necessary to solve this problem and find a way to give a value to these forces and to apply them to the two blades. First, considering that in the simulations the two blades will be considered as rigid bodies, the resistant force on each blade was applied as a single force acting on the center of mass along the ground X axis. Below there is the function used in Adams View to apply the force to the cutting blade:

-(ABS(VX(MARKER_46))/VX(MARKER_46))*(STEP(DX(MARKER_46, MARKER_47),-8.33,0,-7.33,400)-STEP(DX(MARKER_46, MARKER_47),7.2,0,8.2,400))

The same expression without the minus sign was used for the force applied to the teeth blade. The marker 46 is attached to the cutting blade, while the marker 47 to the bar (so it is fixed to the ground); the position of the cutting blade in which start to act the forces are defined using the 3D CAD model created with Solidworks.

So, using this expression, in the model there are two forces that act with the maximum value (400N in the case of the expression above) between the situations represented in the next two figures.

Resistant forces 1Resistant forces 2In the previous two figures the cutting blade is moving to the right and the force acting on it is point to the left (negative), while the teeth blade is moving to the left and the force acting on it to the right (positive). Then, each blade moves in the opposite direction to return to the initial position; so, the mower cut the grass in two steps for each turn of the eccentric shaft. In this second configuration, also the forces acting on the blades change direction, as it is visible on the two graphs.

The explained method to apply the forces could be made more complex and realistic: for example, in the real world, the forces acting on blades probably are not constant while the mower is cutting the grass. However, this way to apply the forces is sufficient to do some considerations about the effects of the resistance forces on the system.

In the force function written above, a force of 400N is assumed; this value can increase only until it is reached one of the two following situations: the 3 belts start to slide, the tractor is providing its maximum power to actuate the mower. If it is supposed that the tractor has enough power, the maximum force that the blades can apply to mow the grass could be found if the maximum torque that the belts can transmit without sliding is known. So, the maximum torque that a B93 belt can transmit was calculated following the steps reported in a V-belts catalogue; the following formula was used:

Formula_1in which Pb is the basic power of the belt, Pd the differential power, Cy and CL two correction factors: the first is for wrap angles smaller than 180° on the smaller pulley and the second according to type and length of the belt. From the use and maintenance manual of the mower, the nominal angular velocity of the power take off is 540rpm; so, if the transmission ratio is 2.37 as said before, the driven pulley has an angular velocity of about 1281rpm. From the V-belt catalogue, for B93 belts type, 2.37 transmission ratio and a smaller pulley with 118mm pitch diameter, 1281rpm angular velocity, 170° wrap angle:

Formula_2that results in a maximum power transmitted from one belt of:

Formula_3Now it is possible to calculate the maximum torque that the driving pulley can transmit to the belts:

Formula_4To determinate the resistant forces that increase the motor torque to this value, some kinematic analysis will be performed, changing each time the value of the resistant forces applied to the blade.

As said before, after doing some kinematic simulations, the coupler joint between the driving pulley and the driven one was substituted with belts modelled with Adams Machinery tool. To understand how to create a belt in Adams and which parameters of pulley and belt affect the results, different belt systems were simulated and compared. First, from the 3D CAD model, the relative positions of the two pulleys and the two tensioners were measured; so, when the coordinates of the driven pulley are determined in the Adams model, it is simple to positioning the other pulley and the tensioners. The following render, created using Solidworks, represents the belt system of the mower that was modelled into Adams.

BeltsSummarizing, this belt system consists of 3 B93 belts wrapped around two pulleys. It is assumed that the driving pulley has a pitch diameter of 280mm and the driven pulley of 118mm, because the only measures found about the pulley are the external diameters from a spare parts website, so, from this information, the pitch diameters were deduced. The external diameter of the two tensioners is assumed of 42mm.

