Equations of Motion for the Spherical Pendulum (2DOF) Using Lagrange's Equations
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- Опубликовано: 28 авг 2019
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Deriving the equations of motion for the spherical pendulum using the method of Lagrange's Equations. Two degree of freedom system.
NOTE: In Eqn 6, I am missing a theta_dot in the first term and in eqn 10, the first term should be theta_dot squared (it's missing the squared).
can you recommend a good book for learning system modeling?
Thank you for the above edit..I went over the end part of the film numerous times and thought I must be tired because it didn't make sense..good, now I feel better; I am now getting the same final answer you have..Thank you for this vid..
I enjoy these videos and was happy to see you release a new one!
Thanks for the videos. It is very easy to follow your explanation. I wish I had a good teacher like you in the intoduction to vibration discipline during my graduation. Geatings from Brazil!
Explained the mathematics excellently! It's very hard to find physics tutors who can teach at an accessible level as many physicists explain beyond the capabilities of their students as they expect people to have the same understanding of them, a fault in today's science teaching departments in my eyes, so this is highly appreciated!
So you explained it so beautifully. Thank you.
This is awesome! I wondered what the equations of motion for a 3d simple pendulum would be and this is the top google result. So it's called the spherical pendulum. Nice.
You're doing great, keep up the good work.
Thank u for videos. Huge fan of your teaching.
truly amazing work
Thank you so much sir ....you explain every problem so nicely...I'm last year student of MSC physics ...and your every video is so much useful for me😇
Perfect video!
Bro, your video i't's so great, thanks, saludos desde México.
Glad it helped
Very helpful!
Explicas muy bien saludos desde Ecuador.
Thanks for the video
very nice tutorial thanks
thanks brother
Alot.
You videos is baie cool en interessant. Dankie!
Dis my plesier. Bly jy het dit geniet!
ya hooked me up, freeball 🤙
the intersection of l*sin(theta)*sin(phi) and l*sin(theta)*cos(phi) should lie within the circle and not outside as shown in the figure, since l*sin(theta)*cos(phi) will not be longer than the diagonal of the circle in horizontal plane. Minor but took me a while to figure out trigonometry.
Hi, your video was very helpful. I am trying to get an equation for a spherical pendulum's position, at any time, based on initial conditions (position and velocity). I've been out of school for a few years, and solving your final equations for t is a bit beyond my level now (Even at my freshest, solving a set of 2nd order non-linear differential equations with trig might not have been in my wheel house)
Could you point me in the right direction for this? Either a video/webpage that is explicitly solving for a pendulums position with respect to time, or just some keywords to google to help me get there? Or is this more complicated than I think it is?
As far as I am aware, the solution does not exist in closed form - ie you cannot write equations for the angles explicitly as a function of time; the solution must be computed numerically. I have shown a few examples of how to do this, including for the case of the double pendulum, which is similar. Using the equations for the spherical pendulum (which I derived above), a slight modification of the model I used in this video will make it suitable for the spherical pendulum. ruclips.net/video/nBBQKZb6JZk/видео.html
Great , Can you recommend a numerical solution for simulation of 3d pendulum with ODE solver
Check out my videos on the Runge-Kutta integrators. This should give you a numerical solution for anything you will need.
ruclips.net/video/r-jWnXjwQvk/видео.html
The rest is just modeling, you should be able to use my Double Pendulum video to get there; it can simply be extended it for one (or more) more additional masses. It's a little tedious to get there, but it's just algebra after all!
ruclips.net/video/tc2ah-KnDXw/видео.html
Is there a source you would recommend to help me find the tension for this problem using newton's second law of motion?
Here's an example of how to do it for the simple 2-D pendulum and the double pendulum. Using this example and my video should get you there. www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf
Hi I really liked your video ! But I'm wondering, what should we add in those equations such that the pendulum is damped ? How to bring the energy losses in those equations ? Thanks
Here is a video that shows how to incorporate damping into Lagrange's equations. ruclips.net/video/NV7cd-7Rz-I/видео.html
If torques terms were assigned at each of the theta and phi axes would it be possible to calculate those torque and corresponding speed values ?
You can calculate one or the other depending on what you’re given. If the torques are given then you can calculate the angular velocities. If you’re given the angular velocities then you can calculate the torques.
If we consider a rod with mass instead of a mass connected to an inextensible string, how does that change things?
I am getting tripped up on writing the kinetic energy. If we have a fixed pivot, we know one term in the kinetic energy is 1/2*Iend*thetadot^2, but this is the same for a simple pendulum. How can we incorporate something to account for the change in phi into the kinetic energy for a rod? Can I just add another moment of inertia term as if the rod was just spinning according to phi dot without swinging like a pendulum?
I showed in a different video, how to treat a compound pendulum, which is what I believe you are describing (ruclips.net/video/eBg8gof1RBg/видео.html). That was for the 2D case, you using that video and this one, you should be able to combine the two to model what you are describing. For the kinematics, use the location of the center of mass (instead of the location of the discrete mass).
Whats the frame of reference where we find the height of the prndulum?
This is a ground-fixed, inertial frame.
Hey, could I describe the motion for spherical pendulum with two angles in xy and yz planes?
Yes, you can do that. Any two coordinates that can uniquely locate the pendulum in any of its configurations can be used.
@@Freeball99 Awesome thanks! Could I also ask, in "Multi-degree of Freedom Systems (MDOF)" video you explain how to write equations in state space form. But how do you approach the case like this where you have angles and their derivatives but also sines and cosines of angles?
for the equation 10, shouldn't the theta (dot) at the beginning be squared?
yes, it's missing the squared
If the origin point of the pendulum was able to move in 3 dimensions, would you add a x(t), y(t), z(t) term to the respective equation of motion and recalculate the Lagrange?
Noooo. This is a common misconception. This additional variables will affect the kinetic and potential energies of the mass. You need to go back to step 1 (ie the kinematics) and add them there.
1. We first need to locate the mass, so add these variables to the appropriate kinematic equations.
2. Take the time derivative to to get the velocity of the mass (this will include time derivatives of x, y & z variables
3. Find the potential and kinetic energies and combine these to form the Lagrangian.
4. Substitute the Lagrangian into Lagrange's equation.
@@Freeball99 Thanks! Using the same coordinate system as in the video, I went back to the beginning and set my kinematics as:
x: x(t) + L sin (theta) cos (phi)
y: y(t) - L cos (theta)
z: z(t) - L sin (theta) sin (phi)
Then followed your steps, which gave me what I was looking for, equations of motion for theta and phi which contained contributions from x, y and z.
How would you factor in drag and air resistance into the code to slow the ball down like in real life?
You can incorporate viscous damping by including Rayleigh's Dissipation Function and using the extended form of Lagrange's equations. I have shown how to do it in this video: ruclips.net/video/NV7cd-7Rz-I/видео.html
Why is PE mglcostheta insteas of mgl(1-costheta)?
This is just a question of where one places the origin of the coordinate system. The origin of my coordinate system is at the hinge while you're placing yours at the mass (in its equilibrium position). These expressions differ by a constant, so when substituted in to Lagrange's equation and differentiated, the results are identical.
what would happen to the equation if the mass is allowed to slide up that rod of length L and the angle theta is fixed?
Unless you attached a spring or similar, the mass would slide off the rod due to the centrifugal force and at that point the rod (which is massless would rotate about the vertical axis at infinite angular velocity - so you wouldn't really have a vibration problem.
On the other hand, of you attached, say a spring to the mass so that it could oscillate back-and-forth along the length of the rod, then you could model it using a similar technique to that which I demonstrated in this video: ruclips.net/video/iULa9A00JpA/видео.html
if I want to use the lagrange equation equal to Qi, does that change anything in the equations of motion?
It means you have a forced vibration problem instead of a free vibration problem. It changes the equation by adding a term(s) to the right-hand side.
Sir, you told us how we can get simple pendulum out of spherical pendulum...could you tell how to get a circular(conical) pendulum?
Just solve as he did and then set theta=constant for a conical pendulum, such that theta-dot and theta-doubledot=0 and you find get the standard result that the angular speed (phi-dot) = sqrt (g/(Lcos(theta))).
How about Newton"s framework?
Did you find anything for this?
