Francesco Belluzzo, firstname.lastname@example.org
This multibody system is a simplified type of machinery commonly used in production lines. Its job consists in picking and depositing a product component from one assembly line to another. It can be used for pick and place devices such as the one shown in this video: https://youtu.be/QfGOnluLyrc.
The CAD model of this mechanism was available in engineer Nguyen Duc Thang YouTube page: “thang010146”. CAD model was then imported in Adams as a Parasolid file.
The original parts the whole model is composed of are:
|Pink Crank||Green LockRingA||Grey Runway||White Bearings|
|Green Slider||Orange Genevadisk||Brown T-bar|
This mechanism is a closed chain system; as input we have the Crank, as output the Tbar, which moves according to Crank and LockRingA rotations.
Its functioning is briefly described: the Crank has a plug that is in contact with the Slider. Slider and Tbar are connected with another plug. A belt connects Crank with LockRingA, with a transmission ratio equal to 2. Tbar is in contact with Geneva Disk, and is responsible for its oscillation. For half a turn, LockRingA makes both Tbar and Geneva Disk oscillate 180 degrees, then the Crank plug lowers both the Slider and the Tbar. Therefore, the overall movement is an up-down motion and reverse 180 degrees rotation.
Some important CAD model changes were introduced to simplify and significantly speed up every simulation:
- Tbar is divided in two parts, the upper rotates while the lower does not: as a consequence, the contact between Tbar and Slider does not interfere with Tbar rotation. This modification allows to extremely decrease contact forces between the Slider and the Tbar and, as a consequence, minor contact forces and torque on the input Crank;
- The original belt is removed; instead of it, a rigid coupler is adopted.The last out of the 4 analyses performed however provides a belt system taken from Adams Machinery;
- Both Bearings and Runway are removed since they play a “dummy” role and do not have structural functions.
The project objectives are the following:
- Tbar kinematic and dynamic analyses: introducing a rigid coupler between Crank and LockRingA, the main aim consists in analysing the influence of contact stiffness and damping on Tbar velocity, acceleration, angular velocity, input torque, as well as on local contact forces and contacts penetration depths;
- Maximum lifted weight: once a constant torque on Crank is setted, the focus is on the detection of the maximum weight the system can lift and move, with and without friction on joints, to vary contact stiffness and damping;
- Efficiency evaluation: the input and output works, represented by two spring dampers, are calculated to vary joint friciton values;
- Belt system: from Adams Machinery, a smooth belt and two pulleys are inserted on the system. A kinematic and kinetic comparison between simulation #1 and #4 is done, as well as the detection of belt tension, axial/normal contact forces and displacements.
The modelling problem
The CAD model is imported assembled to ease the creation of joints between the bodies, as they already are in the correct relative positions. To avoid redundancy of constraints, joints creation must be done carefully.
- Runway: fixed joint
- Boxes: fixed joint
- Crank: rotational joint
- LockRingA: rotational joint
- Geneva: rotational joint
- Runway – Slider: translational joint
- Geneva – Tbar: cylindrical joint
Grubler equation gives 5 DOF, which correspond to: Crank rotation (coupled with LockRingA rotation), Tbar rotation/translation, Geneva Disk rotation and Slider translation:
DOF = 6xN – 6xF – 5xR – 5xT – 4xC = 5 There are no redundant constraints.
There are five DOF because some joints were not introduced. Instead of them, contacts must be introduced. Therefore, even if the system gives 5 DOF, actually it has only 1 DOF.
Furthering with the modelling problem, mass properties must be given to the bodies. Since each component geometry is known, only material density has to be added (steel, ρ=7800 kg/m^3): Adams autonomously provides the computation of the mass of each, as well as inertia and COM location.
Four Solid to Solid contacts are subsequently created in their respective location:
- Crank – Slider
- LockRingA – Geneva Disk
- Slider – Tbar
- Tbar – Geneva Disk
Contacts are defined once stiffness, damping, force exponent and penetration depth are setted. Their values will be changed throughout the simulations (we anticipate that the results will be extremely affected by their values).
Normal force is calculated with impact based (continuous impact) modelling.
A rotational motion is applied on Crank revolute joint so that to have a rotational period equal to T = 5 s. Therefore, the Crank angular velocity is equal to ω = 2*π/T=1.256 rad/s, while LockRingA ω will be halved because of the reduction ratio. Once the motion is created, Adams will remove 1 DOF to the system.
Units and Solver settings
As far as Units and Solver setting are concerned, every simulation is generally performed with the adoption of the following parameters:
(Actually, I would point out that each of the following simulations was run with different integration errors, but there was no distinguishable effect in the results between 1E-05 and 1E-06 error tolerance. For this reason, I preferred to choose 1E-05 error, in order to speed up every simulation)
Simulations and analysis of results
1. Tbar kinematic and kinetic analyses
The first part of the project is focused on Tbar kinematic and kinetic analyses.
As I previously mentioned, in addition to idealised joints, this system is charachterised by four Solid to Solid contacts. Due to their nature, contacts introduce several problems in simulations because it is laborious to calculate deformations geometrically compatible, satisfy the EOM and give equal but opposite reaction forces on the colliding bodies.
It is therefore of paramount importance to underline that, once contacts are created in a multibody system, kinematic analyses alone are not permitted, since contacts precisely introduce local forces which affect both kinematics and kinetics. More precisely, all kinematic quantities depend on stiffness, damping and penetration depth contacts are given.
These considerations brings to investigate the effect of contacts on the system kinematic and kinetic. MSC Software house suggests to adopt a damping value about 1% of the stiffness.
Leaving penetration depth always equal to 1E-03 and neglecting friction on joints, several simulations are run, each time varying contacts stiffness and damping:
|Stiffness [N/mm]||Damping [Ns/mm]|
- Tbar translational stroke never obviously modifies: it is equal to 60 mm;
- Tbar translational velocities present the same overall trend, with relative errors up to 4% between each simulation (see the second image below);
- Tbar angular velocities have the same trend too, but this time high stiffness and damping lead to more consistent peak oscillations when the LockRingA plug enters in the second half of the Geneva Disk slot;
- Instantaneous 1G Tbar accelerations occur with low stiffness; on the contrary, if stiffness increases, they can even soar up to 9G.
As well as kinematics, dynamics significantly change when contact properties are varied. Therefore, it is mandatory to find the best configuration able to guarantee at the same time reasonable levels of motor torque, penetration depth, contact forces, joint reactions. Moreover, contacts must avoid system failure once external loads are acting on it.
It is reasonably conceivable that the higher contacts parameters are, the higher all dynamic quantites will be. It is therefore a matter of quantification of them.
- It is known that if a constant motion is imposed to the Crank, the torque T must vary in order to guarantee the correct functioning. Therefore, T oscillates along time according to the resistant load, which is mainly given by each body weight and contact properties. In the second image below, a focus on the Crank torque with k =1000 N/mm and d = 10 Ns/mm.
- Penetration depth reveals the reliability of the siulation. It could be analitycally calculated as x=Fn/k (Fn is the continous impact normal force while k is the contact stiffness). Fortunately, it is Adams itself that provides its calculation. Here are the ensemble of all the penetration depths detected, for k = 1000 N/mm and d = 10 Ns/mm. We have nearly 0.15 mm max, which is perhaps too much for steel. Therefore, contacts stiffness should be increased.
- In the first picture, contact forces between CrankA and Geneva Disk are shown, since they have the highest values. In the second picture below, a zoom of all four contact forces trend when k = 1000 N/mm and d = 10 Ns/mm.
As it can be seen, the highest contact force (i.e between LockRingA and Geneva Disk) is reached in correspondence with the maximum penetration depth. This agrees with continous impact modelling approach.
In conclusion, it is clear that, regardless of contacts parameters, the overall motion is irregular. Within all the simulations performed, the one that gives the most satisfying results in almost every previously-mentioned term is the one with k = 1000 N/mm and damp = 10 Ns/mm, in which it can be found:
|Max penetration depth||Peak torque||Peak contact force||Peak acceleration|
|0.15 mm||4300 Nmm||171 N||1 m/s^2|
Observation: if I decrease stiffness and damping, these values decrease too, but at the same time the penetration depth increases, affecting the reliability of the simulation. It is mandatory to find the ideal configuration which can satisfy both of them.