At the beginning of the problem modelling, there was some problem during the simulations: sometimes the simulations failed without an obvious reason. So, after a research of the causes that produced this problem, it was discovered that the simulations failed for some reason related to the units of measure. It was discovered that if the unit of angles of the Adams model is set to radian, the simulation could fail. Since the unit of angles of the model has been set to degrees (default if MMKS units are used), the simulations no longer failed. After solving this problem there was another important issue to be resolved: if the 2D link method is used to create the belt system, as recommended by Adams when the axes of rotation of the pulleys are parallel to one of the global axes, the belt is constrained to a specific plane. So, the pulleys and the tensioners can’t be attached directly to the ground with a rotational joint, because after wrapping the belt Adams reports 9 redundant constraints. To solve this problem, each pulley and tensioner was connected with a rotational joint to a small cylindrical body and each of this 4 bodies was connected to the ground with 3 GCON (general constraint) that locked the translation along global X and Y axes and the rotation along global Z axis. At the end of this operation, only the necessary DOFs remain and there aren’t redundant constraints in the model.

However, to model 3 belts in Adams it is necessary to create 3 belt systems each one with its own pulleys and belts because it is not possible to wrap 3 belts to the same group of pulleys. So, in the multibody model used for the analysis there are 3 similar belt systems, each one with one grooved pulleys. The pulleys and tensioners of the belt system in the middle are constrained to a small cylindrical body with a revolute joint and the body is attached to the ground with 3 GCON as described previously; the pulleys and tensioners of the other two belt systems parallel to the first are always constrained with a revolute joint to small bodies that are connected to those of the first belt system using a inline joint and imposing a zero rotation motion along global Z axis in the properties of the primitive constraint. The relative rotation among the driving pulleys and among the driven pulleys is constrained using coupler joints. In this way, only the correct DOFs for the simulation remain in the system and redundant constraints are avoided if a 2D link method to create the belt is used.

The masses and inertias of the real pulleys were calculated using the option “mass properties” in Solidworks that derives mass end inertias of a solid part through its geometry and material properties. The mass and inertia properties so calculated were given to the pulleys of the first belt system, while an almost zero density is assigned to the other pulleys.

The characteristics of the belt section were derived from the belts catalogue [3], while the area and the inertia of the section were calculated drawing the section in Solidworks and using the option “section properties”. The Young’s Modulus used in the simulations is among the typical values for trapezoidal belts, that is from 60MPa to 100MPa [4]. The segment length and the number of segment were set with the main aim of obtaining the right tension on the belt, but also to obtain a number of elements neither too high, to reduce the calculation time, or too low, to have reliable results from the simulations. The correct tension of the belt was calculated using the information reported in the belts catalogue [3]: to verify that a belt has the right tension, it is necessary to apply at the middle of the span length T a perpendicular force F capable of producing a deflection of 1.5mm for every 100mm of T (see the following image).

Correct fitting tensionThe tension of the belt is correct if F measured is between two values F’ and F’’; these depend on the type of belt, the outside diameter of the smaller pulley and its angular velocity. In this case F’=21.0N, F’’=31.0N and from the geometry of the belt system T=880mm and f=13.2mm. So, from simple balance considerations, the tension of the belt must be between 350N and 515N. The tension obtained wrapping the belt is reported in the following image:

Belt tensionAt this point, the geometry of each belt segment is known through its length just defined and its section geometry from the belt catalogue [3]; the density of a trapezoidal belt is about 1.2÷1.3kg/dm3 [4]. Now it is possible to calculate the mass and moments of inertia of each belt segment.

Finally, the static friction coefficient between rubberized fabric and steel or cast iron is 0,35 [4]; so, this parameter was set to this value. Any information is provided about the dynamic friction coefficient; therefore, this parameter was set to 0,21 so that the ratio between the values used for the two coefficients is the same between the default values.

