1. BK50A2200 Design Methodologies and Applications of Machine Element Design
Lecture 2
Introduction to the textbook:
“Norton: Machine Design”
D.Sc Harri Eskelinen

2. Goals of this lecture
Support the contents of the previous lectures dealing with machine design approaches, reliability design and wear phenomena
To get familiar with the main designing and dimensioning criteria of the most important machine elements (according to Norton)
Main consecutive designing steps and aspects
Fundamental dimensioning equations

3. Briefly about the book
The textbook presents an integrated approach to the machine elements by combining the usual set of machine element topics with a series of case studies that illustrate the relationships between force, stress and failure analysis in real-world design.
The book emphasizes the design and synthesis aspects of machine elements but it forms also a good balance between synthesis and analysis.

4. The first part of the book presents the fundamentals of design, materials, stress, strain, deflection, failure and fracture theories.
The second part treats of the aspects of machine element design, such as designing springs, shafts, gears, bearings etc.

5. PART 1. FUNDAMENTALS

6. Chapter 1. Introduction to Design
This chapter partially supports the ideas of systematic design approach (see the yellow items below) : according to Norton the design process consists of the following ten stages:
1 Identification of need
2 Background research
3 Goal statement
4 Task specifications
5 Synthesis
6 Analysis
7 Selection
8 Detailed design
9 Prototyping and testing
10 Production

7. Chapter 2. Materials and Processes
The contents of this chapter will be discussed in details during the university course “Introduction to Material Technology”
Basic definitions of the most common material properties are presented briefly:
Modulus of elasticity
Yield strength
Ultimate tensile strength
Modulus of rigidity
Fatigue strength
Toughness
Hardness
Most typical hardening and surface coating processes are presented briefly
Basic information about some material groups is given:
Steels
Cast iron
Aluminium
Titanium
Copper Alloys
Polymers
Ceramics
Composites

8. Chapter 3. Load Determination
The content of this chapter produce the fundamentals for the further stress, strain and deflection analysis presented in chapter 4.
Main topics are:
Different loading cases
Free-body diagrams
Static loading
Dynamic loading
Vibration loading
Impact loading
Beam loading

9. Classification of loading cases
Constant
Loads
Time-Varying Loads
Stationary elements
Class 1
Class 2
Moving elements
Class 3
Class 4

10. Identification of different loading cases
Identification of different loading cases in necessary to make it possible to use proper material properties as criteria during the material selection process
Tension or compression  tensile or compressive stress
Bending  bending stress
Shear  shear stress
Torsion  torsion stress (shear strass)
Reverced loading  endurance limit (for reverced stress)
Pulsating loading  endurance limit (for pulsating stress)
Pulsating loading
Reverced loading

11. Free-Body Diagrams
Case example: Wire connector crimping tool

12. Chapter 4. Stress, Strain and Deflection
This chapter includes the basic theories of “strength of materials”, the following topics are discussed (the most important items are high-lighted with yellow):
Principal stresses
Axial Tension
Bending stresses of beams
Deflection of beams
Torsion
Combined stresses
Stress concentration
Axial compression
Stresses in cylinders

13. Critical cross-sections due to stress concentrations:
Combined loading:
Axial force
Radial force
Tangential force
Torque
 combined stresses T1 Fa Fr Ft
A B C D
Critical cross-sections due to
stress concentrations:
End of the keyseat at cross-section A
Cross-sections B, C and D of a smaller diameter

14. Chapter 5. Static Failure Theories
Chapter 5 is divided in three main sections:
Failure of ductile materials under static loading
The main failure mode is permanent yield under static loading  yield strength of the material is exceeded
Critical material property is yield strength
Failure of brittle materials under static loading
Instead on yielding brittle materials fracture
Fully hardened steels, cast iron, materials in low temperatures can behave like brittle materials
Critical material property is toughness at certain temperature
Fracture mechanics
This theory presumes the presence of a crack, which starts to grow under the specific loading and finally leas to either ductile or brittle failure

15. Toughness
Temperature
Brittle
behaviour
Ductile
Transition
zone

16. Modes of crack displacement
Loading
Modes of crack displacement
Mode I = load tends to pull the crack open in tension
Mode II = shear crack in-plane
Mode III = shear the crack out-of-plane

17. Chapter 6. Fatigue Failure Theories
The use of typical Wöhler’s strength-life- diagrams is presented
The main principles of the use of Paris-Equation are presented
The use of Goodman’s diagram for fatigue life analysis is presented
Schematic fatigue-fracture surfaces of a shaft cross-sections are presented to support further failure mode analysis

18. Wöhler’s diagram

19. Schematic fatigue-fracture surfaces
Rotating bending
Low nominal stress
Mild stress concentration

20. Paris-equation
The crack growth “speed” is presented as a function of loading cycles:
Where
a = crack width
N = number of cycles
A, n = material coefficients
ΔK = stress intensity factor range

21. D A M G I N S P E
Region I
Crack initiation stage
Region II
Crack propagation
Region III
Unstable fracture No
crack
growth
Stress intensity

22. Chapter 7. Surface Failure
This chapter contains the following topics
Mathematical theory of surface contacts
Characteristics to describe the value of surface roughness
Spherical contact
Cylindrical contact
Dynamic contact stresses
Designing rules to avoid surface failure
Wear phenomena (discussed earlier during this course)
Abrasive wear
Adhesive wear
Fatigue wear
Tribochemical wear or corrosive wear

24. Mathematical definition of Ra:n

25. Designing rules to avoid surface failure
1 Remember the rules which were presented during the special lesson dealing with wear phenomena
2 Choose proper materials
Hardness
Surface roughness
Use of coatings
3 Choose proper lubricants
Take care of EHD- or HD- lubrication (avoid boundary lubrication)
Use EP-lubricants if needed(extreme pressure)
4 Take care of cleanliness
Use proper sealing constructions
Select proper material pairs (e.g. hardness pairs)
5 Avoid and minimize stress concentrations
Select proper stiffness and/or geometry
6 Avoid fretting problems by taking care of possible vibration
phenomena (near joints or fits)

26. Case example: How to minimize the stress concentrations in a cylindrical roller bearing by using a proper geometry of the roller elements.

27. Case example: Fretting wear on a shaft beneath a press-fit hub.

28. PART 2. Machine Design

29. Chapter 8. Design Case Studies
This brief chapter is written just to form “a bridge” between the theories of material science, strength of materials, failure theories (presented in part 1) and practical dimensioning and analysing instructions of some typical machine elements (to be presented in part 2).
The iterative nature of designing process is emphasized.

30. Chapter 9. Shafts, Keys and Couplings
Designing of shafts step-by-step (iterative analysis):
1 Determine the affecting loading cases
E.g. gear forces, torque, forces due to belt drives etc.
2 Collect contacting dimensions from the construction and select possble shaft materials
E.g. shaft-hub joints, diameters of bearing seats, width of gears etc.
3 Produce the free-body diagram and calculate the teaction forces
4 Draw loading (force), shear and moment diagrams
5 Find the critical cross-sections of the shaft
E.g. key seats, changes of the diameters, grooves, threads etc.
6 Calculate the affecting stresses and deflections
E.g. tensile stress, bending stress, shear stress,
7 Calculate the critical rotating speed due to vibration and resonance
8 Calculate safety factors
Constant and time-varying loading

31. The dimensioning procedure of shafts is based on ASME-method:
Soderberg’s hypothesis
(in Finland several hypothesis are used and usually compared in university text books)
Goodman’s line
(in Finland the use of Smith’s diagram is more common)

32. Estimation of shaft diameter:
Some rules of thumbs
Estimation of shaft diameter:
d = required shaft diameter
Tmax = affecting torque
Tsall = allowed shear stress of the material

33. Critical angular velocity:
Bending vibration
ncr = critical angular velocity
δmax = maximum deflection of the shaft

34. Critical angular velocity:
Torsinal vibration
fcr = critical angular velocity
kv = torsional stiffness coefficient
J1 = moment of inertia (input)
J2 = moment of inertia (output)
d = diameter of the shaft
G = modulus of rigidity
L = length of the shaft
m = weight of (each) component
r = rotating radius of (each) component

35. Loading cases of shaft-hub-joints
Torque Fr Ft Fa
Moment
Torque

36. If the joint is able to withstand also axial loading, its torque transmission capacity can be estimated according to the following equation:
Where
Ttheor = theoretical maximum allowed torque which joint could transmit without any axial loading
T = torque, which can be transmitted even though Fa is affecting simultaneously (usually the value which is calculated)
Fatheor = theroretical maximum allowed axial force, which joint could transmit without any torque loading
Fa = axial load, which is decreasing the torque transmission capacity (“the disturbing factor”)

37. Main designing steps are as follows:
Dimensioning of parallel keys is based on SFS-standards (we skip the presentation presented by Norton):
Main designing steps are as follows:
Check the maximum surface stress of the hub
Check the maximum surface stress of the key
Check the maximum shear stress of the key
Ensure that the required torque transmission capapacity is achieved

38. Chapter 10. Bearings and Lubrication
This chapter includes the following important topics:
Lubricants and types of lubrication
Briefly about sliding bearings and their material combinations
Rolling-element bearings
Failure of rolling-element bearings
Selection of rolling bearings

