System Dynamics and Controls

Apply Laplace transform tables to break down complex expressions using linearity, unit steps, and standard forms for sine, cosine, and exponential terms in example 2.

Apply the differentiation theorem of Laplace transforms to convert derivatives into algebraic terms. Use L{f'}=sF(s)-f(0) and L{f''}=s^2F(s)-s f(0)-f'(0) to solve initial value problems.

Utilize partial fraction expansion for a fraction with repeated roots, factor the denominator as s(s+1)^2, determine a, b, c, and rewrite for inversion.

Decompose a complex fraction into partial fractions for inverse Laplace transforms. Solve for constants A, B, and C, handling imaginary roots to reconstruct the original fraction.

Apply Laplace transforms to initial value problem x¨ - 3x˙ + 2x = 4t - 6, use the differentiation theorem and initial conditions, and obtain X(t) = 2t + e^{t}.

Apply the parallel axis theorem to compute the net moment of inertia about the x axis by subtracting the hollow cylinder's inertia from the solid cylinder's, using the given density.

Analyze the rigid bar in example 12 by applying relative motion to find the center-of-mass velocity, acceleration, kinetic energy, and angular momentum about the center.

Draw a free body diagram for a mass-spring system, apply Newton's second law, and derive the differential equation m y'' + k y = F with downward positive.

Analyze a two-mass system with springs and a damper using three free-body diagrams per mass to derive coupled equations of motion via Newton's second law.

Model two elements connected without a mass by inserting a zero-mass point mass to separate the spring and damper, derive equations of motion, and simplify analysis.

Apply the small angle approximation to linearize the system, replacing sin theta with theta, cos theta with one, and 1 minus cos theta with 1/2 theta^2, as shown in example 18.2.

Learn transfer functions as the output-input ratio in the s-domain with zero initial conditions. Derive G(s) from a differential equation and visualize the system with a block diagram.

Derive the transfer function x1(s)/f(s) for a mass-spring-damper system by building free body diagrams, formulating equations of motion, and applying the Laplace transform.

Explore rotational dynamics with torsional dampers and springs, using free body diagrams and angular variables to derive equations of motion, then apply the impedance method to obtain the transfer function.

derive the equations of motion for a two-mass, three-spring system with two inputs, define four state variables, and build the state-space model with outputs as the two displacements.

Show state-space modeling of a mass-spring system in MATLAB, deriving A and B from m x'' + k x = u(t) with x1 displacement and x2 velocity, and plot results.

Demonstrate a four-by-four state-space model for a two-mass system with a two-input b matrix in Matlab, solve with ode45, and plot displacements and velocities.

Derive the transfer function from state-space form by applying the Laplace transform to x dot = A x + B u, yielding G(s) = C (sI - A)^{-1} B.

Extract the system matrices from the state-space representation, compute (sI − A)⁻¹, and form the transfer function J(s) = C (sI − A)⁻¹ B.

Explain the free (natural) response of a first-order system with zero input, and impulsive response for a damped spring and damper under a 10 N·s impulse, yielding X(t)=0.005 e^{-0.05 t}.

Explore the step response of a first-order system in system dynamics and controls, derive the time constant, rise time, and settling time, and connect exponential decay to the final value.

Solve a step input in a tank system from the system dynamics and controls course to determine the steady-state liquid level, step response, and settling time.

Revisit the tank flow example in MATLAB and demonstrate plotting the transfer function response using the step command, inverse Laplace, and settling time to show the final value.

Explore the ramp response for first order systems, derive x(t) from the s-domain using partial fraction expansion, and relate the time-domain behavior to exponential terms.

Turn a damped mass system into a first-order velocity model and solve for v(t) under a step force, yielding v(t) = (F/C)[1 − exp(−(C/M)t)].

Explore a first-order cooling model of temperature change in a soup cooling scenario with T(t) = 5 + 95 e^{-0.05 t}. Visualize its first-order response in MATLAB plots.

Derive the second order mass–spring–damper model with damping, obtain its equation of motion and transfer function, define zeta and omega_n, and analyze pole locations across damping regimes.

Analyze the step response of underdamped second-order systems using the transfer function and unit step. Derive peak time, overshoot, settling time, and rise time with key formulas.

Determine pole locations for an underdamped second-order system from a 12 percent overshoot and 0.6 s settling time; compute zeta and omega_n, then locate poles.

Plot the hand-derived X(t) in MATLAB, compare it to the transfer function response using a step input of 0.0128, and show overlapping plots of oscillation and damping.

Learn block diagrams with a heating system, illustrating a closed-loop with thermostat, temperature sensor, and a summing junction, and apply cascade and parallel transfer functions.

Explore feedback form concepts as the basis of closed-loop control engineering: negative feedback reduces error to keep systems stable, with HVAC, autopilot, and cruise control as examples.

Learn to form equivalent diagrams by reducing subsystems to one block while preserving outputs; move blocks around summing junctions and apply 1/G(s) to maintain equivalence.

Identify the single feedback loop in the three-block H system and convert to the standard form G/(1+GH). Use cascade and parallel reductions to derive the equivalent transfer function (G3*G1*G2)/(1+G1*H1).

Apply the Ruths (Routh-Hurwitz) stability criterion to the characteristic equation of the closed-loop transfer function to determine if any roots lie in the right half-plane.

Apply the routh's criterion to a cubic characteristic equation, build the routh array, count sign changes, and conclude two right-half-plane poles imply instability.

Apply the Routh stability criterion to a cubic equation by forming its coefficient array, counting first-column sign changes, and noting patterns in even-exponent rows for shortcuts.

Explain special case two in the Routh array, including how a row of zeros yields an auxiliary (even) polynomial and how differentiating it fills the zero row for root analysis.

Analyze a long closed-loop example by forming T(s)=G/(1+G), computing the closed-loop denominator, and using the auxiliary polynomial, symmetry, and sign-change analysis to assess stability and pole locations.

Construct a root locus for a transfer function by identifying poles and zeros, setting k to zero, and tracing symmetric branches on the real axis to assess damping regimes.

Master root locus analysis of a transfer function with two poles and two zeros, tracing symmetric branches from poles to zeros and distinguishing underdamped, critically damped, and overdamped responses.

Compute poles and zeros from the transfer function and draw the root locus with symmetry about the real axis. Determine branches and asymptotes, applying step five when zeros are absent.

Explore steady state error by comparing input and output as time grows, using step, ramp, and parabolic inputs to assess position, velocity, and acceleration tracking and system stability.

Derive open-loop and closed-loop transfer functions for a unity-feedback system, then apply the final value theorem to express the steady-state error in terms of T(s) and G(s).

Examine steady-state error for a unit step, showing zero error when the open-loop transfer function has at least one integration in the forward path; otherwise the error is finite.

Define error constants Kepi, KVI, and KA for step, ramp, and parabolic inputs; classify systems as type 0, 1, or 2 by exponent of s, with type reducing steady-state error.

Derive the circuit transfer function with impedance method and mesh analysis, solving three loops to obtain Vout/Vin for the given circuit.