Introduction to Forced Vibration
When you strum a guitar string and hear the melodious music, it is nothing but the vibrational energy turned into sound energy. Vibration is a quick and periodic back and forth movement of an object about a point of equilibrium. Every object, particle, and system oscillate at its natural frequency. The natural frequency is the frequency of an object in the absence of any external force acting on it. This article will explain what is forced vibration, go through the forced vibration equation, explain free and forced vibration and understand the difference between free and forced vibration. We will also discuss types of forced vibration and see some forced vibration examples to clarify the concepts.
What is a Forced Vibration?
The system that is displaced from its equilibrium state tries to restore it by responding to the forces that caused the displacement. When any musical instrument is hit, they are set into vibration at their natural frequency. If you take one string of a guitar and stretch it and ask someone to pluck it, you would hear some noise but that sound would be nothing in comparison to the volume you hear from an acoustic guitar. But if the same string is attached to a sound box, the vibration in the string forces the soundbox to vibrate at the same natural frequency. Due to the vibrating sound box, the air particles inside the box are set into vibrational motion at the same natural frequency. The whole system (string, soundbox, and the air inside) then forces the surrounding air to vibrate at the same frequency. When the vibration of an object forces adjoining objects into vibration is referred to as forced vibration.
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Difference Between Free and Forced Vibrations
When a simple pendulum is pulled and released, the pendulum vibrates even without an external force. This is called free vibration which happens in the absence of any external force or friction when a body is disturbed only initially from its state of equilibrium. Whereas when a body is subject to external force in a periodic fashion it experiences forced vibration.
In free vibration, the frequency of vibration depends on the natural frequency of the body while in forced vibration the frequency of vibration depends on the external force.
Methods of Forced Vibration
When a mechanical system is continuously being acted upon by a repeated force, it causes the system to vibrate with a forced vibration. Some forced vibration examples are:
Pendulum in a clock
When a child on a swing is pushed each time on a downswing
The vibration of a floor when heavy machinery runs over it.
A car’s vibration when running on an uneven road
Vibration seen in reciprocating engines due to periodic reciprocation of pistons.
Let us consider a spring-mass system (representing a single mode of vibration in a real system) as a model to understand the different ways forced vibration can occur.
There are three types of forcing that can be applied to a spring-mass system to cause forced vibration. We will be restricting our analysis to harmonic excitation, in all three cases. It must be noted that excitation can happen in many ways but the forced vibration equation of motion will always reduce to any one of the three cases outlined below.
External Forcing - In this type of forcing the force acting on the system varies with time. An example of such a vibration is wave loading on an offshore structure. The external force applied to the system is given by:
F(t) = Fa * Sin ⍵t. Where Fa is the amplitude and ⍵ is the frequency of oscillation. The force F(t) varies harmonically with amplitude and frequency.
Base Excitation - This type of excitation models the behavior of a vibration isolation system. With a prescribed motion to the base of the spring, the mass is made to vibrate. Such a system can be seen in vehicle suspension systems or the response of a structure to an earthquake. Its equation is y(t) = Y0 Sin ⍵t.
Rotor Excitation - A model of rotor excitation is a rotating machine mounted on a flexible floor. The crank with tiny mass m0 rotates with a constant angular velocity and causes the large mass m to vibrate.
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Types of Forced Vibration
Damping is a restraining force on a vibrating body. Damping happens due to the dissipation of energy with time. For example, a child on a swing needs to keep pumping the swing else its motion would die out due to damping. Due to the damping factor, forced vibration are categorized into two parts:
Undamped Forced Vibration - It is the external harmonic excitation to a freely vibrating object whose equation of motion is given by:
mẍ + kx = P = Po Sin ⍵t, where ⍵ is the frequency, t is time, and Po is a constant. The general solution for this equation is given by two parts; a complementary function and a particular integral and is denoted by x = CF (complementary function) + PI (particular integral).
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Forced Damped Vibration - As we know all real systems are subject to damping. Any oscillating system is always opposed by dissipative forces like a mass oscillating in water or a metal plate oscillating in a magnetic field. Let us find the forced vibration equation for a damped forced vibration. If damping is added to the above figure we will get the modified diagram as depicted below:
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In this case, the equation of motion for the displacement of spring from its equilibrium position is given by mẍ + cẊ + kx = Fo Sin t
FAQs on Forced Vibration
1. Explain the Equation of Motion of Forced Vibration of Cantilever Beam.
Ans - The figure below shows a cantilever beam which is a beam with a rectangular cross-section, fixed at one end and free at the other end.
This beam is subjected to a forced vibration by using an exciter which gives excitation to the entire system. The exciter can generate different kinds of forcing signals like sine, rectangular, swept sine, triangular, etc. The beam is a continuous system which means its mass, as well as stiffness, is distributed equally all through the beam. In 1967, Leonard Meirovitch (a famous educator and researcher) gave the equation of motion of a cantilever beam as:
d2/dx2 {EI (x) (d2y (x, t)/dx2} = m(x) (d2y (x, t)/dx2) + f(x) δ(x - L1)
Where
E - modulus of rigidity of the material of the beam
I - the moment of inertia of the cross-section of the beam
y(x, t) - the vertical displacement of the beam cross-section at a distance x from the fixed end
m - mass per unit length which is equal to ρA(x) where
ρ - material density
A(x) - beam’s area of cross-section
f(t) - force applied to the system at a distance of x = L1
2. What is Meant By Resonance and What is the Formula For Resonance Frequency?
Ans- When the periodicity of the externally applied force is equal to or close to the natural frequency of an object, then the object starts vibrating at an increased amplitude and this is called the resonance effect. The resonance frequency is denoted by f0 and is displayed by objects that are in equilibrium with forces acting on them and they can keep vibrating for long durations when conditions are perfect. The below formula is used to find the resonance frequency of a single continuous-wave:
v = ƛ * f, where v is the velocity of the wave, ƛ is the wavelength.
