Mechanical Springs – Types, Stiffness and Work Calculations

We developed a wide variety of mechanical spring types for different applications. And all of these types have their calculations and characteristics. If we must use a type in an engineering application, the selection and design of this mechanical spring must be done at the engineering level. 

Spiral Springs

Mechanical spiral springs.
Spiral spring that is used in mechanical fire alarm systems.

We produce spiral types in a form of strips that we generally make from metals. These metal strips are wound around an axle, which we attach to one side of the strip. The slack side of the strip generally carries the load or force. And this spiral-shaped strip transmits the loads to axles. 

We generally use spiral types for energy storage. The most important use of the spiral springs is the clock mechanism. In clock mechanisms, we use spiral ones both as energy storage with the setting of the clock mechanism. They use the energy stored in the spiral type to work the clock mechanism. We use spiral ones also for the moving actions inside the clock mechanisms. Also, there are lots of kinds of engineering applications that we use spiral types. 

Maximum Bending Stress Calculation in Spiral Springs

If you think about the general mechanism of the spiral ones, we apply torsional stress to the axle. So the acting load type is bending stress. You can use the calculator below to calculate the maximum bending stress occurrence on your mechanical spring and axle system. 

Maximum Bending Stress On Spiral Spring

The use of the calculator above is very simple. You just need to enter the required values inside the given brackets and click on the ‘Calculate!’ button. The result will appear in the last bracket. If you want to make another calculation, just click on the ‘Reset’ button and then enter the new values inside the brackets. 

The first term is the load. The load is the total force acting on the slack side of the spiral one. 

The second term is the radius that they apply this load to them. It is the radial distance between the axle and the slack side. 

Width is the width of the strip of the spiral type and the thickness is the thickness of this strip. 

In parentheses, we recommend some units. If you enter the values in these units, you will have your result in the units given in the parenthesis of the result.

Strain Energy Stored in Spiral Springs

As we stated above, the total strain energy storage in spiral types systems can be very important for different kinds of engineering applications. You can use the calculator below again to calculate the strain energy storage in your spiral applications. 

Total Strain Energy On Spiral Spring

The use of the calculator above is also very simple. Here, the ‘length’ value is the total length of the spiral spring strip. Elasticity modulus is the modulus of elasticity of the materials. We need to calculate the moment of inertia for the cross-sectional area of the spiral spring strip.

As you see above, we added also the angular deflection of the spiral type to the calculator. You can also see the angular deflection result of your spiral-type system. 

Mechanical Leaf Springs

Mechanical leaf spring application.
An example of a leaf spring application(Image

We know leaf ones also as laminated mechanical springs in general mechanics. We use leaf-type systems generally in heavy-duty mechanisms such as trucks, trains, buses, etc. 

Also, clamp leaf types are mechanical structures with multiple numbers of thin and long plates on each other to carry loads. Because of the deflection of the plates which is bending, we obtain a damping effect. 

In general, we produce mechanical leaf springs in three types; elliptical, semi-elliptical, and cantilever. In general types of machinery, the most used type of leaf-spring is the semi-elliptical one. 

General Mechanics of Springs

To understand the mechanics and design characteristics of leaf types, you need to know some terminology and information about general classes. 

Sometimes, engineers are using the term proof resilience which is the maximum strain energy that a mechanical spring can carry. We calculate the modulus of resilience as the maximum strain energy for unit volume. 

In these systems in mechanisms, designers are utilizing their elastic deformation capacity of them. Mechanical spring is actually, specially designed mechanical elements to carry dynamic or static loads more effectively than bulk shapes. 

In general application, they must carry the static or dynamic load in elastic limits. We do not want to see plastic deformation in thö in the working conditions. So, there must be a load value that mechanical spring can carry in elastic limits, which we call a proof load. And we calculate corresponding stress by using this proof load value called proof stress. 

Their stiffness of them is another important value. We define this as the minimum stress value that can cause unit deformation of them.  

The required deflection of them must be linear because we are using the mechanical spring in the elastic region. After the elastic region, the plastic region starts in which the deformation and stress behavior of them is non-linear.

Explanation of Mechanical Leaf Springs on Technical Basis

The structure of leaf mechanisms is very easy. The first layer of the system is the longest plate, which is curved from both ends. Also, we hinge these curves to the axles. We place other plates below the first plate, which is of the same thickness as the first plate. 

