Ball Screw Selection and Calculations ME EN 7960 – Precision Machine Design Topic 4 ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-1 Ball Screw Selection Criteria Maximum Rotational Applied Load Speed Thrust force Resonance (bending) Required torque of threaded shaft DN value Ball Screw Selection Ball Screw Life Basic dynamic load Accuracy rating Stiffness ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-2 1 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.). – Torque required to generate the thrust force. ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-3 Driving Torque to Obtain Thrust T: driving torque [Nm] Fal T = Fa: thrust force [N] 2πη l: screw lead [m] η: efficiency Source: THK Co., Ltd. ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-4 2 Required Thrust • The thrust is the sum of all forces acting in the axial direction. FM: Machining force [N] F : Frictional force [N] F = F + F + F + F f a M f i g Fi: Inertial force [N] Fg: Gravitational force [N] Source: THK Co., Ltd. ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-5 Stresses from Applied Loads F 2T σ = a axial 2 τ torsional = 3 πrtr πrtr The equivalent (Von Mises) stress: 2 2 σ eq = σ axial + 3τ torsional 2 4Fa 12l → σ eq = 2 1+ 2 2 2 πdtr π dtrη σmax: Permissible stress [147 MPa] ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-6 3 Graphic Solution Von Mises Stress in Shaft (Fa = 2344N) 8 3 .10 1mm lead 2mm lead . 8 2.5 10 4mm lead 5mm lead 8 2 .10 8 1.5.10 max = 147 MPa 8 1 .10 Von Mises Stress [Pa] 7 5 .10 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 1mm lead -> dtr > 4.6mm 2mm lead -> d > 4.8mm Root Diameter [m] tr 4mm lead -> dtr > 5.2mm 5mm lead -> dtr > 5.5mm ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-7 Permissible Compressive Load • Buckling Load P1: Buckling load [N] 2 λπ EI lb: Distance between mounting positions [m] P1 = E: Elastic modulus [Pa] 2 4 lb I: Second moment of inertia [m ] λ: Support factor Fixed – free: λ = 0.25 Fixed – supported: λ = 2.0 Fixed – fixed: λ = 4.0 ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-8 4 Fixed-Free Mount Source: THK Co., Ltd. Inexpensive but only applicable for short ball screws and/or slow speeds. ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-9 Fixed-Supported Mount Source: THK Co., Ltd. Most commonly used mounting setup. ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-10 5 Fixed-Fixed Mount Source: THK Co., Ltd. Overconstrained 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 expansion. ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-11 Basic Static Load Rating Coa – When ball screws are subjected to excessive loads in static condition (non rotating shaft), local permanent deformations are caused between the track surface and the steel balls. – When the amount of this permanent deformation exceeds a certain degree, smooth movement will be impaired. Coa: Basic static load rating [N, kgf, lbf] Coa ≥ fs Fa fs: Static safety factor Fa: Load on shaft in axial direction [N, kgf, lbf] Use conditions fs (lower limit) Normal operation 1.0 – 2.0 Operation with impacts and vibrations 2.0 – 3.0 ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-12 6 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. -1 nc: Critical speed [min ] l : Distance between supports [m] 60λ2 EI b n = E: Elastic modulus [Pa] c 2πl 2 ρA I: Second moment of inertia [m4] b ρ: Density [kg/m3] 15λ2d E A: Root cross sectional area [m2] tr λ: Support factor = 2 2πlb ρ Fixed – free: λ = 1.875 Supported – supported: λ = 3.142 Fixed – supported: λ = 3.927 Fixed – fixed: λ = 4.730 ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-13 Spindle Speed and DN Value • Shaft speed -1 va n: Revolutions per second [s ] n = va: axial speed [m/s] l l: lead [m] • DN Value. Unless specified otherwise: D: Ball circle diameter [mm] DN ≤ 70000 N: Revolutions per minute [min-1] ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-14 7 Dynamic Load Rating Ca and Life • The basic load rating Ca is 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. 