2. Maximum lifted weight
The second part of the project is focused on the maximum weight the mechanism can lift and move from one assembly line to another. If we take a deeper look at this model, it can be seen that it resembles to the slider-crank mechanism, whose analytical expression that that relates the crank input torque and the resistance external force derives from VWP:
This expression points out that, with the assumption of rigid bodies, the maximum lifted force depends on the input torque and the geometry of the system. Since the latter cannot be modified, it is a matter of chosing the torque. But this equation is actually valid only in idealised joint conditions, contacts excluded, therefore it cannot be used for this mechanism. In short, geometry and torque are not the only actors: again, contacts play an important role for this issue, since the stiffer they are, the higher the moved force will be, and viceversa.
A 3 Nm torque motor is setted in Crank marker location as a concentrated torque, to replace the previously rotational motion. Since the motor is in torque-control, this time the motion will not be constant.
Two masses are created and placed on top of Tbar to represent the simplification of the component. Their mass is almost null, since their weight must represented by two forces, applied on their COM.
- The first contact values adopted are:
|Penetration depth [mm]||1E-03|
With a 3 Nm torque, contact forces raises a lot. Since there is an initial impact between Geneva and LockRingA, it should be necessary to calculate the impulse, which is the integral of the force along impact time.
As seen from the video, the peak instantaneous contact force reaches nearly 17.5 kN, but the impulse is only 2 Ns, because of the infinitesimal duration.
Then, the weight force applied on the boxes is varied. If it is too much for the mechanism, animations like this one can occur:
Here is a table representing the relationship between Fmax and joint friction:
|Fmax [N]||Static/dynamic friction on joints|
With the assumption of rigid bodies, the system can lift m = 65.2 kg at most. Despite the small dimension of the whole model, the mass is respectable.
- Now, contact parameters are changed:
|Stiffness [N/mm]||Damping [Ns/mm]|
It can be found that the max lifted force consistently change: m = 91.7 kg. This is the proof that the analytical expression of the slider-crank cannot be used to predict the weight the mechanism can move in the space.
|Fmax [N]||Static/dynamic friction on joints|
(I would like to add that contact penetraton depths were again detected, but this time not presented. In the k=1000 N/mm case, the maximum depth raised to 0.3 mm, which I would say is too much for a reliable simulation. It is advisable to increase LockRingA-Geneva stiffness in order to reduce it).
3. Efficiency evaluation
Efficiency is defined as η=Pout/Pin.
To calculate the system efficiency, two translational and torsional spring-dampers are installed on top of the Tbar to represent the output work.
On the other hand, the input work can be easily calculated since both Crank angular velocity and torque can be directly measured. Work is found by integration along time of powers Pout=Ptrasl+Ptors and Pin=Mω.
Both spring dampers have null stiffness, since values different to zero lead to output work that strongly depend on the spring stiffness. Furthermore, it is also clear that work depends on the value dampers are given: different damping values will bring to different overall works. Another fact that should be taken into account is that, according to the damping value springs are given, there will be different delays in the output response, so the steady state efficiency will take some time to be reached.
The values adopted for the two spring dampers are respectively:
|Torsional spring damp. coeff. [Ns/mm]||10|
|Translational spring damp. coeff. [Ns/mm]||20|
(The following simulations do not neglect gravity, even though the weight force is conservative and its work in a Crank rotational period T is null).
In an idealised joint case, the efficiency should always be equal to one (i.e Pin=Pout). However, if contacts are present, their damping introduce viscous forces which dissipate energy; the higher the damping is, the more the energy dissipated. As a consequence, the efficiency without friction will never be equal to 1, but it will always be slightly minor, due to this dissipation. It could be possible to adopt null damping on contacts, but this would consistently slow down the simulation, and probably could even exhibit some energy dissipation due to the numeric of the integration algorithm. For this reasons, low contacts damping values are chosen:
|Penetration Depth [mm]||1E-03|
Ideal case: null friction on joints
In the first simulation, friction on joints is not considered.
These graphs bring to some comments: torsional work, developed during the first seconds, is extremely inferior with respect to the translational one, which is instead acting from 1.5 seconds on; there always is a consistent gap between output and input work. Furthermore, the most overwhelming issue is that at the end of the cycle it can be seen that the output work results even bigger than the input one!
There may be several causes that influence this outcome:
- Type of solver: SI2 is here adopted. SI2 formulation is robust and stable even at small step sizes, very accurate on velocities and accelerations. However, all motions must be “smooth” and twice differentiable. Probably, I3 integrator should be used. Despite SI2 is slower that I3, SI2 works better with contacts and thus it should be kept.
