Math And Modeling

Reference notes on the math, model structure, balance laws, scaling, and energy-domain models used across engineering analysis.

Mathematical Foundations

Engineering analysis relies on algebra, calculus, linear algebra, differential equations, and numerical approximation.

Common mathematical objects:

Object	Engineering role
Scalar	single magnitude
Vector	magnitude and direction, state, coordinate, or stacked variables
Matrix	linear map, coordinate transform, system operator
Derivative	local rate of change or sensitivity
Integral	accumulation over time, space, area, or probability
Differential equation	rule for how a system evolves
Jacobian	first-order sensitivity matrix for vector functions
Hessian	second-order curvature matrix for scalar functions
Norm	magnitude measure for vectors, matrices, errors, or residuals

Linear systems are often written:

$$Ax=b$$

where $A$ maps unknowns $x$ to known quantities $b$.

Eigenvalues and eigenvectors satisfy:

$$Av=\lambda v$$

In engineering, eigenvalues commonly describe natural modes, stability, frequency, decay rate, stiffness direction, or matrix conditioning.

A first-order Taylor approximation is:

$$f(x+\mathrm{\Delta }x)\approx f(x)+\mathrm{\nabla }f(x{)}^T\mathrm{\Delta }x$$

A second-order approximation is:

$$f(x+\mathrm{\Delta }x)\approx f(x)+\mathrm{\nabla }f(x{)}^T\mathrm{\Delta }x+\frac{1}{2}\mathrm{\Delta }{x}^T{\mathrm{\nabla }}^{2}f(x)\mathrm{\Delta }x$$

Taylor approximations underlie linearization, uncertainty propagation, optimization, and numerical methods.

For a vector function $f(x)$, first-order linearization uses the Jacobian:

$$f(x+\mathrm{\Delta }x)\approx f(x)+J_f(x)\mathrm{\Delta }x$$

This links nonlinear models to local linear models. It appears in state-space linearization, Newton methods, Kalman-style estimation, sensitivity studies, and uncertainty propagation.

Conditioning describes how strongly input errors affect output errors. A linear solve can be mathematically correct but numerically fragile if the matrix is ill-conditioned. Scaling variables so their magnitudes are similar is often as important as the solver choice.

Modeling

An engineering model is a structured representation of a physical, computational, economic, or organizational system. Models are used to predict behavior, explain behavior, compare alternatives, size components, estimate uncertainty, or guide design decisions.

A model usually defines:

• system boundary
• inputs
• outputs
• states
• parameters
• governing equations
• initial conditions
• boundary conditions
• assumptions and validity range

Common model forms include algebraic equations, ordinary differential equations, partial differential equations, state-space models, transfer functions, empirical regressions, lookup tables, finite-element models, multibody models, optimization problems, and discrete-event simulations.

Anything outside the system boundary is treated as an input, ignored, or absorbed into a parameter.

Models can be physical, empirical, data-driven, or hybrid. A physical model uses conservation laws and constitutive relationships. An empirical model is fit to observed data. A useful engineering model often combines both.

Reduced-order models are useful only when they preserve the behavior relevant to the question being asked. Removing states, geometry, or nonlinear effects is not automatically wrong, but the reduced model must still reflect the original system over the claimed operating region.

Model fidelity is not a single ladder from bad to good. A low-order model can be the correct tool for architecture decisions, control intuition, or envelope studies. A high-fidelity model can still be misleading if its parameters, inputs, or validation evidence are weak.

Balance Laws

Many engineering models begin with a balance statement:

$$\frac{d}{dt}(\text{stored quantity})=\text{rate in}-\text{rate out}+\text{rate generated}$$

Examples include:

• mass balance
• linear momentum balance
• angular momentum balance
• energy balance
• charge balance
• species balance

The storage term usually determines the state. Forces, flows, heat rates, currents, torques, and reactions appear as terms that change stored quantity.

For a control volume, the generic mass balance is:

$$\frac{dm}{dt}={\dot{m}}_{\text{in}}-{\dot{m}}_{\text{out}}$$

For a rigid body, the fundamental linear momentum balance is:

$$\sum F=\frac{dp}{dt}$$

where $p$ is linear momentum. For constant mass with inertial-frame velocity, this reduces to:

$$\sum F=ma$$

The angular momentum balance is:

$$\sum M_O=\frac{d{H}_O}{dt}$$

where $H_O$ is angular momentum about point $O$. For rotation about a fixed principal axis with constant inertia, this reduces to:

$$\sum M=I\alpha$$

The main modeling task is identifying what is stored, what crosses the system boundary, and what is generated or dissipated inside the boundary.

