Mech-215

Mech-215

1. 1. The Phasor Diagram

   • Velocity leads displacement by pi/2
   • Acceleration leads displacement by pi

2. 2.Laplace Transforms

   • L[dx(t)/dt] =sx(s)
   • Initaial conditions do not affect the steady sate response, so can be ignored when doing laplace transform

3. 3. First Order Systems

   • Standard transfer function: 1/ 1+ts
   • Reponses
     • Step
     • Impulse
   • Shape of Responses
     • Impluse
     • Ramp
     • Step
   • Rise Time = 2.2T
   • Settling time = 4T

4. 4. Second Order Systems - Free Vibration

   • Simple Harmonic Motion
   • mg = Change in k
   • Pendulum
     • wn =
   • Natural frequency =
   • Mass Spring Pulley
     • wn =
   • Mass Spring
     • wn =
   • Torsional System
     • wn =
   • Damping Coefficient =
     • critical = 1
   • Free vibrations depend on initial conditions

5. 5. Second order systems - TF & Step Response

   • standard form:
   • step Response:
   • Peak time = pi/wd
   • %o/s=
   • Settling time = 4T

6. 6. Analysis in the frequency domain

   • #MasonsRule
   • characteristic equation
   • poles/zeros
   • Dominant poles: those closest to imag axis
   • Free oscillation given by imag part of comples pole
   • Exp decay rate - real part of complex pole
   • Natural frequency: distance complex pole form origin
     • Damping ratio
       • Angle between real axis and line joining the pole
   • Stable system: No pole in right half plane

7. 7. Steady state Errors in Feedback systems

   • system type: number of open loop pole at origin
     • This is different from the system order
     • These have different SS errors for each type of input
   • A fiinite SS error can be removed by increasing system type
     • I term increses system type
       • increases system order also
   • inifinte SS error = unstable stystem
   • D term has no effect on natural frequency
     • Moves poles away from imag axis
     • Indirectly helps make a system stable
       • Allows P controller to reduce SS error

8. 8. Forced Harmonic Vibration

   • Damping = Phase difference
   • Base Excitation
   • Harmonic force applied
   • Resonance w = wn

9. 9. Stability Analysis

   • Routh - Hurwitz Criterion
     • sign change
       • 0 = stable
       • 1 = 2 poles on real axis
       • 2 = 2 poles in RHP
         • Complex conjugate pairs