The other parameters that could be set during the belt creation, are left to the default values. In the following paragraph, the choices of the Young’s modulus and the tension of the belt will be discussed showing the result of the analysis. The values of the parameters used in the simulation are reported in the following images:

Step 3 beltStep 4 BeltAfter the three belt systems were added to the model of the mower to replace the coupler joint between the driving pulley and the driven one used in the kinematic simulation, the model is verified and the information resulting from this operation are shown in the following image. Here there is a list of the constraints of the model and the software also indicates that there are 2545 DOFs in the system; this value can be verified with the following calculation. Each of the three belts is modelled using the 2D link method and has 282 elements as shown before; so, each of these elements is constraint to a plane and for this it has 3 DOFs. At the end, it results in 282·3·3=2538 DOFs; in addition to these, there is the rotational DOF around the Z global axis of the six tensioners, two for each belt systems, and also the rotation around the Z global axis of the driven pulley that isn’t rigid constrained to the rotation of the driving one. So overall there are 2538+6+1=2545 DOFs as the software indicates.

As it will be discussed in the following paragraph, probably this high number of DOFs contributes to extend the duration of the dynamic simulation: depending on the number of steps and the time duration set for the simulation, this may take several hours.

Command verify model - full modelThe information window just presented results from the verification of the full model of the mower showed in the following image.

Full model of the mower with belt systemsBefore discussing the results obtained from the simulations, a final clarification must be made. Looking at the characteristics of the bodies in the multibody model, for all bodies the density of steel (7800kg/m3) is set, except for the teeth blade that has a density of 7200kg/m3. In fact, after created the full model in Solidworks, the mass of some components was compared with the values reported in a catalogue of spare parts [2]. It was discovered that the mass of the teeth blade was too high if the density of the steel was selected for this part. So, given that in the model the teeth blade is a rigid body that can only translate along global X axis, its density was modified as said above to obtain a correct value of the mass. All other bodies have a density of 7800kg/m3.

Simulations and analysis of results

Most of kinematic and dynamic simulations were performed following this scheme:

  • From 0s to 0.4s. As said before, a motor was applied to the revolute joint of the power take off; its angular velocity varies with this step function: step(time,0,0,0.4,56.55). An angular velocity of 56.55rad/s is equivalent to about 540rpm, value of the nominal angular velocity reported in the use and maintenance manual of the mower [1].
  • From 0.4s to 0.8s. The power take off turns with the maximum angular velocity; only a variable torque is applied by the motor to the system to maintain the angular velocity of the power take off constant over time; any other external forces are equal to zero except the force of gravity.
  • From 0.8s to 1.8s. The resistant force described in the following paragraph are applied to the system.

Kinematic analysis

First, the maximum resistant forces that can be applied to the blade was researched. As said previously, supposing that the tractor has enough power, the maximum power that the belts can transmit restricts the value of the maximum resistant forces. The maximum torque that the motor can apply to the system without the sliding of the belts, is about 1.66·105Nmm (see the previous paragraph “The modelling problem”). Starting from a zero resistant force, it was discovered that a force of 1060N produces peak of the motor torque of about 1.65·105Nmm (see the following figure).

Motor torque kin sim

So, at this point it is assumed that highest resistant forces produce the sliding of the belts; thus, 1060N is the value of the maximum resistant forces that can be applied to the blades. This assumption will be discussed when the result of the dynamic analysis will be presented.

In the following image, there is a comparison between the motor torque obtained from two kinematic simulation: one with a zero resistant force and one with a force of 1060N. The peak of the motor torque reaches high values, about 1·105Nmm, also during the simulation without apply a resistant force. In fact, when the mower is moving in a no-load condition, the motor must provide anyway a positive torque to accelerate the blades that have a considerable mass: about 5.3kg for the cutting blade and 6.5kg for the teeth blade. Then the resistant forces produce an increase of the torque at the beginning because in this phase the blade are still accelerating, but then the motor torque decreases; this because more and more the kinetic energy of the blade and of other components is now providing the necessary energy to mow the grass.

In the following plot, it is possible to compare the motor torque with and without the resistant forces acting on the blades.

