# Computer Aided Engineering Techniques

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## Hypoid Gear Contact Ratio

The contact ratio represents the average number of teeth meshing at the same time. The contact ratio refers to ratio of the length of the arc of contact (blue line figure 4) to the circular pitch (orange line figure 4). The benefit of having a higher contact ratio is that the load is shared more equally, resulting in less wear. Furthermore, the average stiffness of the gear is higher. A higher stiffness means better transmission accuracy due to less tooth deflection resulting in a lower TE. Thus, a higher contact ratio results in less noise since TE is the main source of gear noise in driveline parts [7] (see sections 2.1.4 and 2.1.3).

Gear Mesh Stiffness and Gear Mesh Frequency

Gear mesh stiffness is a material property of the gear which resists deformation. The value is dependent on the gear material, tooth curvature, tooth loading, contact ratio and angular position of the gear [1]. Since the gear contact and tooth loading changes with time (due to tooth bending), the gear mesh stiffness is dynamic rather than a constant value. This results in a variation of the force acting on the gear teeth, generating a TE and thus causing vibrations. Gear mesh stiffness can be calculated using equation 1. F Km ˘ el ¡eo (1) where: Km ˘ Gear mesh stiffness F ˘ Contact force in the line of action el ˘ translational loaded TE eo ˘ translational unloaded TE The gear mesh frequency is the rate at which gear teeth mate together. It can be calculated as shown in equation 2. The shaft speed refers to the input shaft when there are multiple gears. The number of teeth refers to the pinion, which in this case is 15. This value is important because it has been shown that the gear mesh frequency and its harmonics are the frequencies which contribute heavily to the noise and vibration of gears, particularly gear whine [3] [8]. Table 1 shows the gear meshing frequencies of the speeds analysed in this thesis.
TE is the main excitation source of gear noise during vehicle operation, particularly gear whine [8]. It produces a particularly aggravating noise due to its tonality. TE arises due to gear teeth imperfections during manufacturing and assembly [9]. The imperfections inhibit the gear from transmitting the rotational input correctly, causing a variation in the speed ratio. Consequently, the gear ratio is altered slightly and varies with time. These deviations are the main cause of oscillating forces on the gear teeth, which result in noise and vibrations. The oscillations produced as a result of TE can range between a high or low frequency, depending on the teeth characteristics, such as friction, teeth deflections and geometry.
TE can be defined as ’the difference between the actual position of the output gear and the position it would occupy if the gear drive were perfectly conjugate’ [10]. This is expressed in mathematical terms in equation 3 and a graphical representation can be observed in figure 5.
T E ˘ £g ear ¡µ Rg ear ¶£pi ni on (3).

Noise, Vibration and Harshness

Noise and vibrations are an inevitable by-product of mechanical machines. NVH, in auto-motive applications, is the study of the sounds and vibrations concerning vehicles and their components.

### Vibrations – The Fundamentals

Vibration is defined as the periodic back-and-forth motion of particles of an elastic body or an elastic medium. Commonly, this is a result of a physical system being displaced from its equilibrium condition. All bodies containing mass and elasticity are capable of experi-encing vibrations [14]. Thus, most machines and structures engineers deal with experience vibrations to some degree [15].
There are two classes of vibrations:
• Free vibration — A system oscillating due to the forces inherent in a system and no external forces are present. A system that vibrates freely will vibrate at one or more of its natural frequencies.
• Forced vibrations — Vibration that is caused by an external force. If the excitation is oscillatory, the system is forced to vibrate at the excitation frequency.
Key terms in vibrations [15]:
• Natural frequency — The frequency at which a system/object oscillates when not subjected to an external force or damping force. All systems and components have at least one natural frequency and it is dependent on its mass and stiffness. Note, sometimes this may be referred to as an eigenfrequency.
• Mode shape — The motion pattern of a system oscillating at a frequency. Commonly bending, torsion or a mixture of the two.
• Resonance — Dangerously large oscillations that occur when a system is excited by a frequency close to its natural frequency. Resonance can be the cause of major structural failure.
• Damping — The energy dissipation due to friction and other resistances. Low damping has only a minor effect on the natural frequency, which is why natural frequency is often calculated without damping. However, high damping can significantly reduce vibration amplitudes.

