Myth, magic, truth...ukulele "opening up"

[70sSanO]

Opening up is not so much about change but all about improvement. The hypothesis is that the instrument is getting better and better with age, and not deteriorating like all other material and live things.

I think most anything with mechanical attributes goes through a bell curve lifecycle.

There might be a certain break-in, then a period of time in the sweet spot where performance is at its best, then a decline with continued use.

John

 

[merlin666]

I think most anything with mechanical attributes goes through a bell curve lifecycle.

There might be a certain break-in, then a period of time in the sweet spot where performance is at its best, then a decline with continued use.

John

This makes really good sense. So now if we have this concept of "peak uke" it can open up a whole range of new discussion topics. For example how do we know when peak uke is reached? What are the implications for buying and selling? Tips for extending peak uke?

Wow this is like breaking new ground.

 

[Poul Hansen]

I just wonder:
1. Why is everybody talking about "opening"( a good thing) as opposed to "closing". IF there is a change/degradation happening in the wood, glue etc. couldn't it happen that the glue and the wood would be softer and therefore NOT vibrating so well and maybe even dampening the vibrations?
2. What are the criteria of a potentially "opening" instrument? This should be a number one criteria then, when buying a new instrument. We can't all buy 5-10 instruments and hope they will "open" in 20 or 30 years!!

With strings f.x. I like the "shimmer" of new strings and think they degrade from day one. They never "open" ;-)

 

[Ukecaster]

It's like many proposed concepts or beliefs...nobody can prove it either way. Until someone does, just find a good sounding uke, enjoy it, and hope it gets better. If it sounds lousy from the get go, move it on down the road. Let your ears be the judge (hopefully in this lifetime)

 

[Ottertail]

That point is well made; whether or not the change is “good” is entirely in the ear of the behearer. Much as any individual instrument may be preferred or not when new. And that impression can be as simple as it matches the sound and tone in a given circle of playing friends, or perhaps provides a contrast to the other instruments, or complements the players voice. Lotsa individual questions here.

 

[Tin Ear]

I think the debate is hot because people really have experienced the gamut of possibilities, and can only speak truthfully to their own experiences. People apply their own experiences to broad expectations. I’m not criticizing, this is step one of the scientific method. Observe, hypothesize, test.

I hypothesize the following:
Do instruments open up? Yes.
Do all instruments open up? No.
Do instruments that open up open to the same degree at the same rate? No.

I like this take on the topic. I agree with this. Nicely put. I see it pretty much the same way !

and your mileage may vary

 

[70sSanO]

With strings f.x. I like the "shimmer" of new strings and think they degrade from day one. They never "open" ;-)

Strings stretching is such an odd phenomenon.

The higher the tension, the more strings stretch and get smaller in diameter.

As strings get thinner they get brighter. And the tension is a little less with the diameter less.

Depending on each person’s tone preference, a string can start off great and get worse. Or start off okay and sound great when it stops stretching.

John

 

[merlin666]

Strings stretching is such an odd phenomenon.

The higher the tension, the more strings stretch and get smaller in diameter.

As strings get thinner they get brighter. And the tension is a little less with the diameter less.

Depending on each person’s tone preference, a string can start off great and get worse. Or start off okay and sound great when it stops stretching.

John

Elasticity is a material property. I think that fluorocarbon is much less elastic than Nylon or Nylgut formulations. Also over time elasticity on a molecular level is diminished as strings are under tension, and the string will become more brittle and prone to breaking

 

[SeemingMusic]

Hello all,

Been following this thread with interest for a while. If I may, and if anyone is interested, I have some understanding of material science as I am an engineer. Though it isn't my specialty. Nonetheless I want to talk about fatigue in materials.

What is fatigue?
A lot of you are quite right in that materials weakens over time. ALL materials (not just our precious soundboards) will undergo fatigue when subjected to cyclic loads. (Cyclic loads are loads that are applied, then removed, then applied again etc.) Whether it be an engine belt, bolts in a bridge, masts on a sailboat... the list goes on.

You may get a very brief summary on fatigue cracking here. To summarize, if we subject stress on a material, over many cycles, (such as the vibration induced into the soundboard), over time, tiny tiny cracks will develop in the material. The higher the stress, the less cycles it takes for the material to fail. Makes sense, if we push hard with a palm into the soundboard (please don't do this), it may only take a few pumps to break it. As the stresses we subject it to are relatively small, the number of cycles to break the material will be a lot.

