Mechanical Vibration

 

Introduction

    Mechanical vibration is the motion of a particle or body which oscillates about a position of equilibrium. Most vibrations in machines and structures are undesirable due to increased stresses and energy losses. It deals with the relationship between forces acting on the mechanical system and the oscillatory motion of the mechanical system about a point within the system. 

The main reasons for vibration are as follows :

1. Unbalanced centrifugal force in the system. This is caused because of nonuniform material distribution in a rotating machine element.
2. Elastic nature of the system.
3. External excitation is applied to the system.
4. Winds may cause vibrations of certain systems such as electricity lines, telephone lines, etc

 

Classification of Vibrations

Free and forced vibration

Free vibration- If a system, after an initial disturbance, is left to vibrate on its own, the ensuing vibration is known as free vibration. No external force acts on the system. For example oscillation of the simple pendulum

Forced vibration - If a system is subjected to an external force (often, a repeating type of force), the resulting vibration is known as forced vibration. For example oscillation that arises in machines such as diesel engines.

 

Damped and Undamped vibration

Damped vibration - If any energy is lost or dissipated in friction or other resistance during oscillation, it is called damped vibration.

Undamped vibration - If no energy is lost, the vibration is known as undamped vibration. 

Fig. Damped and Undamped Vibrations

    In many physical systems, the amount of damping is so small that it can be disregarded for most engineering purposes. However, consideration of damping becomes extremely important in analyzing vibratory systems near resonance.

Linear and nonlinear vibration

    If all the basic components of a vibratory system the spring, the mass, and the damper behave linearly, the resulting vibration is known as linear vibration.
If, however, any of the basic components behave nonlinearly, the vibration is called nonlinear vibration.

    If the vibration is linear, the principle of superposition holds, and the mathematical techniques of analysis are well developed. For nonlinear vibration, the superposition principle is not valid, and techniques of analysis are less well known.

Fig. Linear and Non Linear Vibrations

Types of Free Vibration:

   1.  Torsional Vibration:

Torsional vibration is the angular vibration of an object, commonly a shaft along its axis of rotation. It is often a concern in power transmission systems using rotating shafts or couplings where it can cause failures if not controlled. The second effect of torsional vibrations applies to passenger cars.

Fig. Torsional Vibration

    Torsional vibrations can lead to seat vibrations or noise at certain speeds. Both reduce the comfort.

2.   Axial or Longitudinal Vibrations:

    Axial vibration is a kind of longitudinal shafting vibration which occurs in the crankshaft because of the radial as well as tangential forces. 

Fig. Axial Vibration

3.   Transverse Vibrations:

    A vibration in which the element moves to and fro in a direction perpendicular to the direction of the advance of the wave.

Fig. Transverse Vibration

Vibration Measurement

 

    When faced with a vibration problem, engineers generally start by making some measurements to try to isolate the cause of the problem.  There are two common ways to measure vibrations:

 

1.  An accelerometer is a small electro-mechanical device that gives an electrical signal proportional to its acceleration.   The picture shows a typical 3 axis accelerometer. 

2.  A displacement transducer is similar to an accelerometer but gives an electrical signal proportional to its displacement.

Fig. Vibration Measurement 

 

Displacement transducers are generally preferable if you need to measure low-frequency vibrations; accelerometers behave better at high frequencies.

 

The most common procedure is to mount three accelerometers at a point on the vibrating structure, so as to measure accelerations in three mutually perpendicular directions.  The velocity and displacement are then deduced by integrating the accelerations.

Applications of Mechanical Vibrations:

The applications of Mechanical Vibrations are as follows.

• Identification of the system: If you want to calculate the mass, stiffness, and damping of a vibratory system then you need to do the vibration analysis which is used in structural health monitoring.

• Design of components: When you are designing the components of an automobile, you need to look at the consideration that the vibrations excited by the engine should not match with the other components and if it does, there may be a chance of failure like loosening of bolts, etc.

 

• Vibrations may cause loosening of parts from the machine

• If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is reached, and dangerously large oscillations may occur which may result in the mechanical failure of the system.  

Stages of Vibration Analysis :

Fig. Stages of Vibration Analysis

Free vibration without damping

    To start the investigation of the mass-spring–damper assume the damping is negligible and that there is no external force applied to the mass (i.e. free vibration). The force applied to the mass by the spring is proportional to the amount the spring is stretched "x" (assuming the spring is already compressed due to the weight of the mass). The proportionality constant, k, is the stiffness of the spring and has units of force/distance (e.g. lbf/in or N/m). The negative sign indicates that the force is always opposing the motion of the mass attached to it:

        𝑭s = ‒𝒌𝔁. 

    The force generated by the mass is proportional to the acceleration of the mass as given by Newton's second law of motion: 

        ∑𝑭 = 𝒎а = 𝒎ẍ = (d²𝔁/d𝙩²). 

    The sum of the forces on the mass then generates this ordinary differential equation: 

        𝒎ẍ + 𝒌𝔁 = 0.

Fig. Simple Harmonic Motion for Mass Spring System

    Assuming that the initiation of vibration begins by stretching the spring by the distance of A and releasing, the solution to the above equation that describes the motion of the mass is:

        (𝙩) = A cos(2𝞹𝒇ₙ𝙩).

    This solution says that it will oscillate with a simple harmonic motion that has an amplitude of A and a frequency of f_n. The number f_n is called the undamped natural frequency. For the simple mass-spring system, f_n is defined as:

        𝒇ₙ = (1/2𝞹)√(𝒌/𝒎) .

Parts of a Vibrating System :

    A vibratory system basically consists of three elements, namely the mass, the spring, and the damper. In a vibrating body, there is an exchange of energy from one form to another. Energy is stored by mass in the form of kinetic energy (1/2 mv2), in the spring in the form of potential energy (1/2 kx ), and dissipated in the damper in the form of heat energy which opposes the motion of the system.

Fig. Parts of Vibrating System 

Natural Frequency of Free Longitudinal Vibrations

    The natural Frequency of the Free Longitudinal Vibrations can be determined by the following methods.

1.    Equilibrium Method

2.    Energy method

3.    Rayleigh’s method

Energy methods for analysis

    For undamped free vibration, the total energy in the vibrating system is constant throughout the cycle. Therefore the maximum potential energy V,, is equal to the maximum kinetic energy T,, although these maxima occur at different times during the cycle of vibration. Furthermore, since the total energy is constant,

 T + V = constant,

and thus d/d𝙩 (T + V) = 0.

Applying this method to the case, already considered, of a body of mass m fastened to a spring of stiffness 𝒌, when the body is displaced a distance x from its equilibrium position,

strain energy (SE) in spring = ½𝒌𝔁²

kinetic energy (KE) of body = ½𝒎𝔁²

Hence V = ½𝒌𝔁²

and T = ½𝒎𝔁²

Thus  d/d𝙩  (½𝒎𝔁² + ½𝒌𝔁² ) = 0

that is

ẍ + (𝒌/𝒎)= 0, as before in equation

    This is a very useful method for certain types of problems in which it is difficult to apply Newton's laws of motion.

    Alternatively, assuming SHM, if 𝒙 = 𝒙ₒ cos 𝟂𝙩

        the maximum SE, Vmax = ½𝒌𝒙²ₒ,

and

        the maximum KE, Tmax = ½𝒎(𝒙ₒ𝟂)².

    Thus, since Tmax = Vmax,

        ½𝒌𝒙²ₒ = ½(𝒙ₒ𝟂)²,

or 𝟂 = √(𝒌/𝒎) rad/s.

    Energy methods can also be used in the analysis of the vibration of continuous systems such as beams.

    The frequency of vibration is found by considering the conservation of energy in the system; the natural frequency is determined by dividing the expression for potential energy in the system by the expression for kinetic energy.

Conclusion

• Mechanical Vibration is a measurement of a periodic process of oscillations with respect to an equilibrium point.
• We experience these mechanical vibrations in everyday life
• There are useful as well as harmful vibrations.
• There can be Free/Forced, Damped/Undamped, Linear/Nonlinear  OR Deterministic/Random Vibration OR combinations of these mechanical vibrations.  
• Modeling of systems can be Single Degree of freedom, 2 DOF, Multi DOF, and Continuous Systems.

References

1)     Research on ‘Impacts of mechanical vibrations on the production machine to its rate of change of technical state’; by Štefánia Salokyová, Radoslav Krehel’, Martin Pollák; July 4, 2016.

2)     ‘Mechanical Vibration Sound waves are mechanical vibrations in solid, liquid or gas’; by Techniques and instrumentation in Analytical Chemistry, 2002.

3)     Research on ‘Mechanical vibration monitoring based on wireless sensor network and sparse’; Bayes. Xinjun Lei & Yunxin Wu ; EURASIP;  Journal on Wireless Communications and Networking volume 2020

4)     Research on ‘Impacts of mechanical vibrations on the production machine to its rate of change of technical state’;July, 2016 ,Advances in Mechanical Engineering 

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