System Dynamics and Vibrations (EML3014C)

Course Description

System Dynamics and Vibrations covers system response, frequency analysis, and feedback control strategies. It is the second part of the integrated sequence in dynamics, vibrations, and controls that begins with EML3013.

Within the SCNS taxonomy, EML is the Mechanical Engineering prefix and the C suffix marks an integrated lecture-and-laboratory course. The FAMU-FSU College of Engineering publishes this at 3 credits, giving approximately 60 contact hours at the standard university integrated-course ratio — the value carried by the published EGN3331C and EGN3353C in this repository.

This is the course where mechanical engineering becomes systems engineering. The central realization is that a mass on a spring, an RLC circuit, a thermal mass, and a fluid tank are all described by the same differential equation — and once you accept that, you stop learning separate subjects and start learning one method that applies to all of them. That unification is genuinely powerful, and it is why the transfer function reappears in every subsequent controls, signal processing, and instrumentation course.

Learning Outcomes

Required Outcomes

• Derive equations of motion for mechanical systems using Newtonian and energy methods.
• Model translational and rotational mechanical systems with mass, damping, and stiffness elements.
• Model electrical, fluid, and thermal systems and recognize their analogies to mechanical systems.
• Linearize non-linear system models about an operating point.
• Apply the Laplace transform to solve system differential equations.
• Derive transfer functions and describe their physical meaning.
• Construct and reduce block diagrams.
• Determine poles and zeros and relate them to system behaviour.
• Analyze first-order system response and determine time constants.
• Analyze second-order system response and determine natural frequency and damping ratio.
• Distinguish underdamped, critically damped, and overdamped response.
• Determine transient response characteristics: rise time, overshoot, settling time.
• Analyze free vibration of single-degree-of-freedom systems.
• Analyze forced vibration and determine the frequency response.
• Describe resonance and its engineering consequences.
• Analyze base excitation and rotating unbalance.
• Describe vibration isolation and transmissibility.
• Analyze multi-degree-of-freedom systems and determine natural frequencies and mode shapes.
• Describe vibration absorbers and damping treatments.
• Construct and interpret Bode plots.
• Assess stability from pole locations and describe stability criteria.
• Describe open-loop and closed-loop control and the purpose of feedback.
• Describe proportional, integral, and derivative control actions and their effects.
• Analyze steady-state error and system type.
• Use computational tools to simulate system response and verify analysis experimentally.

Optional Outcomes

• Apply root locus methods to controller design.
• Apply frequency-domain design using gain and phase margin.
• Describe state-space representation.
• Describe digital control and sampling.
• Perform experimental modal analysis.
• Describe condition monitoring and vibration-based diagnostics.

Major Topics

Required Topics

• Deriving equations of motion
• Mechanical system modelling
• Electrical, fluid, and thermal analogies
• Linearization
• Laplace transform methods
• Transfer functions
• Block diagrams and reduction
• Poles, zeros, and system behaviour
• First-order response
• Second-order response
• Damping classification
• Transient response specifications
• Free vibration
• Forced vibration and frequency response
• Resonance
• Base excitation and rotating unbalance
• Vibration isolation and transmissibility
• Multi-degree-of-freedom systems and mode shapes
• Vibration absorbers and damping
• Bode plots
• Stability
• Open- and closed-loop control
• PID control actions
• Steady-state error and system type
• Simulation and experimental verification

Optional Topics

• Root locus design
• Frequency-domain design and margins
• State-space methods
• Digital control and sampling
• Experimental modal analysis
• Condition monitoring

Resources & Tools

• System Dynamics (Katsuhiko Ogata) or System Dynamics (Palm) — the standard texts for exactly this integrated treatment.
• Mechanical Vibrations (Singiresu Rao) — the vibrations reference, and comprehensive.
• Modern Control Engineering (Ogata) or Control Systems Engineering (Nise) — for the feedback content; Nise is the more approachable.
• Engineering Vibration (Daniel Inman) — clear and well suited to a first course.
• MATLAB with the Control System Toolbox and Simulink — the industry standard, and most institutions license it. Python with SciPy and the `control` package is free and fully capable, as is GNU Octave.
• NCEES FE Reference Handbook — free; the dynamics, vibration, and controls sections are examined.
• Brian Douglas on YouTube — free, and the clearest available explanations of transfer functions, poles and zeros, Bode plots, and PID. The single best supplementary resource for this material.
• MIT OpenCourseWare 2.003 and 2.004 — free, complete, and closely aligned to this course's scope.
• An accelerometer and a data acquisition system — the laboratory hardware; many concepts become obvious the moment you see a real frequency response.
• A study group — this course rewards discussing what a pole location means more than it rewards additional algebra.