- Integration error tolerance: 1E-05 integration error was adopted, but perhaps it should be reduced;
- Contact and springs damping: varying it brings to different results, it is to say the input-output work gap enlarges or reduces.
For this reason, the efficiency is not immediately calculated. Instead, some modifications are introduced to check whether the graphs correctly changes or not:
- Contacts stiffness is increased to 1E+04 N/mm;
- Integration error tolerance is reduced to 1E-06;
- Integration time step is further reduced to 5/2500 = 0.002.
The new graphs shows a better input-output trend, where the input work is always higher than the output one:
The efficiency tends to η = 99.5% (indealised conditions):
This plot has a strange trend and some questions could arise . In order to justify the results, four comments must be done:
- The first 1.5 seconds are occupied by the torsional spring action (Tbar only rotates). The simulation #1 demonstrated that, regardless of the choice of contact parameters, Tbar angular velocity always presents oscillations due to contacts, and so ω goes clockwise and counter-clockwise. Therefore, there could be moments in which the green plug is not in contact with the Geneva slot, the input M breaks down and efficiency obviously decreases;
- There is a a huge gap at nearly 0.7 seconds. This is because LockRingA plug is exactly in the middle of the Geneva Disk slot cross, before it has to enter in the second upper slot: at that very moment, Crank work goes down because the plug does not find any “resistance” and, as a consequence, the torque M falls down a lot.
- Once the translational spring is working, from 1.5 seconds on, these issues do not occur anymore and the efficiency normally reaches the unity with some time according to the value spring dampers are given. The picture below wants to highlight that from 2.5 second on the efficiency unity is reached;
- The chance that integration errors are present cannot be excluded.
Friction on joints
Friction on joints is added (static = 0.23, dynamic = 0.16), and input and output works now have this updated trend:
The efficiency tends to be η = 90%.
Double friction on joints
Friction on joints is then doubled, and the graph becomes:
The efficiency tends to be η = 78%.
4. Belt system
As last simulation, the rigid coupler between Crank and LockRingA is removed. Thanks to Adams Machinery, a smooth belt system is setted in order to faithfully reproduce the real mechanism behaviour.
- Pulley P1 is coupled with Crank, pulley P2 with LockRingA; they have the same diameters as the cranks (d1=40 mm, d2=80 mm) and they are both made of steel;
- The belt was chosen from company A-Zeta Gomma belts catalogue. The material adopted is leather (E=200 MPa, σR=60 MPa), the belt height is 3.3 mm, the width is 10 mm and the stiction coefficient is 0.45.
Once belt width and height are chosen, segment area and section inertia can easily be calculated. Below, the parameters adopted for belt:
The belt is then divided in 242 segments, each one 2 mm long and automatically constrained to a plane normal to the pulleys rotation axis. The belt system actuator (motion control) is inserted in pulley P2.
Due to the huge amount of belt segments introduced, it is wise to simply focus the analysis only on two relevant ones: number #1 and #140.
The objectives are now focused on:
- Pulleys/belt kinematic and dynamic analyses, belt tension, deformation and oscillation.
- Kinematic comparison between the adoption of a flexible belt and and a rigid coupler on Tbar.
It is relevant to underline that the simulation starts abruptly, with the pulleys instantaneously rotating at their respective angular velocities ω . A situation like this does not exist in real life, since there always is a unsteady state transient period in which the motor reaches its regime velocity. Representing a real situation could have been possible, but it would have slowed down every simulation. Therefore, the following initial transient values of each graph might be unreliable.
Pulleys kinematics and dynamics
- Angular displacement: though P1 should rotate double with respect to P2 in a period, due to some sliding between belt and P1, its rotation is not exactly the double of P2. This is acceptable, since some companies confirm some slight sliding during the functioning. Probably, this could also be due to the fact that the angular velocity of the pulley P1 takes some time to reach it regime condition (see next image).
- Angular velocity: P2 ω is imposed and equal to 0.628 rad/s. P1 rotates depending on belt-pulley interaction: its angular velocity naturally increases during the initial transient, for then settling to 2ω = 1.256 rad/s, with occasional fluctuations mostly due to the friction.
