2.1 Concept and representation of single degree-of-freedom systems

Single Degree-of-Freedom Systems

Single degree-of-freedom (SDOF) systems are the foundation of vibration analysis. They describe a system's motion using just one independent coordinate, built from three core elements: mass, a spring, and a damper. Nearly every complex vibration problem starts by understanding SDOF behavior first, so getting comfortable here pays off throughout the course.

The equation of motion for an SDOF system is a second-order differential equation that ties together displacement, velocity, and acceleration. Two parameters define how the system behaves: natural frequency (how fast it wants to oscillate) and damping ratio (how quickly oscillations die out). These concepts show up everywhere, from simple pendulums and car suspensions to seismic analysis of buildings.

Single Degree-of-Freedom Systems

Fundamentals of SDOF Systems

An SDOF system is any system whose configuration can be fully described by a single coordinate or variable. Think of a mass bouncing on a spring: you only need the vertical position to know everything about the system's state.

Every SDOF system has three key components:

• Mass element stores kinetic energy and resists acceleration.
• Spring element stores potential energy and provides a restoring force.
• Damping element dissipates energy (usually as heat), causing oscillations to decay over time.

Natural frequency is set by the system's mass and stiffness. Higher stiffness pushes the natural frequency up, while higher mass brings it down. The damping ratio controls how the system loses energy: a low damping ratio means oscillations persist for a long time, while a high damping ratio causes motion to die out quickly.

Based on the damping ratio $\zeta$, motion falls into one of three categories:

• Underdamped ($\zeta < 1$): The system oscillates with gradually decreasing amplitude. This is the most common case in practice.
• Critically damped ($\zeta = 1$): The system returns to equilibrium as fast as possible without oscillating. This is the boundary case.
• Overdamped ($\zeta > 1$): The system returns to equilibrium slowly, with no oscillation at all.

System response is also categorized by what causes the motion:

• Free response: Motion caused by initial conditions alone (an initial displacement or velocity), with no ongoing external force.
• Forced response: Motion driven by an external excitation such as a periodic force or an impulse.

Mathematical Representation of SDOF Systems

The governing equation of motion for an SDOF system is:

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

where:

• $m$ = mass
• $c$ = damping coefficient
• $k$ = spring stiffness
• $x$ = displacement (with $\dot{x}$ = velocity, $\ddot{x}$ = acceleration)
• $F(t)$ = external force as a function of time

Each term has a physical meaning. The term $m\ddot{x}$ is the inertial force, $c\dot{x}$ is the damping force resisting velocity, and $kx$ is the spring's restoring force. Together, they must balance whatever external force $F(t)$ is applied.

Natural frequency:

$\omega_n = \sqrt{\frac{k}{m}}$

This is the frequency at which the system would oscillate if there were no damping and no external force.

Damping ratio:

$\zeta = \frac{c}{2\sqrt{km}}$

This dimensionless number compares the actual damping to the critical damping value $c_{cr} = 2\sqrt{km}$.

General solution for underdamped free vibration ($\zeta < 1$):

$x(t) = Ae^{-\zeta\omega_n t} \cos(\omega_d t + \phi)$

• $A$ = amplitude (determined by initial conditions)
• $\omega_d = \omega_n\sqrt{1 - \zeta^2}$ = damped natural frequency
• $\phi$ = phase angle (also determined by initial conditions)

Notice that the exponential term $e^{-\zeta\omega_n t}$ creates a decaying envelope around the cosine oscillation. The damped natural frequency $\omega_d$ is always less than $\omega_n$ because damping slows the oscillation slightly.

SDOF Systems in Applications

Real-World Examples of SDOF Systems

Many physical systems can be modeled as SDOF when one mode of motion dominates:

• Simple pendulum: The angle $\theta$ is the single coordinate. Used in clocks and seismometers, where predictable oscillation matters.
• Mass on a vertical spring: A classic vibration isolation setup. Vehicle seats and sensitive equipment mounts use this principle to filter out unwanted vibrations.
• Car bouncing on its suspension: The vertical motion of the car body, with the suspension acting as the spring-damper combination. Engineers use this model to design comfortable ride characteristics.
• Single-story building under earthquake loading: Horizontal ground motion drives the structure, and the building's columns act as the spring. This is the starting point for seismic analysis and design.
• Torsional vibration of a shaft with a single disk: The disk's angular displacement is the single coordinate. Critical in rotating machinery like turbines and generators, where torsional resonance can cause failure.
• Tuned mass damper in tall buildings: A large mass on springs placed near the top of a structure to counteract wind-induced sway. Taipei 101's 730-ton pendulum is a well-known example.
• Floating buoy in calm water: Vertical bobbing motion approximates SDOF behavior. Used in oceanographic studies and wave energy converters.

SDOF Systems in Engineering Design

SDOF models guide the design of many practical vibration-control devices:

• Vibration isolators for sensitive equipment (electron microscopes, precision CNC machines) reduce transmitted vibrations from the surrounding environment by tuning the system's natural frequency well below the excitation frequency.
• Shock absorbers in vehicles provide damping that controls body motion after bumps, balancing ride comfort against handling performance.
• Seismic base isolation systems decouple a building from ground motion by introducing a flexible layer with a very low natural frequency, so earthquake energy doesn't pass into the structure efficiently.
• Dynamic vibration absorbers in power transmission systems attach a tuned mass-spring to a rotating shaft, targeting a specific problematic frequency to suppress harmful vibrations.
• Tuned liquid dampers in structures like water towers use the sloshing of a fluid mass to counteract wind-induced oscillations.
• Mass dampers in sports equipment (tennis rackets, golf clubs) reduce vibration transmitted to the user's hand, improving comfort and control.

Spring-Mass-Damper Models for SDOF

Components of Spring-Mass-Damper Model

The spring-mass-damper model is the standard physical representation of an SDOF system. Each component maps to a specific energy role:

Mass element represents the system's inertia. It's depicted as a rigid block or point mass and determines the kinetic energy: $T = \frac{1}{2}m\dot{x}^2$.

Spring element represents stiffness and provides the restoring force. A linear spring follows Hooke's Law:

$F = kx$

It stores potential energy: $U = \frac{1}{2}kx^2$.

Damper element represents energy dissipation. Drawn as a dashpot, a linear viscous damper produces a force proportional to velocity:

$F = c\dot{x}$

This force always opposes the direction of motion, converting kinetic energy into heat.

To derive the equation of motion, you draw a free-body diagram of the mass showing all forces acting on it:

1. Displace the mass by $x$ from equilibrium in the positive direction.
2. The spring pulls back with force $kx$ (opposing displacement).
3. The damper resists with force $c\dot{x}$ (opposing velocity).
4. Any external force $F(t)$ acts on the mass.
5. Apply Newton's second law: $m\ddot{x} = F(t) - kx - c\dot{x}$, which rearranges to the standard form $m\ddot{x} + c\dot{x} + kx = F(t)$.

Advanced Spring-Mass-Damper Models

Real systems don't always behave linearly. Several extensions of the basic model handle more realistic situations:

Nonlinear springs account for large displacements where Hooke's Law breaks down. The force-displacement relationship becomes:

$F = kx + k_2x^2 + k_3x^3$

The higher-order terms capture stiffening or softening behavior depending on their signs.

Combining multiple springs:

• Springs in series (end to end): $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}$. The equivalent stiffness is always less than the smallest individual spring.
• Springs in parallel (side by side): $k_{eq} = k_1 + k_2$. The equivalent stiffness is the sum of the individual stiffnesses.

Alternative damping mechanisms beyond viscous damping:

• Coulomb damping (dry friction): The damping force has constant magnitude $\mu N$ and always opposes the velocity direction: $F = \mu N \cdot \text{sign}(\dot{x})$. This produces a linearly decaying envelope rather than an exponential one.
• Structural (hysteretic) damping: Energy loss is proportional to displacement amplitude but independent of frequency. Often represented using a complex stiffness.

Rotational SDOF systems follow the same structure but use angular quantities:

• Torsional spring: $T = k_\theta \theta$
• Rotational damper: $T = c_\theta \dot{\theta}$
• The equation of motion becomes $J\ddot{\theta} + c_\theta\dot{\theta} + k_\theta\theta = M(t)$, where $J$ is the mass moment of inertia and $M(t)$ is the applied torque.

Degrees of Freedom for Systems

Determining Degrees of Freedom

The number of degrees of freedom (DOF) equals the minimum number of independent coordinates needed to completely describe a system's configuration at any instant.

A free rigid body in 3D space has a maximum of six DOF:

• Three translational ($x$, $y$, $z$)
• Three rotational (roll, pitch, yaw)

Constraints reduce this number. Each constraint equation removes one DOF:

• A fixed support removes all DOF at that point.
• A pin joint allows rotation but restricts translation, removing translational DOF while keeping rotational ones.

To determine the DOF of a system:

1. Count the total possible independent motions for all bodies in the system.
2. Identify all constraints (supports, joints, connections).
3. Subtract the number of independent constraint equations from the total motions.

For planar (2D) problems, a single free rigid body has three DOF: two translational ($x$, $y$) and one rotational ($\theta$). For systems with multiple bodies, sum the individual DOF and then subtract constraints between bodies.

Examples of Degrees of Freedom Analysis

Working through examples is the best way to build intuition for counting DOF:

• Particle on a straight line (1 DOF): Only the $x$-coordinate is needed.
• Simple pendulum (1 DOF): The bob moves along a circular arc, so the angle $\theta$ fully defines its position. The rigid link constrains what would otherwise be a 2-DOF planar particle down to 1 DOF.
• Mass on an inclined plane (1 DOF): The surface constrains motion to one direction, so distance along the plane is the only coordinate.
• Double pendulum (2 DOF): Two angles ($\theta_1$, $\theta_2$) are required. Each link adds one rotational DOF.
• Planar four-bar linkage (1 DOF): Despite having multiple links and joints, the constraints leave only one independent motion. Specifying one link's angle determines the position of every other link.
• Spatial robot arm with 6 revolute joints (6 DOF): Each joint contributes one rotational DOF, giving the end effector full spatial positioning and orientation capability.
• Gyroscope (3 DOF): Three independent rotations describe its motion: precession, nutation, and spin.