Lesson 3: Velocity Analysis and Instantaneous Centers

Modified: Jul. 20, 2026

Published: Sep. 8, 2025

Position analysis found where every link sits for a given input angle. Velocity analysis answers the next question: how fast is each link moving at that instant? The answer comes almost for free, because velocity is the time derivative of position, and you already have the position equations. Differentiate the vector loop from the position analysis and you get a set of linear equations for the velocities. Velocity matters because it sets the kinetic energy of every part, the flow rate of a pump, and the bearing loads, and because a smooth-looking mechanism can still have a sharp velocity peak that drives vibration. In this lesson you draw the velocity polygon, differentiate the loop to confirm it, and read the velocity ratio that becomes mechanical advantage. You also meet the instantaneous center, a neat velocity shortcut, and see why the course builds on the polygon rather than on it. #VelocityAnalysis #VelocityPolygon #MechanicalAdvantage

Learning Objectives

By the end of this lesson, you will be able to:

Construct the velocity polygon with the drawing set and measure the link velocities from it
Differentiate the vector loop to obtain the velocity equations, and solve for piston and angular velocities in closed form
Read the velocity ratio as mechanical advantage and verify every result in a simulator
Recognise the instantaneous center as a velocity shortcut, and know why the course builds on the polygon instead

Real-World System Problem: Piston Velocity in Engines and Compressors

The engine in a car, the compressor in an air conditioner, and the pump in a hydraulic system are all crank-sliders. Each converts steady rotation into a back-and-forth piston motion, and the velocity of that piston is never constant through the stroke. It starts at zero, rises to a peak somewhere past mid-stroke, and falls back to zero, and exactly where the peak lands decides the bearing loads, the lubrication demand, and the vibration the machine produces.

The Velocity Problem

Engineering Question: Given the input link’s angular velocity, how fast is every other point and link moving at this instant?

For the crank-slider the key output is the piston velocity. For the four-bar it is the angular velocities of the coupler and follower. For the scissor lift it is the platform velocity. All of them come from one operation: differentiating the position loop with respect to time.

Why Velocity Analysis Matters

Bearing and lubrication loads

Peak sliding speed sets the oil-film demand and the friction power lost at every joint.

Flow and delivery

In a pump or compressor the piston velocity is the volumetric flow rate, so its profile is the delivery curve.

Vibration

A sharp velocity peak means a large acceleration nearby (acceleration analysis), and that is what shakes the machine.

Mechanical advantage

The ratio of input speed to output speed is the velocity ratio, and it is the reciprocal of the force ratio (force analysis).

Fundamental Theory: Velocity by Differentiating the Loop

Two Conventions to Fix Before Any Numbers

Signs: counterclockwise is positive

The convention set in the position analysis carries straight through: angles and angular velocities are positive counterclockwise. It is repeated here because velocity is the first place negative answers become common, and a sign is a physical statement, not bookkeeping.

rad/s means the coupler turns clockwise while the crank turns counterclockwise.
A negative piston velocity means the piston moves toward the crankshaft, that is, in the direction.

Read every sign out loud as a direction before moving on. When a later result surprises you, the sign is usually where the story starts.

Normalisation: why results are quoted as v/(rω)

Velocities in this lesson are often reported normalised, as rather than in mm/s. Dividing by , the crank pin’s own speed, produces a dimensionless number that answers the question “how fast is this point moving compared with the crank pin?”

The payoff is that the normalised result depends only on the geometry ratio , not on the crank radius or the running speed. One table then covers every engine with that proportion, at every speed. To recover a real velocity, multiply back by : with mm and rad/s, becomes mm/s m/s.

Differentiate the Position Loop

From Position to Velocity

The position loop is a statement that the link vectors close. Differentiating it with respect to time gives a statement that their velocities are consistent. For the four-bar position loop:

Position loop (x, y):

Differentiate (the ground is constant, ):

These are two linear equations in the unknown angular velocities and . The positions from the position analysis are the known coefficients.