Dimensional Analysis And Scaling

Dimensional analysis checks whether equations are physically consistent and identifies nondimensional groups that govern similarity.

Every term in an equation must have compatible dimensions. For example, in:

$$m\ddot{x}+b\dot{x}+kx=F$$

each term has units of force.

Nondimensionalization rewrites variables using characteristic scales. If $x_c$ is a characteristic length and $t_c$ is a characteristic time:

$$x^{\ast }=\frac{x}{x_c}$$$$t^{\ast }=\frac{t}{t_c}$$

This can reveal which effects dominate and which parameters can be grouped together.

Buckingham Pi analysis states that a physical relationship with $n$ variables and $k$ independent dimensions can be written using $n-k$ nondimensional groups.

Scaling matters when you transfer results between models, tests, simulations, and real systems.

Common nondimensional groups include:

Group	Typical role
Reynolds number	inertial versus viscous effects
Mach number	flow speed relative to sound speed
Froude number	inertial versus gravitational effects
Prandtl number	momentum diffusivity versus thermal diffusivity
Nusselt number	convection relative to conduction
Biot number	internal conduction resistance versus surface convection resistance

Nondimensional groups describe which physical similarity conditions must hold before one test, simulation, or scale model can support claims about another.

System Modeling And Energy Domains

System modeling organizes physical systems by energy storage, energy dissipation, and energy transfer.

Across domains, idealized systems can be described by effort and flow:

Domain	Effort	Flow
Translational mechanical	force	velocity
Rotational mechanical	torque	angular velocity
Electrical	voltage	current
Hydraulic	pressure	volumetric flow
Thermal	temperature	heat or entropy flow

Power is the product of effort and flow:

$$P=ef$$

State-space form describes system evolution:

$$\dot{x}=f(x,u,t)$$$$y=g(x,u,t)$$

For linear time-invariant systems:

$$\dot{x}=Ax+Bu$$$$y=Cx+Du$$

The state vector should contain enough information to determine future system behavior when inputs are known.

Bond graphs, block diagrams, free-body diagrams, circuit diagrams, and state-space models are different representations of the same underlying task: track storage, dissipation, transformation, and input-output behavior.

Across energy domains, ideal elements tend to fall into storage, dissipation, source, and transformation categories:

Role	Meaning	Examples
Inertial storage	stores kinetic-like energy	mass, rotational inertia, fluid inertance, inductor
Compliant storage	stores potential-like energy	spring, capacitor, hydraulic accumulator, thermal capacitance
Dissipation	removes usable energy	damper, resistor, flow restriction, friction
Source	imposes effort or flow	voltage source, force input, pump, heat input
Transformer	changes effort-flow ratio	lever, gear, transformer, piston area
Gyrator	swaps effort-flow relationship	motor, generator, electromechanical transducer

A model can imply impossible information flow. For example, commanding both force and displacement of an ideal mass without a compliance or constraint model usually over-specifies the system.

Electrical And Hydraulic Systems

Electrical and hydraulic systems often behave like energy-domain analogs of mechanical systems.

Electrical ideal elements:

Element	Law
Resistor	$v=Ri$
Capacitor	$i=C\dot{v}$
Inductor	$v=L\dot{i}$

Kirchhoff's current law states that current sums to zero at a node. Kirchhoff's voltage law states that voltage sums to zero around a loop. These are charge and energy conservation statements in circuit form.

An RC circuit has a characteristic time constant:

$$\tau =RC$$

An RL circuit has:

$$\tau =\frac{L}{R}$$

Hydraulic analogs:

Element	Typical relation
Resistance	$\mathrm{\Delta }p=RQ$
Compliance	$Q=C\dot{p}$
Inertance	$\mathrm{\Delta }p=I\dot{Q}$

Hydraulic power is:

$$P=pQ$$

Electrical power is:

$$P=vi$$

Electrical and hydraulic systems are frequently coupled to mechanical systems through motors, pumps, pistons, valves, actuators, sensors, and controllers. Important assumptions include linearity, leakage, compressibility, saturation, deadband, inductance, capacitance, and actuator bandwidth.

A common ideal orifice relation is:

$$Q=C_{d}A\sqrt{\frac{2\mathrm{\Delta }p}{\rho }}$$

This square-root relation is nonlinear, so hydraulic actuator behavior can change strongly with pressure, valve opening, fluid density, and operating point.

Hydraulic compliance can come from fluid compressibility, hose expansion, trapped air, accumulator behavior, or structural flexibility. Ignoring compliance can make simulated actuators unrealistically stiff and instantaneous.