Motor torque woBeltsDynamic analysis: only three belts systems

Before substituting the coupler joint between the two pulleys used in the kinematic simulations with the three belts, a new file was created with only the three belt systems. In this way, the belts were simulated with different combinations of parameter to understand their influence on the results. In particular, in each simulation, the maximum torque that the motor can apply to the system was evaluated; to do this, during the simulation an increasing resistant torque was applied to the driven pulley. In these tests, the only parameters that describe the geometry of the section of the belt were kept constant; the length and the number of element were always chosen to obtain a tension of about 500N.

First of all, the simulations started with almost all the parameters set to the default values. It was observed that the maximum torque that the motor could apply without the sliding of the belts was too low compared to the values calculated from the belts catalogue. This suggested that some parameters had to be changed; so, the correct area and inertia of the belt section were set, but the bending stiffness was too high and the belt had an unnatural shape because touched the pulleys in a small contact arc. It seemed that the default value of the Young’s Modulus (1000MPa) wasn’t correct for this type of belts. In fact, for trapezoidal belt the Young’s modulus is between 60MPa e 100MPa [4]. However, with a value in the middle, that is 80MPa, the maximum torque before sliding was still too high. So, this suggested to have a look to another important parameter: the friction coefficient. The default value for the static friction coefficient is 0.5, but for rubberized fabric in contact with steel or cast iron, the friction coefficient is near 0.35. This value was therefore changed, as explain previously. Finally, being again the maximum motor torque too high, to obtain a value more in agreement with what is reported in the belts catalogue, a Young’s modulus of 60Mpa was chosen and the length and the number of segment were fixed to obtain a tension of about 350N: if the tension is lower, also the maximum motor torque is lower.

Model for testing beltsTo verify the belt systems so obtained (see the previous image), the following simulation was performed:

  • From 0s to 0.4s. The motor applied to the driving pulleys moves following the function step(time,0,0,0.4,56.55); so, at 0.4s it reaches the maximum speed of 56.55rad/s (540giri/min).
  • From 0.4s to 0.8s. The motor keeps the angular velocity constant and any resistant torque is applied to the driven pulleys.
  • From 0.8s to 4.8s. To the driven pulleys is applied a torque with the following function: step(time,0.8,0,1,-0.8E5)+ step(time,1.8,0,2,-0.1E5)+ step(time,2.8,0,3,-0.1E5)+ step(time,3.8,0,4,-0.1E5). So, an increasing resistant torque is applied through step of 1000Nmm.

The results of this simulation are represented in the following image:

Motor_torque_MaxTWhen an initial resistant torque of 8000Nmm is applied to the driven pulleys, the motor replies with a torque of about 2.2E5Nmm and the belts transmit all the power necessary to keep the velocity of the driven pulleys constant, as shown in the graph. Then the resistant torque is increased of 1000Nmm and the motor replies with a torque 2.4E5Nmm, but the power related with this torque and transmitted by the belts to the driven pulleys, isn’t sufficient to keep the angular velocity of these pulley constant; so, the velocity starts to decrease. The next increases of the resistant torque make the situation worse: the velocity of the driven pulleys decreases up to zero and then they start to turn in the opposite direction; the motor torque remains constant.

However, the maximum torque transmissible from the driving pulley to the belt, just obtained in this simulation (about 2.4E5Nmm), is almost 40% higher than that found using the data and formulas in the belts catalogue (1.66E5Nmm). The difference between the two values increases if a new simulation will be performed with a Young’s modulus and/or a tension higher than respectively 60MPa and about 350N. So, in the simulation these two parameters were set equal to the lowest limit of their respectively reference interval, described in the previous paragraph, is trying to understand if the value of the torque derived from the belts catalogue can be equal to that resulted from the simulation in some way. The reasons why there is a such difference from the maximum torque values could be the following:

  1. the belt catalogue suggests a calculation of the maximum transmissible power that gives lower values than the real ones for safety reasons: for example, the friction coefficient of 0.35 that was used in the simulation, is valid for clean pulleys, but if these are wet, the friction coefficient could be reduced to 0.20, that is almost half of the value assumed [4].
  2. the Young’s modulus of the belts available in the catalogue is lower than the reference values used in the simulation.

