Encyclopedic Machine Design and Kinematics Study Guide

Principles of Kinematics and Mechanism Theory

• Grashof's Law: For a planar four-bar linkage, if there is to be a continuous relative rotation between two members, the sum of the shortest and longest link lengths ($S+L$) must be less than or equal to the sum of the remaining two link lengths ($P+Q$). If this condition is not met, the linkage will only exhibit a rocking or toggling motion.

• Linkage Inversions:

  • Single Slider Crank Chain Inversions: Includes mechanisms such as the Whitworth quick return mechanism, the reciprocating engine, and the hand pump. If the slider is fixed in a single slider four-bar linkage, it forms the hand pump mechanism.

  • Double Slider Crank Chain Inversions: Includes tools like the Elliptical trammels.

  • Beam Engine: An inversion where a turning pair is fixed.

  • Coupling Rod of a Locomotive: A four-bar chain mechanism consisting of all turning pairs.

• Intermittent Motion Mechanisms: The Geneva Mechanism is used to produce intermittent rotary motion from continuous rotary motion.

• Lawn Sprinkler Mechanism: A four-bar mechanism that converts rotary motion into oscillatory motion.

• Kinematic Pairs:

  • Lower Pair: Surface contact between two elements, such as a shaft revolving in a bearing, an automobile steering gear, or straight-line motion mechanisms.

  • Higher Pair: Point or line contact.

  • Sliding Pair: Rectangular bar in a rectangular hole is a completely constrained sliding pair.

  • Rolling Pair: In a belt drive, the pulley acts as a rolling pair.

• Mobility and Constraints:

  • Mobility ($M$): For a statically indeterminate structure, the mobility is less than or equal to $-1$.

  • Incompletely Constrained Motion: Occurs if links are connected such that motion between them can take place in more than one direction.

• Kinematic Joints: A quaternary joint is kinematically equivalent to three binary joints.

Advanced Kinematics: Acceleration and Followers

• Coriolis Component of Acceleration: In a slotted lever-crank mechanism, the direction of the Coriolis component is along a line rotated $90$ degrees from the sliding velocity vector in the same direction as the angular velocity of the slotted lever.

• Piston Motion: The motion of a piston inside a cylinder of a uniformly rotating cylinder mechanism involves both Coriolis and radial acceleration.

• Coupler Link Motion: In a 4-bar mechanism, the center of gravity of the coupler link experiences both linear and angular accelerations.

• Slider Acceleration: In a slider-crank mechanism, maximum acceleration of the slider is typically obtained when the crank is at the inner dead center position.

• Cam and Follower Systems:

  • Follower Displacement: In a flat-faced follower operating against an eccentric circle cam, increasing the eccentricity increases the magnitude of follower motion.

  • Jerk Reduction: Cycloidal motion produces the least jerk in the system compared to simple harmonic or constant velocity motions.

  • Radial Cam Offset: Providing an offset in a radial cam-translating-follower mechanism serves to decrease the pressure angle during the ascent of the follower.

Mechanics of Materials and Stress Analysis

• Stress-Strain Relationships:

  • Hooke's Law: Within the proportional limit, stress is directly proportional to strain.

  • Saint Venant's Theory: Also known as the maximum-strain theory, it applies specifically within the elastic range of stresses.

  • Maximum Distortion Energy Theory: Postulated by Mises-Henky (also known as the von Mises yield criterion).

• Deformation Characteristics:

  • Low-Carbon Steel: When loaded past its elastic range, the specimen undergoes permanent deformation after the load is removed.

  • Ductility: Mild steel is capable of sustaining large plastic deformation before fracture, unlike brittle materials like glass, concrete, or cast iron.

  • Resilience: The capacity of a material to absorb energy when deformed elastically and recover it upon unloading. Proof Resilience is the maximum strain energy that can be stored.

• Specific Stress States:

  • Pure Shear Stress: For an element in pure shear, the normal stresses acting on planes oriented at $45^{\circ}$ to the original shear planes are equal tensile and compressive stresses.

  • Plane Stress (Biaxial Stress): A condition where one surface is comparatively free of stress.

  • Stress Concentration: Not as harmful in ductile materials as in brittle materials because local yielding helps redistribute the high local stresses. In cyclic loading, however, it remains serious for both.

  • Autofrettage: A method of prestressing thick cylinders to increase their pressure-carrying capacity or fatigue life.

Structural Analysis of Beams and Columns

• Beam Design:

  • Dangerous Section: The section of a beam most critical for design, located where the bending moment reaches its maximum value.

  • Shear Stress Distributions:

  • In a rectangular cross-section, the maximum shear stress is $50\%$ higher ($1.5 \times$) than the average shear stress.

  • In a circular cross-section, the maximum shear stress is $33\%$ higher ($1.33 \times$) than the average shear stress.

  • Cantilever Geometry: For a uniformly distributed load, the shape of the deflected cantilever is parabolic.

• Column Stability:

  • Buckling Load: Depends on the slenderness ratio, the cross-sectional area, and the modulus of elasticity of the material.

  • Buckling Shape: Influenced significantly by End Conditions (e.g., fixed-fixed, hinged-hinged).

Material Science and Engineering

• Hardness Testing:

  • Rockwell Test: Uses the principle of measurement of the depth of indentation.

  • Mohs Hardness Scale: A value of $5$ on the scale corresponds to Apatite.

• Alloys and Elements:

  • Invar: Contains Nickel as its major alloying element ($\approx 36\%$).

