Pendulum

[Ussuri]

I have a 7000kg pendulum on the end of a 5m long wire. The pendulum will be subjected to a lateral load of 62kN. Is there a quick method of detemining maximum amplitude or do I need to go back to harmonic motion and determine the equations from scratch?



---------------------------------------------------------------------------------------------------------------------


"The world keeps turning, it keeps me in my place; where I stand is only three miles from space"
Spiritualized

 

[ivymike]

a constant lateral load?

 

[handleman]

Here's a calculator that also lists the basic equation. Happy Plug'n'Chug!

 

[Denial]

You need to specify the duration of the 62kN load. If the duration is zero, the maximum amplitude will be zero (there's a nice trite statement for you). If the duration is infinite, then the maximum amplitude will be approximately twice the "steady state" amplitude under a gradually applied load of the same magnitude. Your case will lie somewhere between these two extremes, and you'll probably have to solve the fundamental equation of motion.

You talk about solving the "harmonic motion" equation. This assumes that the amplitude is small. If the amplitude can not be assumed small, then the equation you have to solve is not the harmonic equation, but is the equation given at the URL cited by Handleman.

 

[GregLocock]

Any particualr objection to using a work/PE approach?

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[Denial]

None, if it can be done.

But to do this don't you need to know how far the mass moves during the time of application of the force, which has you needing the detailed response to determine the peak response? Or have I missed something blindingly obvious?

 

[GregLocock]

Sure, the problem as stated is open to many interpretations.

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[Ussuri]

Thanks for the responses folks.

A bit more information. The 7000kg weight is hook on a crane. The crane is on one of our vessels. The vessel is expected to roll with a period of 13s, producing a maximum acceleration at the hook of 8.85 m/s2.

I am trying to work out how far the hook will swing in this instance. It is complicated that depending on the relative phases of the motion for the vessel and crane hook one may cancel the other out. But I am just looking for something quick and dirty.

 

[handleman]

Wow, that's a whole lot more complicated! It's almost a double pendulum, which has no steady-state. Obviously the mass of the hook won't have a measurable effect on the motion of the vessel's mass. I'd say you're pretty close to a bad situation, though, as your driving period is close to an integer multiple of the natural period of your pendulum.

 

[zekeman]

This is a simple single degree of freedom pendulum with a first harmonic of W1=0.48. The natural frequency of the pendulum is W= sqrt(g/l)=sqrt(9.8/5)=1.96.For each of the harmonics the attenuation factor for each harmonic component becomes, after solving the differtential equation
Wi^2/(Wi^2-W^2)
The only problem I see is the possibility that the fourth harmonic, W4=4*.48=1.92 is near resonance causing a huge amplification.

 

[zekeman]

Correction,
I meant to say that "This is a simple single degree of freedom pendulum driven by a periodic forcing function whose first harmonic is W1=0.48 rad/sec^2......."

 

[rb1957]

an interesting set of divergent opinions ...

can the movement of the head of the crane, caused by the ship rolling, be represented as a cyclic load applied to the free end of the pendulum ? how would you go about calculating the equivalent force to represent a displacement function ?

set the system up with everything of the CL. as the crane rocks to port i can see the free end of the pendulum staying stationary (vertically) under the crane head ... the pendulum wouldn't be very effective in transmitting lateral loads. this'd imply (rightly or wrongly) that the free end of the pendulum is stationary (to an observer off the ship).

 

[zekeman]

Yes,
that equivalent force for each harmonic is
mlaiWi^2sin(Wit)
where
l=pendulum length
m=mass at the hook
Wi=frequency of ith harmonic
ai=ship roll ith harmonic input
You get this by writing the differential equation
mld^2(@)/dt^2+mg(@-@s)=0
where
operator d^2/dt^2 = second time derivative
@ =angular motion of pendulum
@s= roll angle of ship
subtracting mld^2@s/dt^2 from both sides, I get
mld^2(@-@s)/dt^2+mg(@-@s)=-mld^2(@s)/dt^2

which is recognized as an linear ODE with the known driving force of -mld^2(@s)/dt^2

 

[GregLocock]

Nice try.

