In the introductory tutorial in
this series we considered a harmonic forcing function **F**_{0}**cos(ωt)**
applied to a translational spring and mass
system with one degree of freedom having mass **m** and spring constant** k** shown in Figure 1 below with the spring
extended by distance x .

It should be understood that a completely undamped spring and mass
oscillating system does not exist. Coulomb friction forces will in time dissipate the initial **transient**
response after which the forcing function produces a**
steady state** response. However, analysing the
response to a forcing function of a system with no explicit damping
term has merit in that it introduces important
concepts, particularly **resonance**.

From the dynamic* free body diagram we obtained the following equation of motion in terms of displacement x and time t

* showing forces producing acceleration of mass m, not static equilibrium

We now find a function x(t) that is a general solution to this equation. If you are not familiar with solving second-order non-homogeneous ordinary linear differential equations using the method of undetermined coefficients refer to the maths tutorial.

From a previous tutorial we know
the general solution x_{h}(t) to this equation is:

We now propose a particular solution x_{p}(t) to equation
(1) of the form x_{p}(t) =
[A.cos(ωt) + B.sin(ωt)] where A and B are undetermined
coefficients and ω is the frequency of the harmonic forcing
function.

The general solution to equation (1) is x_{h}(t)
+x_{p}(t) giving:

Equation (3) is a valid solution for all values of ω except ω
= ω_{n} which gives a duplicate solution containing
the function cos(ω_{n}t). The proposed particular
solution for this condition takes the form
x_{p}(t) = [A.t.cos(ω_{n}t) + B.t.sin(ω_{n}t)] which we develop
later. Meanwhile continuing with a particular solution to
equation (1) for ω ≠ ω_{n
}we find constants C_{1} and C_{2} for initial
conditions x_{0} and v_{0} at time t = 0 where
v_{0 }= dx(t)/dt.

Thus the particular solution for equation (1) for initial
conditions x_{0} and v_{0} is:

The first term in equation (5) with cos(ω_{n}t)
expresses the transient response attributable to the free
vibration. In practice this response dissipates because of
friction. The second term with cos(ωt)
derived from the particular solution x_{p}(t)
expresses the steady state response.

Characteristics of the steady state response x(t) are illustrated in Figures 2 - 5 below.

A useful parameter is the **static amplitude δ = (F _{0
}/ k)** of the forcing function expressing the static
displacement of mass m under force F

Figure 2 plots the steady state response of the system when the
angular frequency of the forcing function ω > ω_{n} (ω_{n}
is the natural frequency of the system): note the response is out of
phase with the forcing function by 180°. In this example ω =
1.26. ω_{n} and the amplification factor (X / δ) ≈
1.6.

Figure 3 plots the steady state response of the system when the
angular frequency of the forcing function ω < ω_{n}: note the response is
in
phase with the forcing function. In this example ω =
0.73. ω_{n} and the amplification factor (X / δ) ≈
2.2.

The characteristics illustrated in Figures 2 and 3 can be deduced
from the relative values of ω and ω_{n} and the sign of (ω_{n}^{2}
- ω^{2}) in the coefficient of the cos(ωt)
term in equation (5). Figure 4 below shows amplification factor
(X / δ) ≈
10 when ω = 0.95 ω_{n} .

Figure 5 below shows the relationship between the **
frequency ratio (ω/ω**_{n})
and the absolute value of **amplification factor (X / δ)** . Note:
(i) different characteristics for ω<ω_{n} and ω>ω_{n},
(ii) an exponential relationship as ω approaches the value of ω_{n
}from either direction. This condition is known as
**resonance** where the theoretical response tends to
infinity. In practice physical constraints limit the amplitude of vibration.

Figures 2-5 illustrate the steady state condition of a free
vibrating system subjected to a harmonic forcing function after
transient effects are eliminated by the small amount of damping
inherent in the system. We now consider the response including
the transient term cos(ω_{n}t) in equation (5).

Equation (7) is plotted in Figure 6 below for ω = 40 rad/s and ω_{n
}= 30 rad/s giving:

This response illustrates the phenomenon of **beating**
where the amplitude of displacement varies periodically In
this example the displacement is a function of sin(35t) and the
envelope of the beat is a function of sin(5t). Beating occurs
as an interference phenomenon in sound
waves.

The final stage of this analysis is to find the separate solution to equation (3) for ω = ω_{n
}. As noted above the proposed particular solution for
this condition takes the form*
x_{p}(t) = [A.t.cos(ω_{n}t) + B.t.sin(ω_{n}t)]
------- (9)

* by the reduction of order method - see the maths tutorial

The general solution to equation (1) for ω = ω_{n} is x_{h}(t)
+x_{p}(t) giving:

From equation (11) find constants C_{1} and C_{2}
for initial conditions x_{0} and v_{0} at time
t = 0 where v_{0 }= dx(t)/dt

gives x_{0} = C_{1} + 0 + 0
thus C_{1} = x_{0}

Thus the particular solution for equation (1) when ω = ω_{n}
with initial
conditions x_{0} and v_{0} is:

Figure 7 below plots x(t) as stated in equation (13).

This plot shows the resonant condition induced by forcing
function F_{0}cos(ωt) when ω = ω_{n} implied
by the response for ω ≠ ω_{n}
illustrated in Figure 5. Note that line x(t) = (F_{0}
/ 2mω_{n}).t expresses the increase in amplitude of the
vibration. Note also the harmonic response (sin(ω_{n}t)) and
forcing function (cos(ω_{n}t)) are out of phase by 90°.

The next tutorial examines the response of damped free vibrations to a harmonic forcing function taking account of a damping term in the equation of motion.

I welcome feedback at: