Natural Undamped Frequency, Damping Ratio, and Transfer Function of Mass-Spring Damper System

In this control engineering tutorial, we derive the natural undamped frequency, damping ratio, and transfer function of a mass-spring damper system. The YouTube tutorial is given below.

Consider the mass-spring-damper system shown in the figure below.

In the figure above, is the displacement of the mass, is mass, is the damping constant, is the spring constant, is the external control force, and is the external force disturbance. From Newton’s second law, we obtain

(1)  

where is the acceleration, and is the velocity. Here, for simplicity, we have assumed a positive sign of the force disturbance.

Our goal is to transform the model (1) into the prototype transfer function form of the second-order system:

(2)  

where

• is the complex Laplace variable
• is the Laplace transform of
• is the Laplace transform of the control input . The control input will be defined later on.
• is the natural undamped frequency
• is the damping ratio

From (2), we have

(3)  

By applying the inverse Laplace transform to the equation (3), we have

(4)  

On the other hand, by dividing (1) by , we obtain

(5)  

By comparing (4) and (5), we obtain

(6)  

From the second equation in (6), we obtain the natural undamped frequency as a function of the spring constant and the spring-mass

(7)  

By substituting (7) in the first equation of (6), we obtain

(8)  

Finally, we obtain the damping ratio as the function of the damping constant , spring-mass , and spring constant

(9)  

By using (7) and (9) in (5), we obtain

(10)  

By defining

(11)  

and by substituting (11) in the second equation of (10), we obtain

(12)  

By applying the Laplace transform to (12), we obtain the system description in the Laplace domain

(13)  

where is the Laplace transform of and is the Laplace transform of .