**Mechanical Vibrations**

Contents

**Recall the important terms used in vibratory motions** 2

**List and recall types of Vibrations** 3

**Define transverse vibration and derive an expression for natural frequency of transverse vibration** 5

**Define Free damped vibration and recall its frequency** 7

**Recall ‘Logarithmic decrement’ as applied to damped vibrations** 8

**Recall Forced damped vibration** 9

**Define the term Vibrations**

Vibration is a term used to describe the motion of a physical system that oscillates about a point of equilibrium. Vibrations can occur in any system that has a degree of freedom or a flexible element, such as a spring or a beam. These systems can be subjected to an external force or an internal disturbance, resulting in a displacement or motion away from the equilibrium point.

Vibrations can be characterized by their frequency, amplitude, and phase. Frequency refers to the number of oscillations or cycles that occur in a given period of time, usually measured in Hertz (Hz). Amplitude refers to the magnitude or size of the oscillation, usually measured in units of distance or acceleration. Phase refers to the position or timing of the oscillation relative to some reference point.

Vibrations can be either desirable or undesirable, depending on the context in which they occur. For example, vibrations can be desirable in musical instruments, where they produce sound waves that are pleasing to the ear. In some mechanical systems, vibrations are necessary for the proper functioning of the system, such as in the case of engines or turbines.

On the other hand, vibrations can also be undesirable and potentially harmful in many other situations. Excessive vibrations can cause mechanical failure, fatigue, and wear and tear on a system, leading to reduced performance, increased maintenance costs, and even safety hazards. For example, vibrations in buildings or bridges can cause structural damage and collapse, while vibrations in cars or aeroplanes can cause discomfort or motion sickness in passengers.

Therefore, understanding the fundamental principles of vibrations is essential for engineers and scientists to develop effective strategies to control or eliminate unwanted vibrations and ensure the safe and reliable operation of mechanical systems.

**Recall the important terms used in vibratory motions**

In vibratory motion, there are several important terms used to describe the characteristics of the motion. These terms include:

- Displacement: The distance between the equilibrium position and the position of the vibrating object at any given time.
- Amplitude: The maximum displacement of an object from its equilibrium position during one cycle of vibration. It is measured in units of distance, such as meters or centimeters.
- Frequency: The number of complete cycles of vibration that occur per unit time. It is measured in units of Hertz (Hz), where 1 Hz is equal to one cycle per second.
- Period: The time required for one complete cycle of vibration. It is the reciprocal of frequency and is measured in units of time, such as seconds.
- Phase: The position of the vibrating object at a specific point in time, relative to the starting point of its cycle.
- Damping: The energy dissipation mechanism that reduces the amplitude of vibration over time. It is a measure of the extent to which energy is lost from a vibrating system.
- Resonance: The phenomenon that occurs when the frequency of an external force matches the natural frequency of a vibrating system, leading to a large amplitude vibration.
- Natural frequency: The frequency at which a system vibrates freely, without the application of any external forces.
- Forced vibration: The motion of a vibrating system that is subject to an external force.
- Transmissibility: The ratio of the amplitude of vibration transmitted through a system to the amplitude of the input force. It is a measure of the effectiveness of a vibration isolation system.

Understanding these terms is essential in the analysis and design of vibratory systems, as they allow engineers to predict and control the behavior of these systems under various operating conditions.

**List and recall types of Vibrations**

Vibration is a common phenomenon that occurs in many mechanical systems. There are various types of vibrations, which can be broadly categorised as follows:

- Free vibrations: These vibrations occur when a system is set into motion and left to vibrate without any external excitation. The amplitude of these vibrations will gradually decrease over time due to the energy dissipation mechanisms like damping.
- Forced vibrations: These vibrations occur when a system is subjected to an external force or excitation. The frequency of the external force can be either the same or different from the natural frequency of the system. Forced vibrations can be harmonic, periodic or non-periodic.
- Harmonic vibrations: These vibrations occur when the external force or excitation is sinusoidal and the response of the system is also sinusoidal. The frequency of the excitation is usually close to the natural frequency of the system.
- Periodic vibrations: These vibrations occur when the system repeats its motion after regular intervals of time. The motion can be simple or complex, but it repeats itself after equal intervals of time.
- Non-periodic vibrations: These vibrations occur when the system does not repeat its motion after equal intervals of time. The motion of the system is random and unpredictable.
- Self-excited vibrations: These vibrations occur when a system contains positive feedback, and the energy input to the system amplifies the response, leading to large-amplitude vibrations. Examples of self-excited vibrations include flutter in aircraft wings, whirling in centrifugal pumps, and hunting in governors.
- Transient vibrations: These vibrations occur when a system undergoes a sudden change in its operating conditions, resulting in a temporary disturbance. Examples of transient vibrations include earthquakes and impacts.

Understanding the type of vibration that a system is experiencing is crucial in designing effective vibration control measures. Engineers can use this knowledge to optimize the design of the system to minimize vibrations or design vibration isolation systems to reduce the impact of vibrations on the system and its environment.

**Define Longitudinal vibration and derive an expression for the natural frequency of free transverse and longitudinal vibrations**

Longitudinal vibrations refer to the oscillations in which the particles of the vibrating object move back and forth along the length of the object. For example, when a spring is compressed and released, it undergoes longitudinal vibrations as the particles move back and forth along the length of the spring. Similarly, when a column of air in a pipe is set into vibration, it undergoes longitudinal vibrations.

The natural frequency of free transverse and longitudinal vibrations can be derived using the equation:

f = (1/2L) x sqrt(EI/m)

Where,

f = natural frequency of free transverse or longitudinal vibrations

L = length of the vibrating object

E = Young’s modulus of the material of the vibrating object

I = moment of inertia of the cross-sectional area of the vibrating object

m = mass per unit length of the vibrating object

For longitudinal vibrations, the moment of inertia is replaced by the area moment of inertia of the cross-sectional area. The expression for the natural frequency of free longitudinal vibrations is:

f_{longitudinal} = (1/2L) x sqrt(EA/m)

Where,

f_{longitudinal} = natural frequency of free longitudinal vibrations

A = cross-sectional area of the vibrating object

**Define transverse vibration and derive an expression for natural frequency of transverse vibration**

Transverse vibrations refer to the oscillations in which the particles of the vibrating object move perpendicular to the length of the object. For example, when a guitar string is plucked, it undergoes transverse vibrations as the particles move up and down perpendicular to the length of the string. Similarly, when a beam is loaded and deflects under the load, it undergoes transverse vibrations.

The natural frequency of free transverse vibrations can be derived using the equation:

f = (1/2L) x sqrt(EI/m)

Where,

f = natural frequency of free transverse vibrations

L = length of the vibrating object

E = Young’s modulus of the material of the vibrating object

I = moment of inertia of the cross-sectional area of the vibrating object

m = mass per unit length of the vibrating object

For transverse vibrations, the moment of inertia is used to characterise the stiffness of the object in bending. The expression for the natural frequency of free transverse vibrations is:

f_{transverse} = (1/2L) x sqrt(EI/m)

Where,

f_{transverse} = natural frequency of free transverse vibrations.

**Recall Torsional vibrations**

Torsional vibration is a type of vibration that occurs in a rotating mechanical system, such as a shaft or a flywheel. It is caused by the twisting motion of the system around its axis of rotation, which creates a periodic variation in the torque applied to the system. This type of vibration can be particularly problematic because it can cause fatigue failure of the system components, such as shafts, gears, and bearings.

Torsional vibration can be caused by a number of factors, including imbalances in the system, variation in the stiffness of the components, and fluctuations in the load applied to the system. When torsional vibrations occur, they can create a variety of unwanted effects, including increased noise and vibration levels, decreased efficiency, and premature wear and tear of the components.

To control and mitigate torsional vibrations, engineers may use a variety of techniques, such as adding damping elements to the system, adjusting the stiffness of the components, and carefully balancing the system. They may also use advanced modeling and simulation techniques to better understand the behavior of the system under different conditions, and to optimize the design for maximum performance and durability.

**Define Free damped vibration and recall its frequency**

Free damped vibration is a type of mechanical vibration that occurs when a mechanical system is set into motion and then left to oscillate freely without any external input or force. In this type of vibration, the system oscillates due to its own natural frequency, which is the frequency at which the system tends to oscillate with the maximum amplitude in the absence of any damping force.

The frequency of a free damped vibration can be calculated by the following equation:

f = f_{n} x sqrt(1 – zeta^{2})

where f_{n} is the natural frequency of the system, and zeta is the damping ratio. The damping ratio is a measure of the system’s ability to dissipate energy, and it is defined as the ratio of the damping coefficient to the critical damping coefficient.

