Modeling Transient Response So far our analysis has been purely kinematic: The transient response has been ignored The inertia, damping, and elasticity of the plant have been ignored But no more! Writing the Equations of Motion The prototypical second order system is the mass spring damper. Lets analyze this system: Force exerted by the spring: Force exerted by the damper: Force exerted by the inertia of the mass: Consider the motion of the mass there are no other forces acting on the mass therefore, the equation of motion is the sum of the forces: Why is this a linear system? Writing the Equations of Motion Your finger Suppose that your finger plucks the mass (i.e. applies a dirac delta transient force): Writing the Equations of Motion Write the equation of motion for a torsional damped inertia: For example, a rotating shaft with friction Writing the Equations of Motion: Example 1 Torque due to inertia: Torque due to damping: Write the equations of motion for the following: assume a frictionless joint Writing the Equations of Motion: Example 2 Write the equations of motion for the following: Writing the Equations of Motion: Example 3 The Laplace Transform How does the mass respond to the transient force? You have to solve this differential equation: We will analyze it using the Laplace transform: And the inverse Laplace transform: The Laplace Transform: Properties Linearity: Frequency shift: Time shift: Scaling: The Laplace Transform: Properties Differentiation: Integration: Final value theorem: The Laplace Transform: Examples Step function: Ramp function: Given:, find The Laplace Transform: Examples Given:, find SMD Laplace Transform SMD Transform The Laplace transform of the SMD equation of motion: SMD transfer function: Multiplying by this expression converts forces to displacements in the frequency domain This is called the characteristic equation of the transfer function. SMD Transform System Transform: Example Write the transform for the following system: System Transform: Example Write the transform from to for the following system: Assume: The block experiences friction has no mass or friction Time out for partial fraction expansion! for some values of Suppose you have the following: You can always decompose the denominator as follows: You have to solve for these constants (you dont need to know how for this course). SMD Transient Response: Example Note that the transient response decomposes into two exponentials The one dominates SMD Transient Response: Example If you reduce damping term, then the mass will oscillate: Note that in this case, the characteristic equation has complex roots: SMD Transient Response: Example Remember Eulers identity: The roots have an imaginary component: SMD Transient Response: Example (corrected) Frequency domain: Time domain: SMD Transient Response: Example (corrected) The bottom line: imaginary roots in the characteristic equation indicate that the system will oscillate. How do we characterize the transient response? The poles of are the roots of the denominator The zeros of are the roots of the numerator Pole-Zero Plot The complex plane Imaginary axis Real axis Consider the following transfer fn: Transient Response Different negative real poles correspond to non-oscillatory exponential decay Overdamped Im Re Overdamped Critically damped Transient Response Repeated negative real poles correspond to the fastest non-oscillatory exponential decay possible in a second order system Critically damped Im Re Im Overdamped Critically damped Transient Response Positive real poles correspond to non- oscillatory exponential increase Not BIBO stable Im Re Transient Response If there is an imaginary component to the root, then the system oscillates Underdamped Im Re Transient Response If the real component of a complex root is positive, then the system is not BIBO stable Im Re Transient Response Purely imaginary roots cause the system to oscillate forever undamped Im Re Second Order Transient Response Second order systems can be characterized in terms of the following: Natural frequency: the frequency of oscillation w/o damping Damping ratio: exponential decay frequency / natural frequency Second Order Transient Response The natural frequency is the frequency of pure oscillation: With a zero damping term, this becomes: Consider the transfer fn: Second Order Transient Response The damping ratio is: Therefore, the transfer fn can be re-written as: Damping ratio: exponential decay frequency / natural frequency Second Order Transient Response The damping ratio characterizes whether the second order system is underdamped, overdamped, or critically damped: underdamped critically damped overdamped If, then Second Order Transient Response: Example (corrected) Is the following SMD system over/under/critically damped? Therefore, its critically damped. If instead: then and the roots are: