Dynamics of Structures

Chapter 2. Damped Single Degree of Freedom System

2.2 Developing and Solving the Equation of Motion

Figure 2.2.1 Mass-spring System and Free Body Diagram

The Eq. of motion can be developed using the shown free body diagram. The term mü represents the inertial force developed opposite to the motion of the mass. The term ců represents the damping force while the term ku represents the stiffness of the system. From the dynamic equilibrium.

Eq. 2.2.1

To solve this differential equation, we can assume the solution in the form.

Eq. 2.2.2

Subsituiting the assumed solution, and its derivatives, in the equation of motion, the characteristic equation can be expressed as

Eq. 2.2.6

The parameter λ can be derived in Eq. 2.2.18, where ζ is known as the damping ratio and depend on the damping constant c, the system stiffness k, and the system mass m. To show the full derivation of Eq. 2.2.18, click on the 'Show full derivation' tab below,

Eq. 2.2.18

From the Free Body DIagram, the Eq. of motion can be developed as,

Eq. 2.2.1

To solve this differential equation, we can assume the solution in the form.

Eq. 2.2.2

Differentiate the solution, we get.

Eq. 2.2.3

and.

Eq. 2.2.4

substitute these into the differential equation, we get.

Eq. 2.2.5

The characteristic equation is.

Eq. 2.2.6

The solution of this characteristic equation is.

Eq. 2.2.7

Eq. 2.2.8

Eq. 2.2.9

For the very special case where the value under the square root is equal to zero, the system is called critically damped.

Eq. 2.2.10

Let's call the damping constant at this situation, the critical damping c꜀ᵣ

Eq. 2.2.11

Eq. 2.2.12

Now let's define the damping ratio as ζ

Eq. 2.2.13

Eq. 2.2.14

Eq. 2.2.15

Eq. 2.2.16

Eq. 2.2.17

Eq. 2.2.18

Based on the value of ζ, three cases can happen, Click on each case icon to show the details.

Case 1: ζ = 1
Critically Damped System
Case 2: ζ > 1
Overdamped System
Case 3: ζ < 1
Underdamped System

Now let's examine all the cases side by side, change the parameters to see it in action!

uo

vo

ω

ζover

ζunder

Figure 2.2.5 All Possible Damping Cases

The critically damped case will always be the first one to come to rest.

The underdamped case is the only oscillating case.

The envelop around the underdamping case is defined by the exponential function,

Another way for representing this data is by selecting the correct response based on the value of ζ as follows,

uo

vo

ω

ζ

Figure 2.2.6 All Possible Damping Cases based of the damping ratio