This is the central idea of the lesson. Position analysis was nonlinear and needed the Freudenstein trick to solve. Velocity analysis is linear, because differentiation turns the trigonometric position terms into coefficients that multiply the unknown velocities. Once you have the positions, finding the velocities is just solving two linear equations, and the same step repeated in the acceleration analysis gives the accelerations.

Ways to Get the Velocity

Velocity uses the same methods as position, in the same draw, solve, simulate rhythm: the graphical velocity polygon is the core hand method, the analytical velocity loop is its exact confirmation, and each Application ends by reading the answer off the simulator.

Build the velocity polygon with the drawing set. Draw each velocity vector to scale, head to tail, using the key rule that the velocity of one point relative to another point on the same rigid link is perpendicular to the line joining them. Where the construction lines cross closes the polygon, and you measure the unknown velocities off it with the scale rule. This is the core hand method, worked in every Application below, and the same construction becomes the acceleration polygon in the next lesson.

The symbol is read “the velocity of relative to ” (equivalently, “of with respect to ”): how fast appears to move when you sit on and watch. The full velocity of is then the velocity of plus this relative part:

Two direction rules settle every velocity polygon, and you should commit them to memory:

Turning link. If and are two points on the same rigid link, is perpendicular to the line , with magnitude . (A rigid link cannot stretch, so can only swing around .)
Sliding pair. Where one part slides along another (a piston on its centre-line, a block in a slotted lever), the sliding part of the velocity is parallel to the slide.

When a point both slides along a link and that link turns (a block in a rotating slot, as in the quick-return shaper), the acceleration picks up an extra Coriolis term that has no velocity counterpart; it is defined and used in the acceleration analysis.

The instantaneous center gives velocities only. It has no counterpart for acceleration, so it cannot carry a problem through the full position, then velocity, then acceleration chain that this course follows. The velocity polygon can: it is the same drawing discipline that becomes the acceleration polygon in Lesson 4, so the effort you spend learning it pays off twice. Locating every center by Kennedy’s theorem (for any three links in plane motion, their three instantaneous centers are collinear) also grows quickly with size, since a mechanism of links has centers, while the polygon and the loop stay the same size on any linkage. So we treat the instantaneous center as a fact worth knowing and a one-step check on a velocity ratio, and we build the course on the polygon, the loop, and the simulator.

Velocity Ratio and Mechanical Advantage

Velocity Ratio

The velocity ratio of a mechanism is the output speed divided by the input speed. For a four-bar it is .

By conservation of power (input power equals output power in an ideal mechanism), the force or torque ratio is the reciprocal of the velocity ratio:

This quantity is the mechanical advantage. It is defined here from velocities and used again in the force analysis from forces; the two views are the same number seen from opposite sides. When the output slows to a near stop (a limit position), the velocity ratio approaches zero and the mechanical advantage grows very large.

Application 1: Piston Velocity of the Slider-Crank

This is the central worked example. We differentiate the slider position from the position analysis and find where the piston velocity actually peaks, which is not where intuition first suggests.

Simulator and hands-on lab

Open the Crank-Slider Simulator

Hands-on lab: Continue in the Crank-Slider Experiments lab (siwit.co/CSM). Experiment 1 plots this velocity profile; Experiment 2 explores how an offset turns it into a quick-return.

Step 1: Draw the Space Diagram

Choose and mark a length scale before you start (for a page, 1 cm = 20 mm works well); the scale is yours to pick, but always state it on the drawing. Then construct the space diagram with a set square and compass at the instant you want, here the crank at .

Click to reveal the space-diagram construction

Centre-line and pivot. Draw the cylinder centre-line horizontally and mark the crank pivot on it. ✅
Set the crank. From , draw at above the centre-line, length mm to scale. Point is the crank pin. ✅
Swing the rod. With centre and radius mm, strike an arc cutting the centre-line at . The line is the connecting rod and is the piston. ✅
Measure. The piston sits mm from , matching the position solution. Your drawing should match the figure below. ✅

Slider-crank space diagram: crank OA at 60 degrees, connecting rod AB to piston B on the centre-line

Step 2: Construct the Velocity Polygon

The velocity polygon solves graphically: draw the three velocity vectors head to tail to a chosen scale and read the answer off the figure.

