94985393-62ed-47d3-9b32-8ecff5364008 327 4 5 false false false 0 3 decimal

&[DATE] &[TIME] - &[FILENAME]

References

[1] K. A. Laith, M.-J. Li , and X.-T. Rui - Transfer Matrix Method for the Determination of the Natural Vibration Characteristics of Realistic Thrusting Launch Vehicle—Part I

[2] L. Majkut - Free and Forced Vibrations of Timoshenko Beams Described by Single Difference Equation

Goal of the spreadsheet

The spreadsheet is established in order to set - up a Transfer matrix algorithm with the formulas in Ref.[1]. for the case under the critcal frequency. For the case beyond the critical frequency, we first compare the transfer matrix obtained in ref.[1] and the transfert matrix in ref.[2]; then, "translate" the beyond-the-critical-frequency transfer matrix from ref.[2] in the form of ref.[1]

The present spreadsheet will be recalled in the spreadsheets devoted to numerical tests

Preliminary Formulas

EIz_ad p_ad R_ad m_ad κGA_ad AQ EIz_ad 1 p_ad 2^R_ad 2^* m_ad κGA_ad / (* - (/ - :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQ p_ad 2^EIz_ad m_ad κGA_ad / * (* p_ad 2^R_ad 2^* m_ad * + k_ad 2^- (α 1 p_ad 2^R_ad 2^* m_ad κGA_ad / (* - (* / - :

EIz_ad p_ad R_ad m_ad κGA_ad α Aθ α κGA_ad / EIz_ad p_ad R_ad m_ad κGA_ad AQ * - :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α Bθ 1 α κGA_ad / EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQ * - :

EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad κGA_ad / EIz_ad p_ad R_ad m_ad κGA_ad AQ * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad κGA_ad / EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQ * EIz_ad α / - :

Roots formulas

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α b0 k_ad 2^EIz_ad α * / - (m_ad p_ad 2^* 1 κGA_ad α * / R_ad 2^EIz_ad / + (* + (:

EIz_ad p_ad R_ad m_ad κGA_ad α c0 p_ad 2^EIz_ad α * / p_ad 2^R_ad 2^* m_ad 2^* κGA_ad / (m_ad - (* :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α Δ EIz_ad p_ad R_ad m_ad κGA_ad k_ad α b0 2^4 EIz_ad p_ad R_ad m_ad κGA_ad α c0 * - :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α b0 - EIz_ad p_ad R_ad m_ad κGA_ad k_ad α Δ (sqrt + (2 / sqrt :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α b0 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α Δ (sqrt + (2 / sqrt :

ω function of ω-adimensional

p_ad A0 ρ Lm EIz0 ω EIz0 p_ad 2^* ρ A0 * Lm 4^* / (sqrt 2 π * / (:

Timoshenko Beam Transfer Matrix - under & beyond-critical-frequency cases

Symbolic Verifications

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Bl ch sh c s Aθ λ1 3^* Bθ λ1 * + (sh * Aθ λ1 3^* Bθ λ1 * + (ch * Aθ λ2 3^* - Bθ λ2 * + (s * - Aθ λ2 3^* - Bθ λ2 * + (c * AM λ1 4^* BM λ1 2^* + (ch * AM λ1 4^* BM λ1 2^* + (sh * AM λ2 4^* - BM λ2 2^* + (c * - AM λ2 4^* - BM λ2 2^* + (s * - AQ λ1 3^* BQ λ1 * + (sh * AQ λ1 3^* BQ λ1 * + (ch * AQ λ2 3^* - BQ λ2 * + (s * - AQ λ2 3^* - BQ λ2 * + (c * 44 mat :

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 B0 10100 Aθ λ1 3^* Bθ λ1 * + (0 Aθ λ2 3^* - Bθ λ2 * + (AM λ1 4^* BM λ1 2^* + (0 AM λ2 4^* - BM λ2 2^* + (- 00 AQ λ1 3^* BQ λ1 * + (0 AQ λ2 3^* - BQ λ2 * + (44 mat :

