Error Budgets - Part I and II

Precision Mechanical Design

Error Budgets - Part I

  • Presented by the Johns Hopkins Whiting School of Engineering.

  • Key focus on understanding the Error Budget as a critical tool in precision engineering.

The Error Budget

  • Definition: A deterministic tool used to predict and control the errors in a machine or system.

  • Functionality:

    • Serves as a model of the machine in its operational environment, expressed in terms of cause-effect relationships.

    • Aids in highlighting specific areas to concentrate resources to enhance accuracy in both existing machines and those under development.

    • Offers a structured format to specify subsystem precision requirements, helping achieve an overall balance of difficulty, risk factors, and costs involved.

  • Reference: Hale, L. C. (1999). Principles and techniques for designing precision machines. MIT, Dept. of Mechanical Engineering.

Literature and Reference Sources

  • Donaldson, R. R. et al. (1983), Design And Construction Of A Large, Vertical Axis Diamond Turning Machine, SPIE.

  • Donaldson, R. R. (1980), Error Budgets, Lawrence Livermore National Laboratory.

  • Dornfeld, D., & Lee, D.-E. (2008). Precision manufacturing. Springer, Chapter 6.

Assumptions According to Donaldson

  1. The instantaneous total error in specified directions is the summation of all individual error components in that direction (linear superposition is assumed to be valid).

  2. Individual error components must have identifiable physical causes that can be isolated, measured, and controlled, allowing for both error magnitude reduction and prediction.

    • Important Note: These assumptions form a deterministic approach supported by the discipline of machine tool metrology, which provides means for measuring individual errors.

Steps in Creating an Error Budget

Step 1: List of Error Sources

  • Objective: Compile a comprehensive list of significant error sources.

  • Methodology:

    • Utilize experience and thorough reviews of the machine subsystems and surrounding environment.

    • Engage in brainstorming sessions with colleagues to identify possible factors affecting machine accuracy.

    • It is crucial to include all significant sources with no known tests available for completeness.

Step 2: Identify Coupling Mechanisms

  • Goal: Ascertain the connection mechanisms between identified error sources and the concern point (displacement error).

  • Examples:

    • Ground motions leading to displacement errors through vibration isolators.

    • Angular motion resulting in displacement errors via the Abbe offset.

    • Point of Concern: Displacement error at the workpiece, which can be modeled using a machine error model.

Step 3: Estimate Error Magnitudes

  • Approach:

    • Assess the estimated magnitudes of the sources of error as well as the resulting displacement errors through coupling mechanisms.

    • It is recommended that magnitude estimates are backed by calculations, such as:

    • Back of the envelope calculations (e.g., considering the Abbe offset and thermal expansion).

    • Finite Element Analysis (FEA) to model machine deformations and vibrations.

    • Leverage specification sheets from vendors to obtain values such as straightness, pitch, roll, and yaw errors specified under given conditions.

Step 4: Separate Error Into Error Directions

  • Objective: Divide the total list of displacement errors into their respective directional components along machine axes (X, Y, Z, etc.).

  • Considerations:

    • Directional influences of vibrational sources on surface finish may add complexity, hence assume vibrational sources are omnidirectional.

Step 5: Combine Errors with Combinatorial Rule

  • Procedure:

    • Ensure all errors are expressed in a uniform format (e.g., Peak-to-Valley (P-V), Root Mean Square (RMS), etc.).

    • Utilize an appropriate combinatorial rule to merge errors.

    • Detailed Analysis:

      • Generate a map of resultant displacement error as a function of motion position, time, control settings, and environmental factors (i.e., machine error simulation).

      • RMS: Applies when averaging occurs at the tool/workpiece interface:
        E{tot} = rac{1}{ ext{n}} imes ext{ Sum} igg( Ei^2 igg)

      • P-V: Used when the largest isolated error peak is critical regardless of surrounding surface perfection, making it extremely conservative. The probability distribution of errors can be statistically analyzed with constants:

      • $K = 2.83$ for sinusoidal distribution.

      • $K = 3.46$ for uniform probability density.

      • $K = 4.0$ for ±2s Gaussian distribution.

    • For P-V calculations:
      PVi = K imes RMSi

    • Various combinations of errors might be evaluated:

    • Direct addition (conservative)

    • Arithmetic average

    • Geometric average

    • Root Mean Square (optimistic)

Application Examples and Data

LODTM: Surface-Finish Error Budget

  • Error Source and Peak-to-Valley Amplitude (nm):

    1. External mechanical disturbances - 5.0 (0.20)

    2. Airborne noise - 2.5 (0.10)

    3. Hydraulic vibration - 2.5 (0.10)

    4. Spindle drive - 2.5 (0.10)

    5. Spindle air pressure - 2.5 (0.10)

    6. Theoretical finish - 2.5 (0.10)

    7. Tool-edge/cutting mechanics - 4.0 (0.16)

    8. MCU resolution - 5.0 (0.20)

    9. Servo-controlled tool mount - 10.0 (0.40)

    10. Interferometer phase distortion - 2.5 (0.10)

    11. Index difference-straightness interferometer - 3.5 (0.14)

  • Sum of squares = 215.8 (0.3452)

  • Root Mean Square (RMS) = 4.2 (0.17)

Radial Figure Error Budget

  • Peak-to-Valley Values (nm):

    • In various error sources affecting X and Z axis motion, such as:

    • Position interferometers - 3.0 (0.12)

    • Laser center frequency - 5.5 (0.22)

    • Straightness interferometers - 5.0 (0.20)

    • Additional external factors also considered explaining deviations in measurements due to vehicle thermal influences or other non-machine factors.

    • Sum of squares = 9128 (14.60), RMS = 27.6 (1.10).

Ultra-Precision CNC Measuring Machine

  • Specifications:

    • Designed for inspecting axisymmetric parts, such as hemispherical shells.

    • Can handle diameters of up to 400 mm with an accuracy less than 0.75 mm per surface and 1.75 mm for wall thicknesses.

    • Operates in continuous path contouring mode (Reference: Thompson, D. C., & McKeown, P. A. (1989)).

Comparison of Results

  • Thompson Report: Noted that the combinatorial rule effectively predicted errors that were closely aligned with actual measured errors, generally within a 15% variance (Thompson, D. C., & Fix, B. L. (1995)).

Error Budget Examples

  • References to various case studies and literature that utilize error budgets for their precision analysis and optimization of machining processes have been documented:

    • Estler, W. T. (1989).

    • Shen, Y.-L., & Duffie, N. A. (1993).

    • Cuttino, J. F. et al. (1999).

    • Treib, T., & Matthias, E. (1987).

    • Fesperman et al. (2012).

    • Henselmans, R. (2009).

Project Administration and Templates

Spreadsheet Features for Error Budget Analysis

  • Sections:

    1. Project and report information

    2. Figures/Schematics

    3. Assumptions

    4. Combinatorial rule application

    5. Error sources and their magnitudes

    6. Notes, actions, and references

    7. Signatures as forms of approval and acknowledgment.

Example Inputs for Error Estimation

  • Document various error sources like external mechanical disturbances and their peak-to-valley representation.

  • Specific dimensions, environmental conditions, and logical representations are crucial for accurate output predictions.

Final Notes

  • The importance of comprehensive error budgeting following systematic steps is crucial in advanced precision engineering design.

  • Future references include new methodologies on error budget management and reports advancing from Part II through III discussions.