Designing a Steel Spring: Calculations, Parameters, and Stability

Springs are a favorite topic for the speaker as they provide a foundation for discussing design in engineering. Design involves making choices about parameters for devices. The speaker plans to cover the design of Springs in detail in the next session.

The problem involves supporting a 200-pound load using a steel spring with nine coils. The spring has squared and ground ends, a wire diameter of 0.15 inches, an outer coil diameter of one inch, and a free length of six inches. The focus is on determining stress, spring constant, and deflection.

Springs can buckle, leading to instability. The speaker discusses the stability of the spring, including parameters for achieving unconditional stability. Factors like fractional overrun-closure and spring index are also considered to ensure stability.

Designing a spring involves numerous interdependent constraints such as coil count, wire diameter, coil diameter, and end configurations. Springs may need to fit in holes or around rods, adding to the complexity of design tasks.

To determine the maximum stress in the spring carrying a 200-pound load, reference is made to Chapter 10, specifically page 511, which discusses stress in the wire of a spring.

When squishing a spring downward, it changes the pitch of the spring, which is the number of coils per inch or the length per coil. This action twists the wire composing the spring, generating the largest component of stress in the spring element.

Apart from torsional stress, direct shear is induced in springs. This shear arises from the reaction forces at a cut in the spring, transmitting linear loads between different parts of the spring.

In addition to torsional and direct shear stresses, there is a curvature effect in curved springs. This effect creates a non-uniform stress distribution, necessitating the use of a Bergstresser factor to adjust stress calculations.

To calculate the Bergstresser factor, one must first determine the spring index, which is the coil diameter divided by the wire diameter. The coil diameter refers to the mean coil diameter, obtained by subtracting the wire diameter from the outer coil diameter.

The wire diameter mentioned is 0.46. The calculated spring index is 5.667, which is a unitless quantity. The actual wound up diameter is 0.85, denoted as D, and the spring index is denoted as capital C. The spring index is crucial for determining the Bergstresser factor.

The Bergstresser factor, denoted as K sub B, is calculated as 1.254 using the formula 4*(spring index) + 2 / 4*(spring index) - 3. This factor is essential for determining the shearing stress induced in the spring.

The shearing stress, denoted as tau, is calculated using the Bergstresser factor (1.254), force (200 pounds), mean coil diameter (0.85 inches), and wire diameter (0.15 inches). The resulting shearing stress is 160.877 ksi.

The calculated shearing stress of 160.877 ksi may seem high, but it is within an acceptable range for spring materials made of high-strength steels. Typically, a maximum shear stress of 130-150 ksi is considered safe.

The next step involves finding the spring constant and the deflection of the spring under the given load. A detailed derivation using an energy method on page 512 provides insights into determining these crucial parameters.

The spring constant equation, as per equation ten nine, states that the spring constant is equal to the wire diameter to the fourth power (0.15 inches to the fourth power) multiplied by the modulus of rigidity, denoted as G.

For steel material, the modulus of rigidity value can be found in table ten fifteen, which specifies carbon steel as having a value of eleven point five million pounds per square inch (11.5 M psi).

In the upcoming discussions, both the strength and elastic modulus values of materials will become variable, no longer remaining constant as they are currently assumed to be.

In the design of compression springs, the end preparations play a crucial role. Options like plain ends, squared ends, or closed ends impact the distribution of forces and moments on the spring, influencing its stability and performance.

There are different ways to modify spring ends to distribute force more evenly. One method involves changing the pitch of the last coil to increase contact area. Another method is to grind the last coil to achieve a similar effect. The most effective approach is to use squared and ground ends, which combines changing the pitch and grinding the end to maximize contact area and balance the force distribution.

Having prepared ends on a spring, such as squared and ground ends, is crucial for its effectiveness. Improperly cutting or modifying spring ends can reduce its efficiency. Prepared ends ensure optimal contact area and force distribution, enhancing the spring's performance.

For squared and ground ends, the total number of coils is equal to the number of active coils plus two. By subtracting two from the total number of coils, the active coils for squared and ground ends can be determined. This calculation method provides a value for active coils that optimizes the spring's functionality.

The spring constant is calculated to be 169.285 pounds per inch. This value represents the stiffness of the spring and is crucial for determining its deflection under different loads.

The deflection of the spring under a 200-pound load is calculated using Hooke's Law formula, which states that deflection (Delta) equals force (F) divided by the spring constant (K). The deflection is found to be 1.181 inches.

Fractional overrun-closure is a parameter used to analyze the behavior of a spring. It involves calculating the difference between the spring's free length and its solid height. This metric provides insights into the spring's compression characteristics and is essential for designing and evaluating its performance.

