Pendulum
Pendulum
(OP)
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?
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RE: Pendulum
RE: Pendulum
RE: Pendulum
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.
RE: Pendulum
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: Pendulum
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?
RE: Pendulum
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: Pendulum
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.
RE: Pendulum
RE: Pendulum
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.
RE: Pendulum
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......."
RE: Pendulum
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).
RE: Pendulum
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
RE: Pendulum
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.
RE: Pendulum
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.
RE: Pendulum
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.
RE: Pendulum
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.
RE: Pendulum
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?
RE: Pendulum
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.
RE: Pendulum
RE: Pendulum
I found this for site for the harmonic analysis:
http://math.ut.ee/~toomas_l/harmonic_analysis/
You need the Fourier series harmonics, which will yield
Ak, Bk components where your roll function is represented as
A0+sum AkSin(2pikt/T) +BkCos(2pikt/T).
As far as the solution to the diff equation, any second year book on differential equations has it, or go online to get it. For your equation, there are two solutions, one is the homogeneous where the diff equation has no forcing function and the other called the particular solution. For your problem you need only the latter whose solution is (I previously gave it in another form)
Wi^2/(W^2-Wi^2){(Ak)sin(kWit)+ (Bk)cos(kWit)}
where
Wi is the fundamental frquency of the roll and k is the kth harmonic
W is the natural frquency of the pendulum=Sqrt(g/l)
RE: Pendulum
In other words, there is not much energy coupling between a forcing function with a fundamental at 1/13 Hz, and a pendulum at say 2 Hz. That does not seem unreasonable.
If you come up with a better description of the geometry, I can refine the estimate. I just guessed to meet what you had stated. The most useful missing numbers would be the peak to peak lateral amplitude of the motion of the suspension point on the crane, and its height above the centre of rotation.
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: Pendulum
RE: Pendulum
You may wish to consult the classical textbook, Elliptic Integrals, Hancock 1917, pg 90, example 7. I obtained a copy of the original textbook off eBay several years ago.
Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada