Ball Screw Selection and Calculations - University of Utah

1 1ME EN 7960 Precision Machine Design Ball Screw Calculations4-1 Ball Screw Selection and CalculationsME EN 7960 Precision Machine DesignTopic 4ME EN 7960 Precision Machine Design Ball Screw Calculations4-2 Ball Screw Selection CriteriaMaximum Rotational SpeedResonance (bending) of threaded shaftDN valueApplied LoadThrust forceRequired torqueStiffnessBall Screw SelectionBall Screw LifeBasic dynamic load ratingAccuracy2ME EN 7960 Precision Machine Design Ball Screw Calculations4-3 Based on Load A ball Screw transforms rotational motion into translational motion. As a result, the shaft is subject to loads: Thrust force (the sum of all external forces such as machining load, gravity, friction, inertial forces, etc.)

2 Torque required to generate the thrust EN 7960 Precision Machine Design Ball Screw Calculations4-4 Driving Torque to Obtain Thrust 2lFTa=T:driving torque [Nm]Fa:thrust force [N]l: Screw lead [m] :efficiencySource: THK Co., EN 7960 Precision Machine Design Ball Screw Calculations4-5 Required Thrust The thrust is the sum of all forces acting in the axial +++=FM:Machining force [N]Ff:Frictional force [N]Fi:Inertial force [N]Fg:Gravitational force [N]Source: THK Co., EN 7960 Precision Machine Design Ball Screw Calculations4-6 Stresses from Applied Loads2traaxialrF =32trtorsionalrT =The equivalent (Von Mises) stress: 223torsionalaxialeq +=222221214 trtraeqdldF+= max:Permissible stress [147 MPa]4ME EN 7960 Precision Machine Design Ball Screw Calculations4-7 Graphic Solution0.

3 00 10. 00 20. 00 30. 00 40. 00 50. 00 60. 00 70. 00 80. 00 lead2mm lead4mm lead5mm leadVon Mises Stress in Shaft (Fa = 2344N)Root Diameter [m]Von Mises Stress [Pa]max= 147 MPa1mm lead -> dtr> lead -> dtr> lead -> dtr> lead -> dtr> EN 7960 Precision Machine Design Ball Screw Calculations4-8 Permissible Compressive Load Buckling Load221blEIP =P1:Buckling load [N]lb:Distance between mounting positions [m]E:Elastic modulus [Pa]I:Second moment of inertia [m4] :Support factorFixed free: = supported: = fixed: = EN 7960 Precision Machine Design Ball Screw Calculations4-9 Fixed-Free MountSource: THK Co.

4 , but only applicable for short ball screws and/or slow EN 7960 Precision Machine Design Ball Screw Calculations4-10 Fixed-Supported MountSource: THK Co., commonly used mounting EN 7960 Precision Machine Design Ball Screw Calculations4-11 Fixed-Fixed MountSource: THK Co., mounting setup for applications where high stiffness, accuracy, and high shaft speed is required. Ball Screw needs to be pre-stretched to avoid buckling in the case of thermal EN 7960 Precision Machine Design Ball Screw Calculations4-12 Basic Static Load Rating Coa When ball screws are subjected to excessive loads in static condition (non rotating shaft), local permanent deformations arecaused between the track surface and the steel balls.

5 When the amount of this permanent deformation exceeds a certain degree, smooth movement will be Coa:Basic static load rating [N, kgf, lbf]fs:Static safety factorFa:Load on shaft in axial direction [N, kgf, lbf] with impacts and operationfs(lower limit)Use conditions7ME EN 7960 Precision Machine Design Ball Screw Calculations4-13 Permissible Speed When the speed of a ball Screw increases, the ball Screw will approach its natural frequency, causing a resonance and the operation will become impossible. EldAEIlnbtrbc2222215260==nc:Critical speed [min-1]lb:Distance between supports [m]E:Elastic modulus [Pa]I:Second moment of inertia [m4] :Density [kg/m3]A:Root cross sectional area [m2] :Support factorFixed free: = supported: = supported: = fixed: = EN 7960 Precision Machine Design Ball Screw Calculations4-14 Spindle Speed and DNValue Shaft speed DN Value.

6 Unless specified otherwise:70000 DND:Ball circle diameter [mm]N:Revolutions per minute [min-1]lvna=n:Revolutions per second [s-1]va:axial speed [m/s]l:lead [m]8ME EN 7960 Precision Machine Design Ball Screw Calculations4-15 Dynamic Load Rating Caand Life The basic load rating Cais the load in the shaft direction with 90% of a group of the same ball screws operating individually will reach a life of 106(1 million) revolutions. 6310 =awaFfCLL:Rotation life [rev]Ca:Basic dynamic load rating [N, kgf, lbf]fw:Load factorFa:Load in shaft direction [N, kgf, lbf] with impacts and movement without impactsfwUse conditionsME EN 7960 Precision Machine Design Ball Screw Calculations4-16 Example Mass of axis: 350kg Maximum velocity: 20m/min Acceleration time: Bearing friction factor: Machining force: 500N Length of work piece: 500mm Length of travel at maximum speed: 100mm Orientation of axis.

7 Horizontal9ME EN 7960 Precision Machine Design Ball Screw Calculations4-17 Machining Profile of Making a Slot00010maxmax876543max2max10= = = = =======vvvvvvvvvvvvvvvvvvvrmmmmMachining speedMaximum speedVariable speedWork piecevmax-vmax-vmvmvxME EN 7960 Precision Machine Design Ball Screw Calculations4-18 Load Profile for Making a Slot00010maxmax876543max2max10= = = = =======vvvvvvvvvvvvvvvvvvvrmmmmvmax-vmax -vmvmV,Ft01234567891010123456789 NFFFNFFNFFFNFFFNFFNFFNFFFNFFFNFFNFFF fiafafiafmafafafmaifiafafia2324102344510 1010510232410234410987654321= = = = = = = = = ====+= =+ ====+=10ME EN 7960 Precision Machine Design Ball Screw Calculations4-19 Running Lengths Depending on Usage()221acclaccltvvl+=Running distance during acceleration:Running distance during deceleration:()221decldecltvvl+=ME EN 7960 Precision Machine Design Ball Screw Calculations4-20 Mean Axial ForceDetermine mean axial load in the positive direction by collecting all individual, positive axial loads.

8 33, ++++=iiiiaimeanallFFDetermine mean axial load in the negative direction by collecting all individual, negative axial loads. 33, =iiiiaimeanallFFDetermine mean axial load:2,,, ++=meanameanameanaFFF11ME EN 7960 Precision Machine Design Ball Screw Calculations4-21 Load Profile Based on Utilization3333,bacclaccluniunimmmeanall FlFlFF++=Mean axial force:Fm:Machining forceFuni:Force at constant velocity (not machining)Faccl:Maximum force during acceleration and decelerationlm:Total travel per cycle during machiningluni:Total travel per cycle at constant velocitylaccl:Total travel per cycle during acceleration and decelerationlb:Length of ball screwTotal travel length:acclunimbllll++=Travel length.

9 Bacclacclbuniunibmmlqllqllql===ME EN 7960 Precision Machine Design Ball Screw Calculations4-22 Load Profile Based on Utilization (contd.)qm:Percentage per cycle spent machining (typically )quni:Percentage per cycle spent at constant velocity (typically )qaccl:Percentage per cycle spent during acceleration and deceleration (typically )Utilization:1=++acclunimqqq12ME EN 7960 Precision Machine Design Ball Screw Calculations4-23 Dynamic Load Rating Caand Life When the rotation life Lhas been obtained, the life time can be obtained according to the following formula if the stroke length and the operation frequency are constant:mhnLL60=L:Rotation life [rev]Lh:Life time [hr]nm.

10 Mean rotational speed [min-1]() =iiiiimllnnni:rotational speed at phase i[min-1]li:distance traveled at phase i[m]ME EN 7960 Precision Machine Design Ball Screw Calculations4-24 Axial Rigidity13ME EN 7960 Precision Machine Design Ball Screw Calculations4-25 Axial RigidityHBNskkkkk11111+++=k:Axial rigidity of linear motion system [N/m]ks:Axial rigidity of Screw shaft [N/m]kN:Axial rigidity of nut [N/m]kB:Axial rigidity of support bearing [N/m]kH:Axial rigidity of support housing [N/m]212111111 HHBBN skkkkkkk+++++=Fixed-freeFixed-supportedF ixed-fixedME EN 7960 Precision Machine Design Ball Screw Calculations4-26 Ball Screw Selection Procedure Axial rigidity of shaftFixed-free and fixed-supported:LAEks=Fixed-fixed:abAELk s=LAEks4min,=14ME EN 7960 Precision Machine Design Ball Screw Calculations4-27 Ball Screw Accuracy Applicable if used in combination with rotary : THK Co.