In the introductory tutorial in
this series we considered a free vibrating translational spring and
mass system with one degree of freedom as shown in Figure 1 below.
Mass **m** connected to a spring with spring constant
**k** is free to move
on a horizontal frictionless plane from the static equilibrium
position x = 0. The diagram shows the force F_{s} exerted by the
spring on mass m in extension (i) and compression (ii).

From dynamic* free body diagrams we obtained the following equation of motion in terms of displacement x and time t noting that this equation is an expression of simple harmonic motion (SHM).

* 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 homogeneous linear differential equations using the characteristic equation method refer to the maths tutorial.

Note that** k** has dimensions ML / T^{2}L
(force / displacement). Thus **ω**^{2}
and **ω = √(k / m) ** have dimensions respectively
1 / T^{2} and 1 / T.

From the maths tutorial we see that two complex conjugate roots of the characteristic equation expressed as:

r_{1} =( α + iβ) and r_{2} = (α
- iβ) give the following general solution for x = f(t):

In the case above r_{1, 2 }= ± iω
Thus α = 0 and β = ω giving the general
solution for the equation of motion as:

It is conventional to replace constants C1 and C2 with A and B giving:

x(t) = [Acos(ωt) + Bsin(ωt)] ---------- (3)

We now find a particular solution for (3) using the following initial conditions at time t = 0 :

This means that at time t = 0 the extension of the spring is x_{0
}and the mass is given initial velocity v_{0}.

Inserting initial condition x_{0} into
equation (3) at time t = 0 gives:

x_{0 }= Acos(ωt) + Bsin(ωt)
= (A).(1) + (B).(0) = A thus A = x_{0}

v_{0} = -ωA.sin(ωt) + ωB.cos(ωt) = -ωA.(0) + ωB.(1)
thus B = (v_{0} / ω)

x(t) can now be expressed
in terms of initial conditions x_{0} and v_{0} as;

x(t) = x_{0}cos(ωt) + (v_{0 }/ ω) sin(ωt) -------
(4)

It is convenient to express the equation of motion in terms only of cos(ωt) using the following identity:

Acos(ωt) + Bsin(ωt) = Rcos(ωt - φ)
where R = √(A^{2} +B^{2}) and φ = tan^{-1}
B /A

**R** is the maximum absolute value (or **
amplitude**) of x(t).

**ω **represents the angular frequency of the SHM in
radians/sec, called the **
natural frequency** of the system denoted by

Figure 2 below plots displacement x(t) = R.cos(ωt - φ) against time t for a spring and mass system with the following parameters:

mass m = 1 kg spring constant k = 10 N/m
initial displacement x_{0} = 0.1 m
initial velocity v_{0} = 1 m/s.

In this example ω_{n }= √(k / m) = √(10 / 1) =
√((10) rad/sec equivalent to (ω_{n} / 2π) = 0.503 Hz..
Period T = (2π / ω_{n} ) = (1 / 0.503) = 1.99 s.
Amplitude R = 0.332 m.

Figure 2 also plots x(t) = R.cos(ωt) which represents SHM with
initial conditions x_{0} = R and v_{0} = 0.
Note that amplitude and frequency of both motions are identical.
The phase angle φ between the two oscillations in this example =
72.5° with R.cos(ωt - φ) lagging R.cos(ωt)

In the next tutorial we consider free vibrations with damping.

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