We call this first plate also the master leaf, which other leaves are shorter than this. But we can produce the second and third leaves in the same length as the master leaf, to obtain a more rigid structure. 

We clamp all these leaves from the center of the leaves. This clamp also carries the load and transmits the load to the leaves. 

Between the load and axles, leaves are acting as them for damping the dynamic loads and carrying the static loads. 

Bending Stress Calculation of The Mechanical Leaf Springs

To calculate the deflection of a semi-elliptical leaf system, we need to know some symbolic and geometric variables about the leaf system; 

Illustration of a semi-elliptical system.
Illustration of a semi-elliptical leaf spring system(Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.370).

As you see in the representation of a leaf system above, we denote the span of the biggest leaf as ‘l’.

As we stated above that the thicknesses of each plate are the same and we denoted it as ‘t’.

The number of plates that we use in the leaf system is ‘n’.

The bending stress occurrence on the leaf system is ‘σ’.

We denote the deflection of the master plate with ‘θ’.

Again, we stated that the curvature of each plate in the leaf system is the same. We denote the radius of the curvature of the leaf system as ‘r’.

Each plate in the leaf system has the same width. We denote this width value with ‘w’.

The load acting on the leaf system that we denote with the ‘W’.

By knowing all of these values related to a mechanical leaf spring system, you can calculate the total bending stress on the spring system with this calculator;

Bending Stress Calculator For Leaf Springs

The use of the calculator above is very simple. Just enter the required values inside the brackets. Then click on the ‘Calculate!’ button to calculate the bending stress. If you want to do another calculation, just click on the ‘Reset’ button then re-enter the whole values again. 

Deflection Calculation for the Mechanical Leaf Springs

Also, we can easily calculate the total deflection of the leaf system. Just like above, use the calculator below to calculate the total deflection of your leaf system; 

Deflection Calculator For Leaf Springs

There is an additional variable in this calculator is the elasticity modulus of the mechanical spring material.

Strain Energy Calculation for the Mechanical Leaf Springs

Sometimes, stored strain energy can be a very important parameter for engineers. You can calculate the total stored strain energy in leaf systems by using the calculator below; 

Strain Energy Calculator For Leaf Springs

Quarter-Elliptic Mechanical Leaf Springs

We also call the quarter-elliptic leaf springs cantilevered leaf types. So we can make their numerical analysis by consideration of the cantilevered systems. But we need to define some of the variables and geometric parameters;

Schematic illustration of quarter-elliptic leaf system.
Schematic illustration of quarter-elliptic leaf springs(Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.375).

Our first parameter is the load on the quarter-elliptic leaf system, which we denoted as ‘W’ above. 

‘l’ is the length of the longest plate in the quarter-elliptic leaf system. 

‘t’ is the thickness of each leaf in the system. We consider each leaf as the same thickness. 

‘w’ is the width of each leaf. All the leaves are of the same thickness. 

The number of leaves in the quarter-elliptic leaf system that we denote with ‘n’. 

‘E’ is the elasticity modulus of the mechanical leaf spring material. 

Maximum Bending Stress on Quarter-Elliptic Mechanical Leaf Spring

You can calculate the maximum bending stress on the quarter-elliptic leaf by using the calculator below. 

Bending Stress Calculator For Quarter-Elliptic Leaf Spring Systems

The use of the calculator above is very simple. You just need to enter the required values inside the brackets. Then click on the ‘Calculate!’ button to calculate the maximum bending stress value. If you want to make another calculation, just click on ‘Reset’ and then re-enter each value again. 

You can enter your values inside the brackets in the units recommended inside the parentheses.

Deflection Calculations of The Quarter-Elliptic Mechanical Leaf Springs

Deflection Calculator For Quarter-Elliptic Leaf Spring Systems

The Radius of Curvature Calculation 

Under a specific load, the quarter-elliptic leaf system undergoes a specific bending. And each leaf inside the quarter-elliptic leaf system will have the same radius of curvature because of this bending stress. You can calculate this radius of curvature value of the quarter-elliptic leaf systems. 

Radius Of Curvature Calculator For Quarter-Elliptic Leaf Springs

Stiffness of Mechanical Springs

Illustration of the close coiled helical type.
Illustration of the close coiled helical spring(Image Source: Mikell P. Groover; Fundamentals Of Modern Manufacturing, 4th edition, pg 382).

Helical mechanical springs are one of the most used types in most engineering applications. And in various applications, axial forces are acting on these helical mechanical springs. We define the stiffness values according to these axial forces. And these stiffness values have the formula;

Spring stiffness formula.

In this formula; 

  • ‘G’ is the modulus of rigidity of the material. 
  • ‘d’ is the diameter of the wire in which we make the helical. 
  • ‘R’ is the radius of the helical one. 
  • ‘n’ is the number of turns of the total helical mechanical spring geometry. 

As you see above, the calculation of the mechanical spring stiffness coefficient is very simple, when the axial forces are present on helical mechanical springs. 

What is the Angular Stiffness?

Close-coiled helical springs can undergo angular twist action. In general, we do not utilize this property of helical springs in general mechanical applications. But in some cases, we must consider this physical fact. 

With this twisting action, we must define the ‘angular stiffness’ value of mechanical springs. We defined this angular stiffness value with the equation below;

Angular stiffness formula of the mechanical springs.

With this new formula; 

  • ‘E’ is the elasticity modulus of the mechanical spring material.

Parallel and Serial Combinations of Mechanical Springs

Mechanical springs are generally combined in a serial and parallel manner. And this combination must be considered in the total stiffness value of this system. 

Here we explain how to calculate the stiffness of the parallel and serial combination springs. 

Stiffness Calculation in Mechanical Spring Combinations

As we stated above, these systems are considered serial or parallel. And the calculation of the spring coefficients which are also called stiffness is very simple. 

Parallel Combined Mechanical Springs

Parallel springs.
Parallel spring combination(Image

The parallel systems are like the above as you see in general. If you consider these two mechanical springs are combined as parallel, the total stiffness of this parallel system is calculated below;

Stiffness of the parallel mechanical springs.

As you understand from this equation, the stiffness’ of individual mechanical springs of a parallel system is summed up directly. This means that if you want to obtain a more stiff system in your physical model, you can combine multiple springs as parallel. 

Serial Combined Mechanical Springs

Serial springs.
Serial spring combination(Image

If the individual mechanical springs are combined like above, the total system is called serial springs. The total stiffness calculation of the serial systems is different from the parallel system.

Total stiffness of serial mechanical springs.

As you see in the calculation, the reverse of the total stiffness of the serial system concerning multiplication is equal to the summation of the reverse of individual stiffness’ of the mechanical springs concerning multiplication again. 

Complex Mechanical Spring Combinations

With these two calculation principles, you can easily calculate the total stiffness of complex systems. In the calculation of these systems, you must start the calculation from the small system to the bigger system. Select the small system inside the whole system, which is far away from the load or loads. 

Mechanical Spring Work Calculations

Firstly, mechanical spring work is a type of mechanical work which you calculate as the difference between the first and second squeezing distances of a mechanical spring. You can release a portion of energy to make work such as by pushing a body or box forward. And also, calculate the work with the formulation below;

Work formula.

So in this formula ‘k’ is the stiffness coefficient of the mechanical spring which has the unit of N.m or lb.ft in English units. ‘x1’ and ‘x2’ are the first and second distances the spring is squeezed. The units of these distances are m or ft. The unit of the mechanical work that we calculate for mechanical springs is the W or ft.lbf/s). 

We use them in lots of machinery. And also the clockwork mechanisms are the most important examples of the energy use. And then, the stored energy is released controllably with the clockwork mechanism to give the power of working and showing the correct time. So, we make strict work calculations in the design of clockwork mechanisms. 

While you are making strict calculations with mechanical springs in mechanical designs, you will probably use the work calculations lots of the time. To eliminate this burden and time-consuming hand calculations, we prepared an easy-to-use and useful work calculator. 

Spring Work Calculator

The use of the work calculator is very simple above. You just need to enter the stiffness coefficient of the mechanical spring and squeeze distances inside the brackets. And click on the ‘Calculate!’ button to see the work result. 

If you want to make another calculation, just click on the ‘Reset’ button and re-enter the new values. 

Conclusion on Mechanical Springs

We explained all the important information about the mechanical springs above. You can find all the important points about mechanical spring calculations. And you will not face any problems while you are calculating the mechanical springs.

These are the general points on the mechanical springs. Finally, do not forget to leave your comments and questions below about the mechanical springs. Your precious feedbacks are very important to us.



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