Precision Machine Design Topic 3 Examples of errors Purpose: This lecture provides detailed examples of different types of errors, and shows how estimates can be made of their magnitude. Outline: Geometric errors Surface finish effects Kinematic errors Load induced errors Deflection of a component due to its own weight Errors caused by rapidly accelerating axes Dynamic errors Thermal errors Thermal breaks Estimating magnitudes of thermal errors "Chance is a pseudonym of God when he did not want to sign." Anatole France 3-1
Geometric errors Abbe 1 errors: Abbe error at caliper tips Errors in form: Ideal path High frequency straightness error ('Smoothness') Straightness error Lateral motion sensor's output trace Nonperfect rollers Nonperfect surface Includes linear motion errors and axis of rotation errors. 1 Named after Dr. Ernst Abbe 3-2
Surface finish effects The surface finish of a bearing not only affects life, it also can affect accuracy. Surfaces with positive (left) and negative (right) skewness: Peak Valley Both surfaces have the same R a value: Ra = 1 L 0 L y(x)dx Sliding contact bearings tend to average out surface finish errors and wear less when the skewness is negative. The larger the positive skewness, the greater the wearin period. Hydrostatic and aerostatic bearings are insensitive to surface finish effects. Surface finish should be at least 10x greater (e.g., 1 µm) than the bearing clearance (e.g., 10 µm). 3-3
For rolling element bearings: If the contact area is larger than the typical peak-tovalley spacing: An elastic averaging effect will occur and a kinematic arrangement of rollers will produce smooth motion. The submicron peaks will be flattened or worn-off during break-in period. Heavy preloads may have to be checked after wear-in. Beware of frictional polymers that can form when "wheels" roll on dry surfaces: Organic molecules in the air that land on the surface are compacted with time to form a layer. If numerous rolling elements are used, the effects of elastic averaging can help to smooth out the motion. If the elements are recirculating: Noise may be introduced into the system as the rolling elements leave and enter the load-bearing region. 3-4
Kinematic errors Errors in motion due to alignment. Y R Y R Y R ε 90 ± z Z R X R Z R X R Z R XR ε 90 ± y ε 90 ± y Errors in motion due to shape: Improper offsets (translational) between components. Spindle axis set too high above tailstock axis on a lathe. Improper component dimension. Linkage length. Bearing location on a kinematic vee and flat system. 3-5
Load induced errors Static loads. Dynamic loads. Bending deformations. Shear deformations. Example: Ratio of bending and shear deformations for a rectangular cantilevered beam loaded by a force at its end: 12 Deflection: Bending/shear = 5(L/H)/3.9 11 10 9 8 7 6 5 4 3 2 1 1 2 Beam length/height 3 Use this graph as a scaling guide. 3-6
1.016 m (40 in.) Errors caused by rapidly accelerating axes: Example: A high-speed machine has an X axis carriage that holds two other axes and the spindle: Tool Spindle Z Y 0.762 m (30 in.) X Modular rolling element linear bearing The Z axis (spindle moves in and out of part): Has a mass M of 682 kg. Is supported by bearings with stiffness of 350 MN/m (2x106 lbs/in). Is required to accelerate and decelerate at 0.5 g while drilling or boring a hole. 3-7
The dominant compliance is due to the X axis bearings whose Y deflection results in angular motion about the X axis. With a Z acceleration "a", the Y axis deflection of the rear X axis bearing and the roll of the structure about the X axis are: δ y-rear bearing = -1.016aM 0.726K x axis bearing θ x = δ y-rear bearing - δ y-front bearing 0.762 When the spindle starts accelerating into a part at 0.5 g, the resultant bearing deflections are found to be: δy-front bearing = 12.7 µm (500 µin) δy-rear bearing = -12.7 µm (-500 µin). The pitch εx of the structure is thus -33.3 µrad. 3-8
If the tool tip's coordinate reference frame origin is at the origin of the reference XYZ system, the tool tip's HTM is: R T tool tip = 1 0 0 0 0 1 3.33x10-5 1.27x10-5 0-3.33x10-5 1 0 0 0 0 1 When the tool is at its forward most position, Z = 0.508 m from the front bearing: The HTM is post-multiplied by the toolpoint vector TPT = [0 1.016 0.508 1]. This describes the position of the tool in its own coordinate system. The resultant position vector of the tool in the reference coordinate system is [0 1.0160296 0.5079662 1]. The tool tip error is: δy = 29.6 µm (1165 µin) δz = - 33.8 µm (-1332 µin). These values would scale linearly with the stiffness of the X axis bearings and the acceleration. 3-9
Dynamic errors Static friction causes start-up forces which deform the structure and make control more difficult. Dynamic friction causes forces which deform the structure, but makes control more easy. However, dynamic friction causes heat to be generated. Vibration causes errors throughout the structure. Vibration transmission through the floor. Rotating mechanical components (e.g., motors, transmissions). Rolling bearings (e.g., microstructure making noise as asperities grind together). Limit cycling in servo loops. Turbulence in fluid supply lines. Sound pressure. Aerostatic instability in air bearings (pneumatic hammer). 3-10
A system containing bearings, may have a fundamental frequency of ωn due to the speed of the bearing. There can be higher order harmonics of frequency Nωn. There can be other frequencies due to the elements in the system such as the cage and the balls, and defects. Rolling element bearings are prime sources of harmonic vibration. Frequency spectrum of a rolling element bearing spindle's radial error motion. 500 Displacement (nanometers) 400 300 200 100 Displacement due to machine deformation 0 0 50 100 150 Frequency (Hz) 200 250 The spindle speed was 1680 rpm (28 Hz). 3-11
When two frequencies are close together, they can form a beat: f waveform = f 2 + f 1 2 f amplitude = f 2 - f 1 2 Y t A beat sounds like a slowly varying-in-amplitude wave, even though it is composed of two higher frequencies. The frequency of the varying amplitude envelope in general doesn't excite a machine natural frequency. The sound of a beat can present ergonomic problems, particularly to sales and marketing. Audible sound is NOT a good indicator of the true problem! Use an accelerometer and do the FFT! 3-12
Dynamic errors are part of a detective game: An accelerometer and a signal analyzer are critical tools for gathering clues. Modal analysis software and the Fourier Transform are critical tools for tracking the villains! The best way to avoid dynamic problems is to build in as much damping as possible into the system. 62.5 grams of prevention is worth a kilogram of cure. 3-13
Thermal errors Very troublesome because they are always changing. Very troublesome because components' heat transfer coefficients vary from machine to machine. Design strategies: Isolate heat sources and temperature control the system. Maximize conductivity, OR insulate Combine one of above with mapping and real time error correction. May be difficult for thermal errors because of changing boundary conditions. Combine two of the above with a metrology frame. 3-14
Thermal effects in manufacturing and metrology (After Bryan.): Heat source/ sinks Room environment Heat added or removed by coolant systems Coolants Electronic Hydraulic Frame Cutting Lubricating systems oil stabilizing fluid oil People Electrical and electronic Friction Heat created by the machine Frame stabilization Motors, transducers Amplifiers, control cabnets Spindle bearings Other Hydraulic Miscellaneous Heat created by the cutting process Heat flow paths Conduction Convection Radiation Conduction Convection Radiatio Conduction Convection Radiatio Temperature gradiants or static effects Temperature variations or dynamic effects Temperature field Uniform temperature other than 20 degrees C Nonuniform temperatures Memory of previous environment Affected Structure Part Master Frame Station-change effect Error components Form error Total thermal error Size error 3-15
Primary mechanisms: Conduction: Use thermal breaks (insulators). Keep the temperature the same in the building all year! Channel heat-carrying fluids (coolant coming off the process) away: Workpiece zone Wheel zone Flood coolant Flood coolant Insulation layer (5 mm foam) Sheet metal trough Convection: Use sheet metal or plastic cowlings. Radiation Plastic PVC curtains (used in supermarkets too!) are very effective at blocking infrared radiation. Use indirect lighting outside the curtains. Never turn the lights off! Always ask yourself if symmetry can be used to minimize problems. 3-16
Thermal breaks Motor to shaft: Cooling holes Coupling Leadscrew shaft Motor shaft Cross section Motor Cast iron Reinforced polymer Spindle design for insensitivity to thermal growth: Spindle body Bearing (e.g., hydrostatic) supports Spindle housing Very difficult to implement in practice. Horizontal wings, anchored in the front and supported by flexures in the back: (Courtesy of Rank Taylor Hobson Inc.): Low-cost effective spindle support The nose of the spindle stays fixed. The back is allowed to grow. 3-17
Effects of a temperature gradient acting on a rod: Linear growth of tools and spindles and columns is an important error. At least it does not contribute to Abbe errors. δ = 0.5αh (T top - T base ) For a meter tall cast iron structure in a 1 Co/m gradient, δ= 5.5 µm. This is a very conservative estimate, because the column will diffuse the heat to lessen the gradient. 3-18
Warping of a plate in a gradient: Beam length = l, height = h, section I, gradient T, straightness error: ε T = y ρ = αy T h M = EI ρ δ T = M (l/2)2 2EI = l2 α T 8h Slope error at the ends of the beam (α=m(l/2)/ei): θ T = α Tl 2h For a 1x1x0.3 m cast iron surface plate with T=1/3 Co (1 Co/m), δ = 1.5 µm and θt = 6.1 µrad. This is a very conservative estimate, because the plate will diffuse the heat to lessen the gradient. In a machine tool with coolant on the bed, thermal warping errors can be significant. Angular errors are amplified by the height of components attached to the bed. 3-19
Consider a 5 m long machine on a concrete slab 2 m thick. To save HVAC costs, the air temperature is 32 degrees in the summer and 28 degrees in the winter. The ground temperature is a constant 26 degrees. THERMERR.XLS To determine temperature gradient induced errors in a simply supported beam Written by Alex Slocum. Last modified 5/26/95 by Alex Slocum Only change cells with boldface numbers. Be consistent with units Material properties Modulus of Elasticity: E 3.45E+10 Coefficient of thermal expansion: a 8.00E-06 Cross section properties flange thickness (0 for rect. beam): t 0 Height: h 2 Width: bo 3 Web thickness (bi=bo for rect. beam): bi 3 Moment of inertia: I 2.0000 Loading characteristics Length of beam: L 5 Temp. gradient across beam: DT 4 Results Radius of curvature 62500 Bending moment 1104000 Max. displacement error (microunits) 50.00 Max. slope error (µrad) 40.00 3-20
Thermal gradient errors in machine tools This is a common error in machine tools: The bed may be subjected to a flood of temperature controlled fluid. The base will be at room temperature, which may vary wildly in old buildings. A gradient will exist from the base to the machine's working volume. Evaporative cooling (common on large grinders) can be a cause. Overhead lights can create gradients in sensitive structures. Plastic PVC curtains are extremely effective at reducing infrared heat transmission. A large machine on a deep foundation (relies on the concrete for support), can have problems: Several meters under the ground, the concrete is at constant temperature. The top of the machine and the concrete are at room temperature. 3-21
Deformation of a bimaterial plate moved from one uniform temperature to another: δ = θ = (α 1 - α 2 ) T (l /2) 2 t 1 + t 2 + 4 t 1 (E 1 I 1 + E 2 I 2 ) 1 E 1 A 1 + 1 E 2 A 2 t 1 + t 2 2 (α 1 - α 2 ) T (l /2) + 2 t 1 (E 1 I 1 + E 2 I 2 ) 1 E 1 A 1 + 1 E 2 A 2 Example: 1 mx1 mx0.3 m with 0.03 m wall thickness surface plate. If not properly annealed, after top is machined and the bottom retains a 0.5 cm layer of white iron. δ = 0.10 µm/co, α = 0.41 µrad. Similar effects are incurred by steel bearing rails grouted to epoxy granite structures. Consider using a symmetrical design (steel on the bottom) to offset this effect. Two materials may have similar expansion coefficients, but very different conduction coefficients and density! For a quick estimate of transient effect, assume that the coefficient of expansion of one member is scaled by the ratio of the conduction coefficients. 3-22
Example: Two size 55 linear guides bolted to a granite bed, later used at a different temperature (e.g., in the summer). BIMAT.XLS To determine thermal deformations in a bi-material beam Written by Alex Slocum. Last modified 5/26/95 by Alex Slocum Only change cells with boldface numbers. Material properties Modulus of Elasticity: Eo 2.04E+11 Coefficient of thermal expansion: ao 1.10E-05 Modulus of Elasticity: Et 3.45E+10 Coefficient of thermal expansion: at 6.00E-06 Cross section 1 properties flange thickness (0 for rect. beam): t1 0 Height: h1 0.055 Width: bo1 0.11 Web thickness (bi=bo for rect. beam): bi1 0.1 Moment of inertia: I1 1.53E-06 Area: Ar1 0.0061 Cross section 2 properties flange thickness (0 for rect. beam): t2 0 Height: h2 0.75 Width: bo2 1 Web thickness (bi=bo for rect. beam): bi2 1 Moment of inertia: I2 3.52E-02 Area: Ar2 0.7500 Loading characteristics Length of beam: L 2 Change in beam temperature from that at which it was initially manufactured: DT 5 Results Max. displacement error (microunits) 0.33 Max. slope error 0.66 Bearing rails made from different materials than the bed can have a significant effect on ultra precision machines! Also beware of embedded iron in cast epoxy-granite structures. 3-23
Temperatures of different principle components and locations need to be plotted along side a quality control parameter (e.g., part diameter). In addition, all other functions on the machine should also be plotted. Lubricators that squirt oil to bearings every N minutes can cause a sudden temporary expansion of the machine. Predictions can be made using fundamental theory, as can detailed finite element models. However, nothing beats real data from a real system. The problem lies in interpolating the data. Hypothetical graph of process variables and machine parameters: Temperature 0 0 T environment Part error. T top and bottom structure. etc.. Part error Time Sliding bearing lubricator cycle Invaluable for identification and correction of the problem via a change in machine design. Constant adjustment (via SPC) does not address the problem. 3-24