Leonardo Da Vinci’s self-propelled cart

Rampado Matteo (matteo.rampa@gmail.com) Degree in Mechanical Engineering
Reffo Enrico (enrico.reffo.2@gmail.com) Degree in Mechanical Engineering
Visentin Fabio (fabio.visentin.89@gmail.com) Degree in Mechanical Engineering

Introduction

The topic of the project is the study of the “self-propelled cart”, a machine made by the famous designer and inventor Leonardo Da Vinci around the year 1478 (Fig.1).

Fig.1 - Leonardo Da Vinci's self-propelled cart

Leonardo’s car wasn’t designed to be a passenger car, since it didn’t even have a seat. The vehicle was actually designed as a special attraction for Renaissance festivals, meant to instill wonder and awe in attendees. It’s a vehicle which can move thanks to the energy given by two torsional springs; they are connected to some gear (precursors of modern gears) which give the motion to the front wheels. Hence the transmission consists of two couple of gear wheels, which move the front wheels thanks to a peg transmission system. The technical drawings are part of the Da Vinci’s Atlantic Code, so that our model is a reinterpretation of Leonardo’s idea. The car is designed to have an automated steering wheel, that can be programmed to turn automatically on left side and with a pre-established curve, which depends on the number of cams and their relative angular position. When the steering rod is in the minimum distance contact point, the rear wheel has null steering angle. During the study we made the machine turn right and, in our case, the straight trajectory is given when the steering rod is in the maximum distance point in touch with cams; this behavior is the opposite of what initially conceived by Leonardo. We used LMS Virtual Lab software in order to build the virtual model: moreover we made some simulations to show how the car behaves, how much it curves varying the number of cams, to study the energy dissipated by the escapement systems, to determine the transmission ratio.

Objectives

The objectives of the project are :

• Determine the transmission ratio
• Determine the trajectory of the machine, at constant speed, varying the number of cams
• Study the energy dissipated by the escapement system
• Model of the spring as a multi-body component (Extra work assigned)

The modelling problem

Transmission

The two torsional springs are directly connected to the gear wheels mechanism. It is built using the Gear Train Super Element feature, which allows for the definition of a series of gears assembled in one or more stages. The user can specify parameters for two or several gears in each of the stages: the complete gear train is then automatically created with accurate geometry, assembly constraints and contact force elements in order to transmit rotational motion in the desired way. The data used to build the gear train are:

 Gear 1 Gear 2 Teeth number z 64 20 Normal module 5.75mm 5.75mm Pitch radius 184mm 57.5mm Normal pressure angle 21° 21° Helix angle 0° 0° Axial gear length 10mm 10mm Hollow radius 7.5mm 10mm Young’s modulus 1.7e10N/m^2 1.7e10N/m^2

The most relevant issue of the Gear Train Super Element regards the fact that the motion of the complete gear train is determined assigning a driven gear function. There are several possibilities: we used the polynomial type and assigned a constant angular velocity (100 rpm). In this way, the torque control is not allowed, so that we decided to study the spring motion and also the escapement system separately.

The front wheels’ motion is transmitted by a pin transmission system, which connects the front wheels to the gears, that lie on two orthogonal planes. This kind of system is realized modeling two sets of pins, one consisting of 32 pins, the other consisting of 8 pins: they move together to the front wheel and to the second gear wheel respectively (as it is shown in Fig.2).

Fig.2 - Transmission

Their number is chosen in order to maintain the same spacing value at radius 200mm and 25mm respectively. One “gear pin” engages every two “front wheel pin”. We used the CAD Contact model, assigning these parameters.

 CAD contact parameters Values Spring coefficient 2e+006N/m Damping coefficient 1kg/s Static friction 0.5 Kinetic friction 0.3 DMAX 0.01mm

The CAD contact requires a dense mesh in order to reach accurate results, but the smaller the elements’ dimension used for the mesh, the lower the integration step, so the longer the analysis. The analysis requires an integration step value 1e-004.

