In the introductory tutorial in this series we considered a free vibrating translational spring and mass system with one degree of freedom and no external force applied as shown in Figure 1 below. Mass **m** connected to a spring with spring constant **k** with static equilibrium position **x = 0** is free to move on a horizontal frictionless plane. The diagram shows the force **F _{s}** exerted by the spring on mass m, (i) in extension and (ii) in compression.

The free body diagrams show **F _{s} = -k.x** at all positions of x in the oscillating cycle. This force must equal the acceleration

**a**of mass m by Newton's Law.

This equation is an expression of **simple harmonic motion** in that acceleration is always directed towards the equilibrium point and is directly proportional to the distance from that point.

### Solution to the equation of motion

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 study 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**give the following general solution for x = f(t):

_{2}= (α - iβ)In the case above **r _{1} = +i.ω** and

**r**Thus

_{2}= -i.ω**α = 0**and

**β = ω**giving the general solution for the equation of motion as:

It is conventional to replace constants C_{1} and C_{2} with **A** and **B** giving:

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

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

Meaning at time **t = 0** the extension of the spring is **x _{0}** and the mass has 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**as;

_{0}**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 of x(t) called the **amplitude** of the oscillation.

**ω** represents the **angular frequency** of the oscillation in radians/sec. This is also the **natural frequency** of the free vibrating system denoted by **ω _{n}**.

Note that:

- ω
_{n}/ 2π is the natural frequency measured in Hertz (Hz). - The period T (seconds)of one cycle is the reciprocal of the frequency = 2π / ω
_{n}.

### Example

Figure 2 below plots displacement **x(t) = R _{1}.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

**Figure 2 Plot of free vibrating spring and mass system**

For this system with the above initial conditions:

**natural frequency ω**= √(k/m) = √(10) rad/sec equivalent to (ω_{n}_{n}/2π) = 0.503 Hz.**period T**= (2π/ω_{n}) = 2π/√(10) = 1.99 s**amplitude R**= √([x_{1}_{0}^{2}+ (v_{0}/ω_{n})^{2}] = √([0.1^{2}+ (1/√((10))^{2}] = 0.332 m**phase angle φ**= tan^{-1}(v_{0}/ω_{n}x_{0}) = tan^{-1}3.162 = 72.45°

Phase angle is measured relative to motion of the same spring and mass system where the initial velocity is zero.

Such a condition is shown in Figure 2 by the plot for **x(t) = R _{2}cos(ωt)** with initial conditions:

- x
_{0}= 0.1 m - v
_{0}= 0

giving R_{2} = 0.1 m

_{n}depends only on k and m. The phase angle φ between the two oscillations in this example = 72.5° with R

_{1}cos(ωt - φ) lagging R

_{2}cos(ωt).

Next: Free vibrations with damping

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