Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by: Dave L Moulton

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  30-October-2002

 

Content

 

This lecture takes a deeper look into the system equations derived during lecture 4.  The differential equation for a simple Mass-Spring-Damper system will be broken down to reveal some very useful characteristics that describe the motion of the system. The nature of the system can be determined and predicted by only having to know simple parameters such as Mass, Stiffness and Resistance.

 

I am using the Mechanical model rather than the Acoustic or Electrical model because it is a much easier system to visualize, however the exact same analysis can be done on the other two equivalent domain systems.

 

 On with the Lecture

 

The now familiar simple mechanical Mass-Spring-Damper system is shown in figure 5.1  below:

 

 

Figure 5.1

 

The differential equation for this system can be easily established by equating the Forces:

 

 

 

 

Thus expressed in differential form:

 

 

This equation totally describes the motion of the system.  An in depth analysis of this equation will reveal the true nature of the system, giving an insight into how the system responds to changes in physical parameters such as the Mass, Stiffness and Resistance.

 

The analysis I am about to show you is also applicable to the acoustic cavity system (Helmholtz) and the tuned electrical circuit (Resistor, Inductor, Capacitor). 

 

The importance of this analysis will become more relevant as this set of lectures progresses. In particular for the attenuation of headset earshells and tuned acoustic filters.

 

Now lets get on with the analysis:

 

Firstly you will notice that equation [5.2] comes in two parts.  On the left hand side we have the external applied Force F and on the right hand side we have the Natural restoring forces of the system.  We can gain a lot of  information about the natural response of the system if we apply an initial force and then remove it completely.  The system will do one of three things after the force has been removed:

 

Case 1.  The Mass will  move very slowly back to its original rest position. This is the case for a Heavily Damped System.

 

Case 2.  The Mass will move back to its rest position as quickly as it can without overshooting the rest position. This is the case for a Critically Damped System.

 

 

Case 3.   The Mass will finally reach its rest position after one or more overshoots. In this case the mass will oscillate about its rest position with an amplitude that eventually reduces to zero. This is the case for an Underdamped System.

 

 

Now lets dive into equation [5.2] and see how these three cases can be determined.

 

Firstly we will remove the external force so that we are only dealing with the natural characteristics of the system.

 

 

 

This expression which equates to zero also has every term as a function of x, this is commonly known as the Homogenous Equation.

 

Solving equation [5.3] is very simple. Any engineering degree should have covered this in the first year.

 

Firstly we will write the derivative operator as follows:

 

It is also very reasonable to also write:

 

 

So let us now put this operator into equation [5.3]

 

 

 

This equation is now in the form of a quadratic, with two possible outcomes. Either x=0 or the quadratic in brackets =0. The case where x=0 is of no consequence since we are assuming that the system is experiencing some form of motion over the period of time t. Thus the quadratic must equate to zero.

 

 

Using some very basic O-level/CSE mathematics we can very easily see that this equation has the following solutions:

 

 

A little tidying up gives:

 

 

What’s the point of this you may ask yourselves.

 

Well the point is that the D-notation provides us with a general solution to the differential equation.

 

 

G₁ and G₂ are constants, whereas D₁ and D₂ are as shown below:

 

 

 

We can now re express the general solution given in equation [5.10] as:

 

What we are  now looking at is a very important relationship which governs the motion of the system.  Clearly the term under the square root sign is very deterministic of the system.  Lets take each case one by one.

 

Case 1: (Heavy Damping)

 

 

In this situation the Resistive elements dominate the system and make it very sluggish to any change from its natural rest position.  As an example this is a required case for the natural response of earshells on an attenuating headset.

 

Lets simplify the expression even further by letting:

 

 

Thus:

 

 

For case 1, a is real and positive.

 

We can  now re-express equation [5.16] so that it better illustrates case 1.

 

Let  G₁ + G₂=H,   and  G₁ - G₂=M

 

Thus:

and

 

We can now express [5.15] in the following way:

 

 

Thus

 

Which can be very conveniently written as:

 

 

This is a hyperbolic function and not an oscillatory function. The system will never overshoot or oscillate about its mean position after the external force has been removed. It will simply go back to its rest position after a time t.

 

A simplified situation is when at time t=0, x(t)=0.  In which case we have H=0.