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Acoustics – The Fundamentals

Acoustics is the study of mechanical waves in solids, gases and liquids.
Sound is the oscillation as pressure, stress, particle displacement/velocity is propagated as an acoustic wave through a transmission medium with internal forces, such as a gas, liquid or solid. A wave is a disturbance travelling through a medium from one location to another, transport-ing energy. Waves can be reflected, superposed, refracted and diffracted at boundaries, as specified in figure 7.

#### Computer Aided Engineering Techniques

The task of NVH engineers is to create methods to predict, analyse and reduce the noise and vibration that radiates. Several techniques can be employed to achieve this. For instance, analytically, experimentally or through other means. It is imperative that NVH issues are detected and resolved early in the design phase. Early detection can significantly reduce the number of design iterations, keeping costs low as NVH problems typically require revised designs. Late detection, e.g. during the prototype phase, can cause significant increases in the development time and unsatisfactory solutions being implemented [19].
A particularly effective way of predicting and improving the NVH characteristics of compo-nents is through Computer-Aided Engineering (CAE). CAE is ‘the use of computer software to simulate the performance of a product to improve the design or facilitate solving engineering problems’ in various engineering disciplines [20].
Usually CAE consists of the following [20]:
1. Pre-processing — model the geometry and assign physical properties (e.g. by applying loads and constraints).
2. Solving — model is solved using the appropriate mathematical and physical con-cepts/formulas.
3. Post-processing – Review of the results and further analysis.
As a result of the improving computational power, NVH engineers now have the means to utilise both Multi-Body Dynamic (MBD) simulations and Finite Element Analysis (FEA) techniques in conjunction with acoustic tools. These simulation techniques enable one to predict the vibrational and acoustic behaviour of parts to a high degree of reliability if the models can capture the important physics at play. The key benefit of using CAE techniques for analysis is that no prototypes are needed, cutting vast amounts of time and costs. Furthermore, problem areas can be detected early in the design phase which, as previously mentioned, is paramount to rectifying NVH issues. As a result, CAE techniques are widely used to solve NVH problems.
Despite substantial yearly improvements in computational power, very large models which contain many elements still require significant resources when solving complex engineering problems. Simplifications must be made to these models which can reduce computation time for more information see section 2.3.1 and 2.3.3. Sometimes, the simplifications are made at the expense of their accuracy. Moreover, no model can fully capture reality meaning that some effects may be unaccounted for. Consequently, results obtained via CAE should be validated with experimental tests.

1 Introduction
1.1 Background
1.2 Purpose and Aims
2 Frame of Reference
2.1 Machine Design
2.1.1 Hypoid Gears
2.1.2 Hypoid Gear Contact Ratio
2.1.3 GearMesh Stiffness and GearMesh Frequency
2.1.4 Transmission Error
2.1.5 Bearings
2.2 Noise, Vibration and Harshness
2.2.1 Vibrations – The Fundamentals
2.2.2 Acoustics – The Fundamentals
2.2.3 Airborne and Structure-Borne Noise
2.2.4 Sound Field Definitions
2.2.5 Transfer Path Analysis
2.2.6 Computer Aided Engineering Techniques
2.3 Finite Element Analysis
2.3.1 ComponentMode Synthesis
2.3.2 Modal Superposition
2.3.3 Craig Bampton Analysis
2.4 Acoustic Simulations
3 Methodology
3.1 Overview
3.2 Pre-processing
3.2.1 Meshing
3.2.2 Generating Flexible Bodies
3.3 Multi-body Dynamics
3.3.1 The Program
3.3.2 Flexible Bodies
3.3.4 ContactMechanics
3.3.5 Solver
3.4 Acoustic Simulations
3.4.1 Acoustic Pre-Processing
3.4.2 Creating the Analysis
3.4.3 Analysis Setup
3.4.4 Far Field Setup
3.4.5 Environment Setup
3.4.6 Analysis Parameters
3.4.7 Mesh
4 Results and Analysis
4.1 Model Validation
4.1.1 Transmission Error Validation
4.1.2 Housing Vibration Validation
4.2 Acoustic Simulations
4.2.1 Analytical Validation
4.3 FutureWork
5 Conclusions

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