Click to reveal the velocity-polygon construction

Crank-pin velocity. Pick a pole and a velocity scale. Draw perpendicular to the crank , with length . Taking rad/s gives mm/s. ✅
Direction of the piston velocity. The piston slides along the centre-line, so is horizontal. Draw a construction line through the pole parallel to the slide. ✅
Direction of the relative velocity. is perpendicular to the connecting rod . From , draw a line perpendicular to . ✅
Close the polygon. Where the two construction lines cross is point . Then is the piston velocity and is the relative velocity. ✅
Measure. mm/s, so at . This matches the analytical value in Step 3 within drawing tolerance. ✅

Slider-crank velocity polygon: pole o, with v_A, v_B/A and v_B closing the triangle

Step 3: Confirm by Differentiation

Click to reveal the closed-form piston velocity

Differentiate the slider position () with and . This is the exact result, with no approximation anywhere in it:

✅
Write it as harmonics, the approximate form, obtained by setting : the primary plus secondary terms

The secondary (twice-per-revolution) term from the connecting rod is what engine balancing must handle. ✅
Know which one you are using, and say so. The two disagree slightly. At with :

Form
Exact
Harmonic (approximate)

About apart, and the gap grows as the rod gets shorter (small ). The tables in this lesson use the exact form throughout, so a value differing from yours in the third digit usually means you used the harmonic one, not that you made an error. The harmonic form is the right tool for balancing work, where separating the once- and twice-per-revolution terms is the whole point; the exact form is the right tool for a numerical answer. State which you used. ✅
Tabulate , from the exact form, for :

Crank
0\degree 0.000
30\degree -0.646
60\degree -1.017
73\degree -1.055 (peak)
90\degree -1.000
120\degree -0.715
180\degree 0.000

At the formula gives , confirming the mm/s measured from the polygon. ✅
The peak is about near , not at : the secondary harmonic shifts it earlier in the stroke. The drawing gives one instant; the calculus gives the whole curve and the true maximum. ✅

Form
Exact
Harmonic (approximate)

Crank
0\degree	0.000
30\degree	-0.646
60\degree	-1.017
73\degree	-1.055 (peak)
90\degree	-1.000
120\degree	-0.715
180\degree	0.000

Step 4: Connecting-Rod Angular Velocity

Click to reveal the rod angular velocity

Rod angular velocity from differentiating . Differentiating the vertical closure gives the rate of the rod’s inclination:

The rod’s angular velocity in the counterclockwise-positive convention carries the opposite sign, because the rod leans to the other side of the centre-line from the crank ():

✅
Evaluate at the working instant , with mm, mm, rad/s:

✅

What the negative sign means physically: the crank turns counterclockwise while the rod swings clockwise. The sign is not a slip, it is the answer to “which way”, and it is what makes the crank-pin rubbing velocity in Step 5 an addition rather than a subtraction. Carry it. ✅
Check the extremes. At : , the rod rotating fastest. At : , the rod momentarily not rotating, only translating. ✅
A one-step check. The rod’s instantaneous center is where the crank line and the vertical through the piston cross, the same intersection that fixed in the polygon; dividing by the distance from that center to reproduces the same . It is a quick confirmation, not the working method. ✅

Step 5: Points on the Rod, and Rubbing Velocities at the Pins

In an engine the connecting rod is a real body with mass, and every pin rubs. Two questions follow from the same polygon you already drew: how fast does a chosen point on the rod move (its inertia needs this), and how fast do the pin surfaces rub? The rubbing speed is worth knowing because it sets the friction heat, the oil-film thickness, and the wear life at each bearing: the faster a journal rubs, the more power it dissipates and the harder its lubrication has to work, so the crank pin and main bearing are sized and oiled for their rubbing speeds, not the engine’s output speed.