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Bl ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 B0 1 -^* :

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr expand Maxima ch λ2 2^* BM AM λ2 2^* - (* c λ1 2^* BM AM λ1 2^* + (* + λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + / sh λ2 * BQ - AQ λ2 2^* + (* s λ1 * BQ AQ λ1 2^* + (* + λ1 λ2 * BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - (* / ch c - λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + / sh λ2 * Bθ Aθ λ2 2^* - (* s λ1 * Bθ Aθ λ1 2^* + (* - λ1 λ2 * BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - (* / λ1 λ2 * sh Bθ Aθ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* s λ1 * BM AM λ1 2^* + (* Bθ Aθ λ2 2^* - (* - (* λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + / c BQ AQ λ1 2^* + (* Bθ Aθ λ2 2^* - (* ch Bθ Aθ λ1 2^* + (* BQ AQ λ2 2^* - (* - BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - / s λ2 * Bθ Aθ λ2 2^* - (* sh λ1 * Bθ Aθ λ1 2^* + (* + λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + / ch c - (Bθ Aθ λ1 2^* + (* Bθ Aθ λ2 2^* - (* BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - / ch c - (λ1 2^* BM AM λ1 2^* + (* λ2 2^* BM AM λ2 2^* - (* λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + / sh λ1 * BM AM λ1 2^* + (* BQ AQ λ2 2^* - (* s BQ AQ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* + BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - / - c λ2 2^* BM AM λ2 2^* - (* ch λ1 2^* BM AM λ1 2^* + (* + λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + / sh λ1 * BM AM λ1 2^* + (* Bθ Aθ λ2 2^* - (* s Bθ Aθ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* + BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - / λ1 λ2 * sh BQ AQ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* s λ1 * BM AM λ1 2^* + (* BQ AQ λ2 2^* - (* - (* λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + / c ch - (BQ AQ λ1 2^* + (* BQ AQ λ2 2^* - (* BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - / s λ2 * BQ AQ λ2 2^* - (* sh λ1 * BQ AQ λ1 2^* + (* + λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + / ch BQ AQ λ1 2^* + (* Bθ Aθ λ2 2^* - (* c Bθ Aθ λ1 2^* + (* BQ AQ λ2 2^* - (* - BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - / 44 mat

Symbolic Verifications : comparison with Ref.[1] for free-free boundary conditions

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 31 el expand Maxima ch c - (λ1 2^* BM AM λ1 2^* + (* λ2 2^* BM AM λ2 2^* - (* λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + /

λ1 2^λ2 2^+ (BM AM λ1 2^λ2 2^- (* + (* expand Maxima BM AM λ1 λ2 - (* λ2 λ1 + (* + (λ2 2^λ1 2^+ (*

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 32 el expand Maxima sh λ1 * BM AM λ1 2^* + (* BQ AQ λ2 2^* - (* s BQ AQ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* + BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - / -

ok

BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - factor Maxima AQ Bθ * Aθ BQ * - (λ2 2^λ1 2^+ (*

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 41 el expand Maxima λ1 λ2 * sh BQ AQ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* s λ1 * BM AM λ1 2^* + (* BQ AQ λ2 2^* - (* - (* λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + /

λ2 2^λ1 2^+ (BM AM λ1 λ2 - (* λ1 λ2 + (* + (* Maxima BM AM λ1 λ2 - (* λ2 λ1 + (* + (λ2 2^λ1 2^+ (*

ok

λ1 λ2 * λ1 BM AM λ1 2^* + (* BQ - AQ λ2 2^* + (* s * BQ AQ λ1 2^* + (λ2 * BM AM λ2 2^* - (* sh * + (* expand Maxima λ1 λ2 * s λ1 * BM AM λ1 2^* + (* BQ - AQ λ2 2^* + (* sh BQ AQ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* + (*

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 42 el expand Maxima c ch - (BQ AQ λ1 2^* + (* BQ AQ λ2 2^* - (* BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - /

ok

BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - factor Maxima AQ Bθ * Aθ BQ * - (λ2 2^λ1 2^+ (*

