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

• 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 $f(t) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t))$
• Exponential form of Fourier series $f(t) = \sum_{n=-\infty}^{\infty} c_n e^{in\omega_0 t}$
• Fourier coefficients calculated through integration over one period $a_n = \frac{2}{T} \int_0^T f(t) \cos(n\omega_0 t) dt$ $b_n = \frac{2}{T} \int_0^T f(t) \sin(n\omega_0 t) dt$
• 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 $x(t) = \sum_{n=1}^{\infty} |H(n\omega_0)| F_n \cos(n\omega_0 t - \phi_n)$
• 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
• Resonance 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
• Spectral analysis 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 $f_{beat} = |f_1 - f_2|$ where f1 and f2 are interfering frequencies
• Envelope function of beats expressed as $A(t) = 2A_0 \cos(2\pi f_{beat} t)$
• 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