# Structural dynamics

#### hyper

http://img17.imageshack.us/img17/5025/sd001.jpg [Broken]

This is a forced oscilliation with the ground moving, and a mass inside a structure moving. Here they obtain the equation for the oscilliation by using Newton's in a particular way. But I would like to obtain the equation by looking at the force that the ground excerts on the frame instead. Can you guys please help me?

It might be a little hard to really understand my question, so I will try to explain it with a very similar example. Please look at this rotating mass:

http://img24.imageshack.us/img24/6738/sd002.jpg [Broken]

You can here obtain the equation for the oscilliation allmost the same way as they did in the first picture. But you can also obtain it by looking at the sentripetal force on the rotating mass. By deriving the equation this way I really got more understanding about the problem. And I would like to understand the problem in the first picture better aswell.

Help would be greatly appreciated.

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#### Dr.D

hyper -- these are two entirely different problems, and they call for two entirely different formulations. Don't try to massage one into the other.

For the problem at hand, the first figure, the force is completely unknown, and therefore you make a mistake in wanting to formulate the problem in terms of an applied force. There is no question that a force acts on the frame, but how big is it? It is big enough to make the frame move in the prescribed manner, no matter how big the frame is. This is what happens in an earthquake or similar event. This is a displacement input, rather than a force input to the problem, and it is simply a different class of problems. You need to come to grips with both types.

#### hyper

Thank you for the answer, but let me rephrase the question then:

In the problem I don't really care how big the frame is, and I get your point. Trying to solve the problem this way would be impossible since we don't know the weight of the frame. So how big it is doesn't matter as you say.

But we do know all the properties the the rest of the system(the spring constant etc.) So when they solve the equation they find the the effect on the system is there there is a driving force equal to m *$$\frac{d^2}{dt^2}$$ (Xb). Are you saying that the frame does not excert this force on the spring where it is hinged(don know if this is the right word). I mean there must be a driving force to even get these oscilliations?

So if infact the frame excerts a force of m *$$\frac{d^2}{dt^2}$$ (Xb) on the spring, is there some other way to come to this result by analyzing the frame in another way? Because this is what I don't seem to get: that when the frame is moving the force it excerts on the system is m *$$\frac{d^2}{dt^2}$$ (Xb)?

Maybe I am way off here, I really don't know, but I hope you understand what I mean. :)

#### Dr.D

I have no idea what you are trying to say.

Analyzing the frame in another way, other than what?

#### hyper

Other than how they did it. I mean: how can you explain the the force on the system is m *$$\frac{d^2}{dt^2}$$ (Xb), is there another way you can see this?

#### Dr.D

They did not say any thing at all about a force m*ddxb.

What they did was to say that the total upward motion was the sum of two terms:

xtotal = x + xb

where xb is the base motion and x is measured relative to the frame. Then they differentiated this to get the acceleration. It is just a straightforward determination of the acceleration in a moving frame of reference, the machine frame. You seem to be confused by the kinematics.

#### hyper

I do get the analysis by looking at it in a pureley kinematic view and I assure you I am not confused. And I do agree about how they derive their equation.

But to get a driven oscilliation you DO need a force, and it is this force I want to look at. What I mean is, I want to understand the force that acts on the system. And yes I know you can solve this problem, by only looking at the motions and then applying Newton's 2. law, but I want to solve the problem by analyzing the driving force that acts on the system.

#### Dr.D

But what you evidently do not get is that the force is unknown and unknowable. It is whatever it has to be in order to cause the displacement to occur.

You will only get that force if you know all of the details of the driven mass, the coupling stiffness, etc. And then you will have the force, AS IT APPLIES TO THAT SYSTEM ONLY.

So, by all means, enjoy beating your head against the wall. You will really enjoy the great feeling when you stop.

#### hyper

But what you evidently do not get is that the force is unknown and unknowable. It is whatever it has to be in order to cause the displacement to occur.

You will only get that force if you know all of the details of the driven mass, the coupling stiffness, etc. And then you will have the force, AS IT APPLIES TO THAT SYSTEM ONLY.

So, by all means, enjoy beating your head against the wall. You will really enjoy the great feeling when you stop.
Hehe it seems we don't get anywhere here, it seems like you really don't get the question. Maybe the question is too hard, I asked it in another forum and I got an answer, but to obtain the right equations you have to do a lot of work, or you sbould use something that is called a Laplace-transformation.

#### Dr.D

A Laplace transform changes nothing about the formulation of the problem which is where your question lies.

#### hyper

This answer was way over my head so I can't control if it is correct(I have not learned about Laplace, since I am at my first year at engineering school). But the guy who came with it seems to know what he his talking about:

First he posted the picture:

http://img158.imageshack.us/my.php?image=324342432q.jpg

Then he wrote:

In order to determine the equations as you wish it is necessary to add M1 and Z1. Writing the dynamic equilibrium equations for M1 and M2 and using the laplace transform to obtain a linear system the results will be:

z1= f*P2(s²)/(P1(s²)*P2(s²)-P3(s) ^2)

z2=f*P3(s)/P1(s²)*P2(s²)-P3(s) ^2)

with the notations:

z1 = transform of Z1(t)

f= transform of F(t)

s= laplace operator

P1(s²)= M1*s²+d*s+C; P2(s²)=M2*s²+d*s+c; P3(s)= d*s+c

If you make the inverse transform you get Z1 and Z2 as functions of time. If you want to obtain the amplitudes as functions of the imposed frequency introduce s= J*ω with j=√-1 and ω the imposed pulsation = 2*PI*frequency.

I asked if there was an even easier way and he wrote:

Of course, you write the forces balance for the two bodies M1&M2 using the fact that they depend on the difference Z1-Z2 with respect to the spring and d(Z1-Z2)/dt for the damper. You obtain 2 equations with 2 variables Z1 and Z2 and solve the system for Z2 which is the coordinate you are interested in.

I will look a little closer on the last one.

#### Dr.D

The key to what he has done is in the line, "it is necessary to add M1 and Z1". His Z1 is the thing you have called Xb, but he has assigned a mass M1 to the frame, and then shown the force acting on it. The Laplace transforms bit is just unnecessary confusion at this point.

This makes you happy? Fine, although I have no idea why that is.

It is a useful form for the formulation if the force F(t) is given, but it is totally worthless if the from Xb(t) is given. You originally posted this question as "structural dynamics." One of the most common problems of structural dynamics is the response of a structure to an earthquake, and in that event, you have a displacement excitation, not a force excitation. We can measure how much the earth moves, but we only know that it exerts as much force as is required to make buildings, dams, etc. move with it. The force cannot be quantified, but the displacement driving the system can be described.

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