In the microscopic level, cracks are slowly growing in length as it's subjected to more cycles. As an example, imagine you put a little tear in the middle of an edge of a piece paper. You hold the paper with both hands on either side of the tear. You then slowly move your hands apart and together, each time your hands move apart, the tear grows a little. This is called crack propagation. The crack is growing.

How many cycles to failure?
Fear not, the first graph is a logarithmic graph, it's actually really stretched out if we display the scale in a linear fashion. For the relatively low stresses on the ukes, it'll take A LOT of cycles to failure. No need to worry!

We can take a look at Stradivarius violins and they still exist, though I know many of them have repaired cracks. I don't have data on the cause of the cracking though and can't say for sure it's due to fatigue crack propagation.

I found the observations of @BradDonaldson really fascinating of the few instruments converging over time. I think this makes sense in my mind as you get crack nucleation, then crack growth over time. So the more quiet ukes went through a more steeper crack growth (area I) and then all the ukes end up at different points in a gentler slope in stable crack growth (area II), in which they will sit for pretty much the remaining lifetime of the uke. (This is my conjecture and personal guess though! I can be very wrong! Anyway back to the science.)



Cycles vs Time
When we ask the question, does wood "open up", it might be helpful to define the number of cycles rather than time. A musician playing 4 hrs a day for 20 years will subject the wood to more cycles than another player that strums for 30 min. a day for 6 months. The former instrument will have undergone more cycles and thus, one will expect to see more fatigue cracking in the microscopic level. Please think about whether this factor alone will make everyone have wildly different experiences to each other in this topic.

Stresses make a difference! How hard do you play?
A hard strummer will induce more stresses compared to a gentle fingerstylist. From the first graph, can you see how different uses of the instrument can dictate where you sit on the curve. From the graph below, it is aluminum, but you can see the higher stress will result in higher crack propagation rate!

How does this differ to Ultimate strengths?
There was some confusion and comments before that the forces aren't strong enough to permanently change the wood. Otherwise the wood will break. I understand this line of thinking. This warrants a small discussion.

When talking about materials, let's look at the graph below, stress starts at zero and goes up on the Y axis. If we subject stress on a material, up to the point B (yield strength), the material will snap back with no permanent deformation. This is known as the elastic range. Think of a stretching a rubber band slightly or bending a wooden ruler very gently. The gradient or how steep this curve is called the stiffness. The stiffer the material, the steeper, it is, the B point will be higher.

Woods have varying degrees of stiffness based on species. I should mention that stress in engineering terms means a force divided by an area. You can imagine poking someone will hurt more than pushing someone with both hands with the same amount of force, simply because the area it is applied is smaller (stress = F/A ... If F same... A low... stress high.... if A high e.g. 100, stress low).

Given the same piece of wood, if you make the piece thicker, given the same amount of force, because there's more cross section area (A), the stresses are lower. Luthiers will need to judge to determine the thickness of the soundboard, to ensure it stays in the elastic region.

Once the material exceeds point B, we are outside the elastic region, even if we release the stress, the material will not snap back to its original shape or length, permanent damage is done to the material. (Think stretching a rubber band too hard, it becomes looser once you let go).

When stresses are at point C, the material has reached its ultimate strength, in which it'll start to deform rapidly such as stretching and thinning out, until it breaks at point D.

This is not the normal mode of failure for our instruments. Only luthiers will encounter this. Perhaps there wasn't enough bracing to provide stiffness or the soundboard was way too thin, in which it'll collapse within itself like a dying star as soon as it's strung up. In summary, the forces and thus stresses induced on the wood isn't enough to cause it to fail in the ultimate strength mode. Fatigue cracking occurs at much lower stresses over many cycles. So yes, changes can occur.

Combining Everything
Combining everything, Here is an example for testing on Pinus pinaster Ait. wood. You will see the stress/strain graph with ultimate load at 220 N, the cycles and propagation rates can be modelled. We can get very rough durability indications here. (No acoustic findings to be inferred from this data though!)

Wood is not a homogeneous material
You can consider wood as an organic polymer. It isn't uniform or homogeneous like metals. I guess everyone knows this. Not accounting for the different species, the grain direction of the cut, temperature the tree was grown in, sunlight it received, ...( list goes on), will affect the internal structure of the wood. The internal structure varies a lot even within a single piece of wood. Therefore, the internal cracking, stress planes, direction the cracks will propagate etc. cannot be generalized.



Modelling fracturing in wood is highly difficult, and maybe in the realms of a phD level. However, given we can't really replicate the same piece of wood, or reproduce the results in the models, is there any value in examining this?


What's not in the scope of this lengthy post

• Effect of the build/construction - glue, joints, bracing etc.
• Internal stresses induced by changes in humidity/temperature, but cracks will occur from this as we all know.
• Effects on acoustics

The million dollar questions

• Do fatigue cracks improve the sound? Or does it degrade it?
• Keep in mind these are microscopic cracks! Are fatigue cracks at a length or magnitude where it has an effect?
• Cracks can join up over time - Is there a further change when this happen?
• Do joined up cracks result in a more flexible soundboard or cause the soundboard to act like a thinner one?
• Do laminated woods/double tops etc. slow down fatigue cracks?
• Are the voids from the cracks in the wood creating overtones or 'little echo chambers'?
• Does sound quality deteriorate when cracks get to a certain length?
• Can the human ear perceive the differences if there are any?

Hahaha I guess I've made everyone read a lot with no solid conclusions or solutions to offer. Maybe even more questions to boot.

I've tried my best to segregate my opinions to what is a known. Objective and commonly observed behavior of materials from a material science point of view. Many engineers will have learnt this in their studies. I'm sure there are a few of you in here. Let me know if I am wrong in anything. I don't mind! If you are a material scientist, I would love to know your thoughts! Hope I didn't offend anyone here, just offering my 2 cents.




You can read more about fatigue and fracture of woods in book below. It talks about the macrostructure more in which I am no expert, but it is fascinating!

Fracture and Fatigue in Wood​

Ian Smith, Eric Landis, Meng Gong

Fracture and Fatigue in Wood

Damage in wood is principally the result of fatigue. Fatigue is the process of progressive localised irreversible change in a material, and may culminate in cracks or complete fracture if conditions that initiated or propagated the process persist. Comprehensive understanding of fatigue and...

books.google.com.hk

Fracture and Fatigue in Wood

Damage in wood is principally the result of fatigue. Fatigue is the process of progressive localised irreversible change in a material, and may culminate in cracks or complete fracture if conditions that initiated or propagated the process persist. Comprehensive understanding of fatigue and...

www.wiley.com

 

[TBB]

Hello all,

Been following this thread with interest for a while. If I may, and if anyone is interested, I have some understanding of material science as I am an engineer. Though it isn't my specialty. Nonetheless I want to talk about fatigue in materials.

What is fatigue?
A lot of you are quite right in that materials weakens over time. ALL materials (not just our precious soundboards) will undergo fatigue when subjected to cyclic loads. (Cyclic loads are loads that are applied, then removed, then applied again etc.) Whether it be an engine belt, bolts in a bridge, masts on a sailboat... the list goes on.

You may get a very brief summary on fatigue cracking here. To summarize, if we subject stress on a material, over many cycles, (such as the vibration induced into the soundboard), over time, tiny tiny cracks will develop in the material. The higher the stress, the less cycles it takes for the material to fail. Makes sense, if we push hard with a palm into the soundboard (please don't do this), it may only take a few pumps to break it. As the stresses we subject it to are relatively small, the number of cycles to break the material will be a lot.

In the microscopic level, cracks are slowly growing in length as it's subjected to more cycles. As an example, imagine you put a little tear in the middle of an edge of a piece paper. You hold the paper with both hands on either side of the tear. You then slowly move your hands apart and together, each time your hands move apart, the tear grows a little. This is called crack propagation. The crack is growing.

How many cycles to failure?
Fear not, the first graph is a logarithmic graph, it's actually really stretched out if we display the scale in a linear fashion. For the relatively low stresses on the ukes, it'll take A LOT of cycles to failure. No need to worry!

We can take a look at Stradivarius violins and they still exist, though I know many of them have repaired cracks. I don't have data on the cause of the cracking though and can't say for sure it's due to fatigue crack propagation.

