2.1 Mechanical systems

Mechanical systems form the foundation of dynamic system modeling in control theory. Before you can design a controller for anything physical, you need a mathematical model that captures how forces, masses, and motion interact. This unit focuses on translating real mechanical components into equations you can analyze and control.

Types of mechanical systems

Mechanical systems involve the motion, forces, and energy of solid bodies. Classifying them helps you choose the right modeling approach.

• Translational systems move along a straight line (linear motion)
• Rotational systems rotate about an axis (angular motion)
• Coupled systems combine translational and rotational motion
• Continuous systems have distributed mass and elasticity, like beams and plates

For this unit, you'll mostly work with translational and rotational systems. Coupled systems show up when, for example, a gear converts rotation into linear motion. Continuous systems require partial differential equations and are typically covered in more advanced courses.

Components of mechanical systems

Every mechanical system model is built from a small set of fundamental elements. Identifying these components in a real system is the first step toward writing its equations of motion.

Masses and inertias

Mass resists changes in translational motion. For a translational system, the force required to accelerate a mass $m$ is $F = m\ddot{x}$, where $\ddot{x}$ is acceleration.

For rotational systems, the equivalent property is the moment of inertia $J$, which resists changes in angular motion. The torque required to produce angular acceleration is $\tau = J\ddot{\theta}$.

A flywheel, for instance, has a large moment of inertia, which is why it stores rotational energy effectively and resists changes in speed.

Springs and stiffnesses

Springs are elastic elements that store potential energy and produce a restoring force proportional to their deformation. For a linear spring with stiffness $k$:

• Translational: $F = kx$ (force proportional to displacement)
• Rotational: $\tau = k\theta$ (torque proportional to angular displacement)

The higher the stiffness $k$, the more force is needed to deform the spring by a given amount. A vehicle suspension spring, for example, must be stiff enough to support the car's weight but compliant enough to absorb road bumps.

Dampers and viscous friction

Dampers are dissipative elements that convert kinetic energy into heat. A viscous damper produces a force proportional to velocity:

• Translational: $F = c\dot{x}$
• Rotational: $\tau = c\dot{\theta}$

Here $c$ is the damping coefficient. Shock absorbers in a car suspension are classic examples: they oppose the velocity of the suspension's motion, which is what causes oscillations to die out over time rather than continuing indefinitely.

Modeling of mechanical systems

Modeling means translating a physical system into mathematical equations. For mechanical systems, this process relies on Newton's laws and a systematic approach using free body diagrams.

Newton's laws of motion

Three laws underpin all mechanical system modeling:

1. First law (inertia): A body remains at rest or in uniform motion unless acted on by a net external force.
2. Second law: $F = ma$ for translation; $\tau = J\alpha$ for rotation. This is the law you'll use most when writing equations of motion.
3. Third law (action-reaction): Forces between two interacting bodies are equal in magnitude and opposite in direction.

Newton's second law is the workhorse here. For every mass or inertia in your system, you'll write a second-law equation that sums all forces (or torques) and sets them equal to $ma$ (or $J\alpha$).

Free body diagrams

A free body diagram (FBD) isolates a single body and shows every force acting on it. This is the most important step in deriving equations of motion, and skipping it is where most mistakes happen.

To draw an FBD:

1. Isolate the body of interest from the rest of the system.
2. Identify every external force and moment acting on it: gravity, spring forces, damper forces, applied forces, and constraint/reaction forces.
3. Draw each force as a vector with its correct direction. Spring forces oppose displacement from equilibrium; damper forces oppose velocity.
4. Choose a positive direction for motion and be consistent.

Once the FBD is complete, apply Newton's second law by summing forces (or torques) in the positive direction and setting the sum equal to $m\ddot{x}$ (or $J\ddot{\theta}$).

Equations of motion

The equation of motion describes the relationship between a system's displacement, velocity, acceleration, and the applied forces. For a standard mass-spring-damper system, the process yields:

$m\ddot{x} + c\dot{x} + kx = F(t)$

where:

• $m$ = mass
• $c$ = damping coefficient
• $k$ = spring stiffness
• $x$ = displacement from equilibrium
• $F(t)$ = external applied force

This is a second-order linear ODE. The $m\ddot{x}$ term comes from the mass (Newton's second law), the $c\dot{x}$ term from the damper, and the $kx$ term from the spring. Notice that the spring and damper forces appear on the left side because they oppose motion.

For multi-degree-of-freedom systems, you get one equation per degree of freedom, and these equations are often coupled. They're typically written in matrix form:

$M\ddot{\mathbf{x}} + C\dot{\mathbf{x}} + K\mathbf{x} = \mathbf{F}(t)$

where $M$, $C$, and $K$ are the mass, damping, and stiffness matrices.

Linear vs nonlinear systems

A system is linear if its equation of motion satisfies two properties:

• Superposition/Additivity: The response to $F_1 + F_2$ equals the response to $F_1$ plus the response to $F_2$.
• Homogeneity: Scaling the input by a factor $\alpha$ scales the output by the same factor $\alpha$.

The mass-spring-damper equation above is linear because $m$, $c$, and $k$ are constants and all terms in $x$ and its derivatives appear to the first power.

Nonlinear systems violate one or both of these properties. They can exhibit behaviors that linear systems cannot, such as multiple equilibrium points, limit cycles, and chaos.

A classic example: a pendulum. For small angles, $\sin\theta \approx \theta$, so the equation $J\ddot{\theta} + c\dot{\theta} + mg\ell\theta = 0$ is linear. For large angles, you must keep the $\sin\theta$ term, making the system nonlinear. Most of the analysis tools in this course (transfer functions, Bode plots, root locus) apply only to linear systems, which is why linearization around an operating point is such a common technique.

Single vs multi-degree-of-freedom systems

The degrees of freedom (DOF) of a system is the minimum number of independent coordinates needed to fully describe its motion.

• SDOF systems need just one coordinate. A single mass on a spring is SDOF: you only need $x(t)$.
• MDOF systems need two or more coordinates. A double pendulum needs two angles, $\theta_1(t)$ and $\theta_2(t)$.

SDOF systems produce a single ODE. MDOF systems produce a set of coupled ODEs, one per degree of freedom. The coupling means that motion in one coordinate affects the others, which makes analysis more involved but also more realistic for complex systems.

Time domain analysis

Time domain analysis studies how a system's output evolves over time in response to inputs or initial conditions.

Transient response

The transient response is the system's behavior during the initial period after a disturbance, before it settles. Key metrics include:

• Rise time: How quickly the output reaches the desired value for the first time.
• Overshoot: How far the output exceeds the desired value (expressed as a percentage).
• Settling time: How long it takes for the output to stay within a specified band (typically 2% or 5%) of the final value.

These metrics depend heavily on the system's natural frequency $\omega_n = \sqrt{k/m}$ and damping ratio $\zeta = c / (2\sqrt{mk})$. A higher damping ratio reduces overshoot but increases rise time. This tradeoff is central to control design.

Steady-state response

After the transient dies out, the system reaches its steady-state response. Here you care about:

• The final value of the output
• The steady-state error (difference between the desired and actual output)
• Whether any sustained oscillation remains

For a step input applied to a stable system, the steady-state response is the value the output converges to as $t \to \infty$. Steady-state error is a major concern in control design and is one of the main reasons for adding integral action to a controller.

Frequency domain analysis

Frequency domain analysis examines how a system responds to sinusoidal inputs at different frequencies. This perspective is powerful because any input signal can be decomposed into sinusoids (via Fourier analysis), so knowing the frequency response tells you how the system handles any input.

Transfer functions

A transfer function relates a system's output to its input in the Laplace domain, assuming zero initial conditions. For the mass-spring-damper system:

$G(s) = \frac{X(s)}{F(s)} = \frac{1}{ms^2 + cs + k}$

You obtain this by taking the Laplace transform of the equation of motion $m\ddot{x} + c\dot{x} + kx = F(t)$ and solving for the ratio $X(s)/F(s)$.

The transfer function captures everything about the system's input-output behavior: its poles (roots of the denominator) determine stability and transient behavior, while its zeros (roots of the numerator) shape the frequency response.

Frequency response

The frequency response is what you get when you substitute $s = j\omega$ into the transfer function and evaluate the magnitude and phase as $\omega$ varies. It's typically displayed on a Bode plot:

• Magnitude plot: $|G(j\omega)|$ in decibels vs. frequency on a log scale
• Phase plot: $\angle G(j\omega)$ in degrees vs. frequency on a log scale

From a Bode plot, you can read off the system's bandwidth, gain margin, phase margin, and resonance peak, all of which are critical for control design.

Resonance and anti-resonance

Resonance occurs when the input frequency matches one of the system's natural frequencies. At resonance, the output amplitude is amplified dramatically. The amount of amplification depends on the damping ratio: low damping means a tall, sharp resonance peak.

The Tacoma Narrows Bridge collapse in 1940 is a famous example of resonance: wind-induced oscillations matched the bridge's natural frequency, causing catastrophic failure.

Anti-resonance occurs at frequencies where the transfer function has a zero, causing the output to be strongly attenuated. Tuned mass dampers (vibration absorbers) exploit this: they're designed so that their anti-resonance frequency matches the problematic vibration frequency of the primary structure.

Stability of mechanical systems

A system is stable if its output remains bounded for any bounded input (this is called BIBO stability). Equivalently, a linear system is stable if all poles of its transfer function have negative real parts (they lie in the left half of the complex plane).

An unstable system has at least one pole with a positive real part, causing the response to grow without bound. Stability analysis is non-negotiable before implementing any controller.

Routh-Hurwitz criterion

The Routh-Hurwitz criterion lets you determine stability from the characteristic polynomial without actually solving for the roots. The procedure:

1. Write the characteristic equation: $a_ns^n + a_{n-1}s^{n-1} + \cdots + a_1s + a_0 = 0$
2. Construct the Routh array by arranging coefficients in rows.
3. Check the first column of the array. The system is stable if and only if all entries in the first column are positive (same sign).
4. The number of sign changes in the first column equals the number of roots in the right half-plane.

Example: For $s^3 + 2s^2 + 3s + 4 = 0$:

The $s^1$ row entry is computed as $(2 \times 3 - 1 \times 4)/2 = 1$. All first-column entries are positive, so the system is stable.

Root locus technique

The root locus plots the trajectories of a closed-loop system's poles in the complex plane as a parameter (usually the gain $K$) varies from 0 to $\infty$.

This is useful because it shows you directly:

• At what gain the system becomes unstable (poles cross the imaginary axis)
• How the damping ratio and natural frequency change with gain
• Where to place the gain for desired transient performance

For a second-order system with unity feedback, the root locus starts at the open-loop poles (when $K = 0$) and moves toward the open-loop zeros (as $K \to \infty$). The portions of the locus in the left half-plane correspond to stable operating conditions.

Control of mechanical systems

Control design uses the mathematical model you've built to regulate the system's behavior. The goal is to make the system track a desired reference, reject disturbances, and remain stable despite model uncertainties.

Feedback control

In feedback (closed-loop) control, you measure the output, compare it to the reference, and use the error to generate the control signal. This is fundamentally different from open-loop control, where the control signal is predetermined and doesn't account for what the system is actually doing.

Feedback provides:

• Reduced sensitivity to disturbances and model errors
• The ability to stabilize an otherwise unstable system
• Improved transient and steady-state performance

PID controllers

The PID controller is the most widely used controller in industry. It computes the control signal $u(t)$ as:

$u(t) = K_p e(t) + K_i \int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}$

Each term addresses a different aspect of performance:

• Proportional ($K_p$): Responds to the current error. Higher $K_p$ reduces steady-state error but can increase overshoot.
• Integral ($K_i$): Responds to accumulated past error. Eliminates steady-state error for constant references but can cause overshoot and slow oscillations if set too high.
• Derivative ($K_d$): Responds to the rate of change of error. Provides a damping effect that reduces overshoot and improves settling time, but amplifies high-frequency noise.

Tuning means selecting $K_p$, $K_i$, and $K_d$ to meet your performance specs. Common tuning methods include Ziegler-Nichols and trial-and-error with simulation.

State-space control

State-space representation models the system as a set of first-order ODEs:

$\dot{\mathbf{x}} = A\mathbf{x} + B\mathbf{u}$ $\mathbf{y} = C\mathbf{x} + D\mathbf{u}$

where $\mathbf{x}$ is the state vector, $\mathbf{u}$ is the input, and $\mathbf{y}$ is the output. For the mass-spring-damper, you'd define states $x_1 = x$ (position) and $x_2 = \dot{x}$ (velocity), converting the single second-order ODE into two first-order ODEs.

State-space control designs a feedback law $\mathbf{u} = -K\mathbf{x}$ that places the closed-loop eigenvalues (poles) at desired locations. When not all states are measurable, an observer estimates them from the available outputs. This approach handles MIMO systems naturally and connects to advanced methods like optimal control (LQR) and robust control.

Applications of mechanical systems

Robotics and mechatronics

Robots are modeled as multi-DOF mechanical systems with coupled translational and rotational dynamics. Accurate dynamic models are essential for trajectory tracking and force control. Applications range from industrial manipulators on assembly lines to autonomous drones and surgical robots.

Automotive systems

Vehicle dynamics involve multiple interacting mechanical subsystems: engine, transmission, suspension, steering, and brakes. Active suspension systems, for example, use feedback control to adapt damping in real time based on road conditions. Regenerative braking in electric vehicles recovers kinetic energy during deceleration by running the motor as a generator.

Aerospace systems

Aircraft and spacecraft operate under extreme conditions and tight performance requirements. Control surfaces (ailerons, elevators, rudders) are actuated mechanical systems governed by the same modeling principles covered here. Spacecraft attitude control uses reaction wheels and thrusters, modeled as rotational mechanical systems with applied torques.

Simulation of mechanical systems

Simulation uses computational tools to solve the equations of motion numerically, allowing you to predict system behavior without building a physical prototype. This is especially valuable for testing control designs before implementation.

Numerical methods

When equations of motion can't be solved analytically (which is most of the time for real systems), you use numerical integration. Common methods include:

• Euler's method: Simple but can be inaccurate for large time steps. Updates the state as $x(t + \Delta t) \approx x(t) + \dot{x}(t)\Delta t$.
• Runge-Kutta methods (e.g., RK4): More accurate than Euler for the same step size. RK4 evaluates the derivative at multiple points within each time step to improve accuracy.
• Variable-step solvers: Automatically adjust the time step to balance accuracy and computational cost. Tools like MATLAB's ode45 use this approach.

Choosing the right solver and step size matters. Too large a step and you get inaccurate results; too small and the simulation takes unnecessarily long. For stiff systems (where dynamics occur on very different time scales), specialized solvers like ode15s are needed.