Mechanical Engineering Design and Kinematics Comprehensive Study Guide

Advanced Mechanical Design and Kinematic Analysis Principles

• Grashof's Law of Linkages: For a planar four-bar linkage, the criteria for having a continuous relative rotation between two members is defined by the sum of the shortest and longest link lengths. This sum cannot be greater than the sum of the remaining two link lengths.

• Kinematic Pairings and Constraints:

  • Lower Pair: Example includes a shaft revolving in a bearing, where surface contact occurs.

  • Higher Pair: Contact typically occurs at a point or along a line.

  • Completely Constrained Motion: Motion occurs in only one definite direction (e.g., a rectangular bar in a rectangular hole).

  • Incompletely Constrained Motion: Motion between links can take place in more than one direction.

  • Kinematic Chain Joint Equivalency: A quaternary joint is kinematically equivalent to $3$ binary joints.

• Mechanism Inversions:

  • Single Slider Crank Chain Inversions: Includes the Whitworth quick return mechanism, reciprocating engines, and oscillating cylinder engines.

  • Hand Pump: An inversion formed when the slider in a single slider four-bar linkage is fixed.

  • Double Slider-Crank Chain Inversions: Includes the Elliptic trammels, Scotch Yoke mechanism, and Oldham's coupling.

  • Geneva Mechanism: A specialized mechanism that produces intermittent rotary motion from continuous rotary motion.

  • Lawn Sprinklers: Typically utilize a four-bar mechanism to convert rotary motion into oscillatory motion.

Material Science and Mechanical Properties

• Elastic and Plastic Behavior:

  • Hooke's Law: States that within the proportional limit of a material, stress is directly proportional to strain ($\sigma ∝ ‵$).

  • Deformation of Low-Carbon Steel: When loaded past its elastic range, the specimen undergoes permanent (plastic) deformation upon removal of the load.

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

• Material Isotropic Characteristics:

  • Isotropy: A property where mechanical properties are the same in a particular direction at each point.

  • Homogeneity: Properties are the same throughout the entire volume of the material.

  • Invar: A nickel-iron alloy known for its uniquely low coefficient of thermal expansion; Nickel is the major alloying element ($36\%$).

  • Austenitic Stabilizer: Nickel is commonly used to stabilize the austenite phase in steel.

  • Ferritic Stabilizer: Chromium is used to stabilize the ferrite phase.

• Cast Iron Varieties:

  • Compacted Graphitic Iron: Features unique worm-shaped (vermicular) graphite particles.

  • Malleable Cast Iron: Contains carbon in the form of rosettes or temper carbon.

  • Nodular (Ductile) Cast Iron: Features graphite in spherical nodules.

• Hardness and Energy Absorption:

  • Mohs Hardness Scale: Apatite is designated as number $5$ on the scale.

  • Rockwell Hardness Test: Uses the principle of measuring the depth of indentation to determine hardness values.

  • Resilience: The capacity of a material to absorb energy when deformed elastically and recover it upon unloading.

  • Proof Resilience: The maximum strain energy that can be stored in a body up to the elastic limit.

Stress Analysis and Failure Theories

• Theories of Failure:

  • Maximum Distortion Energy Theory: Postulated by Mises-Henky; used primarily for ductile materials.

  • Maximum Principal Stress Theory: Also known as Rankine's Theory.

  • Maximum Shear Stress Theory: Associated with Tresca.

  • Maximum Strain Theory: Applicable only in the elastic range; also known as Saint Venant's Theory.

• Advanced Stress States:

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

  • Biaxial or Plane Stress: A condition where one surface of an element is comparatively free of stress.

  • Stress Concentration: In ductile materials, local yielding may redistribute stress, making it less harmful than in brittle materials. Stress concentration refers to the increase in stress at abrupt changes in geometry.

• Fatigue and Variable Loading:

  • Fatigue Failure: Results from repeated application of cyclic stresses below the ultimate strength. Failure surfaces often exhibit "beach marks."

  • Fluctuating Stress: A loading cycle where both maximum and minimum stresses remain tensile but vary in magnitude.

  • Repeated Stress: Varies from zero to a maximum value.

  • Completely Reversed Stress: Alternates between equal positive and negative values.

  • Soderberg Relation: Provides the most conservative design estimates for members subject to fluctuating loads when using the same factor of safety.

