Velocity and Acceleration Analysis
Rigidity gives one vector equation per link, and the polygon closing is the answer. Then acceleration adds a term velocity never had, and sliding adds a third that everybody forgets.
Skip to the animationRigidity means two points on a link differ only by a rotation, so each link gives one vector equation with a known direction — and the velocity polygon closing is the solution, after which acceleration adds a centripetal term and sliding adds a Coriolis one.
Why velocity matters
Forces follow directly from velocities, because power in equals power out — so a point moving slowly must be carrying a large force. That is how a mechanism's mechanical advantage is computed at every position.
The relative velocity method
On a rigid link the distance between two points cannot change, so their relative velocity has no component along the link — it is purely perpendicular, with magnitude ω × AB. That gives one vector equation per link with a known direction and unknown magnitude.
Because the mechanism is a closed chain, the velocity polygon must close — and closure is the solution. Drawn to scale, every unknown is read off directly.
Instantaneous centres
At any instant a link's motion is a pure rotation about its instantaneous centre, found where perpendiculars to two known velocities intersect. Every point's speed is then ω × r from that centre.
Kennedy's theorem says the three instant centres of any three bodies are collinear, which locates centres that cannot be seen directly. A four-bar has six centres, four of which are the pin joints; one of the other two fixes the velocity ratio immediately.
Acceleration
| Component | Magnitude | Direction | Present when |
|---|---|---|---|
| Tangential | αr | Perpendicular to the radius | Rotational speed is changing |
| Centripetal | ω²r | Toward the centre | Always, if it is rotating at all |
| Coriolis | 2ωv | Perpendicular to the sliding velocity | A point slides along a rotating link |
The centripetal term exists even at constant speed and grows with the square of it, which is why inertia forces dominate at high rpm. The Coriolis term is frequently larger than the others and omitting it is the classic error in the topic — it appears in every quick-return and slotted-lever mechanism.
What it is for
Accelerations become forces through F = ma, sizing bearings, links and the motor, and shaking the frame. The worst instant in the cycle governs, which is why the analysis is repeated at every crank angle rather than solved once.
The numbers you will be asked for
- Relative velocity
v_B = v_A + v_BA, with v_BA ⊥ AB and |v_BA| = ω·AB
- Instant centre
v = ω × r from the centre
- Number of instant centres
N = n(n − 1)/2
- Centripetal acceleration
a_c = ω²r
- Coriolis acceleration
a_cor = 2ωv
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.