I found the observations of @BradDonaldson really fascinating of the few instruments converging over time. I think this makes sense in my mind as you get crack nucleation, then crack growth over time. So the more quiet ukes went through a more steeper crack growth (area I) and then all the ukes end up at different points in a gentler slope in stable crack growth (area II), in which they will sit for pretty much the remaining lifetime of the uke. (This is my conjecture and personal guess though! I can be very wrong! Anyway back to the science.)

View attachment 152199

Cycles vs Time
When we ask the question, does wood "open up", it might be helpful to define the number of cycles rather than time. A musician playing 4 hrs a day for 20 years will subject the wood to more cycles than another player that strums for 30 min. a day for 6 months. The former instrument will have undergone more cycles and thus, one will expect to see more fatigue cracking in the microscopic level. Please think about whether this factor alone will make everyone have wildly different experiences to each other in this topic.

Stresses make a difference! How hard do you play?
A hard strummer will induce more stresses compared to a gentle fingerstylist. From the first graph, can you see how different uses of the instrument can dictate where you sit on the curve. From the graph below, it is aluminum, but you can see the higher stress will result in higher crack propagation rate!

How does this differ to Ultimate strengths?
There was some confusion and comments before that the forces aren't strong enough to permanently change the wood. Otherwise the wood will break. I understand this line of thinking. This warrants a small discussion.

When talking about materials, let's look at the graph below, stress starts at zero and goes up on the Y axis. If we subject stress on a material, up to the point B (yield strength), the material will snap back with no permanent deformation. This is known as the elastic range. Think of a stretching a rubber band slightly or bending a wooden ruler very gently. The gradient or how steep this curve is called the stiffness. The stiffer the material, the steeper, it is, the B point will be higher.

Woods have varying degrees of stiffness based on species. I should mention that stress in engineering terms means a force divided by an area. You can imagine poking someone will hurt more than pushing someone with both hands with the same amount of force, simply because the area it is applied is smaller (stress = F/A ... If F same... A low... stress high.... if A high e.g. 100, stress low).

Given the same piece of wood, if you make the piece thicker, given the same amount of force, because there's more cross section area (A), the stresses are lower. Luthiers will need to judge to determine the thickness of the soundboard, to ensure it stays in the elastic region.

Once the material exceeds point B, we are outside the elastic region, even if we release the stress, the material will not snap back to its original shape or length, permanent damage is done to the material. (Think stretching a rubber band too hard, it becomes looser once you let go).

When stresses are at point C, the material has reached its ultimate strength, in which it'll start to deform rapidly such as stretching and thinning out, until it breaks at point D.

This is not the normal mode of failure for our instruments. Only luthiers will encounter this. Perhaps there wasn't enough bracing to provide stiffness or the soundboard was way too thin, in which it'll collapse within itself like a dying star as soon as it's strung up. In summary, the forces and thus stresses induced on the wood isn't enough to cause it to fail in the ultimate strength mode. Fatigue cracking occurs at much lower stresses over many cycles. So yes, changes can occur.

Combining Everything
Combining everything, Here is an example for testing on Pinus pinaster Ait. wood. You will see the stress/strain graph with ultimate load at 220 N, the cycles and propagation rates can be modelled. We can get very rough durability indications here. (No acoustic findings to be inferred from this data though!)

Wood is not a homogeneous material
You can consider wood as an organic polymer. It isn't uniform or homogeneous like metals. I guess everyone knows this. Not accounting for the different species, the grain direction of the cut, temperature the tree was grown in, sunlight it received, ...( list goes on), will affect the internal structure of the wood. The internal structure varies a lot even within a single piece of wood. Therefore, the internal cracking, stress planes, direction the cracks will propagate etc. cannot be generalized.

View attachment 152200

Modelling fracturing in wood is highly difficult, and maybe in the realms of a phD level. However, given we can't really replicate the same piece of wood, or reproduce the results in the models, is there any value in examining this?


What's not in the scope of this lengthy post

• Effect of the build/construction - glue, joints, bracing etc.
• Internal stresses induced by changes in humidity/temperature, but cracks will occur from this as we all know.
• Effects on acoustics

The million dollar questions

• Do fatigue cracks improve the sound? Or does it degrade it?
• Keep in mind these are microscopic cracks! Are fatigue cracks at a length or magnitude where it has an effect?
• Cracks can join up over time - Is there a further change when this happen?
• Do joined up cracks result in a more flexible soundboard or cause the soundboard to act like a thinner one?
• Do laminated woods/double tops etc. slow down fatigue cracks?
• Are the voids from the cracks in the wood creating overtones or 'little echo chambers'?
• Does sound quality deteriorate when cracks get to a certain length?
• Can the human ear perceive the differences if there are any?

