Single Degree of Freedom Systems

When it comes to engineering, it is often ideal to simplify a system in a way that makes it easier to work out any necessary calculations. This process of modeling a system is very common in a variety of engineering fields, and quite possibly the simplest system that can be modeled is a single degree of freedom (SDOF) system.

SDOF systems may be simple, but they have actual applicability to various engineering problems, and if an engineer is able to understand how an SDOF system can be described and modeled using an equation, it is not much of a leap to go from there to higher levels of engineering systems.

Single Degree of Freedom Systems

In the most general terms, an SDOF describes the motion of a system that is constrained to only a single linear or angular direction. This means that the system will only move in the x-, y-, or z-direction, or that the system will only rotate about the x-, y-, or z-axis.

To put this in perspective, an airplane moves in six degrees of freedom: it moves forward and backward in the x-direction, right and left in the y-direction, up and down in the z-direction, rolls about the x-axis, pitches about the y-axis, and yaws about the z-axis. But beginning with a system that is constrained to moving in only one of these six ways provides a solid foundation for subsequent modeling of systems that move in multiple ways.

The simplest SDOF system often describes a system that moves only linearly in the x- or y-direction.

Describing a Single Degree of Freedom System

An SDOF is often described using a damped spring-mass system in the x-direction. A mass is attached to a spring and a damper, which are both fixed at the opposite end. The system starts with some initial displacement and velocity, although usually, the initial velocity is zero.

After the mass is released, it will oscillate or vibrate in the x-direction as the spring is stretched and contracts. The damper, however, will reduce the magnitude of this oscillation until the mass is no longer moving. In this system, things like gravity and friction are often ignored, although both can be included.

The way the system moves can be described using an equation of motion which satisfies Newton’s Second Law. This law is simplified to the following:

newton's second law

where F is a force acting on some mass, m, causing some acceleration, a. This equation can be rearranged in countless ways, including to describe the motion of an SDOF system. This equation is referred to as the equation of motion of the system.

Equation of motion for a Single Degree of Freedom System


single degree of freedom system
A mass suspended by a spring is a single degree of freedom system (SDOF).

An SDOF system can be described by a second-order, non-homogenous, ordinary differential equation (ODE). This equation can be simplified as follows:

single degree of freedom system


  • m is the mass,
  • cv is the damping coefficient,
  • k is the spring constant,
  • x is the linear displacement,
  • ẋ is the linear velocity,
  • ẍ is the linear acceleration.

If things like gravity and friction are to be included, they can be incorporated into the values of cv and k.

To solve this ODE, it is necessary to specify the initial conditions, which are usually as follows:

where t0­ is the initial time, ­x0 is the initial displacement, and v0 is the initial velocity. As stated earlier, the initial velocity is usually zero, but it is not completely uncommon to have some initial velocity of the system. The initial displacement can be set to zero, or some other value, usually depending on the engineering preference or the types of values that the engineer will need in subsequent calculations.

By solving this ODE, a plot of the system as a function of the time can be generated, which will show the position of the mass oscillating with the oscillations becoming less and less until the mass stops moving.

Solving the ODE for a Single Degree of Freedom System

There are many ways to solve the type of ODE that is used to describe an SDOF system. With advancements in computers, several solvers, including online examples, have been developed. In fact, most engineers are going to use some type of computer software to solve the ODE that describes their SDOF system.

Whether the ODE is solved by hand or with a computer, the final solution will provide the engineer with a numerical description of what the SDOF system is doing. As discussed, for the simplest case of a damped spring-mass system, as time varies, the linear x-position of the system will change, and these values can be plotted against time.


Just because a general description of an SDOF is a damped spring-mass system does not mean there are not variations. For example, the system may not be damped – in which case the magnitude of the oscillations would never decrease. Or the system may be driven – in which case the magnitude of the oscillations would increase, rather than decrease.

For any system that only moves in a single degree of freedom, a corresponding model can be developed to simplify that system, and from that model, an equation of motion and necessary initial conditions can be specified to solve for the motion of the system.


As stated, with a firm understanding of modeling an SDOF system, it is possible to model systems that have multiple degrees of freedom. The concepts are very similar, but the system may be moving in various combinations of the x-, y-, and z-directions and rotations about the x-, y-, and z-axes.

Real-Life Single Degree of Freedom Systems

While it is great to have a general concept that describes an SDOF system, these mathematical models can be used to describe real systems that have actual purposes.

Example SDOF systems

One such example is an accelerometer, which is used to determine how much an object accelerates in a given direction. As the object moves, the attached accelerometer measures and reports that acceleration. In this case, the equation of motion previously introduced can be reformatted as follows:


is the acceleration of the object and


Using the above equation with the specified coefficients cv and k, the acceleration of the object can be worked out and plotted in a fashion like plotting the response of the SDOF system discussed earlier.

Another good example of an SDOF system in action is a seismometer, which can be used to detect and measure earthquakes. However, in the case of a seismometer, the movement usually takes the form of a sinusoidal function that involves some angular motion. While this can complicate the process of solving the ODE, the result will still be a useful simplification of the system that can be helpful for subsequent calculations and modeling.

Examples of applying SDOF systems

When it comes to applying SDOF systems to actual engineering problems, it must be remembered that what has been designed is a model of the system. Using this model, an engineer will be able to evaluate the motion that the system will undergo under various conditions. With this information, it is possible to determine if the system will experience any undesired behavior.

From the model of an SDOF system, an actual working system can be built. For example, accelerometers are very common in several applications, and they can all trace their operation to a model of an SDOF system. Depending on how accurate the calculations involved in the model are, referred to as the fidelity of the model, the model will be able to mimic the actual system to varying degrees of acceptability.

Tools for Modeling Single Degree of Freedom Systems

As alluded to, in today’s digitally connected world, engineers often take advantage of computers to solve things like the ODE of an SDOF system model. However, the solution to an ODE is not where computers end when it comes to SDOF systems.

Software tools have been written and developed to help engineers build their models of SDOF systems. An example of a widely used piece of software is Mathworks’ Simulink, which is an add-on for their MATLAB software.

Because an SDOF system is a very simple example of a system that an engineer may encounter, it is quite possible for engineers with even modest programming skills to write their own code using Excel VBA to build a model of a single degree of freedom system.


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By Charlie Young, P.E.

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