**Machine foundations and their vibration analysis**

Analysis and the design of the machine foundations involves careful evaluation of the vibration characteristics of the foundation system. This type of foundation undergoes dynamic loads which develop the vibratory motions which are taken by the soil below the foundation. This effect caused by the vibratory motions on the soil is understood by employing the principles of soil dynamics and theory of vibrations.

**Vibrations in Machine Foundation**

There are two types of vibrations in machine foundation which are as follows:

- Free vibration
- Forced vibration

**Free Vibration in Machine Foundation**

Free vibrations occur under the influence of forces in the system only without any external force. To initiate free vibrations, there is a need of an external force or any natural disturbance. Free vibrations are classified into two types as follows:

- Damped vibrations
- Undamped vibrations

** ****Forced Vibration in Machine Foundation**

Unlike free vibrations, Forced vibrations occur with continuous external forces on the system. If F (t) is an exciting force applied to a damped system, its equation of motion can be written as:

**Vibration Analysis of Machine Foundation**

To understand and analyze the vibration theory of machine foundation, it is necessary to assume that the machine foundation has single degree of freedom even though it has 6 degree of freedom. Imagine a machine foundation is resting on a soil mass. The mass of this foundation acts downward at the center of gravity if the system and let’s assume it as m_{f.} There also exists a mass of soil which acts upwards and let’s say its m_{s} which is the elastic action of soil due to the vibration of the system. This elastic action is dependent of the stiffness k. The resistance against motion is dependent of damping coefficient c. All these 3 parameters namely mass, stiffness and damping co-efficient are required for the complete analysis of the machine foundation.

**Determination of the parameters essential for the analysis**

**Mass (m)**

The soil below the foundation vibrates as and when the machine vibrates. The part of the soil below the foundation which vibrates due to the machine vibrations is called as the in-phase soil mass. Hence the total mass (m) can be shown as:

**m = m _{f} + m_{s}**

Where, m- total mass,

m_{f} – mass of foundation

m_{s} – in-phase soil mass

The in-phase soil mass varies from 0 to m_{f}, hence the total mass varies from m_{f }to m_{s}

**Stiffness ( k)**

This stiffness parameter depends on the following:

- Type of soil below foundation
- Embedment of foundation
- Contact pressure distribution between soil and foundation

Stiffness is derived from the following methods.

- Barkan’s Method
- Plate Load Test
- Resonance Test

**Damping Constant (c)**

Damping occurs when the vibration energy dissipates from the soil. The major reasons to develop damping are:

- Internal friction loss due to viscous effects and hysteresis
- Radiational losses due to propagation of waves through soil.

The damping constant varies from 0.01 to 0.1 and is obtained from the area of hysteresis loop of load deformation curve as follows:

Where, ΔW =work lost in hysteresis

*W *= total work done.