Hahaha I guess I've made everyone read a lot with no solid conclusions or solutions to offer. Maybe even more questions to boot.

I've tried my best to segregate my opinions to what is a known. Objective and commonly observed behavior of materials from a material science point of view. Many engineers will have learnt this in their studies. I'm sure there are a few of you in here. Let me know if I am wrong in anything. I don't mind! If you are a material scientist, I would love to know your thoughts! Hope I didn't offend anyone here, just offering my 2 cents.




You can read more about fatigue and fracture of woods in book below. It talks about the macrostructure more in which I am no expert, but it is fascinating!

Fracture and Fatigue in Wood​

Ian Smith, Eric Landis, Meng Gong

Fracture and Fatigue in Wood

Damage in wood is principally the result of fatigue. Fatigue is the process of progressive localised irreversible change in a material, and may culminate in cracks or complete fracture if conditions that initiated or propagated the process persist. Comprehensive understanding of fatigue and...

www.wiley.com

Brilliant! Thank you so much for taking the time to post this

 

[ploverwing]

Hahaha I guess I've made everyone read a lot with no solid conclusions or solutions to offer. Maybe even more questions to boot.

You've not made us do anything - I loved your thorough, detailed and clear explanation. Thank you so much for taking the time to post this, it was so interesting!

 

[rainbow21]

Thanks, SeemingMusic! Good background and treatise of the subject. This is certainly consistent with what I imagined might be occurring.

Your next assignment is to relate this to the production of quality sound on stringed instruments. I assume that the cellular structure and grain make the vibrations directional (mostly in the grain direction). Is there a curve (graphical) where the sound is optimal (crack length on x-axis?)... or even, how does the sound change at different points of your "cycles to failure" graph. I assume the cracks means that the wood can vibrate more easily as its microcracks increase with time (real time or through playing). Perhaps now the debate can shift to whether this is perceivable as an improvement, or an "opening up".

I anxiously await your follow up report. Thanks again for all the work here and the excellence of the post!

 

[TerryM]

Excellent write-up, SeemingMusic. Thanks for taking the time and interest.

 

[surfink]

Hello all,

Been following this thread with interest for a while. If I may, and if anyone is interested, I have some understanding of material science as I am an engineer. Though it isn't my specialty. Nonetheless I want to talk about fatigue in materials.

What is fatigue?
A lot of you are quite right in that materials weakens over time. ALL materials (not just our precious soundboards) will undergo fatigue when subjected to cyclic loads. (Cyclic loads are loads that are applied, then removed, then applied again etc.) Whether it be an engine belt, bolts in a bridge, masts on a sailboat... the list goes on.

You may get a very brief summary on fatigue cracking here. To summarize, if we subject stress on a material, over many cycles, (such as the vibration induced into the soundboard), over time, tiny tiny cracks will develop in the material. The higher the stress, the less cycles it takes for the material to fail. Makes sense, if we push hard with a palm into the soundboard (please don't do this), it may only take a few pumps to break it. As the stresses we subject it to are relatively small, the number of cycles to break the material will be a lot.

In the microscopic level, cracks are slowly growing in length as it's subjected to more cycles. As an example, imagine you put a little tear in the middle of an edge of a piece paper. You hold the paper with both hands on either side of the tear. You then slowly move your hands apart and together, each time your hands move apart, the tear grows a little. This is called crack propagation. The crack is growing.

How many cycles to failure?
Fear not, the first graph is a logarithmic graph, it's actually really stretched out if we display the scale in a linear fashion. For the relatively low stresses on the ukes, it'll take A LOT of cycles to failure. No need to worry!

We can take a look at Stradivarius violins and they still exist, though I know many of them have repaired cracks. I don't have data on the cause of the cracking though and can't say for sure it's due to fatigue crack propagation.

I found the observations of @BradDonaldson really fascinating of the few instruments converging over time. I think this makes sense in my mind as you get crack nucleation, then crack growth over time. So the more quiet ukes went through a more steeper crack growth (area I) and then all the ukes end up at different points in

May I suggest "Left Brain Lutherie" as further reading material.
I do know one thing... It is pure magic when a completed ukulele is strung and strummed for the first time. Some instruments have a certain "Je ne c'est quoi..." and some don't.

