MAE 3319 Appendix J

Questions and Answers from Modeling and Simulations of Dynamic Systems (Robert L. Woods & Kent L. Lawrence)

What are the purposes of modeling, and what can we do with a model of a dynamic system?

For Understanding, Predicting, Design and Control. Some things we can do is simulations, analysis, optimization, training and cost reduction.

Overall is a safe way to do virtual experiments cheaply

What is the functional definition of linearity?

Must be the following:

1. Additivity f(x1+x2)=f(x1)+f(x2)

2. Homogeneity f(ax)=a*f(x)

What are the so-called elementary inputs that are used as inputs to dynamic systems? Are there other typical inputs?

Step Input:

• u(t)=0 for t<0

• Sudden Constant Change

Impulse Input:

• δ(t)=infinite at t=0, zero elsewhere

• Instantaneous spike

Ramp Input:

• u(t)=At for t>=0

• Linearly increasing

Sinusoidal Input:

• u(t)=Asin(wt)

• Periodic oscillation

They reveal fundamental system characteristics that predict response to any input.

Yes there are other input type’s such as exponential, polynomial, random etc.

The solution to a non-homogenous differential equation is composed of two parts. What are they and how are they found?

Made up of a homogeneous solution and a Particular solution.

For the homogeneous solution solve characteristic equations and form solution from the roots.

For the particular solution use the method of undetermined coefficients, apply initial conditions to find constants

Discuss the concept of the time constant of a first-order differential equation. How can the time constant be determined from the linear differential equation? How can the time constant be determined from the time response of the system?

Measures how fast a first-order system responds, it the time for the system to reach 63% of final value, characterizes exponential decay/rise rate.

Solve the standard form (τD+1)x=Ku(t)

Find the time it takes the system to reach 63% of steady-state value

There are two performance factors for a first-order differential equation. What are they? Explain what each factor means in terms of the response of the sytsem.

Time constant (τ):

• Speed of response

• Large τ = slow

Steady-State Gain (K):

• Final output magnitude relative to input

• K>1 amplified input

• Controls final value system reaches

Standard form τ(D+1)=Ku(t), K determines final height, τ determines how fast.

There are three performance factors for a second-order differential equation. What are they? Explain what each factor means in terms of the response of the system.

Natural Frequency (ωₙ):

• Fundamental oscillation frequency of system

• Higher ωₙ = faster response, higher frequency oscillations

• Controls overall speed of system

Damping Ratio (ζ):

• Amount of damping relative to critical damping:

• ζ<1 Underdamped (Oscillatory)

• ζ=1 Critically damped (fastest without overshoot)

• ζ>1 Over-damped (Slow, no oscillation)

Steady-State Gain (K):

• Final output magnitude relative to input

• Controls final value system reaches

Standard form [(1/ωₙ²)D²+(2ζ/ωₙ)D+1]x=Ku(t), ωₙ sets the speed, ζ sets oscillation, K sets final magnitude

What are the characteristics of an undamped, an underdamped, a critically damped, and an over-damped system?

Undamped (ζ=0):

• Pure oscillation at wn, never settles

Underdamped (ζ<1):

• Oscillatory with exponential decay

Critically damped (ζ=1):

• Fastest approach to steady-state without overshoot

Over-damped (ζ>1):

• Slow, exponential approach, no oscillation

Explain what is meant by the systems approach in the treatment of dynamic systems.

Treating complex systems as interconnected components with defined inputs, outputs, and internal relationships, rather than analyzing individual parts in isolation.

Ex: Setting the car as the system vs. studying the engine, transmission, brakes separately

Discuss the meaning of effort and flow variables for dynamic systems. Mention the effort and flow variables in electrical, mechanical, fluid, and thermal systems.

