Equations of motion for SDOF systems are the backbone of vibration analysis. They describe how a system with one degree of freedom moves when forces act on it. Using Newton's second law, we can derive these equations and use them to predict system behavior.

These equations are crucial for understanding how mechanical systems respond to different forces. By analyzing mass, stiffness, and damping effects, we can determine natural frequencies, system responses, and optimize designs for various applications in engineering and physics.

Equations of motion for SDOF systems

Deriving equations using Newton's second law

• Newton's second law states sum of forces acting on a body equals mass multiplied by acceleration (F = ma)
• Single degree of freedom (SDOF) system characterized by one independent coordinate fully describing motion
• Free body diagram of SDOF system includes applied force, spring force, damping force, and inertial force
• Derive equation of motion by summing all forces acting on mass and setting equal to product of mass and acceleration
• General form of equation for linear SDOF system $mẍ + cẋ + kx = F(t)$
  • m represents mass
  • c represents damping coefficient
  • k represents spring stiffness
  • F(t) represents external force
• Free vibration equation becomes $mẍ + cẋ + kx = 0$ with no external force applied
• Resulting second-order ordinary differential equation describes system's dynamic behavior
• Examples of SDOF systems include:
  • Simple pendulum (angle as coordinate)
  • Mass-spring-damper system (displacement as coordinate)

Applications of equations of motion

• Equation of motion used to analyze various mechanical and structural systems (buildings, bridges, vehicles)
• Helps predict system response to different loading conditions (earthquakes, wind loads, vehicle vibrations)
• Enables design optimization by adjusting system parameters (mass, stiffness, damping)
• Facilitates vibration control strategies (passive damping, active control systems)
• Equation forms basis for more complex multi-degree-of-freedom (MDOF) system analysis
• Applications in diverse fields:
  • Aerospace (aircraft wing vibrations)
  • Automotive (vehicle suspension systems)
  • Civil engineering (seismic analysis of structures)

Dynamic response of SDOF systems

Solution methods for equations of motion

• Dynamic response describes system behavior over time when subjected to external forces or initial conditions
• Analytical techniques for solving equations of motion:
  • Laplace transforms convert differential equations to algebraic equations
  • Method of undetermined coefficients for particular solutions
• Numerical methods for complex or non-linear systems:
  • Runge-Kutta method for step-by-step numerical integration
  • Newmark-β method commonly used in structural dynamics
• General solution consists of homogeneous solution (free vibration) and particular solution (forced vibration)
• Undamped free vibration solution: $x(t) = A cos(ω_nt) + B sin(ω_nt)$
  • ω_n represents natural frequency of system
• Damped free vibration solution depends on damping ratio ζ:
  • Overdamped (ζ > 1): non-oscillatory decay
  • Critically damped (ζ = 1): fastest non-oscillatory return to equilibrium
  • Underdamped (ζ < 1): decaying oscillations
• Forced vibration steady-state response determined by particular solution
  • Depends on type of excitation force (harmonic, periodic, transient)
• Transient response combines free and forced vibration, decays over time due to damping

Types of system responses

• Free vibration response occurs when system displaced from equilibrium and released
• Forced vibration response results from continuous external force application
• Harmonic response produced by sinusoidal excitation force
• Periodic response from repetitive non-sinusoidal forces (square wave, sawtooth)
• Transient response caused by short-duration or non-periodic forces (impulse, step input)
• Resonance phenomenon occurs when excitation frequency matches natural frequency
• Beat phenomenon observed when two close frequencies interact
• Examples of different responses:
  • Tuning fork (free vibration after strike)
  • Machine vibrations (forced vibration from rotating imbalance)

Mass, stiffness, and damping effects

Natural frequency and system parameters

• Natural frequency of SDOF system given by $ω_n = \sqrt{k/m}$
  • k represents spring stiffness
  • m represents mass
• Increasing mass decreases natural frequency, resulting in slower oscillation
• Increasing stiffness increases natural frequency, leading to faster oscillation
• Period of oscillation inversely proportional to natural frequency: $T = 2π/ω_n$
• Damping ratio ζ = c / (2mω_n) determines rate of decay in free vibration response
• Critical damping (ζ = 1) represents boundary between oscillatory and non-oscillatory behavior
• Damped natural frequency always less than or equal to undamped natural frequency: $ω_d = ω_n\sqrt{1 - ζ^2}$
• Logarithmic decrement measures rate of decay in underdamped systems: $δ = ln(x_1/x_2) = 2πζ/\sqrt{1 - ζ^2}$
• Examples of natural frequency effects:
  • Guitar strings (higher tension increases frequency)
  • Skyscrapers (taller buildings have lower natural frequencies)

System behavior and parameter relationships

• Quality factor Q = 1/(2ζ) indicates sharpness of resonance peak
• Time constant τ = 2m/c represents time for free vibration amplitude to decay to 1/e of initial value
• Stiffness-to-mass ratio k/m determines system's dynamic characteristics
• Damping-to-critical damping ratio c/c_c used to classify system behavior
• Resonance frequency shifts with damping: $ω_r = ω_n\sqrt{1 - 2ζ^2}$
• Amplitude at resonance inversely proportional to damping ratio
• Phase angle between force and displacement depends on frequency ratio and damping
• Examples of parameter relationships:
  • Automobile suspension (trade-off between comfort and handling)
  • Seismic isolation systems (low stiffness, high damping)

SDOF systems under excitation forces

Harmonic and step excitations

• Harmonic excitation F(t) = F_0 cos(ωt) results in steady-state response with amplitude and phase shift
• Frequency ratio r = ω/ω_n determines system behavior under harmonic excitation
• Magnification factor M = X/X_st represents ratio of dynamic to static displacement amplitude
• Resonance occurs when excitation frequency matches natural frequency, leading to maximum response
• Transmissibility ratio quantifies force transmitted through system relative to applied force
• Step input F(t) = F_0u(t) produces response with both transient and steady-state components
• Dynamic amplification factor for step input depends on damping ratio
• Examples of harmonic and step excitations:
  • Rotating machinery (harmonic)
  • Elevator starting motion (step)

Impulse and arbitrary excitations

• Impulse excitation F(t) = Iδ(t) results in initial velocity response followed by free vibration
• Impulse response function h(t) characterizes system behavior for unit impulse
• Convolution integral determines response of linear SDOF systems to arbitrary excitation forces: $x(t) = \int_0^t h(t-τ)F(τ)dτ$
• Fourier analysis decomposes complex excitations into sum of harmonic components
• Laplace transforms useful for solving SDOF systems subjected to non-periodic excitation forces
• Shock response spectrum (SRS) used to analyze system response to short-duration impulses
• Duhamel's integral provides solution for arbitrary forcing functions
• Examples of impulse and arbitrary excitations:
  • Drop test (impulse)
  • Earthquake ground motion (arbitrary)