Piaggio Ape (Front Suspension)

Luca Clemente - luca.clemente@studenti.unipd.it
-January,2020

Introduction

Piaggio Ape is a three-wheeler vehicle produced by the Italian company Piaggio starting from a Vespa prototype. At the end of the Second World War, vehicles was lacked and Piaggio created this simple vehicle with the aim of allowing the purchase of a means of transport even for the lower social classes.

Controlled with scooters style handlebars, was designed to seat one, but can accommodate a passenger in its cab. Performance offers a unique versatility of use and a large load capacity, on every kind of path. Its very low fuel consumption petrol engine makes it agile and economical.

Since 1948 many models were built, in this case the model TM 703 was considered, one of the most widespread.

Front and rear suspension systems are of great importance for the optimal transport of things. The project consists in the analysis of the front suspension system, composed of a steering tube (fork) fixed on the arm with a front wheel oscillating hub. The suspension is made by helical spring and with hydraulic shock absorber.

Piaggio Ape TM 703

Piaggio Ape TM 703

Piaggio Ape model

Piaggio Ape model

Object

The purpose of this work is to analyze the behavior of the front body of the vehicle.

In the first part of the project the behavior of the suspensions (displacement and acceleration of wheel’s pivots) was observed, to a variation of the Iso road surfaces (B,C and D).

In the second part a comparison between the Piaggio suspension and a traditional fork suspension.

In the third part we focused on the structural integrity of the front suspension following a jump after a hill, and at the end the focus will be the vehicle’s trim and anti-dive angle.

The modelling problem

The model was created with ADAMS View software, on a 1: 1 scale starting from Piaggio’s user manual and direct measuring.

The main part is the chassis formed by the cargo bed and the cab welded together. The back is simplified and consists of mudguards, the rear suspension system, axle shafts and wheels. In the front part, the front suspension system consisting of the fork, a support for the suspension housing and an arm (connected to the frame), the movable arm and the hub connected to the wheel.

The movable arm can rotate with respect to the wheel so I can have the lowering of the suspended mass, with consequent displacement of the suspension.

The assignment of the constraints is explained in the image below:

Bodies and joints of the model

Bodies and joints of the model

Model of the front suspension: fork (cyan),support (magenta), fixed arm (orange), movable arm (red), hub (yellow)

Model of the front suspension: fork (cyan),support (magenta), fixed arm (orange), movable arm (red), hub (yellow)

Front Suspension of Piaggio Ape

Front Suspension of Piaggio Ape

Many elements, that are not important for simulation purposes, have been neglected, to simplify the model and speed up the calculation. The empty weight is around 350kg. The driver was not represented and therefore a 75kg mass was added.

For the spring it was considered:

stiffness coefficient = 35000 N / m

damping coefficient = 3000 Ns / m

spring free length = 400mm

Also the rear suspensions are modelled by two shaft (rear and left on color cyan) that are linked to wheels with a revolute joint. To the opposite end they are linked to the chassis with a revolute joint and the revolute axis is the direction of motion, so they can swing up and down if they will pass over a irregularity.

Rear suspension model

Rear suspension model

For rear springs it was considered:

stiffness coefficient = 42000 N / m

damping coefficient = 3000 Ns / m

spring free length = 350mm

Tyres are 4.00 R12” and have been modeled by taking one of the ADAMS.tir models that best represented our case, pac2002_175_70R13.tir.

A rotative motion has been applied to the vehicle on the rear wheels

 Simulations and analysis of results

First part

The road surfaces were searched among the models proposed by ADAMS and are 2d stochastic road Iso. To the first test a road Iso B, after a road Iso C and at the end a road Iso D

Vehicle on Iso road

Vehicle on Iso road

The vehicle moves in a straight line (the steering in the model has been neglected) accelerates for the first 4 seconds and then goes at a constant speed of about 50km/h for 26 seconds (total of simulation 30s). The vehicle runs about 400m.

