# half pipe motionWatch

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#1
y''+(k/m)y'-(g/r)(cos theta)=0,[email protected](0)=0,theta'@(0)= 0.6, can anyone help point me in the right direction with this bad boy, regards Patrick.
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4 weeks ago
#2
(Original post by Patrick1753)
y''+(k/m)y'-(g/r)(cos theta)=0,[email protected](0)=0,theta'@(0)= 0.6, can anyone help point me in the right direction with this bad boy, regards Patrick.
Looks like the nonlinear (damped) pendulum dynamics?
How are theta and y related, etc? Is there a full description of the question?
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#3
(Original post by mqb2766)
Looks like the nonlinear (damped) pendulum dynamics?
How are theta and y related, etc? Is there a full description of the question?
hi, from the question, @ time t=0, the 0.9 kg particle is given an initial velocity of v(0)= 0.3m/s @ position theta=0, and slides along a circular path the shape of a half pipe of radius 0.5m, a viscous fluid k=3N.s/m drag parameter, acts to oppose motion, determine and plot both theta & theta' as functions of time over the range 0 < t < 5sec or = to, also determine max values of theta & theta' & corresponding values of t, also what time is theta first @ 90 deg, acceleration tangential =g cos theta-k/m(v) hope you can point me in the right direction, kind regards Patrick.
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4 weeks ago
#4
(Original post by Patrick1753)
hi, from the question, @ time t=0, the 0.9 kg particle is given an initial velocity of v(0)= 0.3m/s @ position theta=0, and slides along a circular path the shape of a half pipe of radius 0.5m, a viscous fluid k=3N.s/m drag parameter, acts to oppose motion, determine and plot both theta & theta' as functions of time over the range 0 < t < 5sec or = to, also determine max values of theta & theta' & corresponding values of t, also what time is theta first @ 90 deg, acceleration tangential =g cos theta-k/m(v) hope you can point me in the right direction, kind regards Patrick.
How is y related to theta? What does theta=0 represent? Can you upload a diagram?
If they want you to plot theta and theta', it sounds like you should simulate in xxx software, rather than try and solve the nonlinear ODE?
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#5
(Original post by mqb2766)
How is y related to theta? What does theta=0 represent? Can you upload a diagram?
If they want you to plot theta and theta', it sounds like you should simulate in xxx software, rather than try and solve the nonlinear ODE?
hi, the question is in merriam & kraige dynamics 6th ed, 2/251, a(t)=dv/dt=gcos theta-k/m(v) v=r *theta', d/dt(r *theta') =gcostheta-k/m(r *theta'), or, d^2theta/dt^2+k/m dtheta/dt-g/r*cos*theta=0, switch to 1st order, x(1)=theta & x(2)=theta' x'(1)=x(2), x'(2)=-k/m(x(2)+g/r cos x(1), x(1)@(0)= theta (0) =0, x(2)@(0)=theta'@(0)=v(0)/r, I have tried undetermined coefficients & variation of parameters, need to learn where to go from here, kind regards Patrick.
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4 weeks ago
#6
(Original post by Patrick1753)
hi, the question is in merriam & kraige dynamics 6th ed, 2/251, a(t)=dv/dt=gcos theta-k/m(v) v=r *theta', d/dt(r *theta') =gcostheta-k/m(r *theta'), or, d^2theta/dt^2+k/m dtheta/dt-g/r*cos*theta=0, switch to 1st order, x(1)=theta & x(2)=theta' x'(1)=x(2), x'(2)=-k/m(x(2)+g/r cos x(1), x(1)@(0)= theta (0) =0, x(2)@(0)=theta'@(0)=v(0)/r, I have tried undetermined coefficients & variation of parameters, need to learn where to go from here, kind regards Patrick.
Sorry to be a pain, but could you upload a picture of the question and a picture of the table of contents of that chapter?

As I said, it looks very much like the dynamics of a damped nonlinear pendulum and from memory, the analytic solution certainly isn't trivial unless you linearize or look for points where the velocity is zero or ...
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4 weeks ago
#7
(Original post by Patrick1753)
hi, the question is in merriam & kraige dynamics 6th ed, 2/251, a(t)=dv/dt=gcos theta-k/m(v) v=r *theta', d/dt(r *theta' =gcostheta-k/m(r *theta' , or, d^2theta/dt^2+k/m dtheta/dt-g/r*cos*theta=0, switch to 1st order, x(1)=theta & x(2)=theta' x'(1)=x(2), x'(2)=-k/m(x(2)+g/r cos x(1), x(1)@(0)= theta (0) =0, x(2)@(0)=theta'@(0)=v(0)/r, I have tried undetermined coefficients & variation of parameters, need to learn where to go from here, kind regards Patrick.
You wont have much luck looking for an analytical solution here.

You should solve this numerically via Runge Kutta on MATLAB (or other such program) so that you can plot theta and theta' then read off the times where they are max.
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#8
(Original post by mqb2766)
Sorry to be a pain, but could you upload a picture of the question and a picture of the table of contents of that chapter?

As I said, it looks very much like the dynamics of a damped nonlinear pendulum and from memory, the analytic solution certainly isn't trivial unless you linearize or look for points where the velocity is zero or ...
hi, google Merriam & Kraige engineering dynamics 6th ed si version, it's the 2nd down from the top, kind regards Patrick.
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4 weeks ago
#9
(Original post by Patrick1753)
hi, google Merriam & Kraige engineering dynamics 6th ed si version, it's the 2nd down from the top, kind regards Patrick.
Note the * at the start of the question. * means a computer oriented problem as mentioned by us both above. So you're expected to simulate the ODE in xxx software (matlab, ...). It is a reformulation of a damped, non-linear pendulum dynamics, which is a well-known simple problem, but is hard to solve analytically.
Last edited by mqb2766; 4 weeks ago
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#10
(Original post by mqb2766)
Note the * at the start of the question. * means a computer oriented problem as mentioned by us both above. So you're expected to simulate the ODE in xxx software (matlab, ...). It is a reformulation of a damped, non-linear pendulum dynamics, which is a well-known simple problem, but is hard to solve analytically.
hi yes I understand that, just so used to solving all my stuff with my calc, whiteboard, and pad, thought I might be able to find a way to solve analytically, will have to learn how to enter these sort of problems into software ie matlab, thanks so much, kind regards Patrick.
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4 weeks ago
#11
(Original post by Patrick1753)
hi yes I understand that, just so used to solving all my stuff with my calc, whiteboard, and pad, thought I might be able to find a way to solve analytically, will have to learn how to enter these sort of problems into software ie matlab, thanks so much, kind regards Patrick.
No, you won't/can't. Just search for
nonlinear damped pendulum
There are a few papers on solutions using Jacobi elliptic functions and I've no idea what they are. Apart from that, its simulation or linearization or ...

If this is part of a Uni(?) course, you should have been taught basic simulation? If not, what maths-dynamics software do you have?
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