〰️Vibrations of Mechanical Systems Unit 2 – Single Degree-of-Freedom Systems

Key Concepts and Terminology

• Single Degree-of-Freedom (SDOF) systems are the simplest vibrating systems, consisting of a mass, spring, and damper
• Mass ($m$) represents the inertial properties of the system and is measured in kilograms (kg)
• Spring stiffness ($k$) characterizes the elastic properties of the system and is measured in newtons per meter (N/m)
• Damping coefficient ($c$) represents the energy dissipation properties of the system and is measured in newton-seconds per meter (N·s/m)
  • Viscous damping is the most common type of damping, where the damping force is proportional to the velocity
• Natural frequency ($\omega_n$) is the frequency at which a system oscillates when disturbed from its equilibrium position and is measured in radians per second (rad/s)
  • Calculated using the formula $\omega_n = \sqrt{\frac{k}{m}}$
• Resonance occurs when the frequency of an external force coincides with the natural frequency of the system, leading to large amplitudes of vibration
• Forced vibration is the response of a system to an external force or excitation, while free vibration occurs in the absence of external forces

Mathematical Foundations

• The equation of motion for an SDOF system is derived using Newton's second law, $F = ma$, and is given by $m\ddot{x} + c\dot{x} + kx = F(t)$
  • $\ddot{x}$ represents acceleration, $\dot{x}$ represents velocity, and $x$ represents displacement
• The Laplace transform is a powerful tool for solving differential equations and is used to convert the equation of motion from the time domain to the complex frequency domain
• The transfer function $H(s)$ relates the input (force) to the output (displacement) in the Laplace domain and is given by $H(s) = \frac{X(s)}{F(s)} = \frac{1}{ms^2 + cs + k}$
• The convolution integral is used to determine the time-domain response of a system to an arbitrary input force
• Fourier series and Fourier transforms are used to represent periodic and non-periodic functions as a sum of sinusoidal components
  • Useful for analyzing the frequency content of a signal and determining the steady-state response to periodic excitation
• The principle of superposition states that the response of a linear system to multiple inputs is the sum of the responses to each individual input

Free Vibration Analysis

• Free vibration occurs when a system is disturbed from its equilibrium position and allowed to oscillate without any external forces
• The solution to the homogeneous equation of motion ($m\ddot{x} + c\dot{x} + kx = 0$) depends on the roots of the characteristic equation
• For an undamped system ($c = 0$), the solution is a sinusoidal function with a frequency equal to the natural frequency $\omega_n$
  • Displacement is given by $x(t) = A\cos(\omega_nt) + B\sin(\omega_nt)$, where $A$ and $B$ are determined by initial conditions
• For an underdamped system ($0 < c < c_c$), the solution is a decaying sinusoidal function with a damped natural frequency $\omega_d = \omega_n\sqrt{1-\zeta^2}$
  • Displacement is given by $x(t) = e^{-\zeta\omega_nt}(A\cos(\omega_dt) + B\sin(\omega_dt))$, where $\zeta$ is the damping ratio
• The logarithmic decrement $\delta$ is used to determine the damping ratio from experimental data and is defined as the natural logarithm of the ratio of two consecutive peak amplitudes
• For critically damped ($c = c_c$) and overdamped ($c > c_c$) systems, the motion is non-oscillatory, and the system returns to its equilibrium position without overshooting

Forced Vibration Response

• Forced vibration occurs when an external force or excitation is applied to a system
• The steady-state response of a system to a harmonic excitation $F(t) = F_0\cos(\omega t)$ is given by $x(t) = X\cos(\omega t - \phi)$
  • $X$ is the amplitude of the response, $\omega$ is the excitation frequency, and $\phi$ is the phase angle
• The amplitude of the steady-state response is given by $X = \frac{F_0/k}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}$, where $r = \frac{\omega}{\omega_n}$ is the frequency ratio
• The phase angle $\phi$ represents the lag between the excitation and the response and is given by $\tan\phi = \frac{2\zeta r}{1-r^2}$
• The transient response of a system is the portion of the response that decays with time and depends on the initial conditions
  • The complete response is the sum of the steady-state and transient responses
• The dynamic amplification factor (DAF) is the ratio of the maximum dynamic response to the static response and is used to quantify the effect of resonance
• The frequency response function (FRF) is a plot of the amplitude and phase of the steady-state response as a function of the excitation frequency