Hello sir,, can you do a demonstration why it is 2DOF
It's 2DOF because this is the minimum number of coordinates required to completely describe the location of the mass. In this case we use Φ and θ. If the length of the string could change, then you'd have a 3rd degree of freedom (r).
That eq12 has a nasty singularity, how would you treat that?
This is a seriously good question! It turns out that the singularity isn't a problem, here's why.
From equation 12, you are correct that there is a singularity when the angle, θ = 0. As a result, when θ = 0 the angular acceleration Φ_ddot, becomes infinite.
So, in the degenerate case where the pendulum initial is hanging vertically down, if you give it any angular velocity, Φ_dot then this would become infinite because in this position, the mass has no rotatory inertia (we're dealing with a point mass) - ie for θ to be 0, then Φ_dot MUST also be zero (or else it's infinite). Nothing complicated here...
Now when we have a regular 2-D pendulum, as it swings, the angle θ starts at some initial angle, swings through to the other side. At some point, θ must have been zero. So it would seem that for the 3D pendulum, this would cause the equations to blow up!
HOWEVER, when considering the 3-D pendulum, the swinging pendulum NEVER swings through θ = 0 and θ IS ALWAYS POSITIVE. How can this be? Imagine a situation where the angular velocity, Φ_dot is very high. As a result, the pendulum mass will be flung outwards (at almost 90 degrees if Φ_dot is high enough). This pendulum will never swing though θ = 0.
So what happens in the case of the 3-D pendulum is this. As θ gets smaller, so the angular velocity, Φ_dot gets large. The result of this acceleration, Φ_ddot, the mass is flung back away from the equilibrium point (the angular velocity will start getting infinitely large until this happens) AND THE PENDULUM NEVER SWINGS THROUGH θ = 0.
BTW - If you haven't yet seen it, I have a video showing how to solve the equations of motion and animate the pendulum. This will demonstrate some of this behavior. ruclips.net/video/_RMCZ3yNkPQ/видео.html
Hope I haven't confused you.
@@Freeball99 man, this is really amazing!! Just after write the comment I spended some time looking if the systems has a first integral wich would force phi to be a multiple of theta, as theta=0 is a simple pole for cot(theta), this would regularize the vector field in the submanifold E=constant. But after some calculations it seems that if the system is integrable, the treatment would require some tools of symplectic geometry...
Now I see that you deal with it using only physical principles! I'm really impressed. If I have understood, when you say "As θ gets smaller, so the angular velocity, Φ_dot gets large", is this just the conservation of angular momenta. A very elegant solution to a problem that would be very difficult to solve with only math.
Thank you for the answer. I planned to finish watch all you videos about lagrangean formulation until the midle of next weekend. They are helping me a lot to prepare for a exam that I will do for entry in a doctor program in physics (my backgraund is a master degree in differential geometry).
@@Freeball99 , If we have dissipative forces, θ can go to zero. To solve it I create another function for very small θ replacing sin(θ) = θ, cos (θ) = 1 and θ_d=0. With this the Φ_ddot = 0 Of course this make us lose all the physics, so I use an "if" function in my solver so it changes from the regular equation to this new one when θ approaches 0 (I am using this for -0.001 < θ < -0.001). It is not very elegant, but solve my engineering problem.
What is the Formula in simple form?
Like the Sphere Formula : (x^2 - a) + (y^2 - b) + (z^2 - c) = r^2
I am having a hard time understanding, Thank you!
In this case we have two equations of motion because we have two degrees of freedom (θ and Φ). We have to solve these equations simultaneously in order to find θ and Φ as a function of time (which I have shown in other videos).
Once you have determined θ and Φ, you can then convert it to x, y, z coordinates if you prefer. This is just a straightforward conversion from spherical to cartesian coordinates.
How to solve the Spherical pendulum with Hamiltonian method ???
I will have to make a video on this. It's too difficult to explain in text, but it amounts to making a few substitutions.
@@Freeball99 okk sir g
Equation 12 shld be 2 theta dot phi dot cot(theta)
Yes, the dot on top of the φ is being covered by the red box.
Equation 6, theta(dot) is missing.
yes, it's missing the theta_dot
the y axis points up :((((
Minor Errors
What errors did you find?