  • 18/8 Stainless Steel: Contains $18\%$ Chromium and $8\%$ Nickel.

  • Vanadium: Added to steel to improve hardenability.

  • Austenitic Stabilizer: Nickel.

  • Ferritic Stabilizer: Chromium.

• Cast Irons:

  • Compacted Graphitic Iron: Characterized by worm-shaped graphite particles.

  • Malleable Cast Iron: Contains carbon in rosette form.

• Industrial Materials:

  • Plastics: Bad conductors of both heat and electricity.

  • Inconel X: An alloy with low creep rate, high oxidation resistance, and high strength at elevated temperatures.

  • Kevlar Epoxy: Widely used in Aerospace applications.

  • Whisker: A single crystal grown by vapor deposition.

• Material Properties Definitions:

  • Homogeneity: Measured mechanical properties are the same in a particular direction at each point.

  • Isotropy: Properties are same in all directions at a single point.

Machine Component Design

• Shafting:

  • Axle: A part subjected to transverse loads and bending moments.

  • Permissible Deflection: For machinery shafts, approximately $0.01\,in./ft$.

  • Maximum Shear: In shafts, shear due to bending is maximum at the neutral plane where the normal stress is zero.

  • Hollow vs. Solid Shafts: A hollow shaft has more torsional stiffness than a solid shaft of the same material, weight, and length.

  • Critical Speeds: The number of critical speeds depends on the size and load (theoretically infinite for a continuous system).

• Keys and Splines:

  • Sunk Key: The keyway is cut in both the shaft and the hub. Holding power is due to resistance in shear and crushing.

  • Woodruff Key: A segment of a circular disc; capable of tilting in its recess.

  • Feather Key: Permitting axial movement of the hub; usually tight in the shaft and loose in the hub.

  • Saddle Key: Relies on frictional resistance only; not recommended for heavy torque.

  • Tangent Key: Preferred for large impact-type torque in both directions of rotation.

  • Spline Force Calculation: For a spline rotating at $120\,RPM$, transmitting $20\,kW$ at a pressure of $6.5\,MPa$, with friction coefficient $0.06$, the shifting force is $20.12\,kN$.

• Couplings:

  • Universal Coupling (Hooke’s Joint): Used for intersecting shafts where the angle varies during operation.

  • Oldham Coupling: Used for shafts with parallel misalignment.

  • Flexible Coupling (Pin-bush): Absorbs shock and accommodates both angular and axial displacement through rubber bushes.

  • Muff Coupling: Selected when shafts are collinear, permanently aligned, and manufacture ease is prioritized.

Fasteners, Power Screws, and Riveted Joints

• Power Screws:

  • Trapezoidal Threads: Used in lead screws of lathes; adaptable to split nuts.

  • Buttress Threads: Strongest thread form, used for transmitting force in one direction.

  • Screw Jack Design: End conditions typically assumed as one end fixed and the other free for buckling considerations.

  • Lead and Pitch: Pitch is the axial distance between adjacent threads.

  • Mean Diameter: For a double square thread with root diameter $0.55\,in.$ and pitch $0.2$, the mean diameter is calculated relative to these values ($0.650\,in$).

• Fasteners:

  • Washers: Used to distribute the load over a larger bearing surface.

  • Eye Bolts: Used for lifting and transportation of machinery.

  • Thread Stripping: Most likely when the engaged thread length is insufficient.

  • Fatigue Failure: Indicated by beach marks on fracture surfaces; caused by repeated cyclic stresses below ultimate strength.

• Riveted Joints:

  • A permanent mechanical joint. In double-riveted lap joints under eccentric loading, outer rivets are more stressed because they resist both direct shear and moment effects.

Clutches and Brakes

• Clutch Design Assumptions:

  • Uniform Wear: Applicable after several thousand operating cycles as pressure redistributes.

  • Uniform Pressure: Used for new clutches.

• Clutch Types:

  • Positive Engagement: Jaw clutch (uses interlocking members).

  • Centrifugal Clutch: Engagement speed increases if spring stiffness increases.

  • Cone Clutch: Typical angle between lining and axis is $12.5^{\circ}$.

• Brakes:

  • Caliper Disc Brake: Uses a rotor squeezed from both sides by friction linings.

  • Leading Shoe: Develops self-energizing action.

  • Asbestos: Historically used for linings due to excellent resistance to elevated temperatures.

Dynamics and Vibrations

• Energy in Vibrations: At the instant a body passes through the equilibrium position, kinetic energy attains its maximum value, while potential energy is zero.

• Types of Vibration:

  • Damped Vibration: Gradual reduction in amplitude due to frictional or resistive forces.

  • Resonant Vibration: Occurs when driving frequency matches natural frequency.

• Frequency Determination:

  • Rayleigh’s Method: Based on the Conservation of Energy.

  • Dunkerley Method: Used for finding the natural frequency of transverse vibrations in shafts with several loads.

• Whirling of Shafts: Caused by factors such as the non-homogeneity of the shaft material.

Questions & Discussion

• Question regarding design safety for fluctuating loads: Which relation yields the most conservative estimate?

  • Response: The Soderberg relation is considered the most conservative.

• Question regarding the center of gravity of a coupler link linkage: What acceleration does it experience?

  • Response: It experiences both linear and angular accelerations.

• Question regarding why stress concentration is less harmful in ductile materials:

  • Response: Local yielding may redistribute the stress concentration.

• Question regarding the effect of increasing bolt pitch-circle diameter in a flange coupling:

  • Response: It will decrease the shear stress in the bolts for the same transmitted torque.

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