Pushing solid objects and longitudinal waves

[robertbram]

Hi All,

I am still trying to wrap my head around the five light year stick and the idea that if you push an object, it moves because longitudinal waves of compression force (?) are sent through the medium of whatever the object is made of: https://www.physicsforums.com/showthread.php?t=386687

If I have a ruler that is exactly 30cm long and push it, does that mean that if I could measure such a thing, there would be an instant where the near end is moving but the furthest end isn't? Does that also mean that the object changes length?

I understand that this involves compression of the object, and the rigidity of the material. Will compression waves move faster through material that is more rigid or less? Intuitively, I think I would move the furthest edge of a steel ruler faster than I could move the furthest edge of a sponge.

Does this therefore mean that a more rigid material would move faster than a less rigid material when put under the same pushing force? If so, I would prefer to make a spaceship out of steel than sponge.

Finally, in mechanical engineering, does this ever become a problem when deciding how large you can make components and out of what material?

Thanks for any thoughts!

Rob
:)

 

[Vorde]

Yes in pretty much everything!

I can't vouch for the mechanical engineering part, but everything else you said is pretty much spot on. These compression waves are really just sound waves, so they travel through a material at the speed of sound. To see compression waves in real life you'd need to either look over immense distances or in really short time intervals, which is why we don't notice them.

 

[HallsofIvy]

Or you could use Jello and see the waves easily!

 

[PAllen]

The delay between a hammer hitting one end of a 1 meter, rigid, ceramic rod and the other end moving has been measured in the lab.

In general, more rigidity means faster speed of sound, thus faster propagation of mechanical stress. A sponge can be made to move at any speed. What is slow in a sponge is sending a mechanical signal.

 

[bobc2]

Hi All,

Does this therefore mean that a more rigid material would move faster than a less rigid material when put under the same pushing force? If so, I would prefer to make a spaceship out of steel than sponge.

The speed of the wave is proportional to the square root of the modulus of elasticity and inversely proportional to the square root of the mass density. So, even for a high modulus (related to stiffness) the speed could be relatively slow if the mass density were extremely high. Modulus for steel is pretty high though--about 30 million psi and the transmission speed is certainly orders of magnitude higher than a sponge as you have surmized.

Finally, in mechanical engineering, does this ever become a problem when deciding how large you can make components and out of what material?

This is certainly a factor in design--but the modulus and mass density are very important in view of potential damage from vibration. High modulus and low mass density generally (depending on geometry) result in high resonance frequencies and low vibration displacements (lower stress). The primary design criteria for structures begin with static load considerations, however--it's the material strength that is important (of course there is no comparison between sponge rubber and steel with it's approx. 100,000 psi strength).

An interesting facit of impact loads on a structure is that the dynamic response of the structure may be understood as a superposition of unique deformation patterns (called mode shapes). Those mode shapes include rigid body motion (six degrees of freedom) and another class of modes, the flexibility modes. When considered from this point of view, an impact on one end of the beam is not looked on as directly imparting a traveling wave. Rather, the impact point force is first decomposed into separate modal forces, each modal force inducing a global modal deformation over the entire sturcture--that is, for a given modal force (a unique collection of point forces spread across the entire structure--applied simultaneously), each point on the structure deforms simultaneously with no delay from one end of the structure to the other at all. The superposition of all such instantaneous and simultaneous modal dynamic responses (considering in-phase and out-of-phase motions among the mode shapes) result in the appearance of traveling waves.

 

[Austin0]

The speed of the wave is proportional to the square root of the modulus of elasticity and inversely proportional to the square root of the mass density. So, even for a high modulus (related to stiffness) the speed could be relatively slow if the mass density were extremely high. Modulus for steel is pretty high though--about 30 million psi and the transmission speed is certainly orders of magnitude higher than a sponge as you have surmized.



This is certainly a factor in design--but the modulus and mass density are very important in view of potential damage from vibration. High modulus and low mass density generally (depending on geometry) result in high resonance frequencies and low vibration displacements (lower stress). The primary design criteria for structures begin with static load considerations, however--it's the material strength that is important (of course there is no comparison between sponge rubber and steel with it's approx. 100,000 psi strength).

An interesting facit of impact loads on a structure is that the dynamic response of the structure may be understood as a superposition of unique deformation patterns (called mode shapes). Those mode shapes include rigid body motion (six degrees of freedom) and another class of modes, the flexibility modes. When considered from this point of view, an impact on one end of the beam is not looked on as directly imparting a traveling wave. Rather, the impact point force is first decomposed into separate modal forces, each modal force inducing a global modal deformation over the entire sturcture--that is, for a given modal force (a unique collection of point forces spread across the entire structure--applied simultaneously), each point on the structure deforms simultaneously with no delay from one end of the structure to the other at all. The superposition of all such instantaneous and simultaneous modal dynamic responses (considering in-phase and out-of-phase motions among the mode shapes) result in the appearance of traveling waves.

This is indeed interesting. But what does it mean??

Is this to be interpreted as merely a useful analytical abstraction with no implication of correspondence to reality? or is this implying some kind of quantum non-locality underlying and determining the macro world measurable wave traveling at the speed of sound?

 

[PAllen]

-that is, for a given modal force (a unique collection of point forces spread across the entire structure--applied simultaneously), each point on the structure deforms simultaneously with no delay from one end of the structure to the other at all. The superposition of all such instantaneous and simultaneous modal dynamic responses (considering in-phase and out-of-phase motions among the mode shapes) result in the appearance of traveling waves.

That may be a valid way of modeling it for ordinary circumstances, but it is clearly not what happens microscopically. Let's say, for example, a rod moving at .8c hits a wall. The back end of the rod travels far toward the wall before it can have any causal connection (at all, by any means) to the collision. Until it is within the future light cone of the collision, it cannot behave differently than if there were no collision.

