`color{blue} ✍️`When a system (such as a simple pendulum or a block attached to a spring) is displaced from its equilibrium position and released, it oscillates with its natural frequency ω, and the oscillations are called free oscillations.
`color{blue} ✍️`All free oscillations eventually die out because of the ever present damping forces. However, an external agency can maintain these oscillations. These are called force or driven oscillations. We consider the case when the external force is itself periodic, with a frequency `ω_d` called the driven frequency.
`color{blue} ✍️`A most important fact of forced periodic oscillations is that the system oscillates not with its natural frequency `ω`, but at the frequency `ω_d` of the external agency; the free oscillations die out due to damping.
`color{blue} ✍️`A most familiar example of forced oscillation is when a child in a garden swing periodically presses his feet against the ground (or someone else periodically gives the child a push) to maintain the oscillations.
`color{blue} ✍️`Suppose an external force `F(t)` of amplitude `F_0` that varies periodically with time is applied to a damped oscillator. Such a force can be represented as
`color{blue} {F(t) = F_o cos ω_d t}`
................... (14.36)
`color{blue} ✍️`The motion of a particle under the combined action of a linear restoring force, damping force and a time dependent driving force represented by Eq. (14.36) is given by,
`color{blue} {m a(t) = –k x(t) – bv(t) + F_o cos ω_d t}`
............ (14.37a)
`color{blue} ✍️`Substituting `d^2x//dt^2` for acceleration in Eq. (14.37a) and rearranging it, we get
`color{blue} {m (d^x)/(dt^2) + b (dx)/(dt) + kx = F_o cos omega_d t}`
.........................(14.37b)
`color{blue} ✍️`This is the equation of an oscillator of mass m on which a periodic force of (angular) frequency `ω_d` is applied. The oscillator initially oscillates with its natural frequency `ω`.
`color{blue} ✍️`When we apply the external periodic force, the oscillations with the natural frequency die out, and then the body oscillates with the (angular) frequency of the external periodic force. Its displacement, after the natural oscillations die out, is given by
`color{blue} {x(t) = A cos (ω_dt + φ )}`
................(14.38)
`color{blue} ✍️`where t is the time measured from the moment when we apply the periodic force.
`color{blue} ✍️`The amplitude A is a function of the forced frequency `ω_d` and the natural frequency `ω`. Analysis shows that it is given by
`color{blue} {A = F_o/(m^2 (omega^2 - omega_d^2 )^2 + omega_d^2 b^2 )^(1//2)}`
..........(14.39a)
and
`color{blue} {tan phi = (- v_0)/(omega_d x_0)}`
.................... (14.39b)
`color{blue} ✍️`where `m` is the mass of the particle and `v_0` and `x_0` are the velocity and the displacement of the particle at time `t = 0`, which is the moment when we apply the periodic force. Equation (14.39) shows that the amplitude of the forced oscillator depends on the (angular) frequency of the driving force.
`color{blue} ✍️`We can see a different behaviour of the oscillator when `ω_d` is far from `ω` and when it is close to `ω`. We consider these two cases.
`color{brown}{(a) ul"Small Damping, Driving Frequency far from Natural Frequency :"}`
`color{blue} ✍️` In this case, `ω_d` b will be much smaller than `m(ω_2–ω_2d)`, and we can neglect that term. Then Eq. (14.39) reduces to
`color{blue} { A = F_o/( m ( omega^2 - omega_d^2 ))}`
...................(14.40)
`color{blue} ✍️`Fig. 14.21 shows the dependence of the displacement amplitude of an oscillator on the angular frequency of the driving force for different amounts of damping present in the system.
`color{blue} ✍️` It may be noted that in all the cases the amplitude is greatest when `ω_d //ω = 1`. The curves in this figure show that smaller the damping, the taller and narrower is the resonance peak.
`color{blue} ✍️`If we go on changing the driving frequency, the amplitude tends to infinity when it equals the natural frequency. But this is the ideal case of zero damping, a case which never arises in a real system as the damping is never perfectly zero.
`color{blue} ✍️`You must have experienced in a swing that when the timing of your push exactly matches with the time period of the swing, your swing gets the maximum amplitude. This amplitude is large, but not infinity, because there is always some damping in your swing. This will become clear in the (b).
`color{brown}{(b) ul"Driving Frequency Close to Natural Frequency :"}`
`color{blue} ✍️`If `ω_d` is very close to `ω , m (ω^2– 2_d^2 )` would be much less than ωd b, for any reasonable value of b, then Eq. (14.39) reduces to
`color{blue} {A= F_o/(omega_db )}`
...........................(14.41)
`color{blue} ✍️`This makes it clear that the maximum possible amplitude for a given driving frequency is governed by the driving frequency and the damping, and is never infinity. The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is called resonance.
