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Ball Screw Selection and CalculationsME EN 7960 – Precision Machine DesignTopic 4ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-1Ball Screw Selection CriteriaMaximum RotationalSpeedResonance (bending)of threaded shaftDN valueApplied LoadThrust forceRequired torqueBall ScrewSelectionBall Screw LifeBasic dynamic loadratingAccuracyStiffnessME EN 7960 – Precision Machine Design – Ball Screw Calculations4-21

Based on Load A ball screw transforms rotational motion intotranslational motion. As a result, the shaft is subject toloads:– Thrust force (the sum of all external forces such as machiningload, gravity, friction, inertial forces, etc.).– Torque required to generate the thrust force.ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-3Driving Torque to Obtain ThrustFlT a2πηT:Fa:l:η:driving torque [Nm]thrust force [N]screw lead [m]efficiencySource: THK Co., Ltd.ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-42

Required Thrust The thrust is the sum of all forces acting in the axialdirection.Fa FM F f Fi FgFM:Ff:Fi:Fg:Machining force [N]Frictional force [N]Inertial force [N]Gravitational force [N]Source: THK Co., Ltd.ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-5Stresses from Applied Loadsσ axial Faπrtr2τ torsional 2Tπrtr3The equivalent (Von Mises) stress:22σ eq σ axial 3τ torsional σ eq 4 Fa12l 2 1πd tr2π 2 d tr2η 2σmax:Permissible stress [147 MPa]ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-63

Graphic SolutionVon Mises Stress in Shaft (Fa 2344N)83 .101mm lead2mm lead4mm lead5mm lead8Von Mises Stress [Pa]2.5 .1082 .1081.5 .10max 147 MPa81 .1075 .1000.0010.0020.0030.0040.0050.0060.007Root Diameter [m]0.0080.0091mm lead - dtr 4.6mm2mm lead - dtr 4.8mm4mm lead - dtr 5.2mm5mm lead - dtr 5.5mmME EN 7960 – Precision Machine Design – Ball Screw Calculations4-7Permissible Compressive Load Buckling LoadP1 λπ EI2lb2P1:lb:E:I:λ:Buckling load [N]Distance between mounting positions [m]Elastic modulus [Pa]Second moment of inertia [m4]Support factorFixed – free:λ 0.25Fixed – supported:λ 2.0Fixed – fixed:λ 4.0ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-84

Fixed-Free MountSource: THK Co., Ltd.Inexpensive but only applicable for short ball screws and/or slow speeds.ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-9Fixed-Supported MountSource: THK Co., Ltd.Most commonly used mounting setup.ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-105

Fixed-Fixed MountSource: THK Co., Ltd.Overconstrained mounting setup for applications where high stiffness,accuracy, and high shaft speed is required. Ball screw needs to be prestretched to avoid buckling in the case of thermal expansion.ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-11Basic Static Load Rating Coa– When ball screws are subjected to excessive loads in staticcondition (non rotating shaft), local permanent deformations arecaused between the track surface and the steel balls.– When the amount of this permanent deformation exceeds acertain degree, smooth movement will be impaired.Coa f s FaCoa:f s:Fa:Basic static load rating [N, kgf, lbf]Static safety factorLoad on shaft in axial direction [N, kgf, lbf]Use conditionsfs (lower limit)Normal operation1.0 – 2.0Operation with impacts and vibrations2.0 – 3.0ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-126

Permissible Speed When the speed of a ball screw increases, the ball screwwill approach its natural frequency, causing a resonanceand the operation will become impossible.nc 60λ222πlbEIρA15λ2 d tr 2πlb2Eρnc:lb:E:I:ρ:A:λ:Critical speed [min-1]Distance between supports [m]Elastic modulus [Pa]Second moment of inertia [m4]Density [kg/m3]Root cross sectional area [m2]Support factorFixed – free:λ 1.875Supported – supported:λ 3.142Fixed – supported:λ 3.927Fixed – fixed:λ 4.730ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-13Spindle Speed and DN Value Shaft speedn valn:va:l:Revolutions per second [s-1]axial speed [m/s]lead [m]D:N:Ball circle diameter [mm]Revolutions per minute [min-1] DN Value. Unless specifiedotherwise:DN 70000ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-147

Dynamic Load Rating Ca and Life The basic load rating Ca is the load in the shaft directionwith 90% of a group of the same ball screws operatingindividually will reach a life of 106 (1 million) revolutions.3 C L a 106 f w Fa L:Ca:fw:Fa:Rotation life [rev]Basic dynamic load rating [N, kgf, lbf]Load factorLoad in shaft direction [N, kgf, lbf]Use conditionsfwSmooth movement without impacts1.0 – 1.2Normal movements1.2 – 1.5Movement with impacts and vibrations1.5 – 2.5ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-15Example Mass of axis: 350kgMaximum velocity: 20m/minAcceleration time: 0.05sBearing friction factor: 0.003Machining force: 500NLength of work piece: 500mmLength of travel at maximum speed: 100mmOrientation of axis: horizontalME EN 7960 – Precision Machine Design – Ball Screw Calculations4-168

