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

Single degree-of-freedom systems are the building blocks of vibration analysis. They use one coordinate to describe motion, involving mass, spring, and damping elements. Understanding these systems is crucial for grasping more complex vibration problems.

The equation of motion for SDOF systems relates displacement, velocity, and acceleration. Key concepts include natural frequency and damping ratio, which determine system behavior. These fundamentals apply to real-world applications like simple pendulums, car suspensions, and building seismic analysis.

Single Degree-of-Freedom Systems

Fundamentals of SDOF Systems

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• Single degree-of-freedom (SDOF) system describes motion using one coordinate or variable
• Key components include mass, spring element, and damping element
• Equation of motion relates displacement, velocity, and acceleration through second-order differential equation
• Natural frequency determined by system's mass and stiffness
  • Higher mass decreases natural frequency
  • Higher stiffness increases natural frequency
• Damping ratio influences system response to external forces and energy dissipation
  • Low damping ratio results in prolonged oscillations
  • High damping ratio leads to quick decay of motion
• Three types of motion based on damping ratio
  • Underdamped (oscillatory decay)
  • Critically damped (fastest return to equilibrium without oscillation)
  • Overdamped (slow return to equilibrium without oscillation)
• System response categorized as free or forced
  • Free response determined by initial conditions and system parameters
  • Forced response depends on external excitations (periodic forces, impulses)

Mathematical Representation of SDOF Systems

• Equation of motion for SDOF system $m\ddot{x} + c\dot{x} + kx = F(t)$
  • m: mass
  • c: damping coefficient
  • k: spring stiffness
  • x: displacement
  • F(t): external force
• Natural frequency calculation $\omega_n = \sqrt{\frac{k}{m}}$
• Damping ratio calculation $\zeta = \frac{c}{2\sqrt{km}}$
• General solution for free vibration $x(t) = Ae^{-\zeta\omega_n t} \cos(\omega_d t + \phi)$
  • A: amplitude
  • $\omega_d$: damped natural frequency
  • $\phi$: phase angle

SDOF Systems in Applications

Real-World Examples of SDOF Systems

• Simple pendulum with angle of displacement as single coordinate
  • Used in clocks, seismometers
• Mass suspended on vertical spring for vibration isolation
  • Applied in vehicle seats, sensitive equipment mounts
• Car bouncing on suspension approximated as SDOF
  • Helps in designing comfortable ride characteristics
• Single-story building under horizontal ground motion during earthquake
  • Used for basic seismic analysis and design
• Torsional vibration of shaft with single disk
  • Important in rotating machinery design (turbines, generators)
• Tuned mass damper in tall buildings to reduce wind-induced vibrations
  • Examples include Taipei 101, John Hancock Tower
• Vertical motion of floating buoy in calm water
  • Used in oceanographic studies, wave energy converters

SDOF Systems in Engineering Design

• Vibration isolators for sensitive equipment (microscopes, precision machinery)
  • Reduce transmitted vibrations from environment
• Shock absorbers in vehicles
  • Improve ride quality and handling
• Seismic base isolation systems for buildings
  • Protect structures from earthquake damage
• Dynamic vibration absorbers in power transmission systems
  • Reduce harmful vibrations in rotating shafts
• Tuned liquid dampers in water towers
  • Mitigate wind-induced oscillations
• Mass dampers in sports equipment (tennis rackets, golf clubs)
  • Enhance performance by reducing vibrations

Spring-Mass-Damper Models for SDOF

Components of Spring-Mass-Damper Model

• Mass element represents system inertia
  • Depicted as rigid block or point mass
  • Determines kinetic energy of system
• Spring element represents system stiffness
  • Usually shown as coil spring
  • Stores potential energy
  • Linear spring follows Hooke's Law: F = kx
• Damper element represents energy dissipation
  • Illustrated as dashpot or viscous damper
  • Dissipates energy through heat
  • Linear damper force proportional to velocity: F = cv
• Free-body diagram includes
  • Inertial force (ma)
  • Spring force (kx)
  • Damping force (cv)
  • External forces (F(t))

Advanced Spring-Mass-Damper Models

• Nonlinear springs for large displacements
  • Force-displacement relationship: F = kx + k2x^2 + k3x^3
• Multiple springs in series or parallel
  • Series: 1/keq = 1/k1 + 1/k2
  • Parallel: keq = k1 + k2
• Alternative damping mechanisms
  • Coulomb damping (dry friction): F = μN * sign(v)
  • Structural damping: F = jkx
• Rotational SDOF systems
  • Torsional spring: T = kθ
  • Rotational damper: T = cω
• Two-dimensional SDOF systems
  • Planar motion with coupled x and y coordinates

Degrees of Freedom for Systems

Determining Degrees of Freedom

• Degrees of freedom equal minimum number of independent coordinates to define configuration
• Rigid body in 3D space has maximum six degrees of freedom
  • Three translational (x, y, z)
  • Three rotational (roll, pitch, yaw)
• Constraints reduce degrees of freedom
  • Fixed support removes all degrees of freedom
  • Pin joint allows rotation but restricts translation
• Calculate degrees of freedom
  • Count possible independent motions
  • Subtract number of constraints
• Planar motion of free rigid body has three degrees of freedom
  • Two translational (x, y)
  • One rotational (θ)
• Systems with multiple bodies
  • Sum individual body degrees of freedom
  • Subtract constraints between bodies

Examples of Degrees of Freedom Analysis

• Particle moving in straight line (1 DOF)
  • Only x-coordinate needed to describe motion
• Simple pendulum (1 DOF)
  • Angle θ fully defines position
• Mass sliding on inclined plane (1 DOF)
  • Distance along plane describes motion
• Double pendulum (2 DOF)
  • Two angles required to define configuration
• Planar four-bar linkage (1 DOF)
  • One angle determines position of all links
• Spatial robot arm with 6 joints (6 DOF)
  • Each joint angle contributes one degree of freedom
• Gyroscope (3 DOF)
  • Three rotational degrees of freedom (precession, nutation, spin)