- Joint forces acting on both pulleys are almost equal to 6000 N:
Belt kinematics and dynamics
(x_velocity: velocity of each segment parallel to its symmetry axis; y_velocity: velocity of each segment normal to its symmetry axis)
Segment #1 is moving along x at nearly 24 mm/s. P1 analytical peripheral velocity is vx =(π/T)*DP1/2 = 25 mm/s. There is a 4% error between analytical and numerical solution, which is probably due to some sliding effect. Anyway, this difference can be considered acceptable. The velocity along y is not null because of the centrifugal forces tend to stave the belt off the pulleys. Segment #1 is located in the smaller pulley P1 and therefore it is easier for it to break away.
On the other hand, segment #140 has exactly the same rotational speed, and does not stave (vy = 0 mm/s):
- Axial and normal contact forces: once the belt segments break away each pulley, they obviously become null. The axial force rise from Coulomb friction:
It is interesting to notice the sign of the axial force for the segment #1: at the beginning it is positive since it opposes to the movement the belt is given, then it becomes negative to avoid slip between belt and pulley.
- Belt tension: while segments #1 and #140 are in contact with the pulley, it can be seen that contact forces are constantly changing. Then, at the end of the simulation, they assess to a constant value, different for P1 and P2.
The highest axial tension on the belt is equal to σ = Fbelt/Abelt = 1154/33 = 34.97 MPa. The leather ultimate strengh is σR = 60 MPa, so the solicitation won’t cause belt breakups. However, a fatigue tests should be performed in order to detect the maximum numer of load cycles the belt can endure.
- As far as Tbar kinematic parameters are concerned, it can be noticed that while Tbar angular velocity remains the same as the previous simulation (there is a constant motion), on the other hand the belt flexibility introduces relevant oscillations on Tbar translational velocity, even if the motion seems fluid at first glance. Anyway, the previous Tbar velocity trend can be recognised:
- Finally, even if the same contact parameters as the first simulation are left, this time max penetration depth raises to 0.3 mm.
CAD model of this mechanism was imported in Adams from “thang 010146″ library as a Parasolid file. Then, in order to simplify the model and accelerate the simulations, some important CAD changes were done. Joints and contacts were created in their respective location so that to avoid redundancy of constraints, and mass was added to each body. Finally, Units and Solver settings were setted.
Four simulations were performed:
- The first one focused on the influence of contacts on both kinematics and kinetics of the system, with their quantification. A constant motion was applied to the Crank, and Tbar position/velocity/acceleration was achieved, as well as contact forces/penetration depth/Crank torque detection. As expected, the lower contact stiffness and damping are, the fluider the motion is and the lower the dynamic values are. In order to consider a simulation acceptable, it is always mandatory to verify contacts penetration depth.
- The second one dealt with the maximum weight the system could lift. A 3 Nm torque-control motor replaced the displacement-control one. Even this time, contact parameters strongly affect the simulation results: the stiffer they are, the higher the lifted force will be, and viceversa. Friction on joints strongly decreases the moved weight. As usual, it should be necessary to check for contacts penetration depth.
- The third one was based on the detection of the system efficiency, with and without friction on joints. The output work is calculated by rotational and translational spring damper power integration along time, while the input work by Crank power integration. The efficiency is the ratio between the two, and its strange trend was previously explained.
- The last one adopted a real belt system, as a replacement of the rigid coupler between Crank and LockRingA. The belt main characteristics were chosen from A-Zeta Gomma catalogue. The main aims were to verify the tension on the belt, as well as the pulleys joint reactions, so that to avoid system breakups, and to analyse their principal kinematic and dynamic quantities, in order to establish the reliability of the simulation. Finally, the kinematic comparison between belt and rigid coupler showed that the absence of the latter brings to discrepancies, and therefore this causes an even more irregular motion.
The project mainly wants to show the strong influence contacts generally have on kinematics and dynamics of the mechanism. Furthermore, they are responsible of the system failure whenever their values are not rather high. Overall, even when adopting low contact stiffness and damping, the model presents motion irregularities, evidenced by the Tbar velocity and angular velocity trend. To conclude, smooth motions can be reached only if idealised joints are present, but contacts are fundamental since they represent the reality. Indentify the best configuration of contacts is not possible if their stiffness is not first calculated or measured.
- Real bearings adoption from Adams Machinery;
- Flexible components for durability analysis;
- Belt vibration modes;
- To vary belt geometry/contact/mass, see how kinematics and dynamics change.
- YouTube page https://youtu.be/Kp6IwV_ks70, thang010146
- “Tabelle proprietà fisiche di materiali solidi vari”, Engineerplant