The speaker introduces the concept of fractional overrun closure, which is a valuable tool for optimized spring design. They mention a video game reference to Qbert, explaining how the Greek letter I reminds them of a snake bouncing on a pyramid.

The speaker discusses the equation for fractional overrun closure, which is implicitly defined in equation 1017. They explain that FS represents the force a spring carries when it goes solid, with FMax being the required force, such as 200 pounds in their example.

The speaker explains how to calculate the solid length of the spring, which is crucial for determining the fractional overrun closure. They mention that the solid length (Ls) is calculated as the wire diameter multiplied by the total number of coils, providing a detailed example with specific values.

The speaker elaborates on calculating the deflection when the spring goes solid. They mention the free length of the spring, the solid length, and how to determine the deflection. Specific calculations are provided to illustrate the process.

The speaker demonstrates how to determine the fractional overrun closure using the calculated force when the spring goes solid. They show the equation involving the maximum force and the force at solid state, providing a detailed example with specific numerical values.

The fractional over-under closure parameter, as defined on page 520, is expected to be greater than or equal to 0.15. This parameter indicates that there is approximately 15% more compression available before the spring reaches a solid state. Additionally, the guidelines mention that the number of active coils is typically between 3 and 15, and the spring index should fall between 4 and 12.

The question of whether a spring will buckle under an applied load is crucial, especially for unsupported springs. Methods to prevent buckling include inserting a rod through the spring or placing it in a hole for support. Understanding the potential for buckling is essential to prevent failure in applications where springs are exposed and susceptible to buckling.

Equation 1010, denoted as Delta sub C R, represents the critical deflection at which a spring may buckle. It is calculated as the free length of the spring multiplied by a parameter c1 prime, adjusted by the effective slenderness ratio lambda e FF. This equation provides insight into the amount of deflection a spring can undergo before buckling occurs.

The speaker introduces the concept of the slenderness ratio, denoted as L sub 0, which represents the free length of a spring. The free length mentioned is six inches. Additionally, the speaker mentions the unknown parameters C1 prime and C2 prime that affect the slenderness ratio calculation.

To calculate the slenderness ratio, the speaker explains the formula involving the parameter alpha, which is multiplied by the free length divided by the mean coil diameter of the spring. Alpha accounts for various end conditions on springs and is crucial for determining the slenderness ratio.

The speaker discusses different end conditions for helical compression springs, such as springs supported between flat parallel surfaces, one end supported by a flat surface perpendicular to the spring axis, and the other end pivoted. These end conditions impact the value of alpha and subsequently affect the spring's behavior.

The speaker explains how different end conditions, such as having one end clamped and the other end free, or having no support for the spring, can influence the slenderness of a spring. Less support for the end of a spring can contribute to a higher slenderness effect.

An example calculation is provided where the free length of six inches and the mean coil diameter of 0.85 inches are used to calculate the slenderness ratio, resulting in a value of 3.529. This calculation demonstrates the practical application of the slenderness ratio formula.

The speaker mentions two dimensionless elastic constants, C1 Prime and C2 Prime, which are described as dimensionless elastic constants without specific names or equation numbers. These constants are calculated using the formula E/2 times E minus G and 2 pi squared times E minus G over 2G plus E, respectively.

By plugging in the values into the equation, the calculated value for the spring is 6.89, and the deflection is predicted to be 1.613 inches.

Comparing the predicted deflection of 1.613 inches to the actual deflection of 1.181 inches, it is concluded that the spring is not expected to buckle under the applied load.

There is a concern about overloading the spring, as exceeding the predicted deflection may lead to buckling. It is crucial to ensure the spring's stability under any circumstances.

Equation 1012 provides a method to calculate if a spring will buckle under any circumstances. If the condition is met, the spring will be unconditionally stable.

To find the load that would make the spring buckle, Hookes law is applied. The critical buckling force is calculated to be approximately 273 pounds.

To achieve absolute stability, the minimum outer diameter needed for the spring is calculated to be 1.143 inches. Increasing the diameter would ensure the spring never buckles throughout its entire range.

To calculate the new wire diameter, the spring constant of 169.285 pounds per inch is maintained. Using the equation G/8 * D^4, with a mean coil diameter of 1.143 inches and 7 coils, the new wire diameter is found to be 0.1873 inches, resulting in a new outer diameter of 1.33 inches.

With the updated wire and outer diameters, the new fractional overrun is 2.652, the new spring index is 6.10, the Newburgh stress factor is 1.234, and the shearing stress is 109.28 ksi. The solid length is 1.686 inches, leading to a deflection of 4.314 inches and a force of 730.35 pounds.

By adjusting the wire diameter, the stress in the spring was reduced to 109.28 ksi, showing a slight decrease in stress levels. This adjustment resulted in a lower amount of shearing stress, indicating an improvement in the spring's performance.