Machine Foundation Criteria and its Analysis

Contents

**Define the following terms: i. Vibration ii. Periodic Motion and Cycle iii. Free and Forced Vibration iv. Frequency and Natural Frequency v. Resonance and Damping vi. Degree of Freedom** 1

**List various types of Machine Foundation** 2

**Recall the General Criteria for Machine Foundation** 3

**Derive an expression for Free Vibration Motion** 4

**Derive an expression for Forced Vibration Motion** 5

**Recall the Vibration Analysis of Machine Foundation** 6

**Describe the determination of Natural Frequency** 7

**Define the following terms: i. Vibration ii. Periodic Motion and Cycle iii. Free and Forced Vibration iv. Frequency and Natural Frequency v. Resonance and Damping vi. Degree of Freedom**

i. Vibration: Vibration is the repetitive motion of an object or system. It can be caused by various factors such as mechanical, electrical, or acoustic energy. Vibration can be measured in terms of its amplitude, frequency, and phase.

ii. Periodic Motion and Cycle: Periodic motion is a type of motion in which an object or system repeats its motion in a regular and repeated pattern. A cycle is one complete repetition of this motion. The time it takes for an object or system to complete one cycle is known as the period of the motion.

iii. Free and Forced Vibration: Free vibration is a type of vibration that occurs when an object or system is set into motion by an initial displacement or velocity, but is not subject to any external forces. Forced vibration is a type of vibration that occurs when an object or system is subject to external forces, such as those caused by wind, earthquakes, or machinery.

iv. Frequency and Natural Frequency: Frequency is the number of cycles of vibration that occur in a given period of time. It is typically measured in hertz (Hz). Natural frequency is the frequency at which an object or system naturally wants to vibrate. It is determined by the object’s or system’s mass and stiffness.

v. Resonance and Damping: Resonance is a phenomenon in which an object or system vibrates at a larger amplitude when the frequency of the applied force is the same as the object’s or system’s natural frequency. Damping is the process of reducing the amplitude of vibration over time. It can be caused by friction, viscosity, or other factors.

vi. Degree of Freedom: Degree of freedom is a term used to describe the number of independent coordinates or parameters needed to describe the motion of an object or system. For example, a rigid body has six degrees of freedom, three for translation and three for rotation. A particle has one degree of freedom, which is its position.

**List various types of Machine Foundation**

i. Isolated Footing: Isolated footing is a type of machine foundation that is used to support a single column or a small group of columns. It is typically constructed using reinforced concrete and is designed to transfer the load of the machine to the soil beneath.

ii. Combined Footing: A combined footing is a type of machine foundation that is used to support multiple columns or a large load-bearing structure. It is typically constructed using reinforced concrete and is designed to transfer the load of the machine to the soil beneath.

iii. Raft Foundation: A raft foundation is a type of machine foundation that is used to support a large load-bearing structure, such as a building or industrial facility. It is typically constructed using reinforced concrete and is designed to transfer the load of the machine to the soil beneath.

iv. Pile Foundation: Pile foundation is a type of machine foundation that is used to support a large load-bearing structure, such as a building or industrial facility. Piles are driven into the ground and the load is transferred to the soil beneath through the piles.

v. Mat Foundation: Mat foundation is a type of machine foundation that is used to support a large load-bearing structure, such as a building or industrial facility. It is typically constructed using reinforced concrete and is designed to transfer the load of the machine to the soil beneath.

vi. Grouted Anchor Foundation: Grouted anchor foundation is a type of machine foundation that is used to support a large load-bearing structure, such as a building or industrial facility. It is typically constructed using reinforced concrete and is designed to transfer the load of the machine to the soil beneath.

**Recall the General Criteria for Machine Foundation**

The general criteria for machine foundation are the guidelines that must be followed when designing and constructing a foundation for a machine. These criteria include the following:

- The foundation must be able to support the weight of the machine and any additional loads, such as those from vibration or impact.
- The foundation must be able to transmit the forces from the machine to the underlying soil in a stable and controlled manner.
- The foundation must be able to resist the effects of soil movement, such as settlement or heave, without causing damage to the machine or the foundation itself.
- The foundation must be able to provide adequate isolation from vibration and noise, to prevent damage to the machine and to minimize the impact on the surrounding environment.
- The foundation must be designed and constructed in a way that allows for easy installation, maintenance, and repair of the machine, as well as easy removal if necessary.
- The foundation must be designed and constructed in a way that meets all relevant building codes and regulations, including those related to safety, health, and the environment.