Escapement system

The flexible rods are made using the command “make it flexible”. This command introduces a new environment that uses the finite element method to represent the solid. So a rigid body can be converted to a flexible body. The flexible rods, in the machine, have the function to reduce the kinetic energy transmitted by the spring to the wheels, so that the acceleration is attenuated and  more regular. This is called “escapement system” (see Fig.3).

Fig.3 - Escapement system

A problem given by the flexible body is that the mesh has to be dense to reach more accurate results, but not too dense because it will take several time for computation.

The steps to make a body flexible are:

1. Draft the geometry
2. Import the body in the analysis case
3. Give the constraint to the body
4. Push the command “Make it flexible”
5. Create the mesh
6. Identify the boundary conditions (they are modeled as a rigid mesh)
7. Compute the ortho-normalization solution (it calculates modal matrices of mass and stiffness)

To evaluate the energy dissipated only by flexible rods we must take off the Gear Train Super Element and substitute it with gear joints. This type of joint is ideal, so there isn’t energy dissipation due to the contact between gear teeth. We didn’t put the gear joint between the two big gears, so that we were allowed to estimate at the same time the energy dissipation caused by the subsystem with (escapement system on the right) and without (escapement system on the left) the flexible rods’ influence (see Fig.3).

In the simulation we deactivated the CAD Contact corresponding to the pins and steering rods. We set the value of the initial speed of the shafts carrying the bigger gears. Before doing this simulation we modified the moment of inertia of the disc, just to make it rotating for a couple of turn. Thanks to this method we were able to evaluate the energy dissipated by the escapement system even testing different rods’ materials: pine, titanium, wild cherry, carbon steel (for the plots see the section simulation and analysis of results).

The contact between the flexible rods and the escapement is realized through a “sphere-to-revolved surface” contact (Fig.4). To model it, it was necessary to create a rigid revolved body and connect it at the end of the rod with a bracket joint, because the software needed a contact between two rigid (i.e. not flexible) bodies. The center of the sphere contact was put on the escapement geometry. This type of contact provides a good representation of real conditions impact: moreover, good computation’s results can be reached with a lower degree of precision than the other point-to-point contact, which was the other contact model tested.

Fig.4 - Particular of contact between flexible rod and escapement

Steering system

The steering system is composed by two moving rods, the rear wheel (with the corresponding tire), the springs and the cams, as shown in Fig.5.

Fig.5 - Steering system

The two rods have different functions: the red one controls the steering angle, the yellow one is designed to generate motion useful for further applications. Consequently, only the first rod is interesting for our study. It is directly linked to the swing arm through a bracket joint: when the rod rotates, the rear wheel steers. The rotation of the rod depends on its contact with the cams and on the force provided by the spring, which is attached to the tip of the rod and to the chassis: its scope is to maintain the rod in contact with the cams. While the cams are equally spaced and rotating, the rod remains in contact with them changing its orientation regularly: as a result of this, also the wheel changes its orientation and the vehicle steers regularly, characterized by an alternating motion. The contact between the steering rod and the cams is realized with the CAD Contact model, using these parameters.

 CAD contact parameters Values Spring coefficient 10000N/m Damping coefficient 1kg/s Static friction 0.5 Kinetic friction 0.3 DMAX 0.01mm

The spring characteristics are set in order to have null steering angle when the contact between the rod and the cams is at the maximum distance from the cams’ rotation axis. We expect that the higher the cams number, the less the steering.

Torsional Spring

The torsional spring ideal behavior follows the equation: M=kθ where k is the elastic stiffness [Nm/rad] of the spring and θ its deformation angle [rad]. This is a linear equation between the moment and the angle of charge (see Fig.6). The motion is given by the discharge of the elastic energy stored into the material of the spring:

Fig.6 - Ideal spring behaviour

In Virtual Lab there is the force command TSDA that allows to describe the torsional spring, but this isn’t a body modeling but only a force representation.

