### Instantaneous Center of Rotation

• State Methods for determining the velocity of a point on a link.
• Define Instantaneous Centre of Rotation.
• State properties of the instantaneous centre.
• Calculate velocity of a point on a link by instantaneous centre method.

It denotes the center of rotation of a body at an instant in time. The center of rotation doesn’t necessarily have to lie within the link itself. The instantaneous center between bodies 1 and 2 is defined as $I_{12}$.

• It is a point in one body about which some other body is permanently or instantaneously rotating about.

• It is a point common to two bodies where the velocity of the two bodies are the same. i.e., a point where there is zero relative velocity between the two bodies.

To find the instant center for a single body when the velocities of two points are known we take advantage of the fact that the linear velocities of all points in a rotating body are perpendicular to their radii of rotation. The two points A and B along with their known velocity vectors $v_A$ and $v_B$ determine the I.C. If the two velocity vectors are equal (i.e., same magnitude and direction) then the I.C. is a point at infinity, and the body’s motion is in pure translation.

#### I.C. for a Rigid Body Rotating in a Plane:

If velocities of two points on a rigid body are known, then draw lines perpendicular to the direction of these velocities. The intersection point of these lines is the instantaneous center of rotation.

#### I.C. for a Sliding Body:

It lies on a line perpendicular to the motion at infinite distance. (It may be on either side).

#### I.C. for a Rolling Body:

The instantaneous center for a rolling body occurs at the point of contact between the two bodies.

### Solved Example:

#### 33-4-01

A wheel is rolling without slipping on a plane surface with the center of velocity V. What will be the velocity at the point of contact?

Solution:
For a rotating wheel (rotation without slipping) the instantaneous center is the point of contact. Hence, its instantaneous velocity is zero.

### Solved Example:

#### 33-4-02

Which is the false statement about the properties of instantaneous centre?

Solution:

The following properties of the instantaneous centre are important:

1. A rigid link rotates instantaneously relative to another link at the instantaneous centre for the configuration of the mechanism considered.

2. The two rigid links have no linear velocity relative to each other at the instantaneous centre. At this point (i.e. instantaneous centre), the two rigid links have the same linear velocity relative to the third rigid link. In other words, the velocity of the instantaneous centre relative to any third rigid link will be same whether the instantaneous centre is regarded as a point on the first rigid link or on the second rigid link.

### Solved Example:

#### 33-4-03

Instantaneous center of rotation of a link with respect to its adjoining link in a four bar mechanism lies on:

### Solved Example:

#### 33-4-04

The total number of instantaneous centers for a mechanism of n links is:

Solution:
The number of pairs of links or the number of instantaneous centres is the number of combinations of n links taken two at a time. Mathematically, number of instantaneous centres, $N = \dfrac{n(n - 1)}{2}$

### Solved Example:

#### 33-4-05

A rigid triangular body, PQR, with sides of equal length of 1 unit moves on a flat plane. At the instant shown, edge QR is parallel to the x-axis, and the body moves such that velocities of points P and R are V$_P$ and V$_R$ , in the -x and y directions, respectively. The magnitude of the angular velocity of the body is:

Solution:

This problem can be solved using I.C. (Instantaneous Center) concept. Instantaneous velocities are always perpendicular to I.C. Draw perpendiculars from both velocity vectors, their intersection point is I.C. as shown in the figure. Consider the velocity V$_R$. Distance from I.C. to R = $\dfrac{1}{2}$ as QR = 1 unit (given)

\begin{align*} V &= r \omega \\ V_R &= \dfrac{1}{2} \omega\\ \omega &= 2 V_R \end{align*}

### Solved Example:

#### 33-4-06

When a slider moves on a fixed link having curved surface, the instantaneous center lies on: (MP Sub Engg Mechanical July 2017 - Shift II)

Solution:
When a slider moves on a fixed link having flat surface, it has instantenous center at infinity. However, if the sliding surface has curvature, the slider is effectively undergoing a circular motion. The center of this motion is the center of curvature of the curved surface.

### Solved Example:

#### 33-4-07

A circular object of radius 'r' rolls without slipping on a horizontal level floor with the center having velocity V. The velocity at the point of contact between the object and the floor is: (GATE ME 2014)

Solution:
The key phrase here is 'at the point of contact', where the velocity will always be zero, unless the objects are skidding.