Glossary

4.2 Non-harmonic periodic excitation

4 min read•july 30, 2024

Non-harmonic periodic excitation adds complexity to forced vibrations. Fourier series breaks down these signals into harmonic components, allowing us to apply the principle of superposition. This approach helps us understand how systems respond to more complex inputs.

The steady-state response to non-harmonic excitation combines the effects of individual harmonics. Each component's contribution is determined by the system's frequency response function. Beat phenomena can occur when frequencies are close, causing amplitude fluctuations in the overall response.

Fourier Series for Periodic Excitations

Fundamentals of Fourier Series

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  • Fourier series represents periodic functions as sums of simple sine and cosine functions
  • General form includes constant term, sine terms, and cosine terms with respective coefficients
  • Fourier coefficients (an and bn) determine amplitude of each harmonic component
  • Fundamental frequency relates to the period of the function
  • Higher harmonics occur at integer multiples of the fundamental frequency
  • Series expressed in trigonometric and exponential forms for different applications
  • Gibbs phenomenon causes oscillations in series approximation at function discontinuities

Mathematical Representation and Properties

  • Trigonometric form of Fourier series f(t)=a0+∑n=1∞(ancos⁡(nω0t)+bnsin⁡(nω0t))f(t) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t))
  • Exponential form of Fourier series f(t)=∑n=−∞∞cneinω0tf(t) = \sum_{n=-\infty}^{\infty} c_n e^{in\omega_0 t}
  • Fourier coefficients calculated through integration over one period an=2T∫0Tf(t)cos⁡(nω0t)dta_n = \frac{2}{T} \int_0^T f(t) \cos(n\omega_0 t) dt bn=2T∫0Tf(t)sin⁡(nω0t)dtb_n = \frac{2}{T} \int_0^T f(t) \sin(n\omega_0 t) dt
  • Parseval's theorem relates energy in time domain to energy in frequency domain
  • Convergence of Fourier series depends on function's continuity and differentiability
  • Even functions have only cosine terms, odd functions have only sine terms
  • Fourier series can represent both continuous and discontinuous functions (square wave)

Superposition for Non-Harmonic Excitations

Principle of Superposition

  • Superposition states response to multiple inputs equals sum of individual responses
  • Each harmonic component treated as individual input to the system
  • Total response obtained by summing responses to each harmonic component
  • Amplitude and phase of response determined by system's frequency response function
  • Valid only for linear systems nonlinear systems require different analysis techniques
  • Time-domain and frequency-domain approaches used to apply superposition
  • Higher harmonics typically contribute less to total response

Application to Non-Harmonic Excitations

  • Decompose non-harmonic excitation into harmonic components using Fourier series
  • Calculate system response to each harmonic component individually
  • Combine individual responses to obtain total system response
  • Frequency response function H(ω) used to determine amplitude and phase for each harmonic
  • Total response expressed as sum of harmonic responses x(t)=∑n=1∞∣H(nω0)∣Fncos⁡(nω0t−ϕn)x(t) = \sum_{n=1}^{\infty} |H(n\omega_0)| F_n \cos(n\omega_0 t - \phi_n)
  • Superposition applied in both forced vibration (external force) and base excitation scenarios
  • Method allows analysis of complex periodic signals (sawtooth wave)

Steady-State Response to Non-Harmonic Excitations

Characteristics of Steady-State Response

  • Steady-state response occurs after transient effects dissipate
  • Response becomes periodic matching excitation period
  • Each harmonic component's amplitude determined by magnitude of frequency response function
  • Phase lag for each component given by phase of frequency response function
  • Total steady-state response sums responses to all harmonic components
  • possible if harmonic frequency near system's natural frequency
  • Relative importance of harmonics depends on excitation amplitude and system frequency response

Analysis Techniques

  • Frequency response function H(ω) used to calculate steady-state response
  • Magnitude of H(ω) determines amplification or attenuation of each harmonic
  • Phase of H(ω) determines phase shift of each harmonic in response
  • Harmonic balance method approximates steady-state response for complex systems
  • Numerical integration techniques (Runge-Kutta methods) solve for steady-state response
  • used to visualize frequency content of steady-state response
  • Nyquist and Bode plots aid in understanding system behavior across frequency range

Beat Phenomenon in Non-Harmonic Excitations

Fundamentals of Beat Phenomenon

  • Beats occur when harmonic components with close frequencies interfere
  • Results in periodic variation in amplitude of response
  • Beat frequency equals difference between interfering component frequencies
  • Occurs between fundamental frequency and higher harmonics or between different harmonics
  • Amplitude of beat envelope determined by amplitudes of interfering components
  • Significantly affects perceived characteristics of system response (loudness in acoustics)
  • Can lead to fatigue issues in mechanical systems due to cyclic stress variations

Analysis and Applications

  • Time domain analysis reveals amplitude modulation characteristic of beats
  • Frequency domain analysis shows closely spaced peaks corresponding to interfering frequencies
  • Beat frequency calculated as fbeat=∣f1−f2∣f_{beat} = |f_1 - f_2| where f1 and f2 are interfering frequencies
  • Envelope function of beats expressed as A(t)=2A0cos⁡(2πfbeatt)A(t) = 2A_0 \cos(2\pi f_{beat} t)
  • Beats used in music tuning (piano tuners)
  • Important in vibration analysis of rotating machinery (misaligned shafts)
  • Consideration of beats crucial in designing systems with multiple excitation frequencies

Key Terms to Review (18)

Beam Dynamics: Beam dynamics refers to the study of the motion and behavior of beams, particularly when subjected to external forces and moments. It encompasses the analysis of how beams respond to various types of loading conditions, including non-harmonic periodic excitation, which can cause complex vibration patterns. Understanding beam dynamics is essential in predicting the performance and stability of structures under dynamic loads.
Damping Ratio: The damping ratio is a dimensionless measure that describes how oscillations in a mechanical system decay after a disturbance. It indicates the level of damping present in the system and is crucial for understanding the system's response to vibrations and oscillatory motion.
Finite Element Method: The finite element method (FEM) is a numerical technique used to find approximate solutions to boundary value problems for partial differential equations. It breaks down complex structures into smaller, simpler parts called finite elements, allowing for detailed analysis of mechanical behavior under various conditions.
Fourier series representation: Fourier series representation is a mathematical method used to express a periodic function as a sum of sine and cosine functions. This technique is crucial for analyzing non-harmonic periodic excitation, as it allows complex waveforms to be broken down into simpler components, making it easier to understand their behavior in mechanical systems.
Frequency-domain analysis: Frequency-domain analysis is a method used to analyze signals and systems by transforming them from the time domain into the frequency domain, where the behavior and characteristics of the system can be examined based on frequency components. This approach is particularly useful for understanding the response of mechanical systems to various types of excitations, including non-harmonic periodic inputs and transient vibrations. By representing signals as a sum of sinusoids, this analysis provides insight into how different frequencies affect system behavior.
Logarithmic decrement: Logarithmic decrement is a measure used to quantify the rate of decay of oscillations in a damped system, defined as the natural logarithm of the ratio of two successive amplitudes of vibration. This term helps assess the damping characteristics and energy dissipation in mechanical systems, linking closely with concepts like damping ratio, non-harmonic excitation, and the modeling of viscous damping. It provides essential insights into the behavior of systems undergoing damped free vibrations, allowing engineers to estimate damping from measured data.
Mass-spring-damper system: A mass-spring-damper system is a simple mechanical model that represents the dynamic behavior of a single degree of freedom system under various forces. It consists of a mass (m) attached to a spring (k) and a damper (c), where the spring provides restoring force and the damper provides resistance to motion, allowing for the analysis of oscillations and vibrations in response to external excitations. This system is foundational in understanding how real-world mechanical systems respond to forces, particularly when subjected to non-harmonic periodic excitation.
Matlab simulations: MATLAB simulations are computational models used to analyze and visualize the dynamic behavior of systems through numerical methods. These simulations allow users to explore how mechanical systems respond to various inputs and conditions, including non-harmonic periodic excitation, which involves forces that vary in time but do not follow a simple sinusoidal pattern. MATLAB is particularly valuable for its ability to solve complex equations, simulate real-time responses, and visualize results in a user-friendly environment.
Modal analysis: Modal analysis is a technique used to determine the natural frequencies, mode shapes, and damping characteristics of a mechanical system. This method helps to understand how structures respond to dynamic loads and vibrations, providing insights that are crucial for design and performance optimization.
Resonance: Resonance is a phenomenon that occurs when a system is driven at its natural frequency, leading to a significant increase in amplitude of oscillation. This effect can cause systems to behave in unpredictable and potentially damaging ways, and it's important in understanding how various vibrations interact with materials and structures.
Spectral Analysis: Spectral analysis is a technique used to identify and analyze the frequency content of signals, especially in the context of vibrations. By decomposing a complex signal into its constituent frequencies, it allows for insights into the behavior of mechanical systems under various excitations. This method is essential for understanding dynamic responses, diagnosing faults, and evaluating performance in different mechanical applications.
Square Wave Excitation: Square wave excitation refers to a type of non-harmonic periodic input where the signal alternates between two levels with a rapid transition, creating a waveform that resembles a square. This form of excitation is characterized by its abrupt changes and can generate a wide range of frequency components due to its non-sinusoidal shape, making it particularly useful in testing and analyzing the dynamic response of mechanical systems.
Superposition Principle: The superposition principle states that in a linear system, the total response at a given time or position is the sum of the individual responses from each excitation acting on the system. This principle is crucial as it allows us to analyze complex vibrations by breaking them down into simpler components, enabling us to study harmonic and non-harmonic excitations separately, as well as transient responses resulting from impulse forces.
System Identification: System identification is the process of developing or improving a mathematical model of a dynamic system based on measured data. This technique helps in understanding how systems respond to various inputs and is crucial for controlling and optimizing system performance. By analyzing the relationship between input and output data, it allows engineers to create accurate models that can predict system behavior under different conditions.
Time-domain analysis: Time-domain analysis is a method used to study and understand the behavior of mechanical systems over time by examining how the system's response evolves in relation to time, rather than focusing solely on frequency components. This approach provides insights into non-harmonic periodic excitations, transient vibrations, and impulse responses, as well as informing vibration testing methods and the interpretation of data. It is essential for designing active vibration control systems, diagnosing faults through vibration-based monitoring, and applying these principles in aerospace and marine applications.
Transient Response: Transient response refers to the behavior of a dynamic system as it responds to a change from equilibrium, particularly during the initial moments after an excitation. This phase is characterized by temporary variations before the system settles into a steady-state or periodic motion. Understanding transient response is crucial for analyzing systems under damped conditions, the effects of periodic excitations, and how different damping models influence system behavior, especially in single degree of freedom (SDOF) systems.
Triangular wave excitation: Triangular wave excitation refers to a type of non-harmonic periodic input characterized by a waveform that rises and falls linearly, creating a triangle-like shape. This waveform is important in analyzing mechanical systems as it produces a wide range of frequency components, allowing for the investigation of system behavior under varying conditions. The triangular wave is particularly useful in experiments and simulations to excite systems across multiple modes and understand their response characteristics.
Vibration testing: Vibration testing is a technique used to assess the dynamic behavior of mechanical systems under various types of vibrations. It helps in identifying the natural frequencies, damping characteristics, and responses of structures or components to specific excitations, including non-harmonic periodic inputs. Understanding how a system reacts to these vibrations can help improve design, durability, and performance.
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