Instead of using this single command, we reproduced the body of the spring using a discrete modeling. The spring has been divided in several parallelepipeds connected each other by revolute joints (one joint can be substitute by a cylindrical joint) and putting on them a TSDA to represent the elasticity of the bodies. The pieces have been linked together following a spiral curve that formed the layout of a pre-charged spring. Furthermore the first 10 parallelepipeds were smaller than the others to better simulate the real body of the spring. The first piece is connected with a revolute joint to the ground and the last one is connected by a bracket joint to the spring container. Then we give a pre-charge angle to every TSDA. The parameters assigned in this simulations are:

• Damping coefficient: 1 kgm2/(rad s)
• Pre-charge angle: -40 deg

Simulation and analysis of results

First of all, the simulations were carried out varying the number of cams in order to provide the trajectory of the vehicle. It has been given a constant speed to the gear train to evaluate only the effect of the varying number of cams.

• Numbers of cams used: 8, 9, 10, 11
• Print interval: 10-4
• Integration step: 10-4
• Time: 10s
• Integration tolerance: 10-4
• Solution tolerance: 10-4
• Acceleration tolerance: 10-4

Leonardo Da Vinci\’s self-propelled cart idling

Leonardo Da Vinci\’s self-propelled cart moving

The transmission ratio can be determined combining the transmission ratio of two stages.

Where ωgear1=100rpm, ωgear2=320rpm, ωfront wheel=40rpm (absolute values). In this computation, we used for the ωfront wheel the mean value observed tracking the front wheels’ angular velocity: indeed, this variable is varying close to its mean value, because of the non-perfectly continuous constant between pins.

Varying the cams’ number, trajectories result as depicted in Fig.7.

Fig.7 - Trajectories

From the graph, it is noted that the increase of the number of the cams increases the lateral displacement of the vehicle. This demeanor differs from what initially supposed. One possible reason is due to the fact that the increase of the number of the cams increases the rear wheel steering frequency. Consequently, despite the lower steering angle amplitude (as you can see in Fig.8), the higher frequency makes the vehicle steering more.

Fig.8 - Steering angles

From the simulations carried out, in the initial moments it was observed an undesired behavior of the vehicle: it can be seen that in the earlier moments there is a non-negligible yaw rotation, which is caused by the difference between the two front wheels’ angular velocities. That demeanor is thought to be linked to the pins CAD Contact model. However, this phenomenon lasts few tenths of a second for each configuration analyzed. From the observation of the plots shown in Fig.9, that initial discontinuity can be graphically estimated.

Fig.9 - Yaw angles

Analysis of the escapement system

Fig.10 - Energy dissipation for different materials

The results show that wood dissipate the energy of the system fastest, because of the higher decreasing trends. In addition we can see that wood has higher damping effects than the metallic materials, which reduce the oscillations’ amplitude of the rod: this aspect depends on the organic nature of wood.

Analysis of the torsion spring

Leonardo Da Vinci\’s self-propelled cart – Spring behaviour

The simulation gives some interesting result: the spring output is not a linear trend as the ideal demeanor. This can be explained by the following considerations:

• The TSDA spring works in compression and extension at the same time: so that, if one part is out of the ideal configuration, it might react in opposite sense than the other parts
• The model doesn’t take into account the contact between the parallelepipeds
• Not all springs are working at the same time

Fig.11 - Spring output

Conclusion

After the analysis carried out, it was noted that the transmission mechanism behaves as a reducer, because the global transmission ratio is lower than the unit.

It was noted that the increase of the number of the cams increases the lateral displacement of the vehicle, because despite the lower steering angle amplitude, the higher steering frequency makes the vehicle steering more. Moreover, in most cases an undesired initial yaw rotation of the vehicle was observed, due to the different angular velocities of the front wheels in the earlier moments.

It was noted that the demeanor of the escapement system is different depending on the material used for the flexible elements: the results show that wood dissipate the energy of the system faster that the others.