Error Sources

Error Sources in Precision Mechanical Design

Error Classification

Errors can be classified based on their predictability and dynamic influences:

• Systematic Errors: Errors that remain constant or vary in a predictable manner.

• Random Errors: Errors that vary in an unpredictable manner.

• Static / Quasi-static Errors: Errors with no dynamic influences, such as accelerations of servo systems.

• Dynamic Errors: Errors with dynamic influences, such as acceleration of servo systems and vibrations.

Types of Errors

Key categories of errors encountered in precision mechanical design include:

• Geometric Errors

• Kinematic Errors

• Load Induced Errors

• Positioning and/or Instrument Error (e.g., Abbe Errors & Cosine Errors)

• Thermal Errors

Geometric Errors

Geometric errors relate to the physical form and configuration of components. Examples include:

• Surface Straightness

• Surface Roundness

• Surface Roughness

• Bearing Preload

• Constraints

Moore Tool Example (Foundations of Mechanical Accuracy)

Wayne R. Moore's work highlights several geometric errors:

• Form: Horizontal & Vertical Bow or Straightness

• Lean

• Center Distance

• Parallelism or Twist

• Rolling Contact Bearing (specifically, rolling element motion errors)

Rolling Element Motion Errors

This category of geometric error arises from imperfections in rolling elements and surfaces.

• Ideal Scenario: For pure linear motion with no geometric errors, a functional point traveling on perfect rollers over a perfect surface would follow an ideal path.

• Surface with Geometric Error (Perfect Rollers):

  • When perfect rollers move over a surface with low-frequency straightness error, the linear motion will exhibit a low-frequency straightness error. This error corresponds to the deviation of the surface from ideal straightness.

  • Effect of Shifted Functional Point: If the functional point (the point where the intended action occurs) is offset from the line of measurement or the surface of contact, a new straightness error will be observed. The original straightness error ($E$) at the surface becomes a different error at the shifted functional point.

• Surface with Geometric Error (Non-perfect Rollers):

  • When non-perfect rollers (i.e., rollers with their own imperfections) move over a surface with geometric error, the resulting straightness error will include both low-frequency components (from surface geometry) and high-frequency components (from the smoothness/roughness of the rollers themselves). This leads to a more complex, combined straightness error ($E$).

Kinematic Errors

Kinematic errors refer to inaccuracies in the intended motion or relative positioning of components. These often involve misalignment of axes or components.

• Examples: Squareness, Offset Errors (e.g., $E<em>{xoc}, E</em>{yoc}, E<em>{zoc}, E</em>{AOC}, E<em>{BOC}, E</em>{COC}$ referring to different axis offsets and orientations), Orientation, Parallelism. These errors collectively describe the spatial misalignments of an axis system relative to the ideal coordinate system.

Load Induced Errors

These errors arise from forces acting on the mechanical system.

• Internal or External Loads: Can be due to assembly forces (e.g., bolt forces, overconstraints) or external forces.

• Inertia Forces: The physical property of an object to resist changes in velocity (acceleration or deceleration).

  • Dependent upon the mass and shape of an object.

  • Governed by Newton's second law: $F = ma$.

  • Can cause hysteresis or reversal errors (due to the system reacting differently depending on the direction of motion or previous loading).

• Cutting Forces: Forces generated during machining operations.

  • Can lead to vibration of the workpiece or machine.

  • Can cause tool deflection, leading to inaccurate material removal.

• Gravity Induced Deflections: The deformation of components due to their own weight or the weight of attached components, which varies with position or orientation.

Positioning & Instrument Error

These errors are related to the measurement and control systems.

Instrument Errors

Errors inherent to the measuring device itself:

• Resolution: "The smallest change in a quantity being measured that causes a perceptible change in the corresponding indication or measurement." (JCGM 200: VIM, 2008)

• Measurement Uncertainty: "The non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used." (JCGM 200: VIM, 2008)

• Drift: "Continuous or incremental change over time in indication or measurement, due to changes in metrological properties of the instrument." (JCGM 200: VIM, 2008)

Errors due to Instrument Position and Alignment (Abbe, Cosine, Dead Path)

• Abbé Error: Named after Ernst Abbe, who stated: "If errors in parallax are to be avoided, the measuring system must be placed coaxially with the axis along which displacement is to be measured on the workpiece."

  • Parallax: The apparent displacement or difference in apparent direction of an object as seen from two different points not on a straight line with the object.

  • Definition: The measurement error resulting from angular motion of a movable component and an Abbe offset.

  • Abbé Offset ($h$): The instantaneous value of the perpendicular