Torsion Spring Calculator

Calculate the spring rate and maximum torque of a torsion spring.

Spring Properties

Understanding Torsion Springs and Their Calculations

Torsion springs are mechanical devices designed to store rotational energy and exert or absorb torque (a twisting force). Unlike compression or extension springs that operate along a linear axis, torsion springs function by applying a torque in response to an angular displacement. They are commonly found in applications such as clothespins, garage doors, hinges, and various mechanical linkages where a rotational force is required.

The key parameters for calculating torsion spring performance include the wire diameter, mean coil diameter, number of active coils, and the material's modulus of rigidity. The behavior of the spring is also dependent on the angles through which it is twisted.

Core Formulas

The primary calculations for a torsion spring involve determining its spring rate (or spring constant) and the torque it produces at a given angle.

1. Spring Rate (k)

The spring rate of a torsion spring quantifies how much torque is produced per degree of angular deflection. It is typically calculated using the following formula:

k = (E * I_p) / (N * 360 degrees/revolution)

Where:

• k is the spring rate in units of torque per degree (e.g., N-mm/degree or lb-in/degree).
• E is the Young's Modulus of the material (in units of pressure, e.g., MPa or psi).
• I_p is the polar moment of inertia of the wire's cross-section. For a circular wire, I_p = (π * d^4) / 32, where d is the wire diameter.
• N is the number of active coils.

However, a more common and direct formula for torsion springs, especially when using the Modulus of Rigidity (G) for torsional applications, is:

k = (G * d^4) / (8 * D * N) (This simplified formula assumes a deflection of 360 degrees per coil and uses G directly).

A more precise formula that accounts for the geometry and material properties, relating torque to angular deflection, is:

Torque = (G * d^4 * θ) / (10.8 * D * N)

From this, we can derive the spring rate k = Torque / θ, which leads to:

k = (G * d^4) / (10.8 * D * N) (where k is typically in N-mm/degree or similar units if G and D are consistent)

2. Torque (T)

The torque produced by a torsion spring is directly proportional to the angle through which it is deflected. The formula is:

T = k * Δθ

Where:

• T is the torque (e.g., N-mm, lb-in).
• k is the spring rate calculated above.
• Δθ is the change in angle from the free position (in degrees or radians, depending on the units of k).

Using the more comprehensive formula directly:

T = (G * d^4 * (θ_end - θ_start)) / (10.8 * D * N)

Note: The constant 10.8 is derived from the geometry of a circular wire cross-section and the relationship between material moduli. Units must be consistent (e.g., if G is in MPa and D is in mm, wire diameter d is in mm, then torque T will be in N-mm).

Calculator Inputs Explained:

• Wire Diameter (d): The diameter of the wire used to form the spring coil.
• Mean Coil Diameter (D): The average diameter of the spring coils.
• Number of Active Coils (N): The number of coils that contribute to the spring's deflection. Legs are typically not counted as active coils.
• Material Modulus of Rigidity (G): A material property that describes its resistance to shear deformation. For spring steel, a common value is around 79,300 MPa (or 11.5 x 10^6 psi).
• Starting Angle (degrees): The initial angular position of the spring in its free state or application.
• End Angle (degrees): The final angular position of the spring after deflection.

Example Calculation:

Let's consider a torsion spring with the following properties:

• Wire Diameter (d) = 3 mm
• Mean Coil Diameter (D) = 25 mm
• Number of Active Coils (N) = 12
• Material Modulus of Rigidity (G) = 79,300 MPa
• Starting Angle = 0 degrees
• End Angle = 90 degrees

First, calculate the change in angle: Δθ = 90 - 0 = 90 degrees.

Now, calculate the torque: T = (79300 MPa * (3 mm)^4 * 90 degrees) / (10.8 * 25 mm * 12) T = (79300 * 81 * 90) / (10.8 * 25 * 12) T = 577998000 / 3240 T ≈ 178,394 N-mm

The calculated torque indicates the force required to rotate the spring from its starting to its ending angle. This value is crucial for ensuring the spring can perform its intended function within the application's constraints.

Please enter valid positive numbers for all fields.

End angle is less than start angle. Torque will be negative (or spring is unwinding).

Calculation resulted in an invalid number. Check inputs.

${formattedTorque} N-mm