Natural frequency of beam exposed to sinusoidal force

[marcas3]

Hello, I am taking a vibrations course and we are currently talking about using PDEs to find responses of beams. In the book, there is an equation to find the natural frequencies of a simply supported beam under an axial load:

omega = (pi^2/l^2)*sqrt(EI/roA)*sqrt(n^4 + (n^2*P*l^2)/(pi^2*EI))

However, what if the load (P) is not axial but sinusoidal. For example, P = sin(t). How would we find the natural frequencies then. There are other equations to find the natural frequency but none of them talk about beams subjected to sin or cos loads.

Thanks!

 

[AlephZero]

In mechanical engineering, "natural frequency" usually means the frequency of small amplitude vibrations (or "free vibrations") about a steady state of the structure (independent of time). The equation you give looks like a formula for the frequencies of transverse oscillations of a beam with a constant axial force P applied to it.

The natural frequencies and mode shapes don't tell you the actual motion of the structure in any situation, they only tell you about its possible motions. In particular, they don't tell you anything about the amplitude of the motion.

Calculating the response of a structure to a time varying force with a specific amplitude is a different problem, though some ways to solve it describe the solution as a linear combination of the free vibration modes. You will probably meet that type of problem later on in the course, or you may have already met it in an earlier dynamics course.

If the beam was subject to a sinusoidal axial force (and no other forces) it would move in the axial direction not the transverse direction, so the natural frequencies for transverse vibrations would be irrelevant. The amplitude of the motion would depend on the axial vibration frequences, which are the same the axial vibration of a rod, and are independent of the bending stiffness (or I value) of the beam.

If there were a combination of sinusoidal forces in both the axial and transverse directions, it would be possible to set up the PDE describing the motion but it would be very hard to solve it. In practice you would get the solutions numerically, for example using a finite element model of the structure.

 

[marcas3]

So if a mode only tells you about the possible motion of a structure, how do you know which mode best describes the beam's motion? I know that such a vibrating system can have multiple modes but is it possible for some modes to have little or no effect on the response? How do you know which ones are basically useless, if any?

 

[AlephZero]

So if a mode only tells you about the possible motion of a structure, how do you know which mode best describes the beam's motion? I know that such a vibrating system can have multiple modes but is it possible for some modes to have little or no effect on the response? How do you know which ones are basically useless, if any?

You transform the equations of motion to use the mode shapes as generalized variables, which is a very similar procoess to the general ideas behind the Lagrangian formulation of Newtonian mechanics.

The modal equations of motion are nice to work with, since the "modal stiffness" and "modal mass" matrices of the structure are both diagonal. The transformation of forces from the "real world" coordinate system into modal components is intutively simple to visualize. If you apply a force to a point on the structure, the mode shapes which have large displacements at that point respond more than those with small displacements. That makes sense, because work = force x distance, so the force puts more energy into the modes that make the biggest contribution to the displacement at the point where the forcei is applied.

As a special case, if a mode has zero displacement in a particular direction, a force in that direction will not excite it at all. So in your beam example, an axial force applied to the beam can not excite any of the tranverse vibration modes, and similarly a transverse force can't excite any axial vibration modes.

 

[covboy123]

hello guys
I am a student of mechanical engg and i have taken instrumentation and control as my elective subject . can any1 please help me with the following question please .. i need to submit this by wednesday 12 / 01 / 2011 .

A beam is vibrating sinusoidal at 40 Hz. The displacement at the centre of the beam is set to 0.60 mm peak-to-peak. A piezoelectric accelerometer attached to the centre of the beam, has a sensitivity of 2.5 V/g and an output impedance of 200 . However, there is a requirement to measure the velocity of the motion at the centre of the beam.

(a) Calculate the output of the accelerometer in Volts rms. (6 marks)

(b) Draw a circuit incorporating an operational amplifier, which will allow a velocity signal to be obtained from the accelerometer signal. (6 marks)

(c) Identify appropriate component values for the velocity measurement system to give a sensitivity of 2 Vs/m. Clearly identify how you arrive at your chosen values.

Your help can be of great use to me .. i ll pass my module if u help me with these guys :)

Thanks ..

 

[MechEng2010]

"In the book, there is an equation to find the natural frequencies of a simply supported beam under an axial load:

omega = (pi^2/l^2)*sqrt(EI/roA)*sqrt(n^4 + (n^2*P*l^2)/(pi^2*EI))"

Which book is this from please?