 

[SeemingMusic]

May I suggest "Left Brain Lutherie" as further reading material.
I do know one thing... It is pure magic when a completed ukulele is strung and strummed for the first time. Some instruments have a certain "Je ne c'est quoi..." and some don't.

Wow, that book looks seriously comprehensive!

Your next assignment is to relate this to the production of quality sound on stringed instruments. I assume that the cellular structure and grain make the vibrations directional (mostly in the grain direction). Is there a curve (graphical) where the sound is optimal (crack length on x-axis?)... or even, how does the sound change at different points of your "cycles to failure" graph. I assume the cracks means that the wood can vibrate more easily as its microcracks increase with time (real time or through playing). Perhaps now the debate can shift to whether this is perceivable as an improvement, or an "opening up".

Hahaha . Sorry I will have no idea! Like I said in my post I don't understand acoustics. I just wanted to clarify the material does change over time. As engineers, we want to over engineer things, that's why all these graphs are focused on failure point and strength/durability/life rather than how it affects the tone during its life.

Many of our Luthier friends here say the best instruments are built on the edge of destruction, getting the top as thin as possible and braces as light as possible. Even scalloping them to voice the instrument. If you consider the soundboard as a building slab, supported by beams , and you told the engineer you want the slab to move and you're shaving the structural beams underneath to make it less stable, they would think you have gone insane. It's a completely opposite mindset that I find fascinating!

Consider what I said about stress = Force/Area, just by varying the thickness of 0.1mm, you can wildly different stresses for the same string tension force. So maybe thinner soundboards develop more fatigue cracking earlier? Not getting into how stiffness of braces then influence the stress distributions.... It's complex But I believe most luthiers have a good understanding of a lot of these from experimentation even if they don't know engineering theory.



Maybe as humans we value these imperfections and the degradation of the material. Kind of romantic huh? The softening of leather goods, the patina on other things, we enjoy it more when it's not pristine. Perhaps why wood is popular because the crack patterns are very different to plastics or carbon-fiber.

One example I was thinking as I was writing was the cases of laminate tops. They are made of wood, but we've used a process to alter its characteristic. I can't generalize but I think most(?) people think solid tops sound better and less plastic-y? (Subjective. Some may prefer sounds of laminate!)

Laminates give cross grain stiffness by having pieces running in perpendicular direction for every layer. The glue will slow crack propagation - the wood wants to split open but it's being bonded by a perpendicular blunting element. Fatigue still occurs but at a very slow rate, the glue will eventually break down, the cracks will find the weakest points to grow along. Maybe it'll "open up" in 200 years. Fatigue still occurs but that's why people seem to agree it doesn't get better over (relatively short amount of) time.

Strings experience fatigue too
There were a few comments above about strings too, you can fail the string in two ways: if you keep twisting the tuning peg until it's too tight, it will snap right away (exceeding ultimate strength from above graph), or if you play it and it's subjected to stresses, again and again (cycles), fatigue occurs. Eventually the cracks will reduce the cross sectional area of the string (A goes down), stresses increase, it goes past the Yield strength (point B), we get permanent deformation -stretching and eventually snaps too (point D) if it's left on for that long. That's why some people change strings at 1 months some can change their strings every 6 months etc. Fatigue cracking is a function of over number of cycles and stress (how hard to strum) rather than time. Haha that's why nobody is able to give definitive answers on re-stringing schedules . So many variables.

These are all fun thought exercises. My opinion is we shouldn't try to find a definitive yes/no answer with highly variable materials, construction, environments and way it is used. Expecting the same experience/result for all and trying to generalize things seems like a moot point. Hope we don't get too caught up over the small things. Trust your own ears, trust your luthiers and be happy =)

 

[kissing]

Holy crap @SeemingMusic your posts are an absolute gem to this discussion and provided the kind of physical/engineering information I craved on the subject.

I am especially intrigued by the information that fatigue cracking occurs at much lower stresses over many cycles vs simply exceeding Ultimate Strength.

This has made me view the topic of instruments opening up with playing in a completely different light.
Thanks for taking the time to explain. The diagrams were also very helpful.