*Leonardo Mazzanti, leonardo.mazzanti@studenti.unipd.it*

## Introduction

This project has analysed the way of functioning of a XIX century mechanism used for the motion transmission between two orthogonal rotating shafts. The model analysed is slightly different to the real patent: in particular, the geometry has been downloaded from an opensource website [2] of 3D model printing. However its way of functioning and the same geometry can be seen watching videos of real built machines [3].

Using the sotware MSC Adams View, the analysis has been carried on step by step from a simple way of functioning to a more complex one: from analytical to dynamic simulations, introducing also friction for the joints considered more critical. The aim of the project is knowing how rotational motion and energy are transmitted by the mechanism between the input and output shaft, trying to explain which are the negative aspects that are affecting the Almond Coupling.

## Objectives

In particular, the purposes of the project are:

- Calculating the rotational speed ratio between the output and the input shaft;
- Calculating the efficiency from the ratio between input and output shaft energy after the simulation with different friction coefficient values;
- Looking for discontinuities and impacts during the functioning.

First of all, the components have been imported in the right position and orientation in order to assemble the whole mechanism. Afterwards, various types of analyses have been performed:

**Analytical kinematic analysis:**ideal joints have been applied between the bodies being careful to have zero degrees of freedom (DOF) according to Grubler equation and at the same time no redundant constraints. As input, constant rotation has been applied at the first shaft and the collected data have been elaborated in order to look for eventual impacts and for transmission speed ratios.**Dynamic analysis with contact joints:**instead of ideal joints, most of the constraints have been replaced with contact forces in order to simulate also the friction effect on the mechanism. In particular friction between greasy steel-greasy steel is simulated using friction coefficients from literature [5] and then they have been doubled to pretend bad lubrication. Efficiency and curves of motion have been analysed.**Analytical and Dynamic analyses comparison.****Other 2 short dynamic analyses:**the first one consisted in using a step motion in order to make a gradual start; the second one consisted in applying a constant torque to shaft 1 instead of the constant motion.

The data collected have been elaborated to reach convergency after validating the parameters adopted for the analyses. Finally, the results obtained have been discussed to underline some differences with a modern gear transmission.

## The modelling problem

### The patent

At the end of the XIX century, during the** Industrial Revolution**, new machines and transmissions were invented or improved from the past by using new materials, in particular steel. Industries were adopting new methods to gain more efficiency and to reduce the cost of their products. **William Gleason** (1836-1922), founder of “The Gleason Works”, came to the United States from Ireland in 1851: he was a gifted inventor and a skilled mechanic. In 1874, his invention of the straight bevel gears planer for the production of bevel gears with straight teeth substantially advanced the** progress of gear making**. Industrial world was increasing its demand for bevel gears in particular in automotive to build car transmissions between orthogonal shafts. Early, manufacturing straight bevel gears was a difficult process because the gears were cast in metal and then were finished manually filing each tooth. [6]

However, another system of transmission between normal axes was patented in the same period, it was bulkier and probably less efficient respect to a bevel gear, but probably more economical convenient at that time: it is called Almond Coupling.

In **1884** the Almond Coupling mechanism **was invented and patented by Thomas R. Almond** [1]: the main function of this mechanism is coupling 2 shafts rotating around orthogonal axes. At the same time the mechanism should have been correctly lubricated (reason why there is a cover all around) and easy to use in different configurations of the line of production.

The transmission is intended to be **mounted on a ceiling** so that it does not take up floor space. Two other holes contain the bearings to sustain the shafts (A and B in the figure) in their position. Each shaft is attached by an hinge to a crank (a,e), whose other extremity has got a spherical joint. This joint connects the crank to a little component (b,f) that can slide inside a lever (C). The lever slides and rotates around a central fixed tube (E) so that the motion is transmitted.

Outside the shell, the two shafts were usually attached to other two wheels (G) for the belt transmission: it can be seen in the video linked in the references [4].

### Model Analysed

Looking at the video [3] the functioning of the real built Almond’s Coupling is slightly different: the crank is fixed to the shaft and instead of having the entire piece b, this one consists only in a ball shaped component with a hole in the middle to make it translating on the lever. The same 3D cad model has been **downloaded from Thingverse [2] and then imported with Parasolid** format in Adams View.

### Units

### Parts

Name |
Mass |

Basement | 1474 g |

Cover | 497 g |

Wheel 1 | 414 g |

Wheel 2 | 414 g |

Crank 1 | 46 g |

Crank 2 | 46 g |

Ball 1 | 8 g |

Ball 2 | 8 g |

Lever | 54 g |

### Material

All the parts have been assumed to be homogeneously composed of **steel (steel density = 7801 kg/m ^{3})**.

### Ideal constraints

**The cover has not been considered** in the analysis because it has no contribution on the kinematic chain of the mechanism.

The constraints have been applied so that no redundant constraints could affect the simulation introducing additional errors.

A cylindrical joint between lever and the basement’s cylindrical tube could seem the right logical and immediate solution to represent that contact: however, in that case the system would have redundant constraints.

This is the reason why primitive joints have been adopted.

**POINT_1** (coordinates 0,0,65 mm) has been created at the centre of the vertical tube over the origin of the principal reference frame and with the same absolute height of the balls’ COMs: POINT_1 is on the axis of rotation of the lever around the tube.

Respecting the ideal way of functioning, POINT_1, Ball 1 and Ball 2 centres of mass (COM) must stay on the same plane and the angle formed by their connection must be of 90°.

One of the problems has been the definition of the centres of rotation of the balls about their centres of mass: the three points didn’t form a perfect 90° angle, moreover Ball 1 COM and Ball 2 COM weren’t perfectly at the same height (65 mm) due to the fact that their coordinates were defined by a too large amount of significant digits, not all perfectly equal. Using the perfect calculated COMs, the analysis incurred into numerical error because the kinematic chain appeared blocked.

To solve the problem other **2 auxiliary points (Ball_2_COM and Ball_1_COM)** were created with the same coordinates of the real COMs but using less significant digits: the conditions for the correct functioning of the mechanism were respected.

The **Inline Primitive Joint** has been used to remove two translational DOFs, so that the lever could translate only along the direction linking POINT_1 to the origin.

The **Perpendicular Primitive Joint** has been located in POINT_1 and one rotation has been constrained: it is the rotation along one axis laying on the plane perpendicular to the Inline direction.

The follwoing table sums up all the constraints adopted.

First Body |
Second Body |
Type of Joint |
Position |
# of DOFs removed |

Basement | Ground | Fixed | 6 | |

Basement | Shaft 1 | Revolute | Along shaft COM symmetry axis | 5 |

Basement | Shaft 2 | Revolute | Along shaft COM symmetry axis | 5 |

Basement | Lever | Inline Primitive | From POINT_1 to ORIGIN | 2 |

Basement | Lever | Perpendicular Primitive | Centre in POINT_1 | 1 |

Wheel 1 | Shaft 1 | Fixed | 6 | |

Wheel 2 | Shaft 2 | Fixed | 6 | |

Crank 1 | Shaft 1 | Fixed | 6 | |

Crank 2 | Shaft 2 | Fixed | 6 | |

Ball 1 | Crank 1 | Spherical | Ball_1_COM | 3 |

Ball 2 | Crank 2 | Spherical | Ball_2_COM | 3 |

Lever | Ball 1 | Translational | From Ball_1_COM to POINT_1 | 5 |

Lever | Ball 2 | Translational | From Ball_2_COM to POINT_1 | 5 |

### Motion

A **constant rotational motion of 360 degrees/sec** has been applied to Shaft1, which is constrained with the revolute joint to the basement.

### Grubler’s equation

**Grubler’s equation** is used to calculate the number of degrees of freedom (DOF’s):

DOF = 6xN – 6xF – 5xR – 5xT – 2xIP – 1xPP – 1xM = 0 , **so there aren’t redundant constraints.**

N = n° of components = 13

F = n° fixed joints = 5

R = n° of revolute joints = 2

T = n° of translational joints = 2

IP = n° of inline primitive joints = 1

PP = n° of perpendicular primitive joint = 1

M = n° of motions = 1

### External Forces

**Gravity** has got z positive direction respect to the principal fixed reference frame, following the hypothesis that the basement was originally thought attached on the ceiling according to the patent.

**Torsional spring** has been put on the revolute joint between shaft 2 and the basement: it has no stiffness and the value of **damping is set on 10 (N*sec)/mm.** The torsional spring is useful to calculate the power transmitted in output: if the contact damping and stiffness are rightly calibrated, the mechanism efficiency should converge to 1 cycle after cycle. Therefore, these data have been also used to calibrate simulation parameters and contact constraints.

For friction simulations, **Friction Force** has been modelled as **Coulomb** and **Stiction/Friction Transition Velocity **have respectively the values of 1 mm/s and 1.5 mm/s.

The joints are supposed greasy steel in contact with greasy steel, so according literature [5]:

**Static Coefficient = 0.23
Dynamic Coefficient = 0.08**

The same simulations are then executed doubling these coefficients to predict the behaviour in case of bad lubrication:

**Static Coefficient = 0.46
Dynamic Coefficient = 0.16**

## Simulation and analysis of results

### Analytical Simulation

**Simulation parameters:**

- Step Size = 10
^{-3}sec - End time = 1 sec (1 cycle)
- Kinematic Solver
- Kinematic Solver Error = 10
^{-5}

Firstly, to show that the simulation is running correctly the torsional spring damper is removed so that the effect of energy dissipation is removed and only conservative forces are acting (gravitational force): from the following graph energy distribution is shown about the input and the output shaft. The input energy is oscillating harmonically, but its value is zero at the end of the cycle as it should be theorically.

Then, adding the spring damper to simulate a load at the output shaft, input power curve still oscillates in function of time while output power plot remains constant: the input power has to supply the power generated by gravitational acceleration applied to masses, which oscillates according to the laws about conservative forces.

From the integration of the output and input functions, energy curves are obtained:

With ideal constraints the mechanism has got a **constant speed ratio equal to 1** during all the motion: these data have been obtained using wheels’ rotation in input and in output.

Friction can’t be applied to primitive joints, that is the reason why contact constraints have been adopted for the successive analyses.

### Contact Constraints

The second part of the project consisted in the substitution of some ideal joints with contact joints, in particular the joints sharing the lever, ball 1 and ball 2 bodies: in fact, these are the constraints with predominant friction and characterized by a more complex motion. The joints between the basement and the shafts have been well modelled by revolute joints because of the presence of ball bearings that have low coefficients of friction.

All contacts are **Solid to Solid** and **Normal Force **is set on **Impact**.

Higher the stiffness is, more real the contact is, however also numerical problems increase. For this reason, **stiffness have been calibrated 10 ^{5} N/mm**, controlling each simulation that the penetration depth was not too elevated.

Other parameters used:

**Force Exponent = 2.2**(default);**Damping = 1000 (N*sec)/mm**(calibrated in order to have stable and fast simulations, 1% value respect to Stiffness)

**Penetration Depth = 0.01 mm****Faceting Tolerance contacts solver: 500**

### Dynamic Simulation

In order to set efficient simulation parameter, a lot of contact dynamic analyses have been run changing step size and the solver error. Firstly, **GSTIFF integrator with I3 formulation has been used setting the step size at 10 ^{-4} sec and solver error at 10^{-5}**. Afterwards, using the same type of integrator, error has been

**decreased at 10**. Finally, knowing that all integrators should be convergent with appropriate parameters, the simulation has been run with

^{-6}**GSTIFF SI2 integrator**(step size = 10

^{-4 }sec and error = 10

^{-3}) The results are convergent, so the

**first simulation is validated**.

The results are summarized inside the following graphs:

Comparing the dynamic simulation with the analytical one, a** little dissipation is affecting the dynamic results**, which are influenced also by the first impacts after the start of the motion. Integrating the power curves respect to time, input and output energy are so plotted:

At the beginning of the motion of the contact simulation impact and energy dispersion are noticed: this fact can be justified by the geometry of the model that is characterized by dimensional backlashes that must be recovered before the movements start. In addition to this impact is increased because the model starting position is not stable.

However the curve obtained with ideal joints is still parallel to the curve with contact frictionless constraints for all the motion: this means that the energy surplus only depends on the first impact.

### Final simulation parameters:

- Step Size = 10
^{-4}sec - End time =1 sec (1 cycle)
- Dynamic Solver
- Integrator = GSTIFF
- Formulation = I3
- Error = 10
^{-6} - Hinit = 10
^{-7} - Interpolate = Yes
- Kmax = 6 (Default)
- Maxit =10 (Default)

### Frictionless case

After choosing the good simulation parameters to obtain reliable solutions, analysis have been executed about the model with contact frictionless joints.

Initially,** penetration depths** are investigated to see if they can be considered acceptable and not so large: they have a **10 ^{-3} mm order of magnitude (0.007% ball diameter)**, so simulation is validated from that point of view.

Integrating the input and output power curves, energy is calculated. Input energy is a bit bigger respect to output energy. In addiction to the fact that input power is not constant because it’s affected by the gravitational effect as seen with ideal joints, so energy is oscillating as in the previous ideal simulation, probably the initial impact is causing energy dispersion.

The following diagram shows the efficiency that varies respect to time: until about 0.70 sec there is a transitional phase, followed by the stabilization of the data near the value of 90% efficiency (value 0.9 in the graph).

In all the graphs peaks have been noticed, in particular in the first steps, also using different types of solver integrators: the rapid variation of speed curves of the bodies’ COM is signal of some impacts, specially at the beginning of the motion. In fact, the model has not been assembled in an equilibrium position and there are also dimensional tolerances between parts:the model is made for a functioning 3D printed plastic mechanism.

This is the reason why at the beginning a dimensional tolerances are recovered and **output speed is equal to zero, even if input speed remains constant.**

Despite the first part of the movement, the motion is characterized by an almost constant speed ratio between input and output, equal to 1: the curve has got periodic spikes.