But...

A few complications:

1)The suspension point of the cable moves in arc, in the XZ plane.

2) I'm not entirely convinced a ship actually rolls with a sinusoidal velocity, but lets say it does.

3) even if (2) is true then 1) means that you will get a non linear input.

So how did you get to 8.85 m/s/s?






Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[Ussuri]

I realise this is a far more complicated problem than a simple pendulum but I was trying to get a quick and dirty ball park answer.

The 8.85 m/s^2 is calculated from the RAO's for the vessel. This the value for roll acceleration at a point above the deck of the vessel approximately at the crane hook location. It is completely independent of the hook as it is vessel dependent. To be completely accurate this would be the acceleration if the hook was connected to the crane boom with a stiff member fully restrained in 6DOF at the boom tip.

And you are correct the vessel motion will in reality not be sinusoidal as sea states are irregular by nature. You do get RAO's for irregular waves but the ones we have are based on regular wave theory. Also effect of determining accurate transfer functions to translate sea state to vessel motion is difficult. So RAO data is at best an approximation of roll acceleration.

There are so many complex variables in this that the time taken to get an accurate answer offsets any additional benefit. Rough as guts is the way forward.

 

[GregLocock]

OK, rough as guts. for 8 m/s/s sideways acceleration, and a 10 m mast (mast tip to metacentric centre) at 13 s, and a 5 m wire, I get roughly +/-12 m of maximum lateral motion.

The accurate result looks somewhat chaotic to me.

You'd be much better specifying the period, length and maximum angle of the mast than some ill-defined acceleration.



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[zekeman]

Ussuri,
In formulating the problem for what is assumed to be the "worst" case, you must get the periodic data of the ship's roll and do a harmonic analysis of it. You can then write the differential equations of motion, keeping in mind that the input motion to the pendulum system ( which is single degree of freedom) is at the pivot point of the pendulum, not at the hook location. My guess is that that arc motion, in a first cut would be mostly and therefore approximately in the X direction . You can then write the two equations in the two dependent variables, namely @ and x for each of the harmonics assuming linearity at first. The assumption of linearity depends on the angular amplitude of the pendulum where the assumption of the sine and the angle are nearly the same. If the case for linearity is sustained, then harmonic analysis of the input is valid and your solution is the sum of the responses to each harmonic.

 

[Ussuri]

Greg, +\-12m would be equivalent to a vessel roll amplitude of 45 degrees. If that occured I believe there would be more pressing worries such as hanging onto something. So, point taken my initial approach was flawed.

Zekeman, It has been a long time since I had to do anything with differential equations and harmonic motion so apologies if I appear particularly dense. The periods and amplitudes of the roll are dependent on the sea state, such as wave height and wave heading. Worst case for roll would be a beam sea (wave heading perpendicular to vessel). The harmonics I have come across in my career are of the simple kind (y=Asin(wt)) so I am going to have to do some reading up on the subject. Any suggestions as to a good text?

 

[zekeman]

As a further simplification, the linearized differential equation becomes
md^2(x+l@)/dt^2+mg@=0 which by rearranging becomes
mld^2@/dt^2+mg@=-md^2(x)/dt^2
and
ld^2@/dt^2+g@=-d^2(x)/dt^2
which shows the classical pendulum equation with a forcing function to be the mass at the hook times the the ship's x component of roll acceleration at the mast pivot point.The analysis is similar to that which I outlined in a a previous post where the 4th harmonic is of concern.

 

[handleman]

It sounds to me like you're assuming that worst-case is your maximum expected amplitude of vessel roll. That's not necessarily the the case. Worst case is when the roll frequency/period is driving the pendulum at its natural frequency. Even a very small amplitude of roll at the wrong period (which you've stated as being dependent on sea state) could cause a very large swing of the crane hook. Bringing this into the practical, two things that could decrease the swing amplitude would be to change the length of the pendulum or add significant drag(damping) to the hook in some way. Of course, anything that adds drag will make the hook tend to blow around in the wind. Preventing the hook from blowing around is the reason the hook is so heavy to start with.