Coriolis component of acceleration

Why it matters

Whenever a block slides along a link that is itself rotating, as in a shaper's slotted lever, an oscillating-cylinder engine, a hydraulic cylinder on a swinging boom or the prismatic axis of a rotating robot arm, its acceleration contains an extra term, 2ωv. Leaving it out gives wrong inertia forces and wrong actuator loads, and it is also the physical principle behind every MEMS rate gyroscope in phones, drones and vehicle stability systems.

Key ideas

Where it comes from. Let a slider P move outward along a link that rotates at angular velocity ω. Let Q be the point of the link that coincides with P at this instant. P's velocity is v_P = v_Q + v_PQ, where v_PQ is the sliding velocity along the link. Two things change the direction and size of that velocity as time passes:

• the sliding velocity v_PQ itself turns with the link, giving a transverse change ω·v;
• because P moves outward, it reaches points of the link that move faster (ω·r grows), giving another ω·v. Together they give 2ω·v, the Coriolis component. In mechanism analysis this is a real component of the absolute acceleration of P, not a fictitious one; in a rotating observer's frame the same term appears as the Coriolis "force".

Full acceleration of a sliding point. a_P = a_Q + a_PQ(sliding) + a_cor

• a_Q: acceleration of the coincident point on the link (centripetal ω²r towards the pivot, plus tangential αr);
• a_PQ: the slider's acceleration along the link (zero if it slides at constant speed relative to the link);
• a_cor = 2ω × v_PQ.

Direction rule. a_cor is perpendicular to both ω (the axis) and v_PQ (the sliding direction), so in a planar mechanism it lies in the plane, perpendicular to the link. To find its sense, rotate the sliding velocity vector v_PQ through 90° in the sense of ω. For a link turning counter-clockwise with the slider moving outward, the Coriolis component points in the direction the link is turning.

When it is zero. If the link does not rotate (ω = 0, a pure sliding guide), if the slider does not slide relative to the link (v_PQ = 0), or if v_PQ is parallel to ω (not possible in a planar mechanism). In the ordinary slider-crank the slider runs on the fixed frame, so there is no Coriolis term; in the crank and slotted-lever and Whitworth quick-return mechanisms, and in the oscillating-cylinder engine, there is.

Polar form (robot R-P arm). For a point at radius r on an arm at angle θ:

• radial acceleration r̈ − r·ω²;
• transverse acceleration r·α + 2·ṙ·ω; the second term is the Coriolis part. This is why a rotating arm with a telescoping axis needs extra joint torque whenever the axis extends or retracts while it swings.

MEMS gyroscopes. A tiny proof mass is driven to vibrate along one axis. When the chip rotates, the Coriolis acceleration 2ω × v pushes the mass along the perpendicular axis; that sense motion, measured capacitively, is proportional to the angular rate.

Formulas

a_cor = 2·ω·v (magnitude, planar mechanism)

• ω = angular velocity of the link on which the point slides (rad/s)
• v = sliding velocity of the point relative to that link (m/s)
• a_cor in m/s²; direction perpendicular to the link, sense = v rotated 90° in the sense of ω.

a_cor = 2·ω × v_rel (vector form, general)

a_P = a_Q + a_PQ + 2·ω × v_PQ (acceleration of a sliding point)

a_radial = r̈ − r·ω² ; a_transverse = r·α + 2·ṙ·ω (polar coordinates; ṙ = sliding speed, r̈ = sliding acceleration)

F_cor = m·2·ω·v (force needed, perpendicular to the link, to give a sliding mass its Coriolis acceleration, N)

Worked examples

Example 1 (standard): slotted lever of a shaper. Given: the slotted lever turns at 120 rpm counter-clockwise; at the instant considered the sliding block moves outward along the slot at 0.8 m/s relative to the lever.

1. ω = 2πN / 60 = 2π × 120 / 60 = 12.57 rad/s
2. a_cor = 2·ω·v = 2 × 12.57 × 0.8 = 20.1 m/s²
3. Direction: rotate the outward sliding velocity 90° counter-clockwise; the component is perpendicular to the lever, pointing the way the lever is turning. Answer: a_cor ≈ 20.1 m/s², perpendicular to the lever, in the sense of rotation.

Example 2 (GATE level): extending robot arm. Given: a horizontal R-P arm turns at ω = 2 rad/s with α = 1 rad/s² (both counter-clockwise). A 2 kg gripper on the prismatic axis is at r = 0.5 m and moves outward at ṙ = 0.4 m/s, accelerating outward at r̈ = 0.3 m/s². Find the acceleration and the transverse force the arm exerts on the gripper.

1. Radial: a_r = r̈ − r·ω² = 0.3 − 0.5 × 2² = 0.3 − 2.0 = −1.7 m/s² (1.7 m/s² towards the axis).
2. Transverse: a_θ = r·α + 2·ṙ·ω = 0.5 × 1 + 2 × 0.4 × 2 = 0.5 + 1.6 = 2.1 m/s²; the Coriolis part is 1.6 m/s².
3. a = √(1.7² + 2.1²) = 2.70 m/s², at tan⁻¹(2.1 / 1.7) = 51° from the inward radial line, towards the direction of rotation.
4. Transverse force: F_θ = m·a_θ = 2 × 2.1 = 4.2 N, of which the Coriolis share is 2 × 1.6 = 3.2 N. Answer: a ≈ 2.70 m/s²; transverse force 4.2 N (3.2 N of it due to Coriolis). Without the Coriolis term the joint torque would be underestimated by about 76%.

Common mistakes

• Using the absolute velocity of the slider in 2ωv; use its velocity relative to the rotating link.
• Using the angular velocity of the wrong link: it is ω of the link that carries the slot or guide.
• Writing ω·v instead of 2·ω·v.
• Taking the Coriolis component along the link; it is always perpendicular to the link in a planar mechanism.
• Adding a Coriolis term to an ordinary slider-crank, where the guide is fixed.
• Getting the sense wrong: rotate v_rel by 90° in the sense of ω, not opposite.

For GATE ME

Common questions: magnitude and direction of the Coriolis component in a crank and slotted-lever or a slider on a rotating rod; deciding which mechanisms have a Coriolis component; one-line conceptual statements (zero when ω or v_rel is zero, perpendicular to the sliding direction). Practise the direction rule with sketches and polar acceleration of a bead on a rotating rod.

Quick check

1. A block slides at 1.5 m/s along a link rotating at 4 rad/s. Coriolis acceleration?
2. Does an in-line slider-crank have a Coriolis component at the piston?
3. If ω and v_rel are both doubled, by what factor does a_cor change?
4. Which way does a_cor point for a slider moving inward on a clockwise-rotating link?
5. Name one mechatronic sensor based on the Coriolis effect.

Answers: 1. 2 × 4 × 1.5 = 12 m/s². 2. No; the guide is fixed. 3. Four times. 4. Rotate the inward velocity 90° clockwise: perpendicular to the link, opposite to the direction of rotation. 5. MEMS vibratory gyroscope.