39. Types of lubrication Hydrodynamic lubrication (HD or HL)
HD refers to the supply of oil to the sliding interface to allow the relative velocity of the mating surfaces to pump oil within the gap and separate the surfaces on the dynamic film of liquid.
Elastohydrodynamic lubrication (EHD or EHL)
When the contacting surfaces are nonconforming, as with gears or cam mechanisms, it is difficult to form a full film of oil.The affecting load creates a contact area from the elastic deflections of the surfaces. This area can be large and flat enough to provide full hydrodynamic film if the relative sliding velocity is high enough. This is possible, because the high pressure between the surfaces increase the viscosity of the fluid.
Boundary Lubrication (BL)
Either the insufficient geometry, too high load level, low velocity or insufficient oil quantity may prevent hydrodynamic lubrication and cause metallic contacts between the surfaces (e.g. at the beginning or end of the rolling)

41. Selection of rolling bearings
Allowed dynamic load
Allowed maximum angular velocity
Facilities of the selected bearing type
Ability to withstand axial loads
Ability to withstand axial bending moments or angular assembly errors
Allowed friction
Required stiffness and accuracy
Allowed static load
Required reliability

42. Some examples
A detailed guide to select an appropriate bearing type will be given out as a hand-out…
Spherical roller bearings
Especially for cases in which bending
moment could cause additional loading
on the bearing or where possible
assembly errors may cause some
misaligning of the shaft
Tapered roller bearings
Especially for cases in
which good axial load
standing capacity is
required

43. Basic equations of bearing design
P = combined dynamic equivalent load of the bearing
Fr = applied radial load
Fa = applied axial load
X = a radial factor
Y = an axial factor
p = exponent, the value depends on the bearing type
Ball bearings p=3
Roller bearings p= 10/3
L10h = Nominal life-time (e.g h)
C = Dynamic load rating

44. Chapter 11. Spur Gears
Spur gears are used to present principles of gear dimensioning in general
Specialized terminology is presented in details
Gear tooth theory is discussed
Equations for dimensioning gears are presented
Also gear manufacturing processes are presented briefly
Different casting processes
Machining
Powder metallurgical processes (sintering)
Extruding and cold drawing processes
Different finishing processes
The presentation is based on standards published by the American Gear Manufacturers Association (AGMA)

45. At first the applying forces on gear teeth must be established! T1 T2Fa Fr Ft
At first the applying
forces on gear teeth
must be established!

46. Main dimensioning criteria: Main characteristics:
Equations are based on experimental factors, parameters and characteristics describing e.g.:
Geometric accuracy of gears
Lubrication conditions
Material properties of gears
Stress concentration phenomena
Surface properties of gears
Loading conditions of gears
Main dimensioning criteria:
Bending stresses of the teeth
Surface stresses of the teeth
Main characteristics:
Gear ratio i=Z1/Z2
Module m=d1/Z1
Contact ratio
Functions:
To transimit
Torque
Angular velocity

47. Basic equations for spur gear design
Power transmission capacity according to allowed bending stresses of the teeth
Power transmission capacity according to allowed surface stresses of the teeth

49. Chapter 12. Helical,Bevel and Worm Gears
Helical gears
Teeth are angled with respect to the axis of rotation
Contact surface between teeth is increased
Axial load component is caused
Bevel gears
Shafts are located usually at 90 degrees angle
Worm gears
Shafts are located at crossing position and high gear ratios are achieved

50. The dimensioning of helical gears is based on the equations of spur gears, so-called virtual number of teeth should be established and then the theory of spur gears is applied
For bevel gears either the theories of dimensioning of spur or helical gears can be applied:
Straight bevel gears  spur gears
Spiral bevel gears  helical gears

51. Two main additional aspects of tooth geometry should be considered
Use of single- or double-enveloping tooth forms
Number of teeth in contact with worm and worm wheel
Worm gears are discussed very briefly, however the main dimensioning aspects consist of four main steps:
Durability against pitting (surface fatigue)
Durability against wear (abrasive wear due to different material properties of the worm and the worm wheel)
Durability against overheat
Allowable bending deflection of the worm (shaft)

52. Example of dimensioning equations of worm gear design
Safety factor SW against wear
Wlim = kulumislujuus
WP = kulumisparikerroin (mm. kovuus)
WR = pinnankarheuskerroin
Wv = liukunopeuskerroin
ZE = materiaalikerroin (E1 ja E2)
Z = kosketuskerroin (kierremuoto)
Materials wear strength
Coefficient depending on materials hardness
Coefficient depending on surface roughnesses
Coefficient depending on the sliding velocity
Coefficient depending on modulus of elasticity
Coefficient depending on the tooth geometry

53. Chapter 13. Spring Design Main contents of this chapter:
Definition of the spring rate is presented
Various spring configurations and materials are presented briefly
Dimensioning and designing criteria is presented for the following spring configurations:
Helical compression springs
Helical extension springs
Helical torsion springs
Belleville spring washers