Effort: Driving Force that causes flow

Flow: Rate of transfer through system

Power = Effort x Flow

Electrical:

Effort: Voltage (V)

Flow: Current (I)

Mechanical (linear):

Effort: Force (F)

Flow: Velocity (v)

Mechanical (rotational):

Effort: Torque (T)

Flow: Angular Velocity (ω)

Fluid:

Effort: Pressure (P)

Flow: Flow Rate (Q)

Thermal:

Effort: Temperature difference (ΔT)

Flow: Heat flow rate (q)

Explain the characteristics of the three types of fundamental components that are the building blocks of all systems.

1. Energy storage elements

2. Energy dissipation elements

3. Energy sources

What are examples of each type of component in electrical, mechanical, fluid, and thermal systems?

Electrical:

• Energy storage: Capacitor, Inductor

• Energy dissipation: Resistor

• Energy source: Voltage/Current source

Mechanical:

• Energy storage: Spring, Mass

• Energy dissipation: Damper, Friction

• Energy source: Force/Velocity source, Motor

Fluid:

• Energy storage: Fluid mass

• Energy dissipation: Flow restriction (valve), Friction

• Energy source: Pressure source, pump

Thermal:

• Energy storage: Thermal capacitance

• Energy dissipation: Thermal resistance (conduction, convection, radiation)

• Energy source: Temperature source

Discuss the different types of friction in mechanical systems.

Static Friction (At rest but with applied force, gradually increases with increased force until motion)

Kinematic/Dynamic Friction (In motion, always opposes motion)

Viscous Friction (Motion through fluid, linear relationship with force)

Coulomb Friction (Dry sliding between surfaces)

What is an op-amp, and how can it be used in electronic circuits?

A high-gain differential amplifier with very high input impedance and very low output impedance.

1. Sign Change

2. Differentiator

3. Amplifier

4. Integrator

5. Lag

What different types of fluid resistance are there, and what is the energy loss mechanism in each?

1. Laminar Flow Resistance, Viscous shear between fluid layers causes energy loss

2. Turbulent Flow Resistance, Turbulent mixing, eddies, chaotic motion causes energy loss

3. Orifice/Restriction Resistance, Vena contracta formation, jet expansion causes energy loss in system

4. Entrance/Exit Losses, Flow acc/decelerating, vortex formation causes energy loss

What types of fluid capacitance are there, and what causes each?

1. Gravitational Capacitance, Gravitational potential energy stores energy

2. Elastic Capacitance (Compressibility), Elastic deformation energy can be stored

What is a common explanation for the effect of fluid inductance?

Commonly known as Fluid Inertia, caused by moving fluid which has momentum, kinetically stores energy

What are resistance and capacitance in a thermal system?

Resistance:

• Conduction, Convection, Radiation

• Controls heat transfer rate at steady state

Capacitance:

• Ability to store thermal energy

• Controls temperature change rate

What is a good way to confirm that en equation has been derived using consistent terms?

Dimensional Analysis (Unit Analysis):

• Identify units of each term and confirm consistency

What are the general forms of the Laplace transform of a first-, second-, and higher order derivative?

First Order:

L{y′(t)}=sY(s)−y(0)

Second Order:

L{y′′(t)}=s²Y(s)−sy(0)−y′(0)

Higher Order:

L{y⁽ⁿ⁾(t)}=sⁿY(s)−sⁿ⁻¹y(0)−sⁿ⁻²y′(0)−⋯−y⁽ⁿ⁻¹⁾(0)

Discuss the form of the Laplace transform of a differential equation.

Laplace transform converts a linear differential equation in the time domain t into an algebraic equation in the s-domain

For an n-th order linear ODE with constant coefficients:

a_n​y⁽ⁿ⁾(t)+a_n−1​y⁽ⁿ⁻¹⁾(t)+⋯+a₀​y(t)=f(t)

the Laplace transform yields:

a_n​[sⁿY(s)−∑s^n−1−ky^(k)(0)]+⋯+a_0​Y(s)=F(s)

• Differential operators D become polynomials s

• Initial conditions are automatically incorporated

• The transformed equation is algebraic, making it easier to solve for Y(s)

How can the time domain solution of a differential equation be found by the Laplace transform technique?

1. Take the Laplace transform of the ODE, forcing functions f(t) → F(s)

2. Substitute initial conditions

3. Solve for Y(s)

4. Perform Partial Fraction Decomposition (if needed), breaking Y(s) into simpler terms

5. Take the inverse Laplace Transform, converting back into the t-domain

Discuss what is meant by partial fractions and why partial fraction expansion is necessary in the solution of differential equations.

Partial fraction expansion decomposes a fraction with polynomials, into a sum of simpler fractions with denominators of lower degree

Useful for getting terms in the s-domain into simpler functions that we can relate to already solved equations

What is the transfer function of a dynamic system? How is it usually expressed?

The transfer function G(s) of a linear time-invariant system is:

• The ratio of the Laplace transform of the output Y(s) to the Laplace transform of the input U(s)

G(s)=Y(s)/U(s)

How are state variables defined to convert a classical linear nth-order differential equation to a state space equation without input derivatives?

Convert the following:

y⁽ⁿ⁾+a_n−1​y⁽ⁿ⁻¹⁾+⋯+a₀​y=u

into state-space by defining the state variables as the output y(t) and its derivatives:

x₁=y

x₂=y’

…

x_n=y^(n-1)

and represent using matrix notation.

How are state variables defined to convert a classical differential equation to state-space form with input derivatives? How does this affect the inputs to the derivatives of the state variables?

y⁽ⁿ⁾+a_n−1​y⁽ⁿ⁻¹⁾+⋯+a₀​y=b_m​u^(m)+⋯+b₀​u (m<n)

To convert the above into state-space form, define state variables to absorb the input derivatives:

x₁=y-β₀u

x₂=y’-β₀u’-β₁u

…

x_n=y^(n-1)-β₀u^(n-1)-…-β_n-1u

Is the state-space representation of dynamic systems limited to linear systems? If a system is nonlinear, can it be represented in state-space format? Can a nonlinear system always be reduced to a classical differential equation?

• State space representation is not limited to linear systems.

• Nonlinear systems can be represented in state-space form

• Nonlinear systems can not always be written as a classical ODE

Describe how a state-space differential equation representation can be derived directly from systems circuit modeling equations or from engineering modeling equations.

1. Choose state variables

   1. Capacitor Voltages/Inductor currents

   2. Mechanical Positions x, velocities x’

2. Write dynamic equations

   1. KVL/KCL, C(dV_c/dt)=I, L(dI_L/dt)=V

   2. Newton’s Laws, mx’’=F

3. Solve for derivatives and express as:

x’=f(x,u)

4. Output equation: y=measured quantities

Discuss the general approach to solving, and the format of, the equations used in the numerical integration of a first order differential equation on the digital computer. Name some of the integration methods that could be used.

1. Discretize time into steps

2. Update state x(t) iritevely

   f(x)=f(x)+f(x-1)

Common methods:

• Euler (Simple, Low Accuracy)

• Runge-Kutta (4th Order, Higher accuracy)

• Implicit Methods (Stable for stiff equations)

• Adaptive Step (RK45 adjusted for delta t error control)

What trade-offs are involved in selecting the step size for numerical integration on the digital computer? Are there limitations on how far the trade-offs can be taken?

• Speed vs. Accuracy (Smaller step size means more precise but is slower to compute)

Limitations include:

• Machine precision (Rounding errors)

• Stability

• Real-Time Needs

Discuss the concepts and equations used to select an appropriate step size for Runge-Kutta integration. How would the final time and the print interval be selected?

From textbook

Discuss and compare the time domain solutions of dynamic systems using analytical techniques and digital simulation techniques. Under what conditions would analytical solutions be more desirable than simulation solutions and vice versa?

Analytical Solutions:

• Precise, reveals system properties, no discretization errors

• Limited to linear/time-invariant systems, hard for nonlinear/high order systems

Digital Simulation:

• Approximate step-by-step integration

• Handles nonlinear and complex systems, flexible for real-world applications

• Approximation errors and computationally expensive for small step size.