Click to reveal the velocity image and rubbing velocities

The rod’s velocity image. Every point of the rigid rod maps to a point on the line of the polygon, in the same proportion: a point one-third along from sits one-third along from . The line is the velocity image of the rod, and reading any point’s velocity is just measuring from the pole to its image. ✅
Velocity of the rod’s mid-point . The mass centre sits at the middle of , so its image is the middle of . Measuring gives mm/s, the velocity the rod’s inertia force will use in the acceleration analysis. ✅
The least-velocity point. The slowest point on the rod is the one whose image is the foot of the perpendicular from the pole onto the line (point ). Here it lands about mm from , close to the mid-point, with mm/s. At this instant the rod’s points hardly differ in speed, from mm/s near the middle to about mm/s at the piston. ✅
Rubbing velocity at a pin. At a pin joint the two links turn at different rates, so the journal surface rubs at the relative angular velocity times the pin radius:

Use signed angular velocities inside the bars, then take the modulus. This one rule handles both cases automatically and is the only reliable way to get it right: if the two links turn in opposite senses the difference of the signed values adds their magnitudes, and if they turn the same way it subtracts them. Guessing add-or-subtract from a sketch is where marks are lost.

With rad/s (crank, counterclockwise), rad/s (rod, clockwise, from Step 4), and (the piston translates, it does not rotate):

Pin Links joined (rad/s) (mm) (mm/s)
Main bearing crank, frame 25 25.0
Crank pin crank, rod 20 23.5
Gudgeon pin rod, piston 15 2.6

The crank pin has the highest relative rate because crank and rod turn in opposite senses, so the subtraction of a negative adds their magnitudes; the gudgeon pin rubs slowest because only the rod’s small acts against a non-rotating piston. Note that the crank pin does not rub fastest here despite its highest relative rate, because the main bearing’s larger radius wins: radius and relative speed both matter. Multiplying each rubbing velocity by the friction force at that pin gives the power lost there. ✅

Pin	Links joined	(mm)	(mm/s)
Main bearing	crank, frame	25	25.0
Crank pin	crank, rod	20	23.5
Gudgeon pin	rod, piston	15	2.6

Velocity image of the connecting rod: the line a b carries every point of the rod, with midpoint g and least-velocity point q

Step 6: Verify in the Simulator

Click to reveal the simulator check

Open the simulator (siwit.co/CSM), set , , , and run it. ✅
Read the velocity chart. The reported maximum piston velocity divided by is about , with the peak before mid-stroke, matching the table. The mean of over a cycle is . ✅
Add an offset. Set and the forward and return peaks become unequal (the quick-return of Experiment 2). Mobility stays one; only the velocity profile changes. ✅

Application 2: Angular Velocities of the Four-Bar

For the four-bar we draw the velocity polygon to read the coupler and follower angular velocities, then confirm with the velocity loop and read the velocity ratio that becomes mechanical advantage.

Simulator and hands-on lab

Open the Four-Bar Linkage Simulator

Hands-on lab: Continue in the Four-Bar Linkage Experiments lab (siwit.co/FBL). The angular-velocity charts there plot and against crank angle.

Step 1: Draw the Space Diagram

Construct the four-bar to scale at using the same arc-intersection method you used for position analysis.

Click to reveal the space-diagram construction

Ground and crank. Draw the ground mm horizontally. From , draw the crank at , length mm. ✅
Intersect the arcs. With centre and radius mm, and centre and radius mm, strike two arcs. Their intersection is the coupler-follower joint (the upper intersection is the open assembly). ✅
Measure. The coupler sits at and the follower at , matching the position solution. ✅

Four-bar space diagram at crank angle 120 degrees, found by intersecting the coupler and follower arcs at B

Step 2: Construct the Velocity Polygon

Click to reveal the velocity-polygon construction

Crank-pin velocity. Pick a pole and a velocity scale. Draw perpendicular to the crank , length . With rad/s, mm/s. ✅
Direction of the follower velocity. Point moves perpendicular to the follower . Draw a construction line through the pole perpendicular to the follower. ✅
Direction of the relative velocity. is perpendicular to the coupler . From , draw a line perpendicular to the coupler. ✅
Close the polygon. The two construction lines cross at . Then is the velocity of on the follower and is the relative velocity. ✅
Measure and convert. mm/s and mm/s, so

✅

Four-bar velocity polygon at 120 degrees: pole o with v_A, v_B/A and v_B closing the triangle