Symbolic : characteristic equation for free-free boundary conditions

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 z_free_free ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 31 el ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 42 el * ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 41 el ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 32 el * - :

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 z_free_free expand Maxima λ1 λ2 * sh BQ AQ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* s λ1 * BM AM λ1 2^* + (* BQ AQ λ2 2^* - (* - (sh λ1 * BM AM λ1 2^* + (* BQ AQ λ2 2^* - (* s BQ AQ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* + (* c ch - (ch c - (* λ1 * BM AM λ1 2^* + (* BQ AQ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* BQ AQ λ2 2^* - (* + (* λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + (BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - (* /

Symbolic Verifications : comparison with Ref.[1] for other terms of the Transfer Matrix

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 33 el expand Maxima c λ2 2^* BM AM λ2 2^* - (* ch λ1 2^* BM AM λ1 2^* + (* + λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 33 el factor Maxima c λ2 2^* BM - AM λ2 2^* + (* ch λ1 2^* BM AM λ1 2^* + (* - λ2 2^λ1 2^+ (BM - λ2 λ1 - (λ2 λ1 + (* AM * + (* /

λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + factor Maxima λ2 2^λ1 2^+ (BM - λ2 λ1 - (λ2 λ1 + (* AM * + (* -

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 34 el expand Maxima sh λ1 * BM AM λ1 2^* + (* Bθ Aθ λ2 2^* - (* s Bθ Aθ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* + BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 34 el factor Maxima λ2 λ2 AM s λ2 * Bθ Aθ λ1 2^* + (* Aθ sh * λ1 3^* + (* Aθ BM * sh * λ1 * + (* - Aθ BM * s * λ1 2^* + BM Bθ * s * + (* Bθ sh * λ1 * BM AM λ1 2^* + (* + AQ Bθ * Aθ BQ * - (λ2 2^λ1 2^+ (* /

BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - factor Maxima AQ Bθ * Aθ BQ * - (λ2 2^λ1 2^+ (*

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 43 el expand Maxima s λ2 * BQ AQ λ2 2^* - (* sh λ1 * BQ AQ λ1 2^* + (* + λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 43 el factor Maxima s λ2 * BQ - AQ λ2 2^* + (* sh λ1 * BQ AQ λ1 2^* + (* - λ2 2^λ1 2^+ (BM - λ2 λ1 - (λ2 λ1 + (* AM * + (* /

λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + factor Maxima λ2 2^λ1 2^+ (BM - λ2 λ1 - (λ2 λ1 + (* AM * + (* -

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 44 el expand Maxima ch BQ AQ λ1 2^* + (* Bθ Aθ λ2 2^* - (* c Bθ Aθ λ1 2^* + (* BQ AQ λ2 2^* - (* - BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 44 el factor Maxima λ2 2^Aθ AQ λ1 2^* ch c - (* BQ ch * + (* AQ Bθ * c * - (* λ1 2^AQ Bθ * ch * - Aθ BQ * c * + (* + BQ Bθ * ch - c + (* + AQ Bθ * Aθ BQ * - (λ2 2^λ1 2^+ (* / -

BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - factor Maxima AQ Bθ * Aθ BQ * - (λ2 2^λ1 2^+ (*

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 11 el expand Maxima ch λ2 2^* BM AM λ2 2^* - (* c λ1 2^* BM AM λ1 2^* + (* + λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 11 el factor Maxima ch λ2 2^* BM - AM λ2 2^* + (* c λ1 2^* BM AM λ1 2^* + (* - λ2 2^λ1 2^+ (BM - λ2 λ1 - (λ2 λ1 + (* AM * + (* /

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 12 el expand Maxima sh λ2 * BQ - AQ λ2 2^* + (* s λ1 * BQ AQ λ1 2^* + (* + λ1 λ2 * BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - (* /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 12 el factor Maxima sh λ2 * BQ - AQ λ2 2^* + (* s λ1 * BQ AQ λ1 2^* + (* + AQ Bθ * Aθ BQ * - (λ1 * λ2 * λ2 2^λ1 2^+ (* /