Career Pathways

• Controls engineer — SOC 17-2141 adjacent; a well-paid specialization across manufacturing, aerospace, and process industries.
• Mechanical engineer — system modelling is central to design in any dynamic application.
• Aerospace engineer — SOC 17-2011; guidance, navigation, and control is exactly this subject extended, and Florida's Space Coast is a direct pathway.
• Robotics engineer — manipulator control is this course applied.
• Vibration and noise engineer — a genuine specialization; rotating machinery, structures, and vehicles.
• Mechatronics engineer — the mechanical-electrical-software intersection.
• Automation and process control engineer — Siemens Energy in Orlando, the utilities, and Florida's manufacturing base.
• Test engineer — modal testing and system identification.
• Condition monitoring and predictive maintenance — vibration analysis is the primary diagnostic tool, and certification pathways exist.
• Simulation engineer — multibody and system-level modelling.
• Graduate study in dynamics, controls, or aerospace.

Special Information

⚠ Everything is the same differential equation — that is the point

• The unifying idea is analogy. A mass-spring-damper, an RLC circuit, a fluid tank with resistance, and a thermal mass all produce second-order differential equations of identical form, and solving one solves all of them.
• Learn the element analogies deliberately — mass with inductance, damping with resistance, compliance with capacitance. Once the mapping is fluent you can model a hydraulic system with mechanical intuition.
• The transfer function is the object. It describes what a system does to any input, independent of what the system is made of, and it is what every downstream course uses.
• Poles are physically meaningful. The real part is a decay rate; the imaginary part is an oscillation frequency; a pole in the right half-plane means the response grows without bound. Learning to read a pole-zero plot as a prediction of behaviour is what makes the rest of the course intuitive rather than algebraic.
• Damping ratio and natural frequency describe the whole second-order response. Two numbers determine overshoot, settling time, and oscillation — and knowing that lets you specify a design rather than iterate blindly.
• Sketch Bode plots by hand before computing them. Asymptotic construction — corner frequencies and slopes — is the skill; the software plot is the check.
• Linearization is the assumption that makes all of this work, and knowing when it stops being valid is a real engineering judgement rather than a technicality.
• Simulate everything. Building the model in MATLAB or Python and comparing the simulated step response to your hand analysis is the fastest way to develop intuition.

⚠⚠ Resonance is not an abstraction — it destroys machinery and structures

• A system driven near its natural frequency responds with amplitude limited only by damping, and in a lightly damped system that amplitude can be enormous.
• Rotating machinery must pass through or avoid critical speeds. Shafts, turbines, pumps, and fans all have them, and running at one produces destructive vibration.
• Rotating unbalance is the most common excitation in real machinery, and balancing is a routine industrial procedure precisely because of it.
• Structures resonate too. Bridges, buildings, towers, and aircraft components all have modes, and matching an excitation to one is a recognized failure mechanism — the well-known bridge failures are more complicated than simple resonance, but the design lesson stands.
• Isolation is a design decision with a counterintuitive result. A vibration isolator only isolates above a frequency ratio of about 1.4 — below that it amplifies, so a badly chosen isolator makes the problem worse. This is examinable and it catches people out professionally.
• Damping matters most at resonance and hardly at all away from it, which is why adding damping is the right fix for a resonance problem and the wrong fix for a stiffness problem.
• Tuned vibration absorbers work by adding a degree of freedom that splits the resonance, and they are used from engine mounts to skyscrapers.
• Florida-specific: wind-induced vibration matters in this state's tall structures and coastal infrastructure, and it is a live design consideration rather than an academic one.

⚠ Feedback: what it buys and what it costs

• Feedback is the reason control systems work, and the central idea is simple: measure the output, compare it to what you wanted, and act on the difference.
• Feedback reduces sensitivity to disturbances and to model error, which is why it is used even when the system is well understood.
• And it can destabilize a stable system. That is the trade-off — too much gain, or too much phase lag, and the loop oscillates or diverges. Stability analysis exists because feedback can make things worse.
• Learn what each PID term does physically. Proportional responds to present error and always leaves some steady-state error in a type-0 system; integral eliminates steady-state error and adds phase lag that can destabilize; derivative anticipates and adds damping but amplifies measurement noise.
• Derivative action on a noisy signal is a classic practical failure, and filtering it is standard practice.
• Integral windup occurs when an actuator saturates and the integral term keeps accumulating; anti-windup is a real implementation concern rather than a theoretical footnote.
• Gain and phase margin quantify how close to instability you are, and a controller that meets the performance specification with no margin will fail in service when the plant changes.
• Tune a real system if the laboratory allows it. The gap between a simulated PID loop and a physical one — with saturation, noise, backlash, and delay — is where the engineering actually lives.

