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How do we analyse problems involving harmonic motion?

In our previous article on mechanical resonance, we looked at harmonic motion and how it affects structures.  Now we’re going to jump into how we can solve issues that involve simple harmonic motion.

What is simple harmonic motion?

Many systems consist of components that vibrate in the presence of appropriate forces, which can be desirable, undesirable, or unavoidable.  This is called Simple Harmonic Motion (SMH).

Understanding the nature of vibration means we can analyse and solve a range of engineering problems.  Any system that consists of a mass resting on, or supported by, a spring or flexible structure falls into this category.  One of the most important factors is identifying the natural frequency of the system.  This is the frequency at which resonance occurs.  This is the frequency at which the system will vibrate in the absence of any outside force.

Categorising Vibration

Vibration can be categorised into four different types.  One is free vibration, where there is no force generating the motion.  Another is forced vibration, where a force acts on the system.

The other categories are damped or undamped.  Damping is a force that causes vibrations to die away, and it is present in most systems to some extent.  Undamped is the opposite.

Notation

When we analyse vibration, we are dealing with displacement, velocity, and acceleration.  It’s very common to use this short-hand notation:

, How do we analyse problems involving harmonic motion?

The over-dot represents differentiation with respect to time, t.

Response of a mass spring system

The spring-mass system is simple, but many engineering systems show the same behavior, so it is useful for us to understand.  It provides a great deal of information about a range of systems. Let’s look at a system where the mass is displaced slightly from its equilibrium position and then released.  The only force acting on the mass is the spring force.

spring mass system at its equilibrium position

The diagram shows a spring mass system in its equilibrium position.  The corresponding position, velocity, and acceleration are shown; position and velocity are towards the right and are positive, and acceleration is negative since the mass is decelerating.

Let’s imagine a mass is attached to this spring, and we assume there is no friction or damping force acting.  If we displace the mass from its equilibrium position by a certain distance, x, it will feel a force from the spring equal to kx directed towards its original position.

If the mass is released, this force will accelerate the mass towards its original position with a force that reduces to zero. As the mass passes its original position, the spring force begins to increase in the other direction, decelerating the mass.

The equation can be described as below, assuming the positive is to the right.  If the mass is displaced to the right, it feels a spring force acting to the left, and so our equation is: 𝒎𝒙̈ = −𝒌𝒙

Dividing across by m, we obtain an expression for acceleration: 𝒙̈ = -kmx

This equation allows us to make two important observations about the motion of the mass.  We can see that the acceleration, the second derivative of position, is always equal to minus a constant (k/m) multiplied by position (x). In fact, trigonometric functions have this property, so suppose we assume that: 𝑥(𝑡) = A cos(𝜔𝑡)

Where A is the amplitude (m) of the response and t is time (s) and ω is the frequency (rad/s), then if we differentiate with respect to time:

𝑥̇ = −𝜔𝐴 sin(𝜔𝑡)

𝑥̈ = −𝜔2𝐴 cos(𝜔𝑡) = −𝜔2𝑥

Therefore:

𝒙̈ = −𝝎𝟐𝒙          𝑤ℎ𝑒𝑟𝑒 𝝎𝟐 = k/m

This solution will work for our equation of motion, so we can say that a spring-mass system that has been given an initial disturbance moves in a sinusoidal manner. If we plot the position of the mass against time, it has a sinusoidal path, and the maximum value of this displacement is A, so that the mass moves between maximum displacements of –A and +A from its rest position.

Alternative Equation

Another form of the simple harmonic motion equation can be obtained by noting the relationship between acceleration and velocity:. Since 𝑥̈ = −𝜔2𝑥, we can write:

This equation is valid at any time and is valid for any system that executes simple harmonic motion.

Let’s look at an example.  Say a mass of 5kg is suspended from a spring with a stiffness of 2 kN/m.  We want to calculate the natural frequency of the spring, and then calculate the maximum velocity and acceleration of the mass if the spring is displaced by 25mm.

For the sake of this example, let’s say the spring is initially in equilibrium, so we don’t need to consider the weight of the mass as the spring has already deflected to account for this.

Firstly, the natural frequency and periodic time can be calculated by:

n=km=20005=20rads         fn=n2π=3.18 Hz          T=1fn=0.31 s

Now let’s calculate the maximum velocity and acceleration.  The formula for velocity is 𝑥̇ = −𝜔𝑋 𝑠𝑖𝑛 𝜔𝑡 where A is the amplitude of the position. Since we initially displaced the mass by 25 mm before releasing it, A must be 25 mm. Since the maximum value of sin(ωt) is ±1, then the maximum absolute value of velocity must be ωA:

|𝑥̇|𝑚𝑎𝑥 = 𝜔A = (20)(0.025) = 0.5𝑚/𝑠

A similar situation exists for acceleration, whose formula is 𝑥̈ = −𝜔2𝑋 cos 𝜔𝑡. Again, since the maximum value of the cosine term is ±1, we have a maximum value of acceleration of:

|𝑥̈|𝑚𝑎𝑥 = 𝜔2A = (20)2(0.025) = 10𝑚/𝑠2

We’re going to continue our series with more interesting facts and calculations around vibration and how it affects civil engineering structures.

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