Machining Profile of Making a SlotMachining speedMaximum speedvWork pieceVariable speedvmaxv0 0v1 vmaxv2 vmaxv3 vmv4 vmvmxv5 0v6 vm-vmv7 vmv8 vmaxvr vmaxv1 0 0-vmaxME EN 7960 – Precision Machine Design – Ball Screw Calculations4-17Load Profile for Making a Slot0 12 34 5 67 8910V,Fvmaxv0 0Fa1 Fi F f 2344Nv1 vmaxFa 2 F f 10 Nv2 vmaxv3 vmt v4 vmvmv5 0-vmv6 vmv7 vmv8 vmaxvr vmaxv10 0Fa 3 Fi F f i 2324NFa 4 Fm F f 510 NFa 5 F f 10 NFa 6 F f 10 NFa 7 Fm F f 510 NFa 8 Fi F f 2344NFa 9 F f 10 NFa10 Fi F f 2324N-vmax12345 678910ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-189

Running Lengths Depending on UsageRunning distance during acceleration:laccl (v1 v2 )taccl2(v1 v2 )tdeclRunning distance during deceleration: ldecl 2ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-19Mean Axial ForceDetermine mean axial load in the positive direction by collecting all individual,positive axial loads.Fa ,mean 3 F l l3ai i i iiDetermine mean axial load in the negative direction by collecting all individual,negative axial loads.Fa , mean 3 F l3ai i i liiDetermine mean axial load:Fa ,mean Fa ,mean Fa ,mean 2ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-2010

Load Profile Based on UtilizationMean axial force:Fa ,mean 333Fm3lm Funiluni ng forceForce at constant velocity (not machining)Maximum force during acceleration and decelerationTotal travel per cycle during machiningTotal travel per cycle at constant velocityTotal travel per cycle during acceleration and decelerationLength of ball screwTotal travel length:lb lm luni lacclTravel length:lm qm lbluni quni lblaccl qaccl lbME EN 7960 – Precision Machine Design – Ball Screw Calculations4-21Load Profile Based on Utilization (contd.)qm quni qaccl 1Utilization:qm:quni:qaccl:Percentage per cycle spent machining (typically 0.5 – 0.9)Percentage per cycle spent at constant velocity (typically 0.05 – 0.45)Percentage per cycle spent during acceleration and deceleration (typically 0.05 – 0.1)ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-2211

Dynamic Load Rating Ca and Life When the rotation life L has been obtained, the life timecan be obtained according to the following formula if thestroke length and the operation frequency are constant:Rotation life [rev]Life time [hr]mean rotational speed [min-1]L:Lh:nm :LLh 60nm (n l ) li inmiiini:l i:rotational speed at phase i [min-1]distance traveled at phase i [m]ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-23Axial RigidityME EN 7960 – Precision Machine Design – Ball Screw Calculations4-2412

Axial RigidityFixed-freeFixed-supported1 1111 k ks k N kB kHFixed-fixed1 1111 k k s k N k B1 k B 2 k H 1 k H 2k:k s:kN:kB:kH:Axial rigidity of linear motion system [N/m]Axial rigidity of screw shaft [N/m]Axial rigidity of nut [N/m]Axial rigidity of support bearing [N/m]Axial rigidity of support housing [N/m]ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-25Ball Screw Selection Procedure Axial rigidity of shaftFixed-free and fixed-supported:ks AELFixed-fixed:AELab4 AEk s ,min Lks ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-2613

Ball Screw Accuracy Applicable if used in combination with rotary encoders.Source: THK Co., Ltd.ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-27Overall Error The overall error is the summation of a number of individual errors:––––Non-uniformity of threaded shaft.Resolution/accuracy of rotary encoder.Axial compliance of ball screw assembly.Torsional compliance of threaded shaft.δa δS δa:δs:l:Nrot:Fa:32Tl lFl a 4b N rot k axial πd tr G 2πoverall linear error [m]ball screw non-linearity [m]lead [m]resolution of rotary encoder[pulses/rev]thrust force onto system [N]kaxial:Τ:lb:dtr:G:overall linear stiffness [N/m]applied torque [Nm]ball screw length [m]root diameter of shaft [m]shear modulus [Pa]ME EN 7960 – Precision Machine Design – Ball Screw Calculations4-2814