⚠ An honest account of the workload

• The thermal-fluids and systems sequence is, with mechanics, where engineering programmes lose students. The reason is rarely intelligence — it is that these courses demand sustained daily problem-solving and reward nothing else.
• Budget eight to twelve hours a week outside class. Students who treat it like a lecture course to be revised before the exam fail it.
• Work problems without the solution visible. Reading a worked example produces the feeling of understanding and none of the ability.
• Draw the diagram every time — control volume, system boundary, free-body diagram, or block diagram. Skipping it is the most common cause of wrong answers.
• State your assumptions explicitly. Steady state, incompressible, adiabatic, fully developed, small angle — these courses are largely about knowing which assumptions apply, and writing them down is how you find out whether they do.
• Carry units through every step and check that the answer is physically plausible.
• Form a study group and explain solutions aloud.
• Go to office hours in week two, not week ten. These courses are cumulative.
• If you are struggling, the problem is usually the prerequisite — weak calculus or weak differential equations shows up here as an inability to finish problems you set up correctly.

⚠ The EGN / EML numbering divergence and the 2000/3000 boundary

• Florida teaches much of the mechanical engineering core under two prefixes. Many institutions use the general engineering prefix EGN — EGN3343 for thermodynamics, EGN3353C for fluid mechanics — while the FAMU-FSU College of Engineering and others use EML numbers.
• Under SCNS these are different courses, and equivalency does not cross prefixes automatically or cross a C suffix. Get transfer determinations in writing.
• Lower-division credit generally cannot satisfy an upper-division requirement, and the sophomore/junior pairing documented across the EGN prefix in this repository is the trap: the same subject appears at both levels and programmes use one consistently.
• Florida's 2+2 articulation guarantees admission to the State University System with junior standing, but it does not guarantee that a specific lower-division course satisfies a specific upper-division requirement in a limited-access engineering programme.
• Check with the receiving engineering department, and note that answers can differ between catalog years.
• Plan the mathematics and physics sequence early. Calculus and calculus-based physics are the real gatekeepers, and the engineering-technology variants close the A.B.E.T.-EAC route.

⚠ FE exam and PE licensure — this is the accredited engineering pathway

• This course sits inside an A.B.E.T.-EAC accredited engineering programme, which is the pathway that leads directly to professional licensure. That distinguishes it sharply from engineering technology, where the route to a P.E. is longer and carries additional experience requirements — see this repository's ETG, EET, and ETC guides.
• Florida licenses professional engineers under Chapter 471, Florida Statutes, through the Florida Board of Professional Engineers. The sequence is: A.B.E.T.-EAC accredited degree → Fundamentals of Engineering (FE) examination → qualifying experience → Principles and Practice of Engineering (PE) examination → licensure.
• This subject is directly examined on the FE Mechanical exam. Most students sit the FE in their final year while the material is fresh, and pass rates are markedly higher for recent graduates than for people who wait.
• The NCEES FE Reference Handbook is the only reference permitted in the exam, and it is free to download. Use it as your reference now so that finding an equation in it is automatic by exam day.
• Only a licensed P.E. may offer engineering services to the public in Florida, seal drawings, or use the title in a way implying licensure.
• Rule 11 applies — verify current requirements with FBPE and NCEES directly.

How Florida course levels affect transfer

The first digit of an SCNS number denotes the year of offering, not transferability. Courses at the 1000 and 2000 levels transfer transparently between Florida public institutions, and 3000 to 4000 is unproblematic since both are upper division. The boundary that actually matters is 2000 to 3000, where lower-division credit generally cannot satisfy an upper-division requirement.

EML3014C is 3 credits and approximately 60 contact hours, and the C suffix indicates an integrated laboratory — expect experimental work measuring real system responses alongside the analysis, which is the most valuable part of the course.

It is the second of the three-part dynamics, vibrations, and controls sequence begun in EML3013. Keep the differential equations fresh: MAP2302 material is used continuously here, and students who treated it as a hurdle rather than a tool find this course substantially harder than it needs to be.