What is vibration?

In mechanical engineering, vibration is expressed as the term that occurs due to an object's backward and forward movement about a point of equilibrium. Generally, there are three types of vibrations which are,

  1. Forced vibration
  2. Natural vibration
  3. Damped vibration

What do you mean by forced undamped vibration?

Forced undamped vibration is described as the kind of vibration in which a particular system encounters an outside force that makes the system vibrate. Some of the examples of forced undamped vibration are:

  • Movement of laundry machine due to asymmetry
  • The vibration of a moving transport due to its engine
  • Movement of strings in guitar

The following is the free body diagram of the system, where an additional force is exerted on the block having mass m.

A block of mass m is attached to a wall through a spring and an external force is exerting on it on the other side.
A mass-spring system with an external force

The equation of motion of the above system can be expressed as:


Here, m is the mass of the block, x.. is the compression distance, and F is the external force. 

The steady state solution of force in this case is,


Here, ω is the angular velocity of the block, and t is the time.

Now, the general solution of equation of motion for the system can be re-written in standard form:


The standard solution is not only a particular solution for this system because in this initial conditions can be employed to obtain various other cases.

Amplitude of forced vibration

In the case of forced vibrations, the amplitude of steady state relies on the fraction of the forced frequency with the natural frequency. The forced frequency is ω0 and the natural frequency is ωn.

The magnification factor (MF) is known as the quantity which is the ratio of the amplitude of the steady-state vibration to the displacement covered by the deflection.

The relation of magnification factor in terms of frequency can be given as:


The following is the graph drawn between the magnification factor and ratio of the forced frequency with the natural frequency.

The graph represents the variation in ratio of forced frequency and natural frequency with magnification factor.
Graph plotted between magnification factor and ratio of forced frequency and natural frequency

As seen from the above graph, the following things can be observed while discussing various cases.

  • In the first case, resonance occurs when the natural frequency is equivalent to the forced frequency. In this case, large amplitude vibrations are produced. It is associated with high stress and failure to the system.
  • In the second case, when the forced frequency is approximately equal to zero, and the magnification factor is also approximately equal to zero, the forcing function is nearly static, eliminating the static deflection and limited natural vibration.
  • In the third case, if the magnification factor is more than 1, the forced vibration is greater than the natural vibration. The vibrations are in phase, and the amplitude of the vibration is more than the static deflection.
  • In the fourth case, if the magnification factor is smaller than 1, the forced vibration is more than the natural vibration. The vibrations are out of the phase, along with the motion of forcing function. Also, the vibration's amplitude will be smaller than the static deflection, which is the opposite of the third case.
  • In the fifth case, if the natural vibration is very much less than the forced vibration, then the force will quickly alter its direction for the motion of the block to respond.

Rotating unbalanced for forced vibration

In forced vibrations, one of the most usual causes in a given system is rotating unbalance. When in a mechanical system, the axis of revolution doesn't go through the center of mass of the system, then rotating unbalance will occur. Due to rotating unbalance the axle will vary its direction as the center of mass revolves. Also, in this condition, the angular frequency of the system will be equivalent to the forced angular frequency. Some of the common causes of rotating unbalance are:

  • Blowholes in casting
  • Distortion
  • Eccentricity
  • Corrosion
  • Hydraulic imbalance

Several features of the steady state response of spring mass system to forced vibrations

The free-body diagram of the spring-mass system is illustrated below:

A block is connected with a vertical spring of spring constant k. The other side of spring is connected to a flat plate.
Spring mass system

The following are some of the characteristics of the steady-state response of the spring-mass system to forced vibration.

  • The most essential feature in a spring-mass system to forced vibrations is that, the steady-state response will be harmonic. The frequency will be similar to the frequency of the force.
  • The vibration's amplitude relies on the frequency of excitation, properties of the spring, and mass in the mechanical system.
  • When the frequency of force is approximately equal to the natural frequency of the mass-spring system, the system can be slightly damped, and greater amplitude will occur. This occurrence is also determined as resonance.
  • In a spring mass system to forced vibration the phase lag among the system response and force is dependent on the properties of the system or frequency of excitation.

Difference between free vibration and forced vibration

  • In the free vibration, a particular force is essential to begin the vibration, whereas continuous periodic function is needed to start the forced vibration.
  • The frequency of free vibration relies on the particular object, so it is known as natural frequency, whereas the forced vibration frequency is dependent on the outside exerting force.
  • The free vibration is self sustained vibration but on the other hand the forced vibration is externally sustained vibration.
  • In forced vibration, the amplitude remains unchanged before the periodic force acts. In free vibration, the amplitude does get affected by time and it reduces with time.
  • The energy of the system in forced vibration is preserved to be constant by the force exerted on it. On the other hand, in free vibration the energy remains unchanged in the absence of air and drag coefficient, friction and other resistances. Also, the net energy reduces because of the damping force.

Advantages of vibration

  • Vibrations can be utilized for agriculture purposes in harvesting by forced vibration.
  • Drilling in geotechnical wells
  • In geological investigations, vibrations are used to simulate natural disasters like earthquake.

Disadvantages of vibration

  • Vibrations generates undesired stress and pressure in most of the mechanical devices or parts.
  • In gears, bearings and other parts, the possibility of wear increase rapidly due to vibration.
  • Vibration in excessive amount can cause damage to living organisms.

Common Mistakes

  • Students always get confused about the reduction in total energy of a system in free vibration. The total energy decreases as the damping force reduces.
  • One of the common misconceptions about forces exerting on the system is forced vibration. Students make mistakes and consider the fundamental forces, whereas the forces acting on the mechanical system will be inertial drag forces.
  • Free vibration and natural vibration are similar, but students get confused and make mistakes in problems related to natural vibrations.
  • Always use standard international units for the calculation of frequency, angular frequency, vibration's amplitude and other physical quantities. For example the amplitude will be measured in standard units meter.
  • Don't consider the approximation values after decimal point, it may lead to variation in your answer. Always measure the quantity with full accuracy and precision.

Context and Application

The topic forced undamped vibration is very much significant in the several professional exams and courses for undergraduate, Diploma level, graduate, postgraduate. For example:

  • Bachelor of Science in Physics
  • Bachelor of Technology in Mechanical Engineering
  • Bachelor of Technology in Civil Engineering
  • Masters in Technology in Mechanical Engineering
  • Doctor of Philosophy in Mechanical Engineering
  • Diploma in Mechanical and Civil engineering
  • Free Vibration
  • Damped vibration
  • Underdamped vibration
  • Oscillation
  • Simple pendulum

Practice Problems

Q1. What is the factor behind the large sound produced by a tuning fork placed on a block?

  1. Forced Vibration
  2. Free Vibration
  3. All of the above
  4. None of these

Correct answer: (a)

Q2. What is the phase lag in steady state forced vibration at resonance?

  1. 45 degrees
  2. 90 degrees
  3. 30 degrees
  4. 0 degrees

Correct answer: (b)

Q3. What is one of the essential characteristics of vibration other than amplitude?

  1. beats
  2. Frequency
  3. Wavelength
  4. None of these

Correct answer: (b)

Q4. When is the acceleration of the vibration zero?

  1. Mean position
  2. Actual position
  3. Initial position
  4. All of the above

Correct answer: (a)

Q5. Which of the following parameters indicate vibrations in free damped vibrations?

  1. Rate of decay of amplitude
  2. Natural frequency
  3. Both a and b
  4. None of the above

Correct answer: (c)

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Types of Vibrations

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