Machine Elements: Shafting, Keys, and Couplings

• Shafting Principles:

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

  • Shaft Deflection: Permissible deflection for machinery shafts is approximately $0.01\,\text{in./ft}$.

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

  • Critical Speed: The number of critical speeds for any shaft depends on the number of loads (masses) attached, potentially reaching infinite theoretical values for continuous mass.

• Keys and Keyways:

  • Sunk Key: The keyway is cut into both the shaft and the hub. Its holding power is derived from resistance to shear and crushing.

  • Feather Key: Usually tight in the shaft and loose in the hub; permits axial movement of the hub while transmitting power.

  • Woodruff Key: A segment of a circular disc of uniform thickness that fits into a semi-circular recess; capable of tilting in its recess.

  • Saddle Key: Relies primarily on friction rather than shear to transmit torque; not recommended for heavy torque as no keyway is cut into the shaft.

  • Kennedy Key: Also known as a tangential key.

• Couplings:

  • Flexible Coupling: Used to absorb shock and vibration and accommodate small misalignments.

  • Universal (Hooke's) Coupling: Used when two intersecting shafts must transmit motion while the angle between them varies.

  • Oldham Coupling: Used for connecting two shafts with lateral (parallel) misalignment.

  • Pin-bush Coupling: Accommodates both angular and axial displacement via the deformation of rubber bushes.

Fasteners, Power Screws, and Joints

• Threaded Fasteners:

  • Pitch: The axial distance between adjacent threads.

  • American National Thread: The tool bit for cutting this thread should be ground to a $60^{\circ}$ angle.

  • Proof Strength: In bolt design, the quotient of the proof load and the tensile-stress area.

  • Elastic Nut: A locking device utilizing a hard fiber or nylon insert (cotter) that becomes threaded as the nut is screwed on, creating a tight grip.

• Power Screws:

  • Square Threads: Used for transmitting force and power with high efficiency.

  • Trapezoidal (Acme) Threads: Often used for the lead screw of a lathe; adaptable to split-type nuts.

  • Buttress Threads: Stronger than other threads; specifically used for transmitting force in one direction only.

  • Multiple Threads: Used when high efficiency and rapid axial travel are required.

• Riveted Joints:

  • Permanent Joint: Riveting is a permanent mechanical joining method.

  • Eccentric Loading: In double-riveted lap joints under eccentric load, outer rivets are more highly stressed because they resist both direct shear and the moment.

  • Cold-Riveted Joints: Allowable strength is most often governed by the shear stress in the rivet.

Dynamics: Cams, Vibrations, and Governors

• Cams and Followers:

  • Cycloidal Motion: Follower motion type that produces the least amount of jerk in the system.

  • Offset Cam: An offset in a radial cam-translating-follower mechanism is provided to decrease the pressure angle during the ascent of the follower.

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

• Mechanical Vibrations:

  • Equilibrium Position: When a vibrating body passes through this point, its kinetic energy attains its maximum value.

  • Damped Vibration: Associated with a gradual reduction in amplitude due to frictional or resistive forces.

  • Natural Frequency: Can be computed using Rayleigh's method (based on conservation of energy) or Dunkerley's method.

• Governors:

  • Isochronous Governor: Occurs when the controlling force line, when produced, intersects the y-axis at the origin. This results in the governor maintaining a constant speed for all radii of rotation.

Mechanical Components and Manufacturing

• Clutches and Brakes:

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

  • Uniform Wear Assumption: Used for torque calculations of plate clutches that have seen extended service ($P \times r = \text{constant}$).

  • Centrifugal Clutch: Increasing the spring stiffness increases the engagement speed of the clutch.

  • Leading Shoe: Develops a self-energizing action because the friction force draws the shoe tighter into the drum.

• Pressure Vessels and Columns:

  • Spherical Pressure Vessels: Preferred over cylindrical ones because they have a uniform lower circumferential stress for a given thickness.

  • Thick Cylinders: Under internal pressure, hoop stress ($\sigma_{\theta}$) is maximum at the inner radius.

  • Column Buckling: The buckling load depends on the slenderness ratio ($\frac{L}{k}$), area of cross-section ($A$), and modulus of elasticity ($E$). End conditions significantly influence the shape assumed during buckling.

• Process Equipment:

  • Agitator Ball Mills: Also known as tube mills; commonly used for wet grinding processes.

  • Autofrettage: A manufacturing process used for prestressing thick cylinders to increase their pressure-carrying capacity.