Trying to analyse a steady-state situation, a period of 2 seconds has been simulated with the same parameters and efficiency exceeds also 90 % in the second cycle of rotation of the mechanism, increasing and converging to 100%.

### Friction Greasy Steel-Greasy Steel

Friction inside contacts is implemented and the simulation is executed. These are the friction coeffiecients that have been used:

**Static Coefficient = 0.23
Dynamic Coefficient = 0.08 (mu)
**

As expected, output energy remains the same during the cycle, while input energy increases and efficiency decreases respect to the no friction simulation.

At the steady state functioning the medium efficiency is about 74%.

The curves about rotational speeds are less smooth than the ones in frictionless plots.

The two big peaks at 0.25 sec and 0.75 sec seems to be strictly connected to lever’s minimum and maximum height:probably dimensional tolerances between the tube and the lever are recovered during the motion inversion and other impacts are generated, as it can be seen in the graph below. In fact, the points of the lever’s maximum and minimum height are the ones characterized by the most higher accelerations (inversion of the motion).

### Bad lubrication Greasy Steel-Greasy Steel

The normal friction coefficients are doubled in order to simulate a bad lubrication case:

**Static Coefficient = 0.46
Dynamic Coefficient = 0.16**

The steady state efficiency is about 60% with dooubled friction coefficients.

As before, the speed ratio remains almost equal to 1 despite the increase number of spikes: increasing friction also the effect of impacts becomes more relevant.

### Comparison between different values of friction

The following plot shows the behaviour of input and output energy in a motion cycle lasting 1 second.

Spikes on speed curves increase their number increasing the friction coefficients.

### Other simulations

Other secondary simulations with the same dynamic solver parameters have been carried on without friction:

- the
**law of input motion has been changed, the function step(time, 0, 0, 0.3, 360d) is used**to increase more**gradually**the velocity and avoiding a bad impact. However, the output is not stable and subjected under gravitational force:**shaft 2 moves with opposite rotation**respect the rotation given by the input applied to shaft 1. Then, when the contact between crank 2 and ball 2 is established, there is a big impact followed by energy dispersion.

- a constant torque equal to 70 Nmm has been applied instead of giving as input the constant rotational speed:

**Plotting the rotational speeds of the two wheels, the two curves (about the input and output shafts) oscillate around an average value almost in counterphase.** In addition to this, the first steps are characterized by the opposite output rotation respect to the direction of regular motion: the system isn’t in stable equilibrium starting the simulation and that is the reason why the first impacts are more visible. However, the analysis is run in advantage of safety, simulating bad conditions of functioning.

## Conclusions

Especially at the beginning of the motion, **joint dimensional tolerances** are causing impacts that don’t let a perfectly responsive transmission from input to output. The system isn’t in stable equilibrium starting the simulation and that is the reason why the first **impacts** are more visible. However, the analysis is run in advantage of safety, simulating bad conditions of functioning: in the reality is difficult to see a step constant input motion of 360 degrees/sec from the beginning. After all, the transmission can be considered **homokinetic**, almost in the ideal case.

Simulations obtained about **74% efficiency for a well lubricated mechanism** (dynamic mu=0.08) and about **60% efficiency for a bad lubrication** (double coefficients dynamic mu=0.16).

**Differently from a gear transmission**, the Almond’s Coupling occupies a** wider space**, is more difficult to lubricate and having so many components implies being more expensive about maintanance, having more frictional area and having **more dimensional backlashes**: last one observation is really important because it means that Almond’s Coupling hasn’t precise motion transmission (discontinuities in the speed ratio have been observed).

Finally, the Almond Coupling is characterized by the** oscillation of input power to give a constant power** to the output shaft: bigger are the masses to move in the direction of the gravitational force bigger are the discontinuities of the loads applied to the structure. This a great disadvantage after all and probably the main reason why bevel gears are commonly used instead Almond’s Coupling.

## Future Developments

All considered other types of simulations could be executed, using as input **more realistic laws of motion** or doing more simulations with constant torques. Other simulations can investigate if the starting position of the mechanism parts has some influences on the first impacts.

To avoid impacts, a **more realistic 3D cad model** could be realized with smaller dimensional tolerances.

Finally, no vibrational analysis has been carried on, but should be interesting knowing the resonance frequencies of the mechanism and the acoustic effects of vibrations, comparing the Almond’s coupling to a modern transmission.

## References

[1] Patent US304156A (https://patents.google.com/patent/US304156A/en)

[2] 3D cad models: https://www.thingiverse.com/thing:3413776

[3] Video https://www.youtube.com/watch?v=bRjvfiYZO1Y

[4] Belt functioning video https://www.youtube.com/watch?v=Df2mgjhv1TQ

[5] Friction Coefficients https://web.mit.edu/8.13/8.13c/references-fall/aip/aip-handbook-section2d.pdf