⚙️ Rotational Stiffness Calculator
Calculate rotational spring constant
How to Use This Calculator
Enter Torque
Input the torque (τ) in newton-meters (N·m). This is the rotational force applied to twist the object. Torque is the product of force and the perpendicular distance from the axis of rotation. For example, a 10 N force applied 1 m from the axis creates 10 N·m torque.
Enter Angular Displacement
Input the angular displacement (θ) in degrees. This is how much the object rotates under the applied torque. The calculator converts degrees to radians internally. Angular displacement must not be zero (division by zero).
Calculate and Review
Click the "Calculate" button to compute the rotational stiffness in N·m/rad. Rotational stiffness (k_rot) is the rotational equivalent of spring constant, measuring resistance to angular deformation. Higher stiffness means more torque is needed to cause the same angular displacement.
Formula
k_rot = τ / θ
Where:
• k_rot = Rotational stiffness (N·m/rad)
• τ = Torque (N·m)
• θ = Angular displacement (radians)
Note: The calculator converts degrees to radians: θ_rad = θ_deg × (π/180)
This is analogous to linear spring constant: k = F / x (force per displacement)
Example 1: Simple Rotational Spring
A torque of 20 N·m causes an angular displacement of 30°. Calculate the rotational stiffness.
Given:
• Torque (τ) = 20 N·m
• Angular displacement (θ) = 30°
Solution:
Convert to radians: θ = 30° × (π/180) = π/6 rad ≈ 0.524 rad
k_rot = τ / θ = 20 / 0.524
k_rot ≈ 38.2 N·m/rad
Example 2: Torsional Spring
A torsional spring requires 50 N·m torque to rotate 45°. What is its rotational stiffness?
Given:
• Torque (τ) = 50 N·m
• Angular displacement (θ) = 45°
Solution:
θ = 45° × (π/180) = π/4 rad ≈ 0.785 rad
k_rot = 50 / 0.785
k_rot ≈ 63.7 N·m/rad
Frequently Asked Questions
What is rotational stiffness?
Rotational stiffness (k_rot) is the rotational equivalent of linear spring constant. It measures how much torque is required to produce a given angular displacement. Higher rotational stiffness means the object resists rotation more strongly. The relationship is τ = k_rot × θ, analogous to F = kx for linear springs.
How does rotational stiffness differ from linear stiffness?
Linear stiffness (k) relates force to linear displacement: F = kx (units: N/m). Rotational stiffness (k_rot) relates torque to angular displacement: τ = k_rot × θ (units: N·m/rad). Both measure resistance to deformation, but one applies to translation and the other to rotation. They're related when a rotational spring converts to linear motion.
What affects rotational stiffness?
Rotational stiffness depends on: material properties (shear modulus, Young's modulus), geometry (cross-sectional shape, length, radius), and the specific design of the rotational element. For a torsional shaft, k_rot = (GJ)/L, where G is shear modulus, J is polar moment of inertia, and L is length.
Why is angular displacement in radians?
Radians are the natural unit for angular measurements in physics because they relate directly to arc length: s = rθ. The formula τ = k_rot × θ works correctly when θ is in radians. While the calculator accepts degrees for input convenience, it converts to radians for the calculation. One full rotation = 360° = 2π radians.
What happens if angular displacement is zero?
If angular displacement is zero, the calculation would involve division by zero, which is undefined. The calculator will show an error. Rotational stiffness is defined as the ratio of torque to angular displacement, so a non-zero displacement is required. This is similar to how linear spring constant requires non-zero displacement.
Where is rotational stiffness used in real-world applications?
Rotational stiffness is crucial in: torsional springs and dampers, drive shafts and couplings, steering systems (measuring steering wheel resistance), robotic joints and actuators, suspension systems (torsion bars), clocks and watches (balance wheels), and any system where rotational deformation occurs under applied torque.
About Rotational Stiffness Calculator
The rotational stiffness calculator computes the rotational spring constant from torque and angular displacement using k_rot = τ/θ. Rotational stiffness is the rotational equivalent of linear spring constant, measuring resistance to angular deformation.
This calculator is essential for engineers designing rotational systems, analyzing torsional springs, and anyone working with rotational mechanics. Understanding rotational stiffness helps in designing efficient systems that control angular motion, from steering systems to robotic joints.