Step 3: Confirm by the Velocity Loop

Click to reveal the closed-form angular velocities

Differentiating the position loop gives two linear equations whose Cramer’s-rule solution is:

✅
At (, , ):

✅

These confirm the and measured from the polygon.
Full profile across the crank rotation:

Crank
30\degree -0.262 +0.122
60\degree -0.040 +0.457
90\degree +0.064 +0.539
120\degree +0.139 +0.514

At (the instant used for position analysis) the coupler is almost in pure translation, , so its velocity triangle is very thin. That is why we drew the polygon at instead. ✅

Crank
30\degree	-0.262	+0.122
60\degree	-0.040	+0.457
90\degree	+0.064	+0.539
120\degree	+0.139	+0.514

Step 4: Velocity Ratio and Mechanical Advantage

Click to reveal the mechanical advantage

Velocity ratio at : the follower turns at of the crank. ✅
Mechanical advantage is the reciprocal:

✅

An ideal crank torque appears amplified about twice at the follower here. The value changes through the cycle and grows large near the limit positions, which the force analysis uses.

Application 3: Platform Velocity of the Scissor Lift

The scissor-lift height was a one-line expression in the position analysis, so its velocity is one differentiation away.

Simulator and hands-on lab

Open the Scissor Lift Simulator

Hands-on lab: Continue in the Scissor Lift Experiments lab (siwit.co/SLM). The platform-velocity chart plots the relation derived here.

Step 1: Draw the Space Diagram

Draw the scissor to scale at to fix the geometry before finding the velocity.

Click to reveal the space-diagram construction

Base and arms. Draw the base horizontally. From the fixed bottom pin draw one arm of length mm at ; draw the second arm from the sliding bottom pin so the two cross at their midpoints. ✅
Platform. Join the two upper arm ends with the platform line, which stays parallel to the base. ✅
Measure. The platform height is mm, matching the position solution. ✅

Single-stage scissor lift drawn at 30 degrees: base, crossed arms of length L, and platform at height h = L sin theta

Step 2: Differentiate the Height

Click to reveal the platform velocity

From the platform height , differentiate with respect to time:

✅
Read the behaviour. Near the flat position (), , so the platform rises quickly for a given . Near the top (), , so the platform velocity falls to zero even while the arms keep closing. The lift slows as it reaches full height. ✅
The actuator side. A constant actuator speed does not give a constant platform speed, because the geometry between actuator length and angle is itself nonlinear. The simulator’s velocity chart shows the actual platform-velocity curve for the chosen actuator type. ✅

Step 3: Verify in the Simulator

Click to reveal the simulator check

Open the simulator (siwit.co/SLM), set , one stage, and run it at a fixed actuator speed. ✅
Confirm that the platform velocity is largest at low angle and tapers toward zero near full height, matching . ✅

Application 4: Velocity Ratio of the Toggle Clamp

The toggle clamp shows velocity analysis at its most dramatic: at the toggle position the output velocity ratio collapses to zero, which is the exact mechanism behind self-locking.

Simulator and hands-on lab

Open the Toggle Clamp Simulator

Hands-on lab: Continue in the Toggle Clamp Experiments lab (siwit.co/TCM). Experiment 1 shows the velocity ratio collapsing at top-dead-centre.

Step 1: Draw the Space Diagram at the Toggle Position

Sketch the four-bar skeleton at top-dead-centre, where the geometry behind self-locking becomes visible.

Click to reveal the toggle-position construction

Ground and handle. Draw the base line . From draw the handle to joint . ✅
Collinear main link. Draw the main link from to in line with the handle, so , , and lie on one straight line. This collinear configuration is top-dead-centre. ✅
Clamp arm. Join to . Near this position moves almost perpendicular to the clamp arm, so the pad velocity per unit handle velocity drops toward zero. ✅

Toggle clamp four-bar skeleton at top-dead-centre, with the handle and main link collinear

Step 2: Velocity Ratio Near the Toggle

Click to reveal the vanishing velocity ratio

Apply the four-bar velocity solution. As the handle approaches top-dead-center, the handle link and main link become collinear. In the velocity-ratio expression, the term in the denominator does not vanish, but the geometry drives the output pad velocity per unit handle velocity toward zero: the pad momentarily stops while the handle still moves. ✅
The consequence. A vanishing velocity ratio means, by the power balance of the theory section, that the mechanical advantage grows very large:

✅

A modest handle force produces a very large clamping force. This is the quantitative form of the self-locking seen in the mobility analysis.
Mobility is unchanged. The clamp still has one degree of freedom throughout. What changes at the toggle is the instantaneous velocity ratio, not the number of inputs. This is the difference between a singular configuration and a change in mobility. ✅

Step 3: Verify in the Simulator

Click to reveal the simulator check

Open the simulator (siwit.co/TCM) and drive the handle toward top-dead-center. ✅
Watch the mechanical-advantage chart rise sharply as the links approach collinear, while the pad velocity per handle increment falls toward zero. Past the toggle by the lock margin, the clamp holds itself closed. ✅

Application 5: Ram Velocity of the Quick-Return Shaper

The metal shaper of the position analysis drives its cutting ram through a crank-and-slotted-lever mechanism, an inversion of the slider-crank. It brings in something the engine did not have: the crank pin is a block that slides inside the slotted lever while the lever turns. That sliding-in-a-turning-slot is exactly the joint whose velocity polygon carries a slip vector, and whose acceleration will carry a Coriolis term.

Hands-on lab

Open the Crank-Slider Simulator

Hands-on lab: This inversion has no simulator of its own; the drawing and the calculation confirm each other here. For the related quick-return principle, open the crank-slider simulator’s Quick-Return (offset) preset and watch the velocity curve go asymmetric, then continue in the Crank-Slider Experiments lab.

Step 1: Draw the Space Diagram

Fix the mechanism to scale at the chosen instant so the polygon has real directions to work from. The crank sets where the block sits, and the line through to is the slotted lever.

Click to reveal the space-diagram construction

Choose and note a scale, say 1 cm = 40 mm, and draw the fixed centres and with mm. ✅
Crank and block. Draw the crank mm to the given angle; is the sliding block. ✅
Slotted lever. Draw the lever from through and on to the driving point at mm. Measure the block distance mm and the lever angle. ✅

Quick-return shaper at the cutting instant: crank CB, slotted lever A to P, sliding block at B, ram driven from P

Step 2: Construct the Velocity Polygon

The block is two coincident points at this instant: on the crank and on the lever. They share a position but not a velocity, and the difference is the slip along the slot: .

Click to reveal the velocity-polygon construction

Velocity of the block on the crank. Pick a pole and a velocity scale. Draw perpendicular to the crank , length mm/s. ✅
Direction of the coincident lever point. belongs to the turning lever, so is perpendicular to the lever . Draw that direction through the pole . ✅
Direction of the slip. The block slides along the slot, so is parallel to the lever . Draw that direction through . ✅
Close the polygon. The two construction lines meet at . Then and is the slip. Measuring: mm/s and the slip mm/s. ✅
Lever angular velocity. rad/s. ✅

Quick-return velocity polygon: v(B2) from the crank, v(B3) of the coincident lever point, and the slip v(B2/B3) along the slot

Step 3: Ram Velocity and Analytical Check

Click to reveal the ram velocity and confirmation

From lever speed to ram speed. The driving point is on the same lever at mm, so its velocity is perpendicular to the lever with magnitude

✅
The ram takes the horizontal part. The ram slides horizontally, so its speed is the horizontal component of , about mm/s at this instant. ✅
Analytical confirmation. Resolving the crank-pin velocity mm/s along and across the lever gives the slip and the transverse part ; the transverse part divided by returns rad/s, and the slip returns mm/s, matching the polygon. ✅
Why it matters. The ram speed varies through the stroke, and it is deliberately low and steady over the cutting pass and high on the return. The acceleration analysis takes the next step: because the block slides while the lever turns, its acceleration carries a Coriolis term with no counterpart in the plain slider-crank. ✅