3 L: Rotation life [rev] C : Basic dynamic load rating [N, kgf, lbf] ⎛ Ca ⎞ 6 a L = ⎜ ⎟ ×10 f : Load factor ⎜ f F ⎟ w ⎝ w a ⎠ Fa: Load in shaft direction [N, kgf, lbf] Use conditions fw Smooth movement without impacts 1.0 – 1.2 Normal movements 1.2 – 1.5 Movement with impacts and vibrations 1.5 – 2.5 ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-15 Example • Mass of axis: 350kg • Maximum velocity: 20m/min • Acceleration time: 0.05s • Bearing friction factor: 0.003 • Machining force: 500N • Length of work piece: 500mm • Length of travel at maximum speed: 100mm • Orientation of axis: horizontal ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-16 8 Machining Profile of Making a Slot Machining speed Maximum speed v Work piece Variable speed vmax v0 = 0 v1 = vmax v2 = vmax v3 = vm v4 = vm vm x v5 = 0 v6 = −vm -vm v7 = −vm v8 = −vmax vr = −vmax v10= 0 -vmax ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-17 Load Profile for Making a Slot 01 23 456 78 910 V,F vmax v0 = 0 Fa1 = Fi + F f = 2344N v = v 1 max Fa2 = F f = 10N v2 = vmax F = −F + F = −2324N a3 i f i v3 = vm Fa4 = Fm + F f = 510N v = v vm t 4 m Fa5 = F f = 10N v = 0 5 F = −F = −10N -v a6 f m v6 = −vm Fa7 = −Fm − F f = −510N v7 = −vm Fa8 = −Fi − F f = −2344N v8 = −vmax Fa9 = −F f = −10N vr = −vmax Fa10 = Fi − F f = 2324N v10 = 0 -vmax 1 2 3 4 5 6 7 8 9 10 ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-18 9 Running Lengths Depending on Usage (v + v )t Running distance during acceleration: l = 1 2 accl accl 2 (v + v )t Running distance during deceleration: l = 1 2 decl decl 2 ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-19 Mean Axial Force Determine mean axial load in the positive direction by collecting all individual, positive axial loads. 3 ∑ Fai+li+ i+ Fa,mean+ = 3 ∑li i Determine mean axial load in the negative direction by collecting all individual, negative axial loads. 3 ∑ Fai− li− i− Fa,mean− = 3 ∑li i F + F Determine mean axial load: F = a,mean+ a,mean− a,mean 2 ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-20 10 Load Profile Based on Utilization 3 3 3 Fmlm + Funiluni + Faccllaccl 3 Mean axial force: Fa,mean = lb Fm: Machining force Funi: Force at constant velocity (not machining) Faccl: Maximum force during acceleration and deceleration lm: Total travel per cycle during machining luni: Total travel per cycle at constant velocity laccl: Total travel per cycle during acceleration and deceleration lb: Length of ball screw Total travel length: lb = lm + luni + laccl Travel length: lm = qmlb luni = qunilb laccl = qaccllb ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-21 Load Profile Based on Utilization (contd.) Utilization: qm + quni + qaccl =1 qm: Percentage per cycle spent machining (typically 0.5 – 0.9) quni: Percentage per cycle spent at constant velocity (typically 0.05 – 0.45) qaccl: Percentage per cycle spent during acceleration and deceleration (typically 0.05 – 0.1) ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-22 11 Dynamic Load Rating Ca and Life • When the rotation life L has been obtained, the life time can be obtained according to the following formula if the stroke length and the operation frequency are constant: L: Rotation life [rev] L Lh: Life time [hr] Lh = -1 nm: mean rotational speed [min ] 60nm ∑(nili ) i nm = ∑li i -1 ni: rotational speed at phase i [min ] li: distance traveled at phase i [m] ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-23 Axial Rigidity ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-24 12 Axial Rigidity Fixed-free 1 1 1 1 1 Fixed-supported = + + + k ks kN kB kH 1 1 1 1 1 Fixed-fixed = + + + k ks kN kB1 + kB2 kH1 + kH 2 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] ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-25 Ball Screw Selection Procedure • Axial rigidity of shaft Fixed-free and fixed-supported: AE k = s L Fixed-fixed: AEL k = s ab 4AE k = s,min L ME EN 7960 – Precision Machine Design – Ball Screw Calculations 4-26 13 Ball Screw Accuracy • Applicable if used in combination with rotary encoders.