Control System Analysis, Electrical Engineer

Mass-Spring-Damper System Dynamics

Basic elements of a mechanical system.

The basic elements of any mechanical system are the mass, the spring and the shock absorber, or damper. The study of movement in mechanical systems corresponds to the analysis of dynamic systems. In Robotics, for example, the word Forward Dynamic refers to what happens to actuators when we apply certain forces and torques to them.

The mass, the spring and the damper are basic actuators of the mechanical systems.

Consequently, to control the robot it is necessary to know very well the nature of the movement of a mass-spring-damper system.

In addition, this elementary system is presented in many fields of application, hence the importance of its analysis. Again, in robotics, when we talk about Inverse Dynamic, we talk about how to make the robot move in a desired way, what forces and torques we must apply on the actuators so that our robot moves in a particular way.


I recommend the book “Mass-spring-damper system, 73 Exercises Resolved and Explained” I have written it after grouping, ordering and solving the most frequent exercises in the books that are used in the university classes of Systems Engineering Control, Mechanics, Electronics, Mechatronics and Electromechanics, among others.

If you need to acquire the problem solving skills, this is an excellent option to train and be effective when presenting exams, or have a solid base to start a career on this field. Take a look at the Index at the end of this article. 

Before performing the Dynamic Analysis of our mass-spring-damper system, we must obtain its mathematical model. This is the first step to be executed by anyone who wants to know in depth the dynamics of a system, especially the behavior of its mechanical components.

We will begin our study with the model of a mass-spring system.

This is convenient for the following reason. All the mechanical systems have a nature in their movement that drives them to oscillate, as when an object hangs from a thread on the ceiling and with the hand we push it. Or a shoe on a platform with springs. It is good to know which mathematical function best describes that movement.

But it turns out that the oscillations of our examples are not endless. There is a friction force that dampens movement. In the case of the object that hangs from a thread is the air, a fluid. So after studying the case of an ideal mass-spring system, without damping, we will consider this friction force and add to the function already found a new factor that describes the decay of the movement.


Mass-Spring System.


Figure 5

The dynamics of a system is represented in the first place by a mathematical model composed of differential equations. In the case of the mass-spring system, said equation is as follows:

This equation is known as the Equation of Motion of a Simple Harmonic Oscillator. Let’s see where it is derived from.

If our intention is to obtain a formula that describes the force exerted by a spring against the displacement that stretches or shrinks it, the best way is to visualize the potential energy that is injected into the spring when we try to stretch or shrink it. The following graph describes how this energy behaves as a function of horizontal displacement:

As the mass m of the previous figure, attached to the end of the spring as shown in Figure 5, moves away from the spring relaxation point x = 0 in the positive or negative direction, the potential energy U (x) accumulates and increases in parabolic form, reaching a higher value of energy where U (x) = E, value that corresponds to the maximum elongation or compression of the spring. The mathematical equation that in practice best describes this form of curve, incorporating a constant k for the physical property of the material that increases or decreases the inclination of said curve, is as follows:

The force is related to the potential energy as follows:


It makes sense to see that F (x) is inversely proportional to the displacement of mass m. Because it is clear that if we stretch the spring, or shrink it, this force opposes this action, trying to return the spring to its relaxed or natural position. For that reason it is called restitution force. The above equation is known in the academy as Hooke’s Law, or law of force for springs. The following is a representative graph of said force, in relation to the energy as it has been mentioned, without the intervention of friction forces (damping), for which reason it is known as the Simple Harmonic Oscillator. It is important to emphasize the proportional relationship between displacement and force, but with a negative slope, and that, in practice, it is more complex, not linear.

Source: Física. Robert Resnick

For an animated analysis of the spring, short, simple but forceful, I recommend watching the following videos: Potential Energy of a Spring, Restoring Force of a Spring


Sistema MRA


Going back to Figure 5:

We go to Newton’s Second Law:

This equation tells us that the vectorial sum of all the forces that act on the body of mass m, is equal to the product of the value of said mass due to its acceleration acquired due to said forces. Considering that in our spring-mass system, ΣF = -kx, and remembering that acceleration is the second derivative of displacement, applying Newton’s Second Law we obtain the following equation:

Fixing things a bit, we get the equation we wanted to get from the beginning:

This equation represents the Dynamics of an ideal Mass-Spring System.

Apart from Figure 5, another common way to represent this system is through the following configuration:


In this case we must consider the influence of weight on the sum of forces that act on the body of mass m. The weight P is determined by the equation P = m.g, where g is the value of the acceleration of the body in free fall.

If the mass is pulled down and then released, the restoring force of the spring acts, causing an acceleration ÿ in the body of mass m. We obtain the following relationship by applying Newton:

If we implicitly consider the static deflection, that is, if we perform the measurements from the equilibrium level of the mass hanging from the spring without moving, then we can ignore and discard the influence of the weight P in the equation. If we do y = x, we get this equation again:

Mass-spring-damper System


If there is no friction force, the simple harmonic oscillator oscillates infinitely. In reality, the amplitude of the oscillation gradually decreases, a process known as damping, described graphically as follows:

The displacement of an oscillatory movement is plotted against time, and its amplitude is represented by a sinusoidal function damped by a decreasing exponential factor that in the graph manifests itself as an envelope. The friction force Fv acting on the Amortized Harmonic Movement is proportional to the velocity V in most cases of scientific interest. This force has the form Fv = bV, where b is a positive constant that depends on the characteristics of the fluid that causes friction. This friction, also known as Viscose Friction, is represented by a diagram consisting of a piston and a cylinder filled with oil:

The most popular way to represent a mass-spring-damper system is through a series connection like the following:

Figura 6

As well as the following:

In both cases, the same result is obtained when applying our analysis method. Considering Figure 6, we can observe that it is the same configuration shown in Figure 5, but adding the effect of the shock absorber. Applying Newton’s second Law to this new system, we obtain the following relationship:

This equation represents the Dynamics of a Mass-Spring-Damper System.

