Relation between electrical and mechanical resonant frequency

[mechandmeta]

My first post here. :D

usually this situation does not arise, but I am working on thin piezoelectric films. When a apply a alternating voltage on the electrodes, the film vibrates. Now this constitutes a LCR circuit which has a resonant and anti-resonance frequency. Since it is vibrating, it will have a mechanical resonant frequency at which the amplitude of vibration will be maximum. Obviously, the frequency of vibration will be determined by the frequency of the applied alternating voltage.

What I don't understand is this, is there any relation between the electrical and mechanical resonant frequencies? Or can the electrical and mechanical resonance conditions be achieved at the same applied frequency.

Please correct me if there is any fundamental mistake in my reasoning.

 

[Studiot]

Good morning mechandmeta and welcome to Physics Forums.

Posting is good as the discussion may help others as well as yourself.

You are correct in surmising that there is a link between the (differential) equations controlling the electrical system and those controlling the mechanical one.

You solve these as a pair (set) of simultaneous differential equations to obtain the annswer (model) you are seeking.

If you are familiar with differential equations we can discuss this further.

 

[jsgruszynski]

One way of looking at an electromechanical transducer is to think of it as a 2-port in a lumped model system. Basically you have an electrical circuit on one side, a mechanical circuit on the other side and the two port representing the transducer relationship between the two. This, if you can get away with it, is the simplest model you can use. You have to be able to meet the criteria of lumped models on both sides. Because of the coupling, one side affects the other much like a transformer couples impedances, currents and voltages.

 

[Studiot]

One way of looking at an electromechanical transducer is to think of it as a 2-port in a lumped model system.

I am not familiar with this approach - it sounds interesting. I, at least, would be grateful if you would post the applicable equations. What dimensions would the constants of the system possess?

Two port networks are generally regarded as part of linear theory. How would non linearity be treated?

 

[mechandmeta]

Good morning mechandmeta and welcome to Physics Forums.


You solve these as a pair (set) of simultaneous differential equations to obtain the answer (model) you are seeking.

If you are familiar with differential equations we can discuss this further.

Yes I am familiar with differential equations. I am also aware of the equation for mechanical resonance, but have little clue about electrical circuits beyond the bare basics. Please do help me along.


I looked over the equation for electrical circuits (for series resonance for a start) and that makes sense, but how do I connect these two equations ?

 

[Studiot]

Here is a start

https://www.physicsforums.com/showthread.php?t=421136&highlight=coupled+differential+equations

 

[Studiot]

Google has lots of references to Surface Acoustic Wave (SAW)devices which are very similar.

Here for instance.

http://www.math.missouri.edu/~carmen/pdf/article2.pdf

 

[mechandmeta]

For a piezoelectric crystal with a applied alternating voltage
The mechanical equation is

[tex]m\frac{{{d^2}x }}{{d{t^2}}} + r\frac{{dx }}{{dt}} + sx = F[/tex]

E= -g33.S (S=stress, piezoelectric voltage const, E=electric field)
F= -A.E/g33 (A=Area of resonator, l=hieght)
F= - A.V/(g33 .l)

V=V0sin(ωt)

and the electrical equation is

[tex]L\frac{{{d^2}q }}{{d{t^2}}} + R\frac{{dq }}{{dt}} + q/C = V0sinωt [/tex]

Where

L is the circuit inductance
R is the circuit resistance
D is common to both equations and is dependent on the magnetic flux
q is the charge
x is the displacement of the surface
m is the effective mass in oscillation
r is the mechanical resistance to displacement
s is the stiffness

So what all is wrong in the above formulation?

 

[Studiot]

Hello mechandmeta. I am sorry it has taken so long to reply but I have been busy with other things.

No, there is nothing inherently wrong with you equations, except that they do not form a linked system - they are independent.

You need an assumption of the relationship between force and (in your case) charge or perhaps voltage. That is some equation that relates the amount of voltage generated for a given displacement (x) or the amount of displacement exhibited for a given voltage.

Substituting this relationship will link your equations and allow solution.

Inherent in the magnetic example I showed was the assumption that displacement is proportional to current. I used current, not charge in the electrical differential equation because it leads to a first, rather than second order equation. Also you electrical equation at the moment implies an (additional external) electrical forcing function (applied voltage) rather than one which is the result of the interaction between the electrical and mechanical system.

does this help?

 

[mechandmeta]

Inherent in the magnetic example I showed was the assumption that displacement is proportional to current. I used current, not charge in the electrical differential equation because it leads to a first, rather than second order equation. Also you electrical equation at the moment implies an (additional external) electrical forcing function (applied voltage) rather than one which is the result of the interaction between the electrical and mechanical system

does this help?

Yes, every word helps and thank you so much for helping out.

But in your example you had neglected capacitance which allowed you to get away with using I instead of q.

I do not understand your second part. There IS a electrical forcing function (thats the point of my transducer). As I look at it, the mechanical vibration is influenced by the electric circuit (which leads to the presence of a electrical term in the mechanical DE) but i don't think that the electrical circuit is influenced by the mechanical vibrations, so how can I put the interaction between the electrical and mechanical system into the electrical equation?

 

[Studiot]

If this is a genuine piezo film then the piezo effect is the direct conversion of mechanical energy or force into electrical energy. Basically the harder you hit the crystal the greater the voltage generated.

 

[BFisher]

Hello Mechandmeta. I am not sure if you have already solved your problem, but I also work with piezoelectric substrates and I have spent a significant amount of time evaluating piezoelectric theory. You need to start with the constitutive equations for stress and charge displacement in a piezoelectric given by:

T = cS - eE
D = eS+episonE

where T is stress, S is strain, e is the piezoelectric constant of your material, D is the charge displacement and epsilon is the dielectric constant of the piezoelectric substrate.