**Derive an expression for Free Vibration Motion**

The general equation of motion for a system undergoing free vibration is given by:

m*x”(t) + k*x(t) = 0

where

m = mass of the system

x(t) = displacement of the system at time t

x”(t) = acceleration of the system at time t

k = spring constant, which represents the stiffness of the system

The solution to this differential equation is of the form:

x(t) = A*cos(ωt + φ)

where

ω = natural frequency of the system, given by √(k/m)

A = amplitude of the vibration

φ = phase angle, which represents the initial phase of the vibration

The natural frequency of a system is a measure of how easily the system oscillates or vibrates. A system with a high natural frequency will oscillate more easily and with a larger amplitude than a system with a low natural frequency.

It’s important to note that this equation only applies to systems that are undergoing free vibrations, which means that the only force acting on the system is that of its own weight and the spring force. In cases of forced vibrations, there will be an external force acting on the system in addition to its own weight and spring force. The equation of motion and the solution for forced vibrations will be different from that of free vibrations.

**Derive an expression for Forced Vibration Motion**

The motion of a mechanical system under the influence of an external force is known as forced vibration. The motion of the system will be periodic if the external force is periodic. The equation of motion for a forced vibration system can be derived using Newton’s second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.

The equation of motion for a single degree of freedom system subjected to a harmonic force can be represented as:

F(t) = F0 cos(ωt)

Where:

F(t) = external force acting on the system

F0 = amplitude of the force

ω = angular frequency of the force

t = time

The equation of motion for the system can be represented as:

m*x”(t) + c*x'(t) + k*x(t) = F0 cos(ωt)

Where:

m = mass of the system

c = damping coefficient

k = spring constant

x(t) = displacement of the system at time t

x'(t) = velocity of the system at time t

x”(t) = acceleration of the system at time t

The above equation is a second-order ordinary differential equation, which can be solved to find the displacement, velocity, and acceleration of the system as a function of time. The general solution of the above equation is given by:

x(t) = A cos(ω1t-φ)

Where:

A = amplitude of the forced vibration

ω1 = damped natural frequency of the system

φ = phase angle

The amplitude of the forced vibration will depend on the amplitude of the external force, the damping coefficient and the natural frequency of the system. The phase angle represents the phase difference between the external force and the displacement of the system.

**Recall the Vibration Analysis of Machine Foundation**

The vibration analysis of machine foundations is a process of evaluating the dynamic behavior of a foundation subjected to vibrations caused by the operation of the machine. It involves the study of the amplitude, frequency, and phase of the vibration, as well as the effects of the vibration on the foundation and the surrounding soil. The analysis is typically done using mathematical models and computer simulations, with the goal of identifying any potential issues that may arise from the vibration and determining ways to mitigate them. Factors that are considered in the analysis include the type of machine, the size and weight of the machine, the type of soil, and the presence of any underground structures or utilities. The analysis is critical in the design of machine foundations, as it ensures that the foundation will be able to withstand the vibrational forces generated by the machine, and that the vibration will not cause damage to the surrounding soil or structures.

**Describe the determination of Natural Frequency**

The natural frequency of a system, also known as the resonant frequency, is the frequency at which a system will vibrate at its maximum amplitude when excited by an external force. This frequency is determined by the physical properties of the system, such as its mass, stiffness, and damping. The natural frequency of a machine foundation is an important factor to consider when designing the foundation, as it determines the frequency at which the foundation will vibrate when the machine is in operation.

There are several methods to determine the natural frequency of a machine foundation, including:

- Analytical methods – mathematical equations can be used to calculate the natural frequency based on the physical properties of the system.
- Experimental methods – the natural frequency can be determined by exciting the system with an external force and measuring the resulting vibration.
- Finite element analysis – computer simulations can be used to model the behavior of the system and calculate the natural frequency.

It is important to determine the natural frequency of a machine foundation because if it is too close to the operating frequency of the machine, it can lead to resonance, which can cause excessive vibration and damage to the foundation and the machine. Therefore, it is essential to design the foundation to have a natural frequency that is well separated from the operating frequency of the machine.