 

[lightarrow]

If I have a ruler that is exactly 30cm long and push it, does that mean that if I could measure such a thing, there would be an instant where the near end is moving but the furthest end isn't? Does that also mean that the object changes length?

Imagine every "rigid" body as if actually made of many masses connected with soft springs, and you can immediately visualize what really happens.

 

[bobc2]

Imagine every "rigid" body as if actually made of many masses connected with soft springs, and you can immediately visualize what really happens.

Of course the modal concept applies quite well for this configuration of masses and springs. We develop the mass and stiffness matrices (and include a damping matrix as well if desired) and combine them to form a global dynamical matrix. Now, do an eigenvalue-eigenvector problem, diagonalizing the dynamical matrix using the matrix of eigenvectors (the mode shapes). The set of differential equations can then be set in modal coordinates--uncoupled equations. The mass matrix can be diagonalized using the matrix of mode shapes, yielding the modal masses--and likewise for the stiffness matrix (these matrices transform as operators). So, each mode is treated as a single degree of freedom. The solution for a single modal mass, single modal spring and single modal damper is obtained for each modal single-degree-of-freedom. Again, each modal force is represented as a unique collection of forces instantly and simultaneously applied to each physical mass in the system.

You can easily see what is going on if you perform a coordinate transformation on a column vector (function of time) of the global motion of the system, where each element in the column is the displacement vs. time for a particular mass (the number assignments of the masses from one end of the system to the other could be used as row numbers in the column vector). Using the mode shape matrix (eigenvectors) and its inverse you can transform back and forth between modal coordinates and physical coordinates.

The modal pictures (modal coordinates) are equivalent to the generalized physical coordinate pictures.

 

[bobc2]

This is indeed interesting. But what does it mean??

Is this to be interpreted as merely a useful analytical abstraction with no implication of correspondence to reality? or is this implying some kind of quantum non-locality underlying and determining the macro world measurable wave traveling at the speed of sound?

I'm not sure, Austin0. But you make an intriquing comment about non-locality. Bell would like that, because I think he believed that physics is non-local. And of course Einstein's special relativity uniquely satisfies that concept as well. It's probably the one consideration that identifies Einstein's relativity as the successful theory as compared to the Lorentz ether theory.

I've got to mull this one over after your keen sense of observation.

 

[lightarrow]

Of course the modal concept applies quite well for this configuration of masses and springs. We develop the mass and stiffness matrices (and include a damping matrix as well if desired) and combine them to form a global dynamical matrix. Now, do an eigenvalue-eigenvector problem, diagonalizing the dynamical matrix using the matrix of eigenvectors (the mode shapes). The set of differential equations can then be set in modal coordinates--uncoupled equations. The mass matrix can be diagonalized using the matrix of mode shapes, yielding the modal masses--and likewise for the stiffness matrix (these matrices transform as operators). So, each mode is treated as a single degree of freedom. The solution for a single modal mass, single modal spring and single modal damper is obtained for each modal single-degree-of-freedom. Again, each modal force is represented as a unique collection of forces instantly and simultaneously applied to each physical mass in the system.

You can easily see what is going on if you perform a coordinate transformation on a column vector (function of time) of the global motion of the system, where each element in the column is the displacement vs. time for a particular mass (the number assignments of the masses from one end of the system to the other could be used as row numbers in the column vector). Using the mode shape matrix (eigenvectors) and its inverse you can transform back and forth between modal coordinates and physical coordinates.

The modal pictures (modal coordinates) are equivalent to the generalized physical coordinate pictures.

I had never heard of such description. It seems interesting, can you give a link about the subject?

 

[bobc2]

I had never heard of such description. It seems interesting, can you give a link about the subject?

I first encountered normal mode theory in my first grad school mechanics course using Goldsein's "Classical Mechanics" (there was one chapter on small vibrations). The mode shapes and resonance frequencies (eigenvectors and square root of the eigenvalues scaled in Hz) were routinely measured for various subsystems on the NASA Orbiter (wings, vertical tail, body flap, payload bay, etc.). Googling "NASA SMIS Project" might turn up something on this. Googling "Experimental Structural Dynamics" would probably find engineering books on the subject. I'm on a trip right now and could post more references after returning home.

 

[PAllen]

I first encountered normal mode theory in my first grad school mechanics course using Goldsein's "Classical Mechanics" (there was one chapter on small vibrations). The mode shapes and resonance frequencies (eigenvectors and square root of the eigenvalues scaled in Hz) were routinely measured for various subsystems on the NASA Orbiter (wings, vertical tail, body flap, payload bay, etc.). Googling "NASA SMIS Project" might turn up something on this. Googling "Experimental Structural Dynamics" would probably find engineering books on the subject. I'm on a trip right now and could post more references after returning home.

Thank's for the reference. With that I reviewed this theory and find that:

- it is completely non-relavistic
- further, it assumes disturbance of sufficiently small size that harmonics of the fundamental frequencies are not significant.

Thus, for the topic in this thread its relevance is essentially zero, IMO.

 

[lightarrow]

I first encountered normal mode theory in my first grad school mechanics course using Goldsein's "Classical Mechanics" (there was one chapter on small vibrations). The mode shapes and resonance frequencies (eigenvectors and square root of the eigenvalues scaled in Hz) were routinely measured for various subsystems on the NASA Orbiter (wings, vertical tail, body flap, payload bay, etc.). Googling "NASA SMIS Project" might turn up something on this. Googling "Experimental Structural Dynamics" would probably find engineering books on the subject. I'm on a trip right now and could post more references after returning home.

Thanks.