Laplace Transform of a Mass-Spring-Damper System

A solution for equation (37) is presented below:

Equation (38) clearly shows what had been observed previously. An example can be simulated in Matlab by the following procedure:


The shape of the displacement curve in a mass-spring-damper system is represented by a sinusoid damped by a decreasing exponential factor. It is important to understand that in the previous case no force is being applied to the system, so the behavior of this system can be classified as “natural behavior” (also called homogeneous response). Later we show the example of applying a force to the system (a unitary step), which generates a “forced behavior” that influences the final behavior of the system that will be the result of adding both behaviors (natural + forced). Remark: When a force is applied to the system, the right side of equation (37) is no longer equal to zero, and the equation is no longer homogeneous.

The solution for the equation (37) presented above, can be derived by the traditional method to solve differential equations. However, this method is impractical when we encounter more complicated systems such as the following, in which a force f(t) is also applied:

Figura 7

The need arises for a more practical method to find the dynamics of the systems and facilitate the subsequent analysis of their behavior by computer simulation. The Laplace Transform allows to reach this objective in a fast and rigorous way.

In equation (37) it is not easy to clear x(t), which in this case is the function of output and interest. A differential equation can not be represented either in the form of a Block Diagram, which is the language most used by engineers to model systems, transforming something complex into a visual object easier to understand and analyze.The first step is to clearly separate the output function x(t), the input function f(t) and the system function (also known as Transfer Function), reaching a representation like the following:

r(t)=f(t), c(t)=x(t)

The Laplace Transform consists of changing the functions of interest from the time domain to the frequency domain by means of the following equation:

The main advantage of this change is that it transforms derivatives into addition and subtraction, then, through associations, we can clear the function of interest by applying the simple rules of algebra. In addition, it is not necessary to apply equation (2.1) to all the functions f(t) that we find, when tables are available that already indicate the transformation of functions that occur with great frequency in all phenomena, such as the sinusoids (mass system output, spring and shock absorber) or the step function (input representing a sudden change). In the case of our basic elements for a mechanical system, ie: mass, spring and damper, we have the following table:

That is, we apply a force diagram for each mass unit of the system, we substitute the expression of each force in time for its frequency equivalent (which in the table is called Impedance, making an analogy between mechanical systems and electrical systems) and apply the superposition property (each movement is studied separately and then the result is added).

Figure 2.15 shows the Laplace Transform for a mass-spring-damper system whose dynamics are described by a single differential equation:



The system of Figure 7 allows describing a fairly practical general method for finding the Laplace Transform of systems with several differential equations. First the force diagram is applied to each unit of mass:

For Figure 7 we are interested in knowing the Transfer Function G(s)=X2(s)/F(s).

Arranging in matrix form the equations of motion we obtain the following:

Equations (2.118a) and (2.118b) show a pattern that is always true and can be applied to any mass-spring-damper system:

The immediate consequence of the previous method is that it greatly facilitates obtaining the equations of motion for a mass-spring-damper system, unlike what happens with differential equations. In addition, we can quickly reach the required solution. In the case of our example:


These are results obtained by applying the rules of Linear Algebra, which gives great computational power to the Laplace Transform method.

Application examples ...under construction

Example 1.

Exercise B318, Modern_Control_Engineering, Ogata 4t p 149 (162),




Answer Link: Ejemplo 1 – Función Transferencia de Sistema masa-resorte-amortiguador

Example 2.

  1. Control Systems Engineering, Nise, p 101

Answer Link: Ejemplo 2 – Función Transferencia de sistema masa-resorte-amortiguador

Rotational Case

So far, only the translational case has been considered. In the case that the displacement is rotational, the following table summarizes the application of the Laplace transform in that case:


The following figures illustrate how to perform the force diagram for this case:



We observe that again it is true that:



  1. Robert Resnick, tomo1
  2. Dinamica_de_Sistemas, Katsuhiko Ogata
  3. Control Systems Engineering, Norman Nise
  4. Sistemas de Control Automatico, Benjamin Kuo
  5. Ingenieria de Control Moderna, 3° ED. – Katsuhiko Ogata

I recommend the book “Mass-spring-damper system, 73 Exercises Resolved and Explained” I have written it after grouping, ordering and solving the most frequent exercises in the books that are used in the university classes of Systems Engineering Control, Mechanics, Electronics, Mechatronics and Electromechanics, among others.

If you need to acquire the problem solving skills, this is an excellent option to train and be effective when presenting exams, or have a solid base to start a career on this field. 

Preface ii
Introduction iii
 Chapter 1———————————————————- 1
o Mass-spring-damper System (translational mechanical system)
 Chapter 2———————————————————- 51
o Mass-spring-damper System (rotational mechanical system)
 Chapter 3———————————————————- 76
o Mechanical Systems with gears
 Chapter 4———————————————————- 89
o Electrical and Electronic Systems
 Chapter 5——————————————————— 114
o Electromechanical Systems – DC Motor
 Chapter 6——————————————————— 144
o Liquid level Systems
 Chapter 7——————————————————— 154
o Linearization of nonlinear Systems
 References——————————————————- 164


If what you need is to determine the Transfer Function of a System … We deliver the answer in two hours or less, depending on the complexity. In digital Contact us, immediate response, solve and deliver the transfer function of mass-spring-damper systems, electrical, electromechanical, electromotive, liquid level, thermal, hybrid, rotational, non-linear, etc. Optional, Representation in State Variables. Simulation in Matlab, Optional, Interview by Skype to explain the solution.

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Written by Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer

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