Damping Effects and Types

• Damping is the dissipation of energy in a vibrating system, which leads to the decay of the vibration amplitude over time
• Viscous damping is the most common type of damping, where the damping force is proportional to the velocity of the system
  • The damping coefficient $c$ is used to quantify the amount of viscous damping in a system
• Coulomb (dry friction) damping is caused by the friction between sliding surfaces and is independent of the velocity
• Hysteretic (structural) damping is caused by the internal friction within the material and is proportional to the displacement
• The quality factor $Q$ is a measure of the sharpness of the resonance peak and is inversely proportional to the damping ratio $\zeta$
  • Higher $Q$ values indicate less damping and sharper resonance peaks
• The half-power bandwidth method is used to determine the damping ratio from experimental frequency response data
  • The damping ratio is related to the frequencies at which the response amplitude is $1/\sqrt{2}$ times the peak amplitude
• Damping reduces the peak response at resonance and increases the bandwidth of the system

Resonance and Natural Frequency

• Resonance occurs when the frequency of an external excitation coincides with one of the natural frequencies of the system
• At resonance, the vibration amplitude reaches a maximum value, which can lead to excessive vibrations and potential damage to the system
• The natural frequency of an undamped SDOF system is given by $\omega_n = \sqrt{\frac{k}{m}}$ and depends only on the mass and stiffness of the system
• For a damped system, the damped natural frequency $\omega_d$ is slightly lower than the undamped natural frequency and is given by $\omega_d = \omega_n\sqrt{1-\zeta^2}$
• The resonance frequency $\omega_r$ is the frequency at which the maximum amplitude occurs in a damped system and is given by $\omega_r = \omega_n\sqrt{1-2\zeta^2}$
• The peak amplitude at resonance is limited by the amount of damping in the system
  • Higher damping results in lower peak amplitudes and a broader resonance peak
• In practice, resonance is often avoided by designing systems with natural frequencies far from the expected excitation frequencies or by adding damping to limit the peak response

Practical Applications and Examples

• Vibration isolation is used to reduce the transmission of vibrations from a source to a sensitive component or structure
  • Example: Automotive suspension systems isolate the passenger compartment from road irregularities
• Vibration absorption is used to reduce the vibration amplitude of a primary system by attaching a secondary system tuned to the same frequency
  • Example: Dynamic vibration absorbers (DVAs) are used in tall buildings to reduce wind-induced vibrations
• Vibration measurement and analysis are used to monitor the health of machines and structures and to diagnose faults
  • Example: Accelerometers and data acquisition systems are used to measure vibrations in rotating machinery (turbines, pumps, etc.)
• Modal analysis is used to determine the natural frequencies, mode shapes, and damping of a system from experimental data
  • Example: Finite element analysis (FEA) is used to predict the modal properties of complex structures (aircraft, spacecraft, etc.)
• Active vibration control uses sensors, actuators, and feedback control to actively suppress vibrations in real-time
  • Example: Active engine mounts in vehicles use piezoelectric actuators to cancel engine vibrations

Problem-Solving Techniques

• Free body diagrams (FBDs) are used to identify all forces acting on a system and to derive the equation of motion using Newton's second law
• The Laplace transform is used to convert the equation of motion from the time domain to the complex frequency domain, simplifying the solution process
  • The inverse Laplace transform is then used to obtain the time-domain response
• The convolution integral is used to determine the time-domain response of a system to an arbitrary input force
  • The impulse response function $h(t)$ is convolved with the input force $f(t)$ to obtain the output response $x(t)$
• Fourier series and Fourier transforms are used to represent periodic and non-periodic functions as a sum of sinusoidal components
  • The steady-state response to a periodic excitation can be determined by evaluating the transfer function at the excitation frequencies
• Numerical methods, such as Runge-Kutta and Newmark-Beta, are used to solve the equation of motion when an analytical solution is not possible
  • Example: MATLAB and Python are commonly used to implement numerical methods for vibration analysis
• Experimental modal analysis (EMA) is used to determine the modal properties of a system from measured frequency response functions (FRFs)
  • The natural frequencies, mode shapes, and damping ratios can be extracted from the FRFs using curve-fitting techniques
• The principle of superposition is used to analyze systems subjected to multiple inputs by considering the response to each input separately and then summing the results