 

Thus:

 

 

This type of heavily damped motion can be seen in figure 5.2 below:

 

 

Figure 5.2

 

Now lets move on to  case 2.

 

Case 2  (Critical Damping)

 

Again let us consider the nature of equation 5.13, only this time we will look at what happens when the Resistive forces and the spring Stiffness forces balance out.

 

Thus:

 

 

 

 

This results in a special case solution to the original differential equation. In this case the quadratic has repeated roots since a=0.

 

We could be tempted to write the solution as:

 

 

 

However, this is not the complete solution since we must have at least two constants in our solution.  I will leave it to the class to prove this to themselves (its not difficult).

 

The actual solution takes the form:

 

 

Where K is the second constant.

 

Once again we can take a typical case where x(t)=0 when t=0.

This reduces the equation down to:

 

 

 

[5.25] is not an oscillatory function and has no overshoot, however when the damping is critical the Mass will return to its rest position in the shortest possible time without overshooting the position.  You could say that the system is on the brink of overshoot, hence the damping is at a critical value.

 

 

Figure 5.3

 

Now for the final case.

 

Case 3  (Under Damped System)

 

Now lets consider what happens when the Spring Stiffness Force (Low Compliance) dominates the Resistive Force.

 

This condition is clearly achieved when:

 

 

This condition makes the expression inside the square root sign negative, thus the whole expression becomes complex.

 

 

This leads to the solution to our differential equation taking the form:

 

 

Applying the same reasoning as used for equations [5.17, 5.18, 5.19 and 5.20] we can easily see that:

 

 

If we now let H =P and Mi=Q, we get:

 

 

 

Which conveniently changes to:

 

 

We are now dealing with a system that has overshoot, the system will oscillate about its rest position before finally coming to rest.

 

Once again the typical case would be x(t)=0 when t=0. Thus reducing the equation to:

 

 

A typical example as given in lecture 4, was that of a system with no resistive forces producing Simple Harmonic Motion (SHM).

 

This case can clearly be shown in equation [5.32] by making R_m=0. The system then oscillates continuously as sine function with an angular frequency a.

 

Lets look a bit more closely at the angular frequency of oscillation and re-express it in a more usual form w.  Where w=2pf

 

 

Now remember from lecture 4 where we talked about the natural frequency of the system being:

 

 

Thus:

 

A little more manipulation:

 

Leading to:

 

 

Where z is known as the damping coefficient.

 

 

Using this expression for the damping coefficient we can now simplify equation [5.31] down to:

 

 

 

From [5.39] it is very clear that the oscillation frequency of the system is significantly influenced by the damping coefficient.

 

[5.39] can be used as the general solution in all three cases:

 

Case 1  Heavy Damping

 

z>1.  The Damping coefficient is greater than one.  The greater the value, the more heavily damped the system is.  The term under the square root sign becomes negative converting the sine and cosine functions into non-oscillatory Hyperbolic functions.

 

 

Case 2  Critical Damping

 

z=1.  The sine and cosine functions disappear, leaving a basic non-oscillatory relationship. (remember that the function has to be modified to take into account the second coefficient).

 

Case 3  Under Damping

 

z<1.  The function remains oscillatory and can be described by the periodic sine and cosine functions.

 

Quality Factor Q

 

Now lets take a quick look at another important feature known as the Quality Factor or Q-factor.

 

Firstly lets just consider the part of the general solutions which causes the amplitude of the system to decay down to zero. If we assume that the system amplitude at time t=0 is A₀, then the amplitude after time t will have decayed to a new lower level ‘A’.

 

Thus:

 

 

 

It is often more useful to look at the energy decay in the system, as a reducing amplitude is directly related to energy being lost from the system due to Resistive Friction.

 

Now the energy is proportional to the square of the amplitude:

 

 

A special case occurs when t=1/(2zw₀).

 

Under this condition the amplitude decays to

 

 

Now if we consider the system to be oscillating as a sinusoid with decaying amplitude.

 

 

Thus for our special time constant  we will have:

 

 

We define Q as the number of radians that the system will move through before the amplitude reaches e^-1 times the initial peak amplitude.

 

 

Q will always be a positive value (greater than zero). This is the condition for an oscillatory system (with overshoot).  We can make a simplification to equation [5.45] if we assume that z² <<1, Thus we can approximate the Q-value to be:

 

 

End of Lecture