54. Example: Designing steps of helical compression springs
1 Decide spring configuration
1.1 Length
1.2 End details and number of active coils
1.3 Tentative material selection
2 Establish functional properties
2.1 Sprind index C = D/d (coil diameter/wire diameter)
2.2 Spring deflection y
2.3 Spring rate k = F/y
3 Loading cases and stress analysis
3.1 Shear stress
3.2 Torsional shear stress
3.3 Stress concentrations
3.4 Residual stresses due to manufacturing stages (e.g coiling into the form of helix causes tensile stress)
3.5 Buckling
3.6 Vibration and resonance phenomena
3.7 Fatigue analysis
3.8 Final material selection

55. Chapter 14. Screws and Fasteners
This chapter deals with the following topics:
Standardized thread dimensions
Power screws
Screw fasteners
Stresses in threads
Tensile, torsion and shear
Joint stiffness
Preloaded fasteners

56. A bolted assembly

57. A bolted assembly
The clamped construction may include two or more pieces and they may be of different material.
Also a long bolt usually has threads over only a portion of its length having at least two different cross-sectional areas.
These different stiffness-sections act as springs in series and their function can be described according to the following equation:
When we know the dimensions and geometry of the bolt and pieces to be joined it is possible to calculate the partial spring rates and combine them according to this simple equation.
Analogical with spring analysis we can now write the relationship between the total spring rate, deflection and applying force.

58. Other dimensioning aspects of bolted joints
1 Washers
Decrease the surface stresses at the joint
Ensure the tightness of the joint
Prevents the possible bending moment of the bolt due to
slant surface
2 Bolt’s straight length without threads
The stress concentration at the end of the straight length before the first thread could be critical
The straight length is planed to function “as a spring”
3 Parts to be clamped together
In many cases the friction coefficient between
the parts to be clamped together is in key-role
There should be enough distance between the mounting holes of the screws and edges of the parts to be clamped to avoid fractures of the base material

59. Possible failure modes of bolted joints
1 The bolt breaks under the static tensile loading
Tensile stress exceeds the ultimate tensile strength of the bolt
The first thread of the bolt is cut off
The first thread of the nut is cut off
2 The fatigue strength of the bolt is exceeded
Typically the fatigue limit is only 10% of screw material’s yield strength
3 The failures of the parts to be clamped
The shear stress is too large near the edges

60. Means to improve the fatigue strength of the bolted joint:
Select a taller nut (to increase the number of load standing threads)
Select more suitable material pair for the nut and the bolt combination (lower coefficient of elasticity for the nut compared to that of the screw’s e.g. aluminium or cast iron for nuts with and steel for bolts)
Use sufficient pre-loading of the bolts (equalize the stresses applying at each thread of the bolt and nut)
Improve the surface quality of the bolt,
Select the bolt (and thread) geometry with the smallest stress concentration coefficient
Select the advantageous manufacturing technology of the bolts (usually cold forming is recommended)

61. Chapter 15. Clutches and Brakes
This chapter forms only a brief overall picture of various types of clutches and brakes to give the reader a sight of possible constructions.
Classification according to the actuation
(impulse to start the function)
Electrical (press the button)
Mechanical (push the bedal)
Pneumatic or hydraulic
Automatic
Classification according to the function
(what phenomenon the contact is based on)
Friction between (two) surfaces
Locked geometry (e.g. toothed components)
Magnetic
Fluid coupling

62. SUMMARY : STRESS AND DEFLECTION ANALYSIS
Failure theories
• Static failure theories
- ductile materials
- brittle materials
• Fracture mechanics theory
• Fatigue failure theories
- Wöhler
- Paris –Eguation
- Soderberg
- Goodman
• Surface failure theories
- wear
- surface contact theories
Load determination and
different load cases
• utilization of free-body diagrams
• static or dynamic loading
• vibration loading
- bending vibration
- torsional vibration
• impact loading
• tension, compression, bending shear or torsion
• reversed or pulsating loading
Pulsating loading
Reverced loading
Stress, strain and deflection
• allowed stress and deflection
• due to affecting load cased
• combined stresses
• stress concentrations
Material properties
• metallic materials
- steels
- aluminium
- cast iron
• polymers
• ceramics
• composites
• nanomaterials

63. Exercise 2 Exercise 2A Exercise 2B
Present typical applications of power transmission or guiding shafts under different loading cases. Explain the reasons for the affecting combined loading. Use illustrative figures. Present at least the following loading cases:
Tension or compression + bending
Tension or compression + bending + torsion
Bending + torsion
Shear + any other loading case
Reversed loading
Pulsating loading
Exercise 2B
Compare different fatigue failure theories and approaches by presenting typical applications of different types of mechanical components or constructions. Compare at least the following approaches:
Wöhler’s strength-life-theory
Goodman’s theory
Paris-equation
Utilization of schematic fatigue-fracture surfaces
Fracture mechanics