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ENGINEERING CALCULATORS

Ball-screw shaft buckling and critical speed calculator

Engineering calculators (Beta)

Compare load and speed using separate compression and rotating spans, core diameter and ideal end constraints.

Inputs

Choose units and enter numbers. Full-width digits and fractions such as 1/8 or 1 3/8 are accepted. Changing a unit converts the entered value.

Results

Reference calculation only. Results do not certify suitability for an actual design or its safety.

Diagram

Inputs: select a blue symbol to edit Results: red-orange values and lines

Related: ball-screw theoretical thrust and drive torque

Formula and applicability

A=πd₁²/4; I=πd₁⁴/64; Pcr=π²EI/(KLb)²; Pb=Pcr/S; σ=F/A; ωcr=β²√(EI/(ρA))/Lr²; ncr=60ωcr/(2π); nlim=min(c·ncr,nprod) when nprod>0, otherwise c·ncr.

Displayed values are rounded. Diagrams illustrate the calculation conditions. Initial material and fluid properties are examples; replace them with values for the actual material, temperature and conditions.

Reference
Using this information as a reference

Equations assume particular loads, supports and deformation ranges. Check inputs, units and the applicability of each equation. Diagrams explain conditions; they are not manufacturing drawings or proof of safety.

Material values are typical values or reference ranges, not certified product properties or ready-to-use design allowables. They vary with standards, product form, dimensions, temper and temperature. Check the specifications and inspection certificates for the actual product.

Displayed safety factors and checks apply only to the supplied inputs and calculation model. They do not automatically establish the safety of a complete design, including code compliance, service life, fatigue or joints. For actual design, manufacture and use, separately verify applicable regulations and standards, product specifications, and the necessary tests and engineering review.

Calculations do not submit input values to a server. Results are not an automatic code-compliance or safety determination.

この計算で求められること

圧縮区間と回転区間を分け、谷径・支持条件・長さから荷重と回転数の比較を行います。

入力・結果・適用条件を確認する

選択した形状・計算方法により、使う項目は変わります。実際の計算は上の図と入力欄で行えます。

主な入力

  • 圧縮区間の端部条件 (—)
  • 回転区間の端部条件 (—)
  • ねじの谷径 d₁（呼び径ではない） (mm / cm / m)
  • 圧縮される計算区間 Lb (mm / cm / m)
  • 回転する計算区間 Lr (mm / cm / m)
  • 最大の軸方向圧縮荷重 F (N / kN / mN)
  • 使用回転数 n (rpm / rad/s / °/s)
  • ヤング率 E（入力例） (GPa / MPa / Pa)
  • 密度 ρ（入力例） (kg/m³ / g/cm³ / g/L)
  • 材料条件の弾性上限応力 σe（入力例） (MPa / GPa / Pa)
  • 軸方向の許容応力 σa（入力例） (MPa / GPa / Pa)
  • 理論座屈荷重に対する比較安全率 S (—)
  • 危険速度に掛ける比較割合 c（0より大、1以下） (—)
  • 製品固有の回転数上限 nprod（未確認は0） (rpm / rad/s / °/s)

得られる結果

  • 理論座屈荷重÷比較安全率 Pcr/S (N)
  • 回転数の比較上限 nlim（軸力なし） (rpm)
  • 使用荷重による軸応力 σ (MPa)
  • Euler理論座屈荷重 Pcr (N)
  • 入力した許容応力からの荷重上限 (N)
  • 理論危険速度 ncr（軸力なし） (rpm)
  • 理論座屈時の軸応力 (MPa)
  • 圧縮区間の有効細長比 KLb/r (—)
  • 座屈比較の使用率 (%)
  • 軸応力の使用率 (%)
  • 回転数の比較使用率 (%)

計算式の概要

A=πd₁²/4; I=πd₁⁴/64; Pcr=π²EI/(KLb)²; Pb=Pcr/S; σ=F/A; ωcr=β²√(EI/(ρA))/Lr²; ncr=60ωcr/(2π); nlim=min(c·ncr,nprod) when nprod>0, otherwise c·ncr.

谷径を持つ均一な中実丸棒に置き換えた理想弾性モデルです。上段は圧縮区間、下段は軸力なしの回転区間で、同一部品を二つの観点で描いています。固定は曲げの回転を拘束、支持は横変位のみ拘束、自由はどちらも拘束しません。固定でも軸まわりには回転できます。Lbはナット位置を含む実際の圧縮経路、Lrは振れを拘束する間隔から最も不利な区間を指定します。ナットを自動で横支持とみなしません。初期物性・許容値・比較安全率は入力例で、製品の保証値ではありません。Euler応力が入力した弾性上限を超える場合は座屈の比較を採用できません。回転式は軸力を含まないEuler–Bernoulli梁の一次固有振動数です。圧縮荷重による固有振動数の低下、段付き軸・ねじ溝・軸受剛性・付加質量・不釣合いは別途評価します。製品固有のDN等の上限は製品仕様から入力してください。寿命、支持軸受、ナットの強度、精度・潤滑・発熱は別の確認です。

現在選択中の方法には、操作欄の「計算式・適用条件」を使ってください。参考計算であり、設計全体の安全性を保証するものではありません。

関連する計算

What this calculator does

Compare load and speed using separate compression and rotating spans, core diameter and ideal end constraints.

Inputs, outputs and applicability

The selected model determines which of these inputs and outputs apply. Use the working diagram and input panel above to calculate.

Inputs

  • Compression-segment end conditions (—)
  • Rotating-span end conditions (—)
  • Screw root diameter d₁ (not nominal diameter) (mm / cm / m)
  • Compressed calculation segment Lb (mm / cm / m)
  • Rotating calculation span Lr (mm / cm / m)
  • Maximum axial compressive load F (N / kN / mN)
  • Operating speed n (rpm / rad/s / °/s)
  • Young’s modulus E (example) (GPa / MPa / Pa)
  • Density ρ (example) (kg/m³ / g/cm³ / g/L)
  • Elastic-limit stress σe for the material (example) (MPa / GPa / Pa)
  • Allowable axial stress σa (example) (MPa / GPa / Pa)
  • Comparison factor S on Euler load (—)
  • Comparison fraction c of critical speed (0<c≤1) (—)
  • Product speed limit nprod (0 if unconfirmed) (rpm / rad/s / °/s)

Calculated results

  • Euler load / comparison factor Pcr/S (N)
  • Speed comparison limit nlim (no axial load) (rpm)
  • Axial stress under entered load σ (MPa)
  • Euler critical load Pcr (N)
  • Load at entered allowable axial stress (N)
  • Theoretical critical speed ncr (no axial load) (rpm)
  • Axial stress at Euler load (MPa)
  • Compression effective slenderness KLb/r (—)
  • Buckling comparison utilization (%)
  • Axial stress utilization (%)
  • Speed comparison utilization (%)

Formula overview

A=πd₁²/4; I=πd₁⁴/64; Pcr=π²EI/(KLb)²; Pb=Pcr/S; σ=F/A; ωcr=β²√(EI/(ρA))/Lr²; ncr=60ωcr/(2π); nlim=min(c·ncr,nprod) when nprod>0, otherwise c·ncr.

Ideal uniform solid-core elastic shaft. The upper drawing is the compressed segment and the lower the no-axial-load rotating span: two views of the same component. Fixed restrains lateral displacement and bending slope; pinned restrains only lateral displacement; free restrains neither. Fixed does not prohibit axial spin. Select the worst compressed load path Lb, including nut position, and lateral-support span Lr. The nut is not automatically treated as a lateral support. Initial properties, allowables and comparison factors are examples, not product ratings. If Euler stress exceeds the entered elastic limit, do not adopt that buckling comparison. Rotation uses the first Euler–Bernoulli bending frequency without axial load. Separately assess compression-induced frequency reduction, steps, thread grooves, bearing stiffness, added masses and imbalance. Enter product-specific DN or other speed limits from the actual product specification. Life, support bearings, nut strength, accuracy, lubrication and heating remain separate checks.

The displayed formula and conditions in the working panel apply to the currently selected model. Reference calculation only; it does not certify a complete design.

Related tools

入力する単位を確認・換算する

Check and convert input units

目的から順に確認する設計ガイド

図で条件を整理してから計算し、最後に実際の部品仕様を確認します。

Purpose-based design guides

Understand conditions with diagrams, calculate, then verify actual specifications.

See the motion in an animation

Compare ball-screw size and speed in an animation →

Calculation inputs are not transferred automatically. Enter the required dimensions and speed in the animation.