The dissertation is about the modeling and dynamic analysis of the gear. In this for the accurate diagnostics techniques, this study examines the dynamics response of the gear, which is based on the model involving the effect of friction for the tooth mesh process. The 6 DOF model is developed to involve the effects on not only the gear dynamic but is also supporting the driving motor as well as the loading systems. This dissertation investigates a relevant aspect which are according to the dynamics modeling for the transmission of gear starting with simplest 1DOF, then also develop the 3DOF along with finishing with the 6DOF. The complete study is provided, which influence the design of the parameters, like the number of teeth, gear ratio, pressure angle, an addendum coefficient. The detailed explanation of the gear dynamics model is discussed in detail in the below section introduction. The thesis paper will critically analyze on the bending stresses features that relate to the spear gear when in its static state. The generation of the tooth profile occur through the Catia, and the finite element method of analysis will be used with the help of the MATLAB software. The project will evaluate the stresses that are in the root of the tooth through the various theoretical models.

Introduction of Modelling and Dynamic Analysis of Gear

Gears that is a very important element in the mechanical system of transmission. Which study on the ways the gears will transmit the power in the parallel shafts. The gears are a simple machine element that contains some teeth that are arranged like the cylinder and have equal distance between them. These elements are playing a major role in making different machines like clock and gearbox of the car. The main function of these gears is to transmit forces and as well as the forces from one part to the other part of the driving shaft. The Gears are classified on the basis of their shape and figures. These gears are also divided on the basis of position of shaft that include shaft gears, non-parallel, interesting shaft gears and also non intersecting shaft gears. The gear is one of the oldest machine elements that has been used by the ancient peoples. The thesis paper will critically analyses on the bending stresses features that relate to the spear gear when in its static state. The generation of the tooth profile occur through the Catia and the finite element method of analysis will be used with the help of the MATLAB software.

The project will evaluate the stresses that are in the root of the tooth through the various theoretical models. In this case, the results that will come from the Finite Element and the Theoretical methods will be closely compared to get the deviations. The report will also seek to introduce displacement and the stress features of the tooth while on the dynamic states. The paper will therefore do analytical calculations with the use of relevant software. According to the shape of the tooth pair as well as disposition gears is classified into helical, spur, straight bevel, double helical as well as worm and spiral gears. The fundamental law of gearing is defined as the ratio of angular velocity among the set of gears throughout the mesh. By input and output fluctuations of torque is under idling or lightly loaded conditions gear rattle as well as the tooth impacts caused by the presence of gear backlash.

In the vehicle interior, this dynamic behavior can increase the noise. There are many papers in the literature that concerned with the dynamic gear; eventually to compute the dynamic transmission error it’s very important to predict the noise of gear as well as increase the life of gear. In the mechanical power transmission system gear is one of the important machine elements. To transmit power among parallel shafts Spur gear is most basic gear that is used. By bending failure or contact failure spur gear generally fails.

Research Questions of Modelling and Dynamic Analysis of Gear

· What are the existing models describing gear’s dynamics?

· What are the main parameters that are playing an essential role in gear’s dynamics?

· How it is proven the role of each parameter in gear’s dynamics? (Examples of transient responses to justify it).

Parameters in gear’s dynamics

In the gear dynamics, the lumped parameters are incorporating effects of the asymmetric mesh as well as time-varying stiffness along with the non-linearity of backlash which is formulated for the Spur gear in rattle response below the idling conditions [1]. In the vibrating systems, the lumped parameters are coupled in the dynamic model of gear by the lateral torsional of the Six degrees of freedom (6 DOF). The lumped parameters for the idling spur gear systems involving the backward as well as forward tooth contacts which are formulated as in below section. By using the formulation of mesh stiffness as well as the mesh stiffness, for the dynamics model of the 6DOF as the lumped parameters are also developed for the spur gearbox systems . The figure of the dynamic gear model of the 6 DOF is also shown in appendix 1. Now the equation of motion for the dynamic systems is derived here. And an equation of the motion for the torsional vibrations is also represented by the below equation,

Design parameters for Spur gear system

Parameters	Pinion	Gear
Module (mm)	2
Pressure angle ( degree)	20
Contact ratio	1.6456
Number of teeth	19	48
Mass moment of inertia
Mass	0.96	2.88
Bearing radial stiffness
Bearing damaging

For spur gear systems, the design parameters which is used in the above table. In the gear dynamics, the mesh stiffness K which is the Key parameters to obtain the numerical identification for the finite model in the teeth contact. For the Spur gear systems, the resultant formulation gives the response under the idling condit6iobns for the numerical simulations of the design parameters [2].

The proven parameter in gear’s dynamics

In the Spur gear systems the lumped parameters model for the 2 degrees of freedom is generalized by the meshing stiffness of the transmission error, backlash, external periodic excitation is established. By solving the differential equations for the responses of vibration analysis, the Network Method is used, where the characteristic of the transient’s response effects the rotational speed in the dynamic of the Spur systems. The influences of the gear systems in the bearing coupling for the gear system used the single DOF model as well as the multi DOF model to suppose the parameters of a time-varying in the gear system. In the lumped parameter, the 2 degree of freedom is purely torsional to investigate the gear bearing of spur system [3].

The effects of the excitation for the internal as well as external is the strongly non-linear for the backlash effects. Among the driven gear along with the driving gear, the dynamic model of the gear is shown in appendix 2 . Now the discussion is about the geometric parameters, where the assumption are shown for simple the dynamics model of the gear friction forces at mesh is neglected; And in the appendix 3 the mechanical model which is the retaining characteristics to arise the interaction of spur gear mesh along with the non-linearity’s of bearing [4]. The Shafts as well as bearings, which support the gears to, modeled through the equivalent elements by the different viscous damping coefficients as well as the liner springs.

In the explicit formulations the torque input, as well as the torque output, is assumed through the lower frequency which is named as;

Now take the backlash, the meshing stiffness, as well as transmission error along with the fluctuation torque. The relation of the force-displacement at the bearing is taken as a linear. When the kinetic energy, plus the potential energy U along with the dissipation of the function R is recognized. To prove all parameters of the dynamic model of the gear system, by using the Lagrange equation the 2 DOF in the torsional motion for the non-linear geared system is explained the appendix 2

In the above two equation, the dot means that take the derivative with respect to t, as well as the number of the over dots represent the order of the differentiation with respect to time t; for the torsional displacements driven/driving for the dynamic meshing of force acting in a meshing points

Literature Review of Modelling and Dynamic Analysis of Gear

Gears

Gears are the basic components of various complex and normal machines, gears are used for the transmitting motion, and depends on shifting torque from the shaft to another. However, every invention having the rotation machine have gears. Gears first used in 2600 BC Chinese used them for measuring the speed of chariots, Archimedes used in 250 BC, and in 4^th BC Aristotle used. Greeks and Romans and Greeks used it widely in buildings ecclesiastical and in the clocks [1].

In early centuries the gear was built or manufactured by stones or latterly by wood and stone tooth were installed in wood, to increase the life and stability of gears. However, from last few centuries, the gears advancement took placed, and changed from wood and stone era to metal, like iron and its substitutes to overcome the rusting issue. Until 1835 there was no standard procedure to follow to make gears later English inventor patented the gear systems [2]. The first gear machine Feature patented by Germany in 1897 and introduced NC machine in 1975, and later they introduced 6-axis gear hobbing machine.

Figure 1: Wooden Clock

However, various machines and gear model were built in the early stages, but the stress between the gear systems remained there until 1892. Then later Philadelphia engineer, in 1892 recognized this issue and developed a research on stresses of the gear tooth, which is still used in the various model for gear stress[3].

Spur Gears

Such gears that contain cylindrical pitch surfaces are known as cylindrical gears. The simplest form of the gear is known as a spur gear and used to transmit the rotational motion between the parallel shafts; this is used for all common machines and on all aspects however this gear system is hard to be used in the extensive load, ratio or the speed directed towards another point. The other gear types also remain comfortable with Low vibration operations were demanded. The ratio a single spur gear requires is 1:1 or 1:6 minimum, with a velocity of 25 milliseconds it has efficiency around 98%, it is recommended to have a high number of teeth’s in the gears to strength. The minimum requirement of teeth is twenty degrees. Spur gear is low cost easy to manufacture this increased the use of the spur gear and still they are being used in various machines.

The spur gear is being used from very basic applications like a bicycle to aero-plane hence proved as the best solution.

Special features of Modelling and Dynamic Analysis of Gear

- Centre Distance:

The center point distance of two spur gears is the distance from the center shaft point of one spur gear to another center shaft point. Centre to centre distance limit for two gears in mesh can be assessed with this formula.

Figure 3: Rotation [5]

Spur gears in a 2-gear drive system Gear 1 and 2 will rotate opposite direction.

The relationship between the rotation gears drive system can be stated as follows:

Two meshing gears (Gear sets are comprised of 2 or more gears fixed to the same shaft) rotate in opposite directions. Each odd-numbered gear lead to turns in the same direction.

Application of Spur Gear requirements:

The workload and environment of the gear set in mind.

1. Power, velocity, torque mechanical requirements need consistency and output peaks so that the gear meets

2. The apathy of the gear via accelerations and de-acceleration

3. Precision including gear pitch, shaft diameter, pressure angle, and tooth layout.

4. Gear lubrication requirements. Some gears require lubrication for smooth, temperate operation.

5. Noise Spur gears are especially noisy in operation

6. Corrosive Environments. Gears revealed to weather or chemical which must overprotected from corrosion

7. Temperature exposure. The temperature may react on gears in a way which may wrap or become brittle

Materials which in the Spur gear

1. Cast iron gives durability & simplicity of manufacture.

2. Alloy steel gives superior durability and corrosion resistance.

Chemical composition could be added to the alloy to harden the gear

3. Cast steel vibration resistance, also have decent fabrication and works with strong loads

4. Aluminum is used when low gear inertia with some resiliency is required.

5. Brass is cheap, easy to create

6. Copper is easily shaped, conductive and corrosion resistant. However, for gears potency would increase if bronzed.

7. Plastic is cheap, corrosion resistant, quiet operationally and can overcome missing teeth

Types of gears:

- External spur gears: these gears are the most common of all the types. Two cylindrical wheels, a large one, called gear and a small one known as the opinion form them, straight teeth couples them.

- Inner spur gear: its design is similar to the common spur gear. The difference is basically in how the gears are placed; they have an internal coupling. This type of gear is also known as ring gears. The internal spur gears can be associated with a system of planetary gears, in which three or four gears (planets) are spinning around a pinion (sun)

- Pinion and rack gear: This is constituted by a ribbon that has straight teeth along it, and make contact with a pinion that does have the shape of a normal gear. This type of gear helps to transform a rotary movement into a linear one and vice versa.

- Helical gear: it has a cylindrical structure like the spur gear, but the shape of the teeth has a helical curve. It can transmit more power and produce less vibration and noise than spur gears. However, it generates an additional thrust that requires bearings to hold it.

- Conical gears: it has the same teeth as the spur gears, the main difference is the working on conical surfaces between intersecting axes. They can produce high loads and speeds. There is a variety of conical gears: spur, spiral and angular.

- Worm Screw: it is formed by a screw that has helical teeth, which is mounted on the drive shaft and a crown that is the driven element, which transmits their movement between axes perpendicularly making an angle of 90 °. [13]

Spur gears transmission design

Procedure [14]:

1. Identify the inlet and outlet velocity of the pinion and the gear respectively.

2. Select the material of the gears.

3. Establish the overload factor of the system based on the driver machine and the driven element.

4. Define the approximated size of the diametric pitch.

5. Specify the face wide.

6. Calculate the transmitted load, the speed of the line of passage, the number of qualities, geometric factors and other necessaries factors to find the stress of flexion and contact.

7. Calculate the stress.

8. Iterate the design process to optimize it