Uchanganuzi wa frequency response¶

Uwekaji wa tatizo¶

Frequency response analysis hutathmini amplitude na phase ya response ya mfumo katika steady state kwenye frequency domain wakati external force hubadilika kwa harmonic ya muda. Kwa kuwa equation of motion ina framework tofauti ikiwa damping haipo au ipo, sura hii kwanza hutoa eigenmodes kutoka undamped free vibration, kisha hutumia eigenmodes hizo kama modal basis kupanua harmonic response yenye damping.

Undamped free vibration na eigenmodes¶

Equation of motion bila kuzingatia damping ni kama ifuatavyo.

\[\begin{equation} M \ddot{U} + K U = 0 \label{eq:2.6.1} \end{equation}\]

Ikiipanua kwa kila eigenmode,

\[\begin{equation} U = U_j e^{i \omega_j t} \label{eq:2.6.2} \end{equation}\]

hupatikana. Tukiiingiza katika equation \(\eqref{eq:2.6.1}\), tunapata

\[\begin{equation} K U_j = \omega_j^2 M U_j \label{eq:2.6.3} \end{equation}\]

Njia za nambari za kutatua generalized eigenvalue problem hii (shifted inverse iteration na Lanczos method) zinaelezwa katika Uchanganuzi wa eigenvalue. Hapa tunaweka wazi sifa za natural frequencies na eigenmodes zinazotumiwa baadaye katika harmonic-response expansion.

Uhalisia wa natural frequency¶

Uhalisia wa natural frequency unaweza kuthibitishwa kama ifuatavyo. Tukiweka \(\omega_j^2 = \lambda_j\) na kuchukua complex conjugate ya equation \(\eqref{eq:2.6.3}\), tunapata equation \(\eqref{eq:2.6.4}\).

\[\begin{equation} K U_j = \lambda_j M U_j K \overline{UJ} = \overline{\lambda_J} M \overline{U_J} \label{eq:2.6.4} \end{equation}\]

Tukizidisha kwa \(\overline{U}_J^T\), tunapata

\[\begin{equation} U_j^T K \overline{U}_J = \overline{\lambda}_J U_j^T M \overline{U}_J \overline{U}_J^T K U_j = \lambda_j \overline{U}_J^T M U_j \label{eq:2.6.5} \end{equation}\]

Kutoka equation \(\eqref{eq:2.6.5}\),

\[\begin{equation} 0 = ( \lambda_j - \overline{\lambda_J} ) \overline{U_J}^T M U_j \label{eq:2.6.6} \end{equation}\]

Kwa kuwa mass matrix ni positive-definite symmetric, kwa eigenvector isiyo zero vector tunayo

\[\begin{equation} \overline{U_J} M U_j > 0 \label{eq:2.6.7} \end{equation}\]

Kwa hiyo,

\[\begin{equation} \lambda_j =\overline{\lambda_J} \label{eq:2.6.8} \end{equation}\]

na hivyo \(\omega_j^2 = \lambda_j\) ni real number.

Orthogonality na normalization ya eigenmodes¶

Sasa tuchunguze mode mbili tofauti.

\[\begin{equation} K U_i = \lambda_i M U_i K U_j = \lambda_j M U_j \label{eq:2.6.9} \end{equation}\]

Kutokana na hizi,

\[\begin{equation} ( \lambda_i - \lambda_j ) U_j^T M U_i = 0 \label{eq:2.6.10} \end{equation}\]

hupatikana; ikiwa eigenvalues ni tofauti,

\[\begin{equation} U_j^T M U_i = 0 \label{eq:2.6.11} \end{equation}\]

Hivyo eigenmodes tofauti ni orthogonal kwa mass matrix. Kwa mode ileile, normalization kwa mass matrix (equation \(\eqref{eq:2.6.12}\)) hurahisisha ushughulikiaji.

\[\begin{equation} U_i^T M U_i = 1 \label{eq:2.6.12} \end{equation}\]

Harmonic response yenye damping¶

Sasa tunaonyesha formulation ya frequency response analysis ikiwa damping inazingatiwa. Equation of motion inayolengwa imeonyeshwa katika equation \(\eqref{eq:2.6.13}\).

\[\begin{equation} M \ddot{U} + C \dot{U} + K U = F \label{eq:2.6.13} \end{equation}\]

Damping term hapa inadhaniwa kuwa ya Rayleigh type na inaweza kuonyeshwa kama \(\eqref{eq:2.6.14}\).

\[\begin{equation} C = \alpha M + \beta K \label{eq:2.6.14} \end{equation}\]

Kwa eigenvectors zilizopatikana katika eigenvalue analysis, displacement vector inaweza kupanuliwa wakati t kama equation \(\eqref{eq:2.6.15}\).

\[\begin{equation} U(t) = \sum_i b_i(t) U_i \label{eq:2.6.15} \end{equation}\]

Wakati external-force term ni harmonic oscillator form

\[\begin{equation} F(t) = ( F_R + i F_I )e^{i \Omega t} \label{eq:2.6.16} \end{equation}\]

tunaamua \(b_{j}(t)\). Kwa kuwa equation of motion \(\eqref{eq:2.6.13}\) huwa forced-vibration form,

\[\begin{equation} b_j (t) = (b_{jR} + b_{jI}) e^{i \Omega t} \label{eq:2.6.17} \end{equation}\]

hutimizwa. Real part na imaginary part za expansion coefficient ya \(b_{j}(t)\) ni kama equations \(\eqref{eq:2.6.18}i\) na \(\eqref{eq:2.6.19}\).

\[\begin{equation} b_{jR} = \frac{ U^T_j F_R (\omega^2_j - \Omega^2) + U^T_j F_I (\alpha + \beta \omega_j^2) \Omega}{ (\omega^2_j - \Omega^2)^2 + (\alpha + \beta \omega_j^2)^2 \Omega^2} \label{eq:2.6.18} \end{equation}\]

\[\begin{equation} b_{jI} = \frac{ U^T_j F_I(\omega^2_j - \Omega^2) - U^T_j F_R(\alpha + \beta \omega_j^2) \Omega}{ (\omega^2_j - \Omega^2)^2 + (\alpha + \beta \omega_j^2)^2 \Omega^2} \label{eq:2.6.19} \end{equation}\]

hupatikana.

Vipengee vinavyohusiana¶

Uchanganuzi wa eigenvalue — Formulation ya eigenvalue problem inayotangulia
Njia za dynamic analysis — Dynamic analysis katika time domain
Aina ya uchanganuzi — Muhtasari wa kipengele cha frequency response analysis