Vehicle's law of motion

Vehicle’s law of motion

 

Vertical and lateral displacements and accelerations of three particular points, were analyzed for each type of road:

  • Front wheel pivot (cm_Hub)
  • Rear wheel pivot (cm_shaft_right)
  • A point near the driver (saddle: loc XYZ 0.9, 0.85, 0.74)

The rms for accelerations and displacements was then calculated. Rms is Root Mean Square and it is the square root of the arithmetic mean of the squares of the values. It is a typical parameter for quantifying a vibration because it gives an amplitude value directly related to the energy content of the vibration:

 

RMS equation

RMS equation

In the calculation of the rms, the initial data (up to 4 s) were removed because they are conditioned by the acceleration of the motion. For each measure, the data was exported to a .txt file and processed with matlab to calcolate a “clear” rms.

For road Iso B:

Hub vertical position on Iso B

Hub vertical position on Iso B

Hub lateral position on Iso B

Hub lateral position on Iso B

Hub vertical acceleration  on Iso B

Hub vertical acceleration on Iso B

Hub lateral acceleration on Iso B

Hub lateral acceleration on Iso B

shaft vertical position on Iso B

shaft vertical position on Iso B

shaft lateral position on Iso B

shaft lateral position on Iso B

shaft vertical acceleration  on Iso B

shaft vertical acceleration on Iso B

shaft lateral acceleration  on Iso B

shaft lateral acceleration on Iso B

sella point vertical position on Iso B

sella point vertical position on Iso B

sella point lateral position  on Iso B

sella point lateral position on Iso B

sella point vertical acceleration  on Iso B

sella point vertical acceleration on Iso B

sella point lateral acceleration  on Iso B

sella point lateral acceleration on Iso B

It can be seen from the graphs on the position that the spring has a considerable excursion caused by a state of squat due to acceleration from stand. There is an extension in the front part and a compression in the back. And this have be eliminated in the calculation of the rms.

The SI2 solver was used for the simulations because with the I3 could be spiky problems with accelerations. SI2 formulation is more robust and stabler than I3 becouse accelerations are velocity aren’t back-calculated.
The table summarizes the rms calculated by the MatLab script

ISO B

Hub_posY 9,8883E-04 Hub_accY 0,4057
Shaft_posY 1,0715E-03 Shaft_accY 0,2657
sella_posY 1,0297E-03 sella_accY 0,0877
Hub_posZ 7,7655E-04 Hub_accZ 0,0323
Shaft_posZ 6,8356E-04 Shaft_accZ 0,0189
sella_posZ 4,4325E-04 sella_accZ 0,0119

The same method was used for road Iso C and road Iso D (graphs of behavior on isoC and isoD are in appendix):

ISO C

Hub_posY 1,9136E-03 Hub_accY 0,8111
Shaft_posY 2,1412E-03 Shaft_accY 0,5334
sella_posY 2,0494E+00 sella_accY 0,1754
Hub_posZ 1,5406E-03 Hub_accZ 0,0607
Shaft_posZ 1,3275E-03 Shaft_accZ 0,0378
sella_posZ 8,9493E-04 sella_accZ 0,0239

 

ISO D

Hub_posY 3,8305E-03 Hub_accY 1,6171
Shaft_posY 4,2788E-03 Shaft_accY 1,0744
sella_posY 3,9159E-03 sella_accY 0,3495
Hub_posZ 3,1167E-03 Hub_accZ 0,1214
Shaft_posZ 2,6510E-03 Shaft_accZ 0,0756
sella_posZ 1,8119E-03 sella_accZ 0,0478

 

Observing the graphs and the results obtained, there is an increase in vibrations passing from a less rough road (IsoB) to a series of more rough roads (Iso C and D). Moving at a constant speed of 50km / h, there is an increase in the rms value in all points of interest, as could be expect. Thus the model and the approximations made were verified.

