Newton's second law and pressure wave propagation

[Jpcgandre]

TL;DR Summary

Newton's second law and pressure wave propagation

Imagine a long deformable rod which has just been hammered on the top end (the bottom end is clamped to Earth). Consider a time interval $dt = t_{2} - t_{1}$ in which the pressure wave is traveling somewhere within the length of the rod (meaning some portion of the object has already "felt" the impact whereas the remaining portion is still at rest (assuming the whole rod was at rest in the start)).

My question concerns how to apply $F=M*a$.

Specifically, I am using the following equation:
$$F_{net}|t_{2}*dt = (M*v)|dt$$
where $F_{net} = F_{impact} + M*g - F_{K*du}$
and $F_{impact}$ represents the hammer impact force, $K$ is the stiffness of the rod and $d_{u}$ is the relative displacement between the top end and the end of the affected part of the rod (== displacement of the top end since the displacement at the end of the affected part of the rod = 0).

For my example, the mass density of the rod decreases with height, say linearly.
My question is what $M$ and $v$ should I use:
1) The affected mass, i.e. the mass of the part of the rod which the pressure wave already travelled, at time instants $t_{2}$ and $t_{1}$, and the corresponding velocities? Here masses will differ as well as the velocities.
$$F_{net}|t_{2}*dt = (M*v)|t_{2} - (M*v)|t_{1}$$
2) The affected mass at time instant $t_{2}$ and the velocities at time instants $t_{2}$ and $t_{1}$? Here mass will be the same, multiplied by the velocities' values existing at time instants $t_{2}$ and $t_{1}$ for the affected length of the rod at $t_{2}$?
$$F_{net}|t_{2}*dt = M|t_{2}*(v|t_{2}- v|t_{1})$$
It's not quite like this because a part of the $M|t_{2}$ has zero velocity at $t_{1}$ but it was easier to write it like this.
3) Other?

A similar setting is the long rod hitting the ground after being dropped from a height $h$ above ground.

 

[Dale]

You can apply $F=ma$ to the entire rod at all times where $F$ is the net external force on the whole rod, $m$ is the mass of the whole rod, and $a$ is the acceleration of the center of mass. There is no need to worry about what portion is affected at a given time.

You can also choose some portion of the rod as your system, as in your 2, but that often changes the forces you need to consider. Also, you do not want to change the mass that is included in your system as in your 1. It can be done, but it is definitely more complicated.

 

[Chestermiller]

Are you familiar with the wave equation in terms of the local displacement?

 

[Jpcgandre]

Just to add another possible use of F=m*a:
3) The affected mass at time instant $t_{2}$ and the velocities at time instants $t_{2}$ and $t_{1}$
Here mass will be the same, multiplied by the velocities' values existing at time instants $t_{2}$ and $t_{1}$
$$F_{net}|t_{2}*dt = M|t_{2}*(v|t_{2}- v|t_{1})$$
This is 2) but here we simply multiply the mass at $t_{2}$ with the velocity at $t_{1}$, i.e. we neglect the fact that at $t_{1}$ part of the mass at $t_{2}$ has zero velocity.

This is a simplification of a problem I've been working on trying to come up with an analytical model to study MDOF collisions between deformable bodies. I want to simulate what happens during impact, I want to go beyond the simple determination of the average impact force.
So I need to use different masses at different times. That's well understood on my part. The reason for this post was to see for a specific time instant and time interval which values of mass and velocities to use...
Also my option 2) applied to this example will lead to (M*v)|t2 being possibly quite different than for |t1 since part of the rod's mass affected at t2 is not affected yet at t1, so even if I plug in the same mass M at t2 and t1 the result will end up equal to use different masses since part of M|t2 will be multiplied by 0.

@Dale: Can you give more information on your assessment of my case or provide any relevant references, or similar examples? Thanks!
@Chestermiller: No I'm not familiar with the wave equation in terms of the local displacement. Can you please elaborate? Thanks. I was just using the speed of sound to determine which parts of the rod become affected by the impact, thus changing the Fnet side of Newton's 2nd Law.

 

[Dale]

I want to go beyond the simple determination of the average impact force.

Let’s start here. Can you clarify what you do want to accomplish? What does “go beyond” imply for you?

FYI, on this site inline latex is delimited by two # signs, not a single $ sign.

 

[Jpcgandre]

@Dale I want to compute the deformation, displacement and impact force over time.

 

[Dale]

Ok, this will be way beyond introductory $F=ma$. Are you familiar with multivariate calculus, partial differential equations, vectors, and tensors? There can be some simplifications if you want to consider a one dimensional object, but this is not a trivial undertaking at all. It will definitely help if you have the requisite background.

If not, then do you have programming experience?

 

[Jpcgandre]

@Dale yes I have an engineering background. I've a python code which I've been working for some days now which I want to use. It's "working" without computer errors but it's not working because of issues in the physical modelling of the problem.

 

[Dale]

Ok, so if you have a rod with modulus of elasticity $E$ and density $\rho$ then the displacement $u(x,t)$ of the rod is given by $$\frac{\partial^2}{\partial t^2} u(x,t)=\frac{E}{\rho}\frac{\partial^2}{\partial x^2}u(x,t)$$

You can solve this partial differential equation using your preferred Python library for numerical methods. The derivation can be seen here: https://en.m.wikipedia.org/wiki/Wave_equation

For 1D problems specifically, this does admit a nice easy class of solutions $$u(x,t)=f(x-ct)+g(x+ct)$$ for any arbitrary functions $f$ and $g$ with $c=\sqrt{E/\rho}$

 

[Jpcgandre]

@Dale Ok, thanks. This my starting point. Just one follow up question. This equation remains valid as long as not all rod is affected by the wave, or even after this occurs it's still the equation to solve to get the displacements? In other words, when do I have to use F=m*a? Thanks

 

[Dale]

Both this equation (the wave equation) and the second law ($F=ma$) are valid the entire time. They tell you different things. The wave equation tells you about the deformation $u$ and the second law tells you about the acceleration $a$ of the center of mass. Both work at the same time and throughout the motion.

 

[Arjan82]

@Dale I want to compute the deformation, displacement and impact force over time.

This is not specific enough. Do you mean the local deformation? or the deformation of the entire rod at a certain time instance? Same with displacement. You talk about *the* velocity and *the* deformation, while for a local deformation and displacement this becomes a local value. So you have $u = u(x,t)$, $v = v(x,t)$ and so on.

Also note that if you want to do this, you need to specify the hammer blow exactly. Either with a specified force in time, or with some initial condition like a velocity or acceleration at time 0 at the free end of the rod. This is an input to your problem (unlike the other you mentioned, where you drop the rod from a certain height).

And, just for clarity's sake since you haven't mentioned this explicitly, you do mean longitudinal waves in the axial direction of the rod right? Otherwise you need a completely different approach.

As a last remark, in your code you specify time intervals somewhere (the $dt$ in your equation), you need to make sure that the time intervals are much smaller than the speed at which the wave travels through the rod, otherwise you will miss the entire event in your simulation.

 

[Jpcgandre]

Both this equation (the wave equation) and the second law ($F=ma$) are valid the entire time. They tell you different things. The wave equation tells you about the deformation $u$ and the second law tells you about the acceleration $a$ of the center of mass. Both work at the same time and throughout the motion.

Thanks Dale, is there a way to get from the wave equation the classic motion equation of an object under constant acceleration i.e. $u = u0 + v0*t + 1/2*a*t**2$ ?

 

[Jpcgandre]

This is not specific enough. Do you mean the local deformation? or the deformation of the entire rod at a certain time instance? Same with displacement. You talk about *the* velocity and *the* deformation, while for a local deformation and displacement this becomes a local value. So you have $u = u(x,t)$, $v = v(x,t)$ and so on.

Also note that if you want to do this, you need to specify the hammer blow exactly. Either with a specified force in time, or with some initial condition like a velocity or acceleration at time 0 at the free end of the rod. This is an input to your problem (unlike the other you mentioned, where you drop the rod from a certain height).

And, just for clarity's sake since you haven't mentioned this explicitly, you do mean longitudinal waves in the axial direction of the rod right? Otherwise you need a completely different approach.

As a last remark, in your code you specify time intervals somewhere (the $dt$ in your equation), you need to make sure that the time intervals are much smaller than the speed at which the wave travels through the rod, otherwise you will miss the entire event in your simulation.

Thanks for the inputs. Yes, I meant local deformations and displacements.
Yes, I know the initial conditions of the hammer or the drop of the rod and I was using them in my model.
Yes, I am talking of longitudinal waves :) and finally yes my $dt$ is very very small just for that reason.
I'm just a bit puzzled why I never heard of this wave equation (or at least never used it) during my high school, undergraduate and postgraduate studies in the civil engineering field... this is the origin of my issues with tackling the problem I want to solve.
For example, in the freefalling rod case; once it hits the ground, waves start to propagate in the rod's medium and so this takes some time. During the time waves are propagating, there will be a part of the rod which is yet to "feel" the impact. Does the wave equation apply to this portion also? or just the F=m*a?
Thanks.

 

[Arjan82]

So, we've got that sorted then :)

The wave equation will not apply to a problem with a single wave 'somewhere' within the rod, obviously it will apply where the wave is, but not outside of that. Besides that, it is hard to apply arbitrary boundary conditions (the excitation by the hammer blow for example) to that equation.

[edit] the above is a bit too strong, you can apply it to a single wave in a rod as long as you already know the shape of that wave (my bad...) but my point was that it is not a general solution to a force applied to a rod[/edit]

What you actually need to do is divide the rod in some small parts and do a force balance over each part. Then solve the set of linear equations you've acquired using linear algebra. This is called the Finite Element Method. It is not so trivial how to do this.

 

[Jpcgandre]

So, we've got that sorted then :)

The wave equation will not apply to a problem with a single wave 'somewhere' within the rod, obviously it will apply where the wave is, but not outside of that. Besides that, it is hard to apply arbitrary boundary conditions (the excitation by the hammer blow for example) to that equation.

[edit] the above is a bit too strong, you can apply it to a single wave in a rod as long as you already know the shape of that wave (my bad...) but my point was that it is not a general solution to a force applied to a rod[/edit]

What you actually need to do is divide the rod in some small parts and do a force balance over each part. Then solve the set of linear equations you've acquired using linear algebra. This is called the Finite Element Method. It is not so trivial how to do this.

THanks Arjan82 for the clarifications! I will have to go deeper into the wave equation and ways to solve it for my problem which is the impact between two deformable objects (the rod example was just a simple proxy).

 

[Dale]

Thanks Dale, is there a way to get from the wave equation the classic motion equation of an object under constant acceleration i.e. $u = u0 + v0*t + 1/2*a*t**2$ ?

No. The situation you are describing is not a constant acceleration, so that equation is not useful.

undergraduate and postgraduate studies in the civil engineering field

Hmm, that is odd. Did you not take a statics course and afterwards a dynamics course? I would think that a civil engineering curriculum would require a dynamics course.

During the time waves are propagating, there will be a part of the rod which is yet to "feel" the impact. Does the wave equation apply to this portion also? or just the F=m*a?

As I said before, both apply the whole time.

 

[Jpcgandre]

No. The situation you are describing is not a constant acceleration, so that equation is not useful.

Hmm, that is odd. Did you not take a statics course and afterwards a dynamics course? I would think that a civil engineering curriculum would require a dynamics course.

As I said before, both apply the whole time.

@Dave my point on the constant acceleration equation was for me to get a first insight of how to work with the wave equation with a simple case. If a reference is available can you point it to me? Thank you very much

 

[Dale]

@Dave my point on the constant acceleration equation was for me to get a first insight of how to work with the wave equation with a simple case. If a reference is available can you point it to me? Thank you very much

Sure. Wikipedia is a decent place to start: https://en.m.wikipedia.org/wiki/Wave_equation

 

[nasu]

@Jpcgandre The wave equation shown by Dale can be obtained by applying Newton's second law to the specific case of a wave propgating in a 1D medium. The equation $u=u_0 + v_0 t +\frac{1}{2} at^2$is also a result of using Newton's second law for motion under a constant force along a straight line. They are not alternatives to Newton's laws but results of the above, for various conditions. You should have seen the derivation of the simplest case of elastic wave in an introductory physics course. Unfortunately, like at some universities, first year engineering students focus on statics and they may well finish the introductory courses without dealing with the equations of motion.

 

[Jpcgandre]

Thank you for all the replies.
One thing not clear yet is that it is my interpretation that I can divide the determination of the motion of rod/hammer or rod/ground sort of problems in two components:
1- Wave component, where the $u_{w}(x,t)$ from the wave is given by the wave function.
2- The non-wave component, where the $u_{nw}(x,t)$ that can be found from $F=ma$ neglecting the existence of $u_{w}(x,t)$, but not of the wave (i.e. the existence of the wave is used to determine which part of the rod has been affected by the wave).
If this is true than the total displacement could be obtained by: $$u_{tot}(x,t) = u_{w}(x,t) + u_{nw}(x,t)$$

Also, I'm unsure how to construct the $f$ and $g$ functions of the wave equation. For the case the rod hits the ground, the boundary conditions are $u(0, t) = \frac{\partial}{\partial t} u(0,t) = 0$, plus the curvature of the rod is zero, no moment. Therefore $\frac{\partial^2}{\partial x^2} u(x,t) = 0$ thus $\frac{\partial^2}{\partial t^2} u(x,t) = 0$ which implies that the velocity of the wave is constant and equal to $c$. So $f$ and $g$ cannot be sine or cosines, since they have non zero second derivatives. If this is true (?) I'm not quite sure what $f$ and $g$ can look like.

 

[Dale]

2- The non-wave component, where the unw(x,t) that can be found from F=ma

$F=ma$ doesn’t have enough information to give you $u_{nw}(x,t)$. All it can give you is $x_{CoM}(t)$, the position of the center of mass.

the curvature of the rod is zero

$u$ would be a longitudinal displacement, so $\frac {\partial}{\partial x}u(x,t)\ne 0$ in general

 

[Jpcgandre]

$F=ma$ doesn’t have enough information to give you $u_{nw}(x,t)$. All it can give you is $x_{CoM}(t)$, the position of the center of mass.

$u$ would be a longitudinal displacement, so $\frac {\partial}{\partial x}u(x,t)\ne 0$ in general

But from $F_{net}$ I can also get the displacement of the top and bottom of the section I'm analysing by the material deformation of the rod. I can use Runge-Kutta or finite difference methods to estimate this motion. This may be sufficient for me.
About the velocity $\frac {\partial}{\partial x}u(x,t)\ne 0$, I agree, what I argued was that the velocity of the pressure wave would be constant (?)...

The main point is for me to know if my simplified idea of breaking the motion into two components is workable or it is totally wrong for cases such as the rod/hammer rod/ground problem, where I could expect the deformation of the wave to be small (I guess it is small!)? If it is not usable, I'm unsure how to determine the overall motion from just the wave function with the given boundary conditions.

 

[Dale]

But from Fnet I can also get the displacement of the top and bottom of the section

No, at least not if Fnet is the sum of the external forces acting on the rod.

I am sorry, I don't feel like I am helping you. I have repeated the same point in different ways multiple times. Good luck, hopefully someone else can phrase it right for you to understand.

 

[Jpcgandre]

But from $F_{net}$ I can also get the displacement of the top and bottom of the section I'm analysing by the material deformation of the rod. I can use Runge-Kutta or finite difference methods to estimate this motion. This may be sufficient for me.

No, at least not if Fnet is the sum of the external forces acting on the rod.

I am sorry, I don't feel like I am helping you. I have repeated the same point in different ways multiple times. Good luck, hopefully someone else can phrase it right for you to understand.

Ok, thank you for the help. $F_{net}$ for me is the resultant in a segment of the rod. I can make it as small as I wish to that the approximations I assume in this segment (namely velocity profiles) don't affect the overall accuracy of the motion.

 

[Arjan82]

The way you split up the problem in a wave component and a non-wave component is not possible. Also, $\frac{\delta}{\delta x}u(x,t)$ is not the wave velocity (which is indeed constant). It is the local velocity of the material. This is zero before the wave arives, then positive (say) when the compression part of the wave comes in, then negative when the rarefaction part of the wave comes in, then zero again.

For the wave function $f$ and $g$ are input. You would need to get them via some other way. The easiest would be to assume that the hammer just applies a certain known displacement of the end of the rod in a certain known time. But if you want to start out with the force of the hammer, then you are out of luck here.

Also, $F=ma$ cannot be used as such in this problem. The problem is that both $F(x,t)$ and $a(x,t)$ are continuous functions over the rod and in time. You need to properly discretize the problem, which is not at all trivial. You are trying to re-invent the finite element method here (or discrete mathematics). You would be best suited to read some literature on that subject. If you understand the proper way to approach this problem, it is really not that difficult, but I'm not going to write-out the whole FEM approach for you here (and I would have to refresh my knowledge on the details as wel).

Actually, probably the finite difference method (as opposed to FEM) is a better place to start. You also need to find the correct partial differential equation that you need to solve for this problem. Normally that is found by doing a force balance over a infinitesimally small part of the rod (this is continuum mechanics). I do not know from the top of my head what the differential equation is that you need to solve, but it is not the wave equation, since then you miss the relation with the force, and it is not $F=ma$ as such, it is however some form of the equation of motion:
$$
\Sigma F = m\ddot{x} + c\dot{x} + kx
$$
(I hope I got that correctly from the top of my head...) where $m$ is the mass, $c$ is the damping coefficient (which could be set to zero in your case) and $k$ is the stiffness. Also, the dots show a time derivative, two dots a second time derivative. Here $\Sigma F$ means the sum of all external forces. So maybe gravity, maybe the force by the hammer.

Lastly you need to apply the correct boundary and initial conditions. If you did everything right you have a matrix that you need to solve for each timestep.

As I said earlier, this is not at all trivial and there are many unexpected ways in which this could go wrong. I would really grab a book which explains the finite difference method, discrete mathematics and solid mechanics. It is much too hard to explain in a simple post here on PF.

 

[Chestermiller]

For a constant force F applied at x = 0 at all times $t\geq 0$, according to the wave equation, the displacement u is given by:

$u=\frac{F}{AE}(ct-x)$ for $x\leq ct$

and

$u=0$ for $x\geq ct$

 

[bob012345]

There are whole books on this subject. Depending on what your purpose is you may find a source that describes the solutions for different types of impulses and materials. I found this resource online;

https://www.lps.ufrj.br/~caloba/cenpes_docs_gilberto/Wave%20Propagation%20in%20Elastic%20Solids.pdf