Understanding the Instant Center of Rotation (ICR)
In kinematics, the Instant Center of Rotation (also known as the instantaneous center or pole) is a point in a plane of a rigid body undergoing general planar motion that, at a specific instant, has zero velocity. For all intents and purposes, the entire body appears to be rotating around this fixed point at that exact moment.
How to Use This Calculator
To find the ICR of a moving link or object, you need the coordinates of two points on that body and the direction (angle) of their respective velocity vectors. This calculator uses the intersection method:
- Coordinates: Enter the (X, Y) positions of Point A and Point B.
- Angles: Enter the angle of the velocity vector (in degrees) for each point relative to the positive X-axis.
- The Geometry: The ICR always lies on a line perpendicular to the velocity vector of any point on the rigid body. The calculator computes the intersection of these two perpendicular lines.
Formula and Methodology
The calculation follows these geometric steps:
- Calculate the angles of the lines perpendicular to the velocities:
α = θ + 90°.
- Convert these angles to radians and determine the slopes (
m = tan(α)).
- Solve the linear system for the intersection (x, y):
x = (m₁x₁ - m₂x₂ + y₂ - y₁) / (m₁ - m₂)
y = m₁(x - x₁) + y₁
Real-World Example
Imagine a 10-unit long horizontal bar. Point A is at (0,0) and Point B is at (10,0). If Point A is moving straight up (90°) and Point B is moving at a 45° angle, where is the rotation centered? By entering these values into the calculator, you will find the precise coordinates of the pivot point that would produce this specific motion profile.
Note on Parallel Velocities: If the velocity vectors are parallel, the perpendicular lines are also parallel. In this case, the Instant Center is said to be at infinity, indicating the body is in pure translation.
function calculateInstantCenter() {
var x1 = parseFloat(document.getElementById('x1').value);
var y1 = parseFloat(document.getElementById('y1').value);
var ang1 = parseFloat(document.getElementById('v_angle1').value);
var x2 = parseFloat(document.getElementById('x2').value);
var y2 = parseFloat(document.getElementById('y2').value);
var ang2 = parseFloat(document.getElementById('v_angle2').value);
var resultDiv = document.getElementById('ic-result-display');
var resX = document.getElementById('res-x');
var resY = document.getElementById('res-y');
var statusTxt = document.getElementById('ic-status');
if (isNaN(x1) || isNaN(y1) || isNaN(ang1) || isNaN(x2) || isNaN(y2) || isNaN(ang2)) {
alert("Please enter valid numeric values for all fields.");
return;
}
// The IC lies on lines perpendicular to the velocity vectors
var alpha1 = (ang1 + 90) * (Math.PI / 180);
var alpha2 = (ang2 + 90) * (Math.PI / 180);
// Handle the slopes (m = tan(alpha))
// Check if angles result in vertical lines (slope = infinity)
var m1, m2;
var isV1 = Math.abs(Math.cos(alpha1)) < 0.000001;
var isV2 = Math.abs(Math.cos(alpha2)) < 0.000001;
var icX, icY;
var parallel = false;
if (Math.abs(ang1 % 180 – ang2 % 180) < 0.0001) {
parallel = true;
}
if (parallel) {
resX.innerHTML = "∞";
resY.innerHTML = "∞";
statusTxt.innerHTML = "Velocity vectors are parallel. The body is in pure translation; IC is at infinity.";
} else {
if (isV1) {
m2 = Math.tan(alpha2);
icX = x1;
icY = m2 * (icX – x2) + y2;
} else if (isV2) {
m1 = Math.tan(alpha1);
icX = x2;
icY = m1 * (icX – x1) + y1;
} else {
m1 = Math.tan(alpha1);
m2 = Math.tan(alpha2);
if (Math.abs(m1 – m2) < 0.000001) {
parallel = true;
} else {
icX = (m1 * x1 – m2 * x2 + y2 – y1) / (m1 – m2);
icY = m1 * (icX – x1) + y1;
}
}
if (parallel) {
resX.innerHTML = "∞";
resY.innerHTML = "∞";
statusTxt.innerHTML = "Slopes are identical. Instant Center is at infinity.";
} else {
resX.innerHTML = icX.toFixed(4);
resY.innerHTML = icY.toFixed(4);
statusTxt.innerHTML = "Intersection found successfully based on orthogonal velocity vectors.";
}
}
resultDiv.style.display = "block";
resultDiv.scrollIntoView({ behavior: 'smooth', block: 'nearest' });
}