`color{blue} ✍️`In our daily life we encounter phenomena which involve resonance. Your experience with swings is a good example of resonance. You might have realised that the skill in swinging to greater heights lies in the synchronisation of the rhythm of pushing against the ground with the natural frequency of the swing.
`color{blue} ✍️`To illustrate this point further, let us consider a set of five simple pendulums of assorted lengths suspended from a common rope as shown in Fig. 14.22.
`color{blue} ✍️`The pendulums 1 and 4 have the same lengths and the others have different lengths. Now let us set pendulum 1 into motion. The energy from this pendulum gets transferred to other pendulums through the connecting rope and they start oscillating.
`color{blue} ✍️`The driving force is provided through the connecting rope. The frequency of this force is the frequency with which pendulum 1 oscillates. If we observe the response of pendulums 2, 3 and 5, they first start oscillating with their natural frequencies of oscillations and different amplitudes, but this motion is gradually damped and not sustained.
`color{blue} ✍️`Their frequencies of oscillation gradually change and ultimately they oscillate with the frequency of pendulum 1, i.e. the frequency of the driving force but with different amplitudes. They oscillate with small amplitudes. The response of pendulum 4 is in contrast to this set of pendulums.
`color{blue} ✍️` It oscillates with the same frequency as that of pendulum 1 and its amplitude gradually picks up and becomes very large. A resonance-like response is seen.
`color{blue} ✍️`This happens because in this the condition for resonance is satisfied, i.e. the natural frequency of the system coincides with that of the driving force.
`color{blue} ✍️`We have so far considered oscillating systems which have just one natural frequency. In general, a system may have several natural frequencies. You will see examples of such systems (vibrating strings, air columns, etc.) in the next chapter.
`color{blue} ✍️`Any mechanical structure, like a building, a bridge, or an aircraft may have several possible natural frequencies.
`color{blue} ✍️`An external periodic force or disturbance will set the system in forced oscillation. If, accidentally, the forced frequency `ω_d` happens to be close to one of the natural frequencies of the system, the amplitude of oscillation will shoot up( resonance), resulting in possible damage.
`color{blue} ✍️`This is why soldiers go out of step while crossing a bridge. For the same reason, an earthquake will not cause uniform damage to all building in an affected area, even if they are built with the same strength and materials.
`color{blue} ✍️`The natural frequencies of a building depend on its height, and other size parameters, and the nature of building materials. The one with its natural frequency close to the frequency of seismic wave in likely to be damaged more.
`color{blue} ✍️`When a system (such as a simple pendulum or a block attached to a spring) is displaced from its equilibrium position and released, it oscillates with its natural frequency ω, and the oscillations are called free oscillations.
`color{blue} ✍️`All free oscillations eventually die out because of the ever present damping forces. However, an external agency can maintain these oscillations. These are called force or driven oscillations. We consider the case when the external force is itself periodic, with a frequency `ω_d` called the driven frequency.
`color{blue} ✍️`A most important fact of forced periodic oscillations is that the system oscillates not with its natural frequency `ω`, but at the frequency `ω_d` of the external agency; the free oscillations die out due to damping.
`color{blue} ✍️`A most familiar example of forced oscillation is when a child in a garden swing periodically presses his feet against the ground (or someone else periodically gives the child a push) to maintain the oscillations.
`color{blue} ✍️`Suppose an external force `F(t)` of amplitude `F_0` that varies periodically with time is applied to a damped oscillator. Such a force can be represented as
`color{blue} {F(t) = F_o cos ω_d t}`
................... (14.36)
`color{blue} ✍️`The motion of a particle under the combined action of a linear restoring force, damping force and a time dependent driving force represented by Eq. (14.36) is given by,
`color{blue} {m a(t) = –k x(t) – bv(t) + F_o cos ω_d t}`
............ (14.37a)
`color{blue} ✍️`Substituting `d^2x//dt^2` for acceleration in Eq. (14.37a) and rearranging it, we get
`color{blue} {m (d^x)/(dt^2) + b (dx)/(dt) + kx = F_o cos omega_d t}`
.........................(14.37b)
`color{blue} ✍️`This is the equation of an oscillator of mass m on which a periodic force of (angular) frequency `ω_d` is applied. The oscillator initially oscillates with its natural frequency `ω`.
`color{blue} ✍️`When we apply the external periodic force, the oscillations with the natural frequency die out, and then the body oscillates with the (angular) frequency of the external periodic force. Its displacement, after the natural oscillations die out, is given by
`color{blue} {x(t) = A cos (ω_dt + φ )}`
................(14.38)
`color{blue} ✍️`where t is the time measured from the moment when we apply the periodic force.
