Inverted pendulum attached to cart, cart on frictionless surface & axes

AI Thread Summary
The dynamics of an inverted pendulum attached to a frictionless cart reveal that, without any external forces, the pendulum swings back and forth indefinitely, returning to its initial height at a symmetrical angle. The cart experiences acceleration in opposite directions depending on the pendulum's position—accelerating in one direction while the pendulum falls and in the opposite direction as it rises. This motion creates different perceptions based on the observer's frame of reference; from the cart, the pendulum appears to swing in a circular path, while from a stationary lab frame, the motion appears distorted. The discussion emphasizes the importance of understanding the kinematic laws governing the system. Overall, the findings suggest a consistent behavior of the pendulum and cart system under the specified conditions.
Number2Pencil
Messages
204
Reaction score
1

Homework Statement



As a small verification of a larger problem, I'm trying to determine how the dynamics of an inverted pendulum attached to a frictionless cart would behave.

The cart has no input force applied, so all horizontal movements would be due purely to the centrifugal+normal force of the pendulum, being driven by gravity.

The initial conditions of the system would be that the initial cart speed/position = 0, the initial angular rate of pendulum = 0, and the initial positional angle of the pendulum = 0.01 radians, and just let to fall


Homework Equations





The Attempt at a Solution



After solving through all the kinematic laws, and trying to simulate the system, what I am seeing is that without friction, the pendulum continues to swing perpetually back and forth, always reaching the initial height at a angle symmetrical to the starting angle.

Does this sound feasible?
 
Physics news on Phys.org
It would help if you provide an image of the system you have in mind. The images from a google search 'inverted pendulum' suggest that while the pendulum is falling, the cart will be accelerated in one direction and while the pendulum is rising, the cart is accelerating in the opposite direction.

If that interpretation is correct, the appearance of the motion of the pendulum will depend on the observer. If you are sitting on the cart, I believe the pendulum will appear to swing in a circle but from the lab frame, it would make a distorted shape (I can't imagine it right away, but it should not be hard to find mathematically)
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
View full post »

Similar threads

Simulating a Rotary Inverted Pendulum in Python
Replies
15
Views
1K
Foucault Pendulum at the North Pole
Replies
9
Views
2K
Help with Inverted Pendulum on Cart EoM
Replies
1
Views
2K
Impulse and Momentum: Ball and cart connected by a cord
Replies
24
Views
3K
Acceleration of a cart immediately after the system is released
Replies
15
Views
4K
Modelling Inverted Pendulum With 2 Propellers (Control Engineering)
Replies
4
Views
1K
Impulse and Momentum Conservation for a Frictionless Cart
Replies
5
Views
1K
Differential Equation for a Pendulum
Replies
21
Views
2K
Bob on Cart Problem: Max Speed Calculation
Replies
10
Views
2K
Double Cart Inverted Pendulum Mechanics
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top