Instantaneous Center Of Zero Velocity

Instantaneous Center of Zero Velocity (ICZV)

This section presents relative motion using vector notation.

Instantaneous Middle of Aught Velocity is a graphical technique that tin can assist simplify relative motility problems. At whatsoever instant in time, a rigid trunk undergoing general planar motility may appear to be rotating effectually a particular bespeak. Finding this signal allows you to use simpler equations for fixed rotation to analyze the motion of that body. In some cases, this simplifies the solution.

Finding the Instantaneous Centre of Zero Velocity (ICZV)

To find the ICZV (also denoted "IC") for a rigid body, you need to know the direction (but not the magnitude or the sense) at two points on the same rigid body. If the velocities at the ii points are parallel, then you need additional information. Nosotros volition starting time with the simpler example of non-parallel velocities at ii points.

A rigid body with two point velocities — This rigid body is undergoing general plane motility, and nosotros can identify the direction of velocity at 2 points.

Once y'all have identified 2 not-parallel point velocities, describe lines perpendicular to those velocity directions. The betoken where the perpendiculars run into is the ICZV. At this betoken, the velocity of the body is zero. Note that the IC can be on or off the trunk. If the IC is off the body, you tin imaging massless rods connecting the IC to the rigid body.

A rigid body showing the intersection of perpendiculars to find the IC — The intersection of two lines perpendicular to two point velocities defines the ICZV.

If this was actually pure rotation, with a physical pivot at the location of the IC, we'd wait all points on the body would have velocities perpendicular to the position vector betwixt the pin and the point of interest. The IC gives the states the same event.

A rigid body showing how the IC is equivalent to a pin at this instant — The ICZV can exist treated as a pin for analysis at this instant but.

However, unlike a concrete pin, in the next instant the ICZV location tin can change. The path of the ICZV is called a centrode.

A rigid body showing the centrode — The ICZV follows a path chosen a centrode over fourth dimension.

Since many rigid body kinematics problems involve finding a velocity at a particular instant, the ICZV tin can be a useful tool. In many cases, it is used to quickly notice the velocity at a third point on a body undergoing general plane motion. This can be washed using the fixed axis rotation velocity equation:

Velocity using ICZV:	\[\vec{v}_{C}=\vec{\omega} \times \vec{r}_{C/IC}\]

A rigid body showing three velocities with respect to the ICZV — The velocity at each point on the rigid body can be divers using the fixed axis rotation equation and the position of the point with respect to the ICZV.

Finding ICs when velocities are parallel

When the two velocities are parallel, nosotros need some additional information near sense and/or magnitude, and we find the IC in a slightly different mode.

To detect the IC when velocities are parallel, draw a line that connects the two vector heads, and another line that connects the two vector tails.

Three rigid bodies showing cases for finding ICs when velocities are parallel — Demonstrating the three cases possible when the ii velocities you select are parallel.

Case 1: Opposite senses

When the two velocities are parallel but accept opposite senses (due east.g. one is directed toward positive x and the other toward negative ten), the object is rotating and yous will find an IC. The IC volition exist located on the line betwixt the 2 points. The location of the IC volition depend on the magnitudes of the vectors.

Case 2: Same sense, different magnitude

When the vectors accept the same sense (east.yard. both directed toward positive 10), then you lot also demand magnitude to know if an IC exists. If the magnitudes are different, the object is rotating. The IC will be located on the line passing through both points, past the indicate with lower velocity.

Instance 3: Same sense, same magnitude

If two points on a rigid body accept the aforementioned velocity vector, so the body is not rotating. If we draw the lines connecting the 2 heads and the two tails, we will go parallel lines. The intersection of parallel lines is at infinity. At that place is no useful IC in this instance.

There are a few important things to call up when using the ICZV:

The ICZV is but a model - there is no actual pin.
The ICZV typically changes location over time. An ICZV is only useful at this instant.
The ICZV is for velocity but. The ICZV will not necessarily have naught dispatch.

Worked Problems:

Question 1:

A ladder is propped upward against a wall every bit shown below. If the base of the ladder is sliding out at a speed of 2 m/s, what is the velocity of the COG of the ladder (at the mid-point)?

Problem 1 Diagram

Solution:

Question two:

2 rods, AB and BC, are connected and moving. A pin at betoken A follows the vertical slot shown. Find the ICZV for each rod at the instant shown. If rod BC has an athwart velocity of 3 rad/s, find the athwart velocity of rod AB.

Problem 2 Diagram

Solution:

Question 3:

The rod is rotating with an angular velocity of ω = 2 rad/due south nearly C. If the angle θ = threescore◦ and the length of the rod L = 1 chiliad, find the velocities of the points A and B.

Problem 2 Diagram

Solution:

Practice Problems:

Practice Problem 1: