Angular Acceleration Differential

[godfather]

HI,

I would appreciate if someone could provide me some suggestions. I am trying to figure out angular acceleration and angular velocity of opening a car hood. I think I have the sum of the torques correct however I am having a tough time with speed because the torque is not contant. I do not think I can use the standard equations and I am a little rusty of my differential equations.

The push force is assumed constant but the main issue is that I am also using a torsion spring that decreased with the opening of the hood. I am not sure how to introduce this into the differential A) over the entire angle span B) or at each discrete angle.

Integrating Velocity is straight forward but integrating the sum of the torques with respect to angle is giving me the problem. The force from the CG is dependant on the angle where the handle force and spring is not.

Once, I have the speed and acceleration the time should fall out.

Maybe I am totally off base and there is an easier solution. Any help would be very appreciated.

Thanks

Godfather

 

[godfather]

Any help form anyone? Even if you just help on the differential equation?

Thanks

 

[K^2]

Because of the Cos(phi) in your differential equation, you'll have an elliptic integral as a solution. Look up high-angle solutions to period of a pendulum for a method of solving such an equation. Other terms in the equation are linear, so you should be able to modify the pendulum solution to incorporate these. Good luck. This isn't a simple problem.

 

[godfather]

thanks for the feedback. Do you think this will also accommodate a variable rate spring spread out linearly over the full angular span?

 

[K^2]

Elliptic integral method? It should.

 

[wheelatharm]

Could you use conservation of energy to find the final velocity? If the change in height of the CG of the hood is known, then it plus the K.E. of the hood should be equal to the change in energy of the coil spring. There is an equation for a coil spring that is the equivalent to 1/2 Kx^2, where theta replaces x, and you have a different type of constant, but the concepts are the same.

 

[K^2]

If you want just the final velocity, yes. But if you want to know how long it takes to get to that point, you have to integrate over velocity. That's exactly how you solve the problem. The difficulty is in evaluating an integral. It cannot be done algebraically.

 

[wheelatharm]

Hi again,

I can't pull up the diagram of the car hood problem. It sounds like at least the for the work being done by the force on the handle you would need to integrate to make an algebraic expression for the work done, but the conservation of energy can give you an answer for the speed of the hood at any point, not just for the final velocity. Most of the terms should be algebraic functions of the angle of the hood.
The thing that you won't get from COE is the instantaneous acceleration at any time, so maybe you are stuck doing kinematics.

Marc

 

[K^2]

At any point in space, yes. But not at any point in time. In order to know angular velocity as function of time, you must integrate over velocity at previous points.

 

[godfather]

would you guys mind showing the equations i am stumped

 

[wheelatharm]

Sorry about the earlier reply K^2, I sometimes don't read the posts fully. Yes, you would need to generate a formaula, then integrate to get the time.

I can't pull up the diagram. There is some issue with my work computer opening it.
Is there a word version that could be posted?

Marc