Understanding Inertia Force in the Vibration Theory of a Spring Mass System

[chandran]

in the vibration theory of a spring mass system the text says that the mass is acted upon by three forces

1)spring force-i understand this

2)damping force-in understand this

3)inertia force-i don't understand this

From where this inertia force


We need to apply force to accelerate a mass that is called inertia. But what is this "inertia force" concept

 

[FredGarvin]

Think about what the meaning of inertia is. What does Newton's first law say? The inertial force is going to be the forces due to the mass of the object and it's resistance to changes in its motion. The initial acceleration term in the ODE is the inertial term.

 

[vanesch]

in the vibration theory of a spring mass system the text says that the mass is acted upon by three forces

1)spring force-i understand this

2)damping force-in understand this

3)inertia force-i don't understand this

From where this inertia force


We need to apply force to accelerate a mass that is called inertia. But what is this "inertia force" concept

I remember having had that conceptual problem too when I was young and charming

In fact, the idea was to use the concepts of STATICS to introduce dynamics. In statics, the forces on an object have to balance. That means: you sum all forces, and you have to find zero. It's how you calculate, say, the load on a pillar holding a bridge or something.

If you want to use that same rule with objects in motion, it doesn't work anymore. Well, it only works for objects in uniform, straight motion, but not for any accelerated motion. In that case, the sum of forces didn't sum to 0 anymore. People used to statics found that mind shaking. So a trick was invented: introduce an "inertia force" which is equal to MINUS the mass times the acceleration. NOW, the sum of all forces, including this inertia force, equals zero (haha).

Of course, for people not having done a lot of statics calculations before being introduced to Newton's dynamics, this is a totally incomprehensible act, because for them, "forces" are a way to calculate accelerations.

It's a book keepers' device, in order not to have to write out Newton's second law, but to be able to use the STATICS equation F_tot = 0 in a situation where it is not really appropriate anymore.

 

[pmb_phy]

in the vibration theory of a spring mass system the text says that the mass is acted upon by three forces

1)spring force-i understand this

2)damping force-in understand this

3)inertia force-i don't understand this

From where this inertia force


We need to apply force to accelerate a mass that is called inertia. But what is this "inertia force" concept

The author may be speaking of the inertial force which is defined here

http://www.geocities.com/physics_world/gr/inertial_force.htm

Inertial Force: A force which must be added to the equations of motion when using a non-inertial frame of reference.

Pete

 

[vanesch]

The author may be speaking of the inertial force which is defined here

http://www.geocities.com/physics_world/gr/inertial_force.htm

Well, the way the OP was formulated, I didn't think of it as an inertial force, because no mention is made of the mass-spring system being fixed on an accelerating system wrt an inertial frame. I think what is called "inertia" force is simply d'Alembert's force (namely - m a).

However, it can be seen of course as an inertial force in a frame FIXED TO THE MASS (so that the mass is always at rest). This is nothing else but d'Alembert's force of course (or Newton's 2nd law): static equilibrium *in the frame in which the object is at rest* needs the application of an inertial force in order to make the static equilibrium come out correctly.

However, things are upside-down now, because usually, an inertial force is a fictive term one has to add to the right side of "Newton's second law" when applied in a non-inertial frame (where it shouldn't be applied - so this fictive term corrects for the error in the first place of trying to apply Newton's second law in a non-inertial frame), WHEN KNOWING THE MOTION OF THE FRAME WRT AN INERTIAL FRAME. So it is a property of the frame (which has been defined wrt an inertial frame) that the transformation of Newton's second law takes up extra terms on the right side. But in this particular case, *we don't know how the frame (fixed to the mass) is going to move*. We adapt its movement, in such a way that the inertial force will compensate perfectly all other forces, so as to KEEP the object at rest in the frame.

 

[FredGarvin]

In basic vibration theory, the spring-mass-damper systems FBDs that I have ever seen were not referenced to the mass, they referenced the Earth. I don't think the non-inertial frame angle is what is being asked about here.

 

[pmb_phy]

Well, the way the OP was formulated, I didn't think of it as an inertial force, because no mention is made of the mass-spring system being fixed on an accelerating system wrt an inertial frame.

The OP is seeking to understand what the author meant by "inertia force." As such we should seek to respond to what the author meant, not what the OP asked.

Pete

 

[vanesch]

The OP is seeking to understand what the author meant by "inertia force." As such we should seek to respond to what the author meant, not what the OP asked.

Well, I DID have a course back when I was a student including this "force of inertia", after that I DID have a course on statics and on kinematics. The statics course was full of (old-fashioned but fun) geometrical constructions to find equilibrium, and the trick was to transpose these methods to *dynamics* by simply introducing a force of inertia equal to - m.a. I remember being very confused over that because I already knew Newton's equation m.a = F, so this was a very weird way of dealing with the m.a term: bring it over to the other side : 0 = -m.a + F, call it d'Alembert's force of inertia F_Al = - m.a and now require EQUILIBRIUM of the total force equal to F + F_Al = F_tot = 0. As if we were in statics.

But a typical application is this:

Consider a block B1 with mass m1 on a frictionless horizontal surface, connected with a rope to a block B2 with mass m2. The rope is guided over a pulley, and the block B2 is hanging in mid-air.
Let it go. What is the force acting on block B1 through the rope ?

Second question: instead of hanging a block B2 on the rope, pull with a force F = g m2 on the rope. Is the situation different ?

Someone trained in statics would think that both situations are identical: the force of gravity acting on block B2 is g m2, and this is the force on the rope, and hence block B1 feels (through the rope) a force m2.g.
But BZZZT. The block B2 is FALLING, so it is accelerating, with an acceleration a, downward. (a is not equal to g, because it is held back somewhat by the rope) And here comes our force of inertia of d'Alembert: the block B2 accelerating downwards with acceleration a, one has to introduce an extra force on it, equal to m2.(-a). So there are now 3 forces acting on the block B2: gravity (m2.g), the rope, F_rope, and d'Alemberts' force, m2 (-a). And these have to be in "equilibrium, so:
F_rope + m2.g - m2.a = 0, or F_rope = m2 (a - g).
This F_rope is the force of the rope on B2, so the force on the rope is minus this, m2 (g - a).
So the force on block B1 is smaller than m2.g.

Eh.

What a clumsy way of dealing with the problem !
But it illustrates that the force in the rope is not equal to m2.g.

 

[pmb_phy]

...
3)inertia force-i don't understand this...

Can you post the exact phrase in which the term "force of inertia" appears in your text? Thanks.

Well, I DID have a course back when I was a student including this "force of inertia", after that I DID have a course on statics and on kinematics. The statics course was full of (old-fashioned but fun) geometrical constructions to find equilibrium, and the trick was to transpose these methods to *dynamics* by simply introducing a force of inertia equal to - m.a.

Goldstein's Classical Mechanics refers to that force as the reversed effective force. However Lanczos' The Variational Principles of Mechanics does refer to the vector I = -mA as the "force of inertia" on page 88.

Thanks for reminding me vanesch.

And yes, this is used when deriving the Lagrange equations when using d'Alembert's Principle

Pete

ps - I recommend that you don't place words in all capitals when you wishe to emphasize something. Rather italisize it by placing the term inbetween the italics delimeters. Otherwise you come across as yelling.

 

[Office_Shredder]

ps - I recommend that you don't place words in all capitals when you wishe to emphasize something. Rather italisize it by placing the term inbetween the italics delimeters. Otherwise you come across as yelling.

Even ignoring that... the reason lower case letters are used as opposed to upper case letters is because lower case letters are easier to read... so by capitalizing a word, you very well run the risk of having people skip over it and move on to the easier word next to it

 

[desA]

D'Alembert created a fictitious force (m.a, for constant mass) which is placed in a direction opposite to the direction of motion. This allows the body to be considered in equilibrium, & for a free-body-diagram to be constructed. This is called the 'inertia force'. Divide by mass & we have inertial acceleration.

For fluid elements, this acceleration is the 'substantial derivative', or 'total derivative'. Alternative forms can be considered for variable mass.

A neat connection between kinetics & kinematics.

 

[zina]

I remember having had that conceptual problem too when I was young and charming

In fact, the idea was to use the concepts of STATICS to introduce dynamics. In statics, the forces on an object have to balance. That means: you sum all forces, and you have to find zero. It's how you calculate, say, the load on a pillar holding a bridge or something.

If you want to use that same rule with objects in motion, it doesn't work anymore. Well, it only works for objects in uniform, straight motion, but not for any accelerated motion. In that case, the sum of forces didn't sum to 0 anymore. People used to statics found that mind shaking. So a trick was invented: introduce an "inertia force" which is equal to MINUS the mass times the acceleration. NOW, the sum of all forces, including this inertia force, equals zero (haha).

Of course, for people not having done a lot of statics calculations before being introduced to Newton's dynamics, this is a totally incomprehensible act, because for them, "forces" are a way to calculate accelerations.

It's a book keepers' device, in order not to have to write out Newton's second law, but to be able to use the STATICS equation F_tot = 0 in a situation where it is not really appropriate anymore.

Very well said..! could you recommend a book for structural engineers that has explanations simple and easy like yours? (american books) have you published any? i have Dunamics of structural systems (Boswell and C'D'Mello) but it doesn't explain in details. Thank you.

 

[Bob S]

This reminds me of the question on my graduate student Prelim Orals to solve the problem of two masses, M1 and M2, connected by a spring of spring constant k and spinning in free space. So there was also another force. Was it centrifugal or centripetal? I forget.

 

[zina]

centrifugal was for hydraulics if i remember right..so i think it might be centripetal..

 

[Doc Al]

centrifugal was for hydraulics if i remember right..so i think it might be centripetal..

Centripetal force is the real force that produces centripetal acceleration (viewed from an inertial frame). In the case of the spinning masses connected by a spring, the spring tension provides the centripetal force.

Centrifugal force, on the other hand, is an inertial pseudo-force that is introduced when viewing things from a rotating (and thus non-inertial) frame so that Newton's laws may be applied, as explained by vanesch (almost three years ago!). So if you analyze the spinning masses from the rotating frame, you would have a centrifugal force acting outward on each mass.