However, any information was found to verify the first possibility, while to validate the second hypothesis some experimental tests on the belts would be necessary.

Dynamic analysis: full model

The belts modelled as in the last simulation presented (the details about the modelling of the belts in Adams are in the previous paragraph), were added to the model of the mower to replace the two pulleys and the coupler joint that in the kinematic simulations links them together. Then a dynamic simulation was performed, following the same scheme described at the beginning of this paragraph.

About the resistant forces applied to the cutting blade and the teeth blade during this simulation, the law that describes how they change over time is always the same presented in the previous paragraph; the maximum value that they can assume is set to 1060N for the reasons explain when the results of the kinematic simulation was discussed.

The motor torque and the angular velocity of the driven pulley changing over time, resulting from the dynamic simulation, are represented in the following figure.

Motor_torque_fullMComparing these results with the results of the kinematic simulation, there are some important differences. First the angular velocity of the driven pulley isn’t constant as in the kinematic simulation but oscillates around 7660 deg/s, that is near the nominal value ωD that can be calculated as follow:

Formula_5in which nM is the nominal angular velocity of the power take off in rpm and TR the transmission ratio. Then the motor torque still oscillates also in this dynamic simulation but with an amplitude smaller than that was observed in the kinematic one. This is mainly due to the axial stiffness of the belts that is introduced modelling the belts with the tool available in Adams: increasing this stiffness, the amplitude of the oscillation also increases and the results are nearer those obtained in the kinematic simulation in which the joint that link together the rotation of the two pulley is rigid. This was observed simulating a simplified model in which the axial stiffness of the belts is reproduced with a rotational spring-damper.

So, to understand more in detail the results just presented and the parameters that influence them, a simplified model was created in which the three belts are substituted by only one rotational spring-damper; its stiffness and damping coefficient were calculated so that it gives the same torque provided by the three belts if the same displacement is applied. To do this calculation, only the axial stiffness and damping of the belts are considered and their values were calculated using the formulas described in the Adams help. Here it is shown how the software evaluates the axial stiffness and the damping coefficient of each segment of the belt:

Formula_6

in which:

E=Young’s modulus

A=area

L=segment length

D=damping ratio

But these are values related only to the single segment of the belt; so drawing a scheme of one of the three belt in Solidworks, it was possible to evaluate the length of the portion of the belt between the two pulley that is about 867mm and then the stiffness of this portion of the belt as follow:

Formula_7After simple steps, the following relation between the axial stiffness of the belt and the stiffness of the rotational spring-damper can be found:

Formula_8in which:

n=number of belts

R=radius of the pulley at which the rotational spring-damper is attached

Considering that in the simplified model created, the rotational spring-damper is attached between the driving pulley and a dummy pulley and then this one is connected to the driven pulley using a coupler joint that also impose the correct transmission ratio, in this formula the pitch radius of the driving pulley is used. Finally, the damping coefficient of the equivalent rotational spring-damper was evaluated as follow:

Formula_9Summarizing, this simpler model derives from the model used in the kinematic simulation with the addition of a rotational spring-damper to considering the influence of the axial stiffness and damping of the belts. This model was created mainly for two reasons: instead of the 2545 DOFs of the full model, it has only 1 DOF and the simulations performed take only few minutes, not some hours as the full model required; so the influence of the stiffness and damping could be easier evaluate. Then it is created with the purpose of understand which are the advantage and disadvantage of simulate the belts with the Adams Machinery tool instead of a simpler spring-damper.

So, if the stiffness and the damping coefficient just calculated are set, it is obtained the following curves for the motor torque and the angular velocity of the driven pulley:

Motor_torque_TorsSpring_K20320_C20Changing the stiffness of the belt, the amplitude of the oscillation of the motor torque increases if the stiffness of the spring is set to higher value and viceversa.