 

Second part:

We wanted to make a comparison between the Piaggio Ape fork and a traditional telescopic fork, (going to analyze the behavior of the suspension on the same roads previously considered. The traditional fork is created by a red cylinder fixed on the chassis and a green cylinder that can move along suspension axis. Red and green cylinders are connected by a traslational joint.

Piaggio Fork

Piaggio Fork

Traditional Fork

Traditional Fork

To correctly compare the two forks we need to set parameters that are the same in both cases:

  • Mass of vehicle
  • Load distribution
  • Fork inclination
  • Equivalent vertical Stiffness (stiffness related to the vertical displacement)
  • Equivalent vertical Damping (dumping related to the vertical velocity)

Mass could be considered costant because when the suspension was modified, the mass of whole vehicle remains more or less the same

The load distribution is related to the center of mass of the vehicle which has similar coordinates in both cases and can therefore be assumed to be constant (G with fork Piaggio: XYZ (1.7407,0.7721,0.7397) – G with traditional fork: XYZ (1.7408,0.7721,0.7399))

The wheelbase is the same.

To have the same equivalent vertical stiffnes, proceeded as follows: the Piaggio Ape model was considered and a constraint was placed on the front wheel pivot. After that, zero damping and zero gravity was imposed, because only stiffness is to be analyzed. A vertical displacement was applied and the force was measured as a function of the displacement Fy on y (y is the vertical direction of the fixed frame).

Fy (y) curve was obtained, in which the slope represents the vertical stiffness (Ky) of the Piaggio fork.

fz su spost forc piaggio

In particular, for comparison, the vertical stiffness around the static trim have to be the same (FyStatic = 930N) because velocity ratio is not constant and the slope varied. Green line represent the tangent to the curve in static trim, so from the incremental ratio near the point there is Ky = 36200 N / m

So now, the stiffness of the suspension spring has been changed in the traditional fork model to ensure that the vertical stiffness around the static trim is the same also in this case.

fz su spost forc trad

Comparing two graphs, there is the same slope around the static load. This led to determine the stiffness of the suspension of the traditional fork K = 19400 N / m

To check the equivalent vertical damping, the procedure was similar: a vertical displacement with constant speed was applied and zero stiffness and zero gravity were imposed. It is important to apply constant speed because if it wasn’t constant, we would have accelerations and that lead to a wrong measurement. So, with constant speed of 1m/s (y_dot), the Fy=Cy, because:

formula damp
As previously done, it is necessary to observe Fy around of static trim. Given that, the suspension desplacement is -25mm in static conditions, so the Fy at the the same displacement is 2736N.

Damping_piaggio_mod

 

The damping of the suspension has been changed in the traditional fork model to make sure that the vertical damping (Cy=2736N*s/m) around the static trim is the same also in this kind of fork. To satisfy that condition C=1460N*s/m

damping_trad_modifiche

Sure that the two supsensions respect the parameters, they can be compared. The vehicle with the traditional fork is going to the three different surfaces (iso B, C and D) for 30s, like the procedure done in the first part, and vertical and lateral displacements and accelerations of three points considered before (were analyzed for each type of road. That are the graphs to behavior on Iso B road (graphs of isoC and isoD are in appendix):

Hub vertical position on Iso B (trad_fork)

Hub vertical position on Iso B (trad_fork)

Hub lateral position on Iso B (trad_fork)

Hub lateral position on Iso B (trad_fork)

Hub vertical acceleration  on Iso B (trad_fork)

Hub vertical acceleration on Iso B (trad_fork)

Hub lateral acceleration on Iso B (trad_fork)

Hub lateral acceleration on Iso B (trad_fork)

shaft vertical position on Iso B (trad_fork)

shaft vertical position on Iso B (trad_fork)

shaft lateral position on Iso B (trad_fork)

shaft lateral position on Iso B (trad_fork)

shaft vertical acceleration  on Iso B (trad_fork)

shaft vertical acceleration on Iso B (trad_fork)

shaft lateral acceleration  on Iso B (trad_fork)

shaft lateral acceleration on Iso B (trad_fork)

sella point vertical position on Iso B (trad_fork)

sella point vertical position on Iso B (trad_fork)

sella point lateral position  on Iso B (trad_fork)

sella point lateral position on Iso B (trad_fork)

sella point vertical acceleration  on Iso B (trad_fork)

sella point vertical acceleration on Iso B (trad_fork)

sella point lateral acceleration  on Iso B (trad_fork)

sella point lateral acceleration on Iso B (trad_fork)

 

The table summarizes the rms calculated by the MatLab script

ISO B – Traditional Fork

Hub_posY 9,9662E-04 Hub_accY 0,3719
Shaft_posY 1,0721E-03 Shaft_accY 0,2709
sella_posY 1,0370E-03 sella_accY 0,0884
Hub_posZ 7,7377E-04 Hub_accZ 0,0336
Shaft_posZ 6,8283E-04 Shaft_accZ 0,0192
sella_posZ 4,4169E-04 sella_accZ 0,0118

The same method was used for road Iso C and road Iso D (graphs of behavior on isoC and isoD are in appendix):

ISO C – Traditional Fork

Hub_posY 1,9137E-03 Hub_accY 0,7411
Shaft_posY 2,1423E-03 Shaft_accY 0,5335
sella_posY 2,0295E-03 sella_accY 0,1767
Hub_posZ 1,5365E-03 Hub_accZ 0,0606
Shaft_posZ 1,3655E-03 Shaft_accZ 0,0376
sella_posZ 8,8230E-04 sella_accZ 0,0237

ISO D – Traditional Fork

Hub_posY 3,8305E-03 Hub_accY 1,4836
Shaft_posY 4,2823E-03 Shaft_accY 1,0663
sella_posY 3,8276E-03 sella_accY 0,3533
Hub_posZ 3,1060E-03 Hub_accZ 0,1217
Shaft_posZ 2,7278E-03 Shaft_accZ 0,0750
sella_posZ 1,7855E-03 sella_accZ 0,0477

It is observed, how it could be expected, that the intensity of the vibrations increases with the increase of the roughness of the road. There is a rms of vertical vibrations on the front wheel hub that is less than the Piaggio fork (both for moving and accelerating) and this means that the suspension react better to road roughness at this point. However, the rms of vertical vibrations on the saddle and of vertical vibrations on the shaft is as higher as the Piaggio case, so the comfort of the driver is about the same. On the rear wheel axle, as you could guess, the measurements are very similar, because the rear part has remained the same (same kinematic suspension). The vibrations in the lateral direction also remain very similar to the initial case.

 

Third part:

To simulate the fall of the vehicle, the road model in the figure was taken with the following parameters:

Ascent length: about 11.3 m

Slope: 15%

Maximum height: about 3 m

The motion given to the vehicle is always of a straight accelerated type, with acceleration of 1,41 m /s^2, so at the highest point the vehicle will have a speed of 20 km / h.

Jump road

Jump road

Simulation link: https://youtu.be/5aiPA_C5Qcw

The data obtained for elastic force of suspension is the following:

SpringForceJump

There is a short stretch where the spring returns to the rest position during the fallen, at the end when the wheel has the contact with the ground under the weight of the entire vehicle, the suspension contract a lot.

It is also noted that the forces involved are one order of magnitude higher than a normal trip, which suggests that structural integrity is not guaranteed.

Moreover, it is possible to analyze the stresses on the revolute joint connecting the movable arm to the wheel hub. Below are the graphs relating to the force in the vertical direction and the magnitude force:

VerticalForce_joint_jump MagForce_joint_jump

In the two graphs we can see how the stresses involved are high and not sustainable by these types of vehicles. This excessive force (peaks of over 15000N) almost certainly leads to failure of the joint and the damage of the suspension. Being the graphs similar we can say that the major component of the force on the joint is the vertical one.

Fourth part:

Piaggio Ape is a RWD vehicle so, under breaking, in this kind of vehicle the front suspension compresses.

The dive line will lie between the front-wheel ground-contact point and the velocity centre of the front assembly with respect to the chassis. A two typical configurations for motorcycles are the front telescopic fork (like the fork in the second part of the project) and the four bar linkage. In the first case the velocity center is at an infinite distance and the dive line runs along the sliding axis of the suspension. In the case of the four bar linkage the velocity center is at the intersection of the two rocker extensions.

When the load transfer is along the support line, there is no suspension’s compression under breaking (it’s called anti-dive behaviour). The anti-dive ratio is defined as A=tanβdiveAngle/tanβloadTransfer. If A<1 the suspension will be compressed under breaking, if A>1 reverse.

To know the support line in Piaggio Ape, a torque was applied to the movable arm (which produces also a deformation of the spring) and the tyre contact patch was observed (down white line). https://www.youtube.com/watch?v=dWTp76HHAsM&feature=youtu.be

Once the arc of the tyre contact patch was found with the ADAMS function “Trace”, it was enough to trace the straight line tangent to the curve (blue line) and its perpendicular will be the searched support line (cyan line).

 

Geometrical explain of Anti-dive angle

Geometrical explain of Anti-dive angle

It can therefore be deduced that the antidive line can never be found above the load transfer line, therefore the vehicle will always tend to sink in the event of braking. Measuring the incremental ratio, finds that the angle between antidive line (cyan) and the ground in -10 degrees. To change this behavior could changing the geometry of the suspension, for example by extending one of the arms. Motorbike’s suspensions such as Duolever can be designed for good antidive behaviour, but in this types of vehicles economy and load capacity are preferred then comfort.

Conclusions

From the analyzes, it is easy to understand how the suspension system of this vehicle was designed without too many needs. Compered with the traditional fork observed that it is not more comfortable and the front wheel’s pivot is very stressed during the tests. So fatigue feilure in Piaggio forks could be more frequent than in traditional forks. It also doesn’t guarantee safety in the event of risked maneuvers. However, lateral positioning and a not excessive length favor a rapid and economic replacement in case of failure. Piaggio Ape could be considered an agile work vehicle on every surface and with a low production cost, to the detriment of comfort and performance, which has not prevented it from spreading to the present day.

Appendix

Piaggio Fork Iso C measure:

Measurement about hub on iso C

Measurement about hub on iso C

Measurement about shaft on iso C

Measurement about shaft on iso C

Measurement about sella point on iso C

Measurement about sella point on iso C

 

Piaggio Fork Iso D measure:

Measurement about hub on iso D

Measurement about hub on iso D

Measurement about shaft on iso D

Measurement about shaft on iso D

Measurement about sella point on iso D

Measurement about sella point on iso D

 

Traditional Fork Iso C measure:

Measurement about hub on iso C (trad_fork)

Measurement about hub on iso C (trad_fork)

Measurement about shaft on iso C (trad_fork)

Measurement about shaft on iso C (trad_fork)

Measurement about sella point on iso C (trad_fork)

Measurement about sella point on iso C (trad_fork)

Traditional Fork Iso D measure:

Measurement about hub on iso D (trad_fork)

Measurement about hub on iso D (trad_fork)

Measurement about shaft on iso D (trad_fork)

Measurement about shaft on iso D (trad_fork)

Measurement about sella point on iso D (trad_fork)

Measurement about sella point on iso D (trad_fork)

 

Sources

  • Ape Piaggio’s user manual
  • https://www.mondoape.com/tecnica-ape-piaggio-tm-p703-atm2-atm3.htm
  • David J. N. Limebeer & Matteo Massaro: Dynamics and optimal control of road vehicles



 



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