Removing a function from a system

Mar 2019
1
0
Denmark
[HR][/HR]Hi.


I am a bit stuck with an assignment and i hope that someone can give me a notch in the right direction.


I got a model for a robot arm which is made up of 2 differential equations:


d^2x/dt^2 + a*dx/dt = j (1)
dj/dt = -b*j - dx/dt + u(t) (2)


j = j(t) is the power through the motor x = x(t) is the position of the load, both functions of time. The voltage on the system u = u(t) controls the robot arms movement adx/dt is the mechanical loss in rotor aswell as movement of the arm with a load. bj descripes the electrical resistance in the motors electrical circuit. a and b are positive real constant which is far below 1, meaning that |a-b| < 2.


I have to remove j(t) from the system (1-2) and show that:
d^3x/dt^3 +(a+b)d^2x/dt^2 + (1 + ab)*dx/dt = u


My initial approach was to diffentiate equation 1 which led to:
d^3x/dt^3 + a* d^2x/dt^2 = dj/dt


i then substituted the left side of equation 1 aswell as the j on the right side of equation 2 leading to:
d^3x/dt^3 + a* d^2x/dt^2 = -b(d^2x/dt^2 + a*dx/dt) - dx/dt + u(t)


Since that i have tried a couple of diffentent approaches, but i can't seem to figure it out. Can anyone help me here?
Thanks!