The critical damping coefficient is the minimum amount of damping required to bring the system to rest without overshooting its equilibrium position. When the damping ratio is less than one, the system is said to be underdamped, and the amplitude of the oscillation decays over time. When the damping ratio is greater than one, the system is said to be overdamped, and the system returns to its equilibrium position without any oscillation. When the damping ratio is equal to one, the system is said to be critically damped, and the system returns to its equilibrium position as quickly as possible without overshooting.

**Recall ‘Logarithmic decrement’ as applied to damped vibrations**

The logarithmic decrement is an important parameter used to quantify damping in a vibrating system. In free damped vibrations, the amplitude of the vibration decays exponentially with time due to the presence of damping forces in the system. The logarithmic decrement is defined as the natural logarithm of the ratio of the amplitude of any two successive peaks or troughs in a damped vibration.

If ‘A’ is the amplitude of the first peak, and ‘B’ is the amplitude of the second peak, which occurs after ‘n’ cycles of vibration, then the logarithmic decrement is given by the following formula:

Logarithmic Decrement = ln(A/B) / n

The value of the logarithmic decrement depends on the amount of damping present in the system. A highly damped system will have a higher value of the logarithmic decrement, while a system with lower damping will have a lower value.

The logarithmic decrement can be used to determine the damping ratio of the system, which is the ratio of the actual damping coefficient to the critical damping coefficient. The critical damping coefficient is the minimum amount of damping required to prevent oscillation in the system. If the damping ratio is less than one, the system is said to be underdamped, and if it is greater than one, the system is overdamped.

**Recall Forced damped vibration**

Forced damped vibrations occur when an external force is applied to a system undergoing damped vibrations. This external force can be either periodic or non-periodic.

The equation of motion for a damped vibration under a periodic external force is given by:

m(d^{2}x/dt^{2}) + c(dx/dt) + kx = F sin(ωt)

Where,

m = mass of the system,

c = damping coefficient,

k = spring constant,

F = amplitude of the applied force,

ω = frequency of the applied force,

x = displacement of the system from its equilibrium position.

The solution to the above equation is a combination of the homogeneous solution (which describes the natural response of the system to its initial conditions) and the particular solution (which describes the response of the system to the applied force).

If the frequency of the external force is equal to the natural frequency of the system, then resonance occurs, and the amplitude of the response becomes very large. This can be dangerous for the system and can lead to its failure.

The amplitude of the forced vibration can be determined using the concept of resonance. At resonance, the amplitude of the vibration is maximum, and it is given by:

A = (F/m) / √[(ωn^{2} – ω^{2})^{2} + (2ο ζωων)^{2}]

Where,

ωn = natural frequency of the system,

ζ = damping ratio (ratio of actual damping to critical damping),

ω = frequency of the applied force,

F = amplitude of the applied force,

m = mass of the system.

The above equation shows that the amplitude of the forced vibration depends on the frequency of the applied force, the natural frequency of the system, and the damping ratio.

**Recall vibration isolation**

Vibration isolation is the process of reducing or eliminating the transmission of vibrations from one structure to another. The main objective of vibration isolation is to isolate a system from its environment or protect the environment from the effects of the system’s vibration. Vibration isolation is important in various fields such as automotive, aerospace, and machinery to reduce noise, improve performance and increase the lifespan of the system.

There are several techniques used for vibration isolation, including the use of isolators, dampers, and absorbers. An isolator is a device that physically separates the system from its environment and reduces the transfer of vibrations. A damper, on the other hand, reduces the amplitude of the vibration by dissipating the energy of the system. An absorber absorbs the energy of the system and converts it into another form, such as heat or electrical energy.

The effectiveness of a vibration isolation system is usually measured by the transmissibility ratio, which is the ratio of the amplitude of vibration transmitted to the isolated system to the amplitude of vibration applied to the system. A good vibration isolation system will have a low transmissibility ratio.

Some common applications of vibration isolation include:

- Isolation of machinery: In industries such as manufacturing and construction, machinery can produce significant levels of vibration that can be transferred to the structure of the building. By using vibration isolation techniques, the vibration can be reduced, thereby protecting the structure and improving the performance of the machinery.
- Isolation of electronic equipment: Electronic equipment can be sensitive to vibrations, and any vibration can affect its performance. Vibration isolation is used to protect electronic equipment from the effects of vibration, ensuring that it functions properly.
- Isolation of buildings: Buildings can be exposed to vibrations from nearby sources, such as traffic, construction activities, and nearby industrial facilities. Vibration isolation can be used to reduce the transfer of these vibrations to the building, thereby improving the comfort of occupants and reducing the risk of damage to the building.