Programming Velocity Analysis

The whole lesson reduces to differentiating the loop and solving a linear system, which is a few lines of Python.

import numpy as np

def crank_slider_velocity(theta, r, l, omega):
    """Piston velocity for an in-line slider-crank (e = 0)."""
    phi = np.arcsin((r/l)*np.sin(theta))
    return -r*omega*(np.sin(theta) + (r*np.sin(theta)*np.cos(theta))/(l*np.cos(phi)))

def four_bar_omega(a, b, c, theta2, theta3, theta4, omega2):
    """Coupler and follower angular velocities from the velocity loop."""
    w3 = a*omega2*np.sin(theta4 - theta2) / (b*np.sin(theta3 - theta4))
    w4 = a*omega2*np.sin(theta2 - theta3) / (c*np.sin(theta4 - theta3))
    return w3, w4

# Slider-crank: locate the peak (l/r = 3)
r, l, omega = 0.050, 0.150, 1.0
th = np.linspace(0, 2*np.pi, 100000)
v = crank_slider_velocity(th, r, l, omega)
i = np.argmax(np.abs(v))
print(f"peak |Vp|/(r*omega) = {abs(v[i])/(r*omega):.3f} at {np.degrees(th[i]):.1f} deg")
# peak |Vp|/(r*omega) = 1.055 at 73.2 deg

# Four-bar at theta2 = 60 deg (positions from the position analysis)
w3, w4 = four_bar_omega(40, 120, 80,
                        np.radians(60), np.radians(18.4), np.radians(64.9), 1.0)
print(f"w3/w2 = {w3:.3f}, w4/w2 = {w4:.3f}")   # w3/w2 = -0.040, w4/w2 = 0.457

Design Guidelines for Velocity Analysis

Differentiate, don't restart

Velocity equations are the time derivative of the position loop. Reuse the positions from the position analysis rather than setting up a new problem.

Find the real peak

Do not assume the maximum speed is at mid-stroke. The secondary harmonic shifts the peak, and components must be sized for the true maximum.

Draw first, then confirm

The velocity polygon gives the velocities geometrically and the velocity loop confirms them exactly. Two independent routes agreeing is your check that both are right.

Watch the limit positions

Where the velocity ratio approaches zero, mechanical advantage grows large. Place these positions deliberately, as in a toggle clamp.

Summary and Next Steps

Key Concepts Mastered

Velocity polygon: drawn with the drawing set, using the rule that relative velocity between two points on a link is perpendicular to the link; it is the core hand method and becomes the acceleration polygon next lesson.
Velocity loop: differentiating the position loop gives linear equations for the unknown velocities, the exact confirmation of the polygon.
Piston velocity: the slider-crank peak is about near for , ahead of mid-stroke, because of the secondary harmonic.
Instantaneous center: a velocity-only shortcut for the velocity ratio, worth knowing as a quick check; the course builds on the polygon because that construction also carries into acceleration.
Mechanical advantage: the reciprocal of the velocity ratio, large near limit positions, the basis of self-locking.

Velocity Results at a Glance

Mechanism	What you solve for	Key relation	Simulator
Slider-crank	piston velocity	exact: ; approx:	siwit.co/CSM
Slider-crank	rod angular velocity	(negative: rod swings opposite the crank)	siwit.co/CSM
Any pin joint	rubbing velocity	(signed , then modulus)	drawing calc
Four-bar	,	velocity loop (linear)	siwit.co/FBL
Scissor lift	platform velocity		siwit.co/SLM
Toggle clamp	velocity ratio	at the toggle	siwit.co/TCM
Quick-return (shaper)	ram velocity, slip	velocity polygon with slip vector	drawing calc

A Note on Tools

Every velocity here was found by hand and reproduced with a few lines of Python (NumPy). The simulators confirm the same profiles interactively. No specialised motion software is involved; differentiating the loop is the whole method.

Next, Acceleration Analysis and Dynamic Forces differentiates once more. The velocity equations become acceleration equations, the piston acceleration reveals the primary and secondary inertia forces that shake an engine, and Newton’s second law turns those accelerations into the dynamic loads on bearings and links.

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