If a comparison is done between the last plot just presented and the plot of the motor torque resulted from the dynamic simulation of the full model, it is clear that the amplitude of the oscillation is higher in the simplified model than in the full one. This difference is probably due to some features of the belts that are considered in the full model and are not considered in the simplified one: for example, the bending stiffness and damping of the belts and the friction between pulleys and belts. The influence of these features of the belts on the results of the simulations can be seen also observing the mean value of the motor torque both from 0.4s and 0.8s while the resistant forces aren’t operating on the blades and from 0.8s and 1.8s while they are operating. If in these two period of time the average power spent by the motor to keep constant the angular velocity of the power take off is evaluated, for the simplified model the following values are obtained:

Formula_10

while for the full model:

Formula_11in which with P1 is indicated the average power in the first time interval (0.4s-0.8s) and with P2 the average power in the second time interval (0.8s-1.8s). These values confirm what was already clear observing the plots of the motor torque: the power spent by the motor is higher in the full model than in the simplified one. More in detail, while in the model with the rotational spring-damper in the first time interval the average power is about zero (the motor torque oscillates around zero), in the full model assumes a positive value, about 1.5kW. So, modelling the belts with the Adams Machinery tool, the influence of dissipative forces as the friction forces can be seen on the results of the simulations. This effect is visible also in the second time interval in which the power consumption passes from 1.96kW to 2.90kW.

Going back to the plot of motor torque derived from the full model, after the initial transient it doesn’t exceed the value of 75E3 Nmm; so, the mower can probably contrast resistant forces higher than 1060N as it is assumed at the beginning looking at the results of the kinematic simulation.

Finally, to better understand the oscillation that characterized the motor torque derived again from the full model, an analysis in the domain of frequency was performed. If this output of the simulation is transformed using the FFT between 0.8s and 1.8s, time interval in which the resistant forces are operating on the blades, the results in the following plot are obtained.

FFT_MotorTorque_fullM_3In addition to the initial peak that represent the average value, there are other two peaks: the first at a frequency of about 10Hz and the second at about 43Hz. This second frequency is linked to the motion of the two blades: each of them is accelerated and decelerated two time every one turn of the shaft connected with the driven pulley, so with a frequency that is twice the rotation frequency of the driven pulley:

Formula_12The other oscillation of the motor torque displayed in the FFT plot can be observed also in the plot of the motor torque: looking at the time interval between 0.8s and 1.8s, it is visible another oscillation, but belongs to a transition phase because its amplitude decreases over time.

Conclusion

Observing the results of the simulations performed, it can be seen how the stiffness and damping of the belts have an important influence. More in detail, the flexibility of belts reduces the amplitude of the oscillation of the motor torque and produces an oscillation in the angular velocity of the driven pulley; then, introducing the belts created with the Adams Machinery tool into the model, also the effects of dissipative forces as the friction forces can be seen, especially on the power consumption. However, some of the considerations doing observing the results of the simulations performed on the full model could be done simulating a simplified model in which the three belts are substituted with only one rotational spring-damper, with a stiffness and damping coefficients equivalent to the axial stiffness and damping of the 3 belts. So, if only the effects of these two features of the belts are of interest, they could be modelled with an equivalent rotational spring-damper. In this way, also the duration of the simulation decreases: the simulation of the model with the 3 belts created with the Adams Machinery tool could take also several hours.

References

[1] Maschio Gaspardo S.p.A. (2010-12), Gaspardo FBR Plus: use and maintenance – spare parts.

[2] The catalogue is available at the adress: www.losilvb.it/catalogo2008.pdf

[3] Poggi trasmissioni meccaniche S.p.A., Cinghie e pulegge trapezoidali. The catalogue is available at the adress: www.poggispa.com/pict/cataloghi/IT/080_Cinghie%20e%20pulegge%20trapezoidali.pdf

[4] Mario Cicognani (1990), Trasmissioni con cinghie: piatte, trapezoidali, sincrone, U. Hoepli Milano, p.398.

 

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