`color{blue} ✍️`The amplitude A is a function of the forced frequency `ω_d` and the natural frequency `ω`. Analysis shows that it is given by
`color{blue} {A = F_o/(m^2 (omega^2 - omega_d^2 )^2 + omega_d^2 b^2 )^(1//2)}`
..........(14.39a)
and
`color{blue} {tan phi = (- v_0)/(omega_d x_0)}`
.................... (14.39b)
`color{blue} ✍️`where `m` is the mass of the particle and `v_0` and `x_0` are the velocity and the displacement of the particle at time `t = 0`, which is the moment when we apply the periodic force. Equation (14.39) shows that the amplitude of the forced oscillator depends on the (angular) frequency of the driving force.
`color{blue} ✍️`We can see a different behaviour of the oscillator when `ω_d` is far from `ω` and when it is close to `ω`. We consider these two cases.
`color{brown}{(a) ul"Small Damping, Driving Frequency far from Natural Frequency :"}`
`color{blue} ✍️` In this case, `ω_d` b will be much smaller than `m(ω_2–ω_2d)`, and we can neglect that term. Then Eq. (14.39) reduces to
`color{blue} { A = F_o/( m ( omega^2 - omega_d^2 ))}`
...................(14.40)
`color{blue} ✍️`Fig. 14.21 shows the dependence of the displacement amplitude of an oscillator on the angular frequency of the driving force for different amounts of damping present in the system.
`color{blue} ✍️` It may be noted that in all the cases the amplitude is greatest when `ω_d //ω = 1`. The curves in this figure show that smaller the damping, the taller and narrower is the resonance peak.
`color{blue} ✍️`If we go on changing the driving frequency, the amplitude tends to infinity when it equals the natural frequency. But this is the ideal case of zero damping, a case which never arises in a real system as the damping is never perfectly zero.
`color{blue} ✍️`You must have experienced in a swing that when the timing of your push exactly matches with the time period of the swing, your swing gets the maximum amplitude. This amplitude is large, but not infinity, because there is always some damping in your swing. This will become clear in the (b).
`color{brown}{(b) ul"Driving Frequency Close to Natural Frequency :"}`
`color{blue} ✍️`If `ω_d` is very close to `ω , m (ω^2– 2_d^2 )` would be much less than ωd b, for any reasonable value of b, then Eq. (14.39) reduces to
`color{blue} {A= F_o/(omega_db )}`
...........................(14.41)
`color{blue} ✍️`This makes it clear that the maximum possible amplitude for a given driving frequency is governed by the driving frequency and the damping, and is never infinity. The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is called resonance.
`color{blue} ✍️`In our daily life we encounter phenomena which involve resonance. Your experience with swings is a good example of resonance. You might have realised that the skill in swinging to greater heights lies in the synchronisation of the rhythm of pushing against the ground with the natural frequency of the swing.
`color{blue} ✍️`To illustrate this point further, let us consider a set of five simple pendulums of assorted lengths suspended from a common rope as shown in Fig. 14.22.
`color{blue} ✍️`The pendulums 1 and 4 have the same lengths and the others have different lengths. Now let us set pendulum 1 into motion. The energy from this pendulum gets transferred to other pendulums through the connecting rope and they start oscillating.
`color{blue} ✍️`The driving force is provided through the connecting rope. The frequency of this force is the frequency with which pendulum 1 oscillates. If we observe the response of pendulums 2, 3 and 5, they first start oscillating with their natural frequencies of oscillations and different amplitudes, but this motion is gradually damped and not sustained.
`color{blue} ✍️`Their frequencies of oscillation gradually change and ultimately they oscillate with the frequency of pendulum 1, i.e. the frequency of the driving force but with different amplitudes. They oscillate with small amplitudes. The response of pendulum 4 is in contrast to this set of pendulums.
`color{blue} ✍️` It oscillates with the same frequency as that of pendulum 1 and its amplitude gradually picks up and becomes very large. A resonance-like response is seen.
`color{blue} ✍️`This happens because in this the condition for resonance is satisfied, i.e. the natural frequency of the system coincides with that of the driving force.
`color{blue} ✍️`We have so far considered oscillating systems which have just one natural frequency. In general, a system may have several natural frequencies. You will see examples of such systems (vibrating strings, air columns, etc.) in the next chapter.
`color{blue} ✍️`Any mechanical structure, like a building, a bridge, or an aircraft may have several possible natural frequencies.
`color{blue} ✍️`An external periodic force or disturbance will set the system in forced oscillation. If, accidentally, the forced frequency `ω_d` happens to be close to one of the natural frequencies of the system, the amplitude of oscillation will shoot up( resonance), resulting in possible damage.
`color{blue} ✍️`This is why soldiers go out of step while crossing a bridge. For the same reason, an earthquake will not cause uniform damage to all building in an affected area, even if they are built with the same strength and materials.
`color{blue} ✍️`The natural frequencies of a building depend on its height, and other size parameters, and the nature of building materials. The one with its natural frequency close to the frequency of seismic wave in likely to be damaged more.