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 13 el expand Maxima ch c - λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 13 el factor Maxima ch c - λ2 2^λ1 2^+ (BM - λ2 λ1 - (λ2 λ1 + (* AM * + (* / -

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 21 el expand Maxima λ1 λ2 * sh Bθ Aθ λ1 2^* + (* λ2 * BM AM λ2 2^* - (* s λ1 * BM AM λ1 2^* + (* Bθ Aθ λ2 2^* - (* - (* λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 21 el factor Maxima λ1 λ2 * λ2 λ2 AM sh λ2 * Bθ Aθ λ1 2^* + (* Aθ s * λ1 3^* - (* Aθ BM * s * λ1 * - (* - Aθ BM * sh * λ1 2^* + BM Bθ * sh * + (* - Bθ s * λ1 * BM AM λ1 2^* + (* + (* λ2 2^λ1 2^+ (BM - λ2 λ1 - (λ2 λ1 + (* AM * + (* /

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 22 el expand Maxima c BQ AQ λ1 2^* + (* Bθ Aθ λ2 2^* - (* ch Bθ Aθ λ1 2^* + (* BQ AQ λ2 2^* - (* - BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 22 el factor Maxima λ2 2^AQ Aθ λ1 2^* ch c - (* Bθ ch * + (* Aθ BQ * c * - (* λ1 2^Aθ BQ * ch * - AQ Bθ * c * + (* + BQ Bθ * ch - c + (* + AQ Bθ * Aθ BQ * - (λ2 2^λ1 2^+ (* /

BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - factor Maxima AQ Bθ * Aθ BQ * - (λ2 2^λ1 2^+ (*

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 23 el expand Maxima s λ2 * Bθ Aθ λ2 2^* - (* sh λ1 * Bθ Aθ λ1 2^* + (* + λ2 2^BM AM λ2 2^* - (* λ1 2^BM AM λ1 2^* + (* + /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 23 el factor Maxima s λ2 * Bθ - Aθ λ2 2^* + (* sh λ1 * Bθ Aθ λ1 2^* + (* - λ2 2^λ1 2^+ (BM - λ2 λ1 - (λ2 λ1 + (* AM * + (* /

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 24 el expand Maxima ch c - (Bθ Aθ λ1 2^* + (* Bθ Aθ λ2 2^* - (* BQ AQ λ1 2^* + (Bθ Aθ λ2 2^* - (* Bθ Aθ λ1 2^* + (BQ AQ λ2 2^* - (* - /

ok

ch sh c s Aθ Bθ AM BM AQ BQ λ1 λ2 Tr 24 el factor Maxima ch c - (Bθ Aθ λ1 2^* + (* Bθ - Aθ λ2 2^* + (* AQ Bθ * Aθ BQ * - (λ2 2^λ1 2^+ (* / -

Common terms

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 (2^EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 (2^+ :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_min_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 (2^EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 (2^- (:

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AMλ1sq_min_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_min_λ2sq * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α DenCol1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AMλ1sq_min_λ2sq + (* :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AθBQ EIz_ad p_ad R_ad m_ad κGA_ad α Aθ EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQ * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AQBθ EIz_ad p_ad R_ad m_ad κGA_ad AQ EIz_ad p_ad R_ad m_ad κGA_ad k_ad α Bθ * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AθBQ_minus_AQBθ EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AθBQ EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AQBθ - :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α DenCol2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AθBQ_minus_AQBθ * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMplusAMλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 2^* + (:

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQplusAQλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQ EIz_ad p_ad R_ad m_ad κGA_ad AQ EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 (2^* + (:

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad cminch EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * cos EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * cosh - :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMminAMλ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 2^* - (:

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQminAQλ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQ EIz_ad p_ad R_ad m_ad κGA_ad AQ EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 (2^* - (:

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθplusAθλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α Bθ EIz_ad p_ad R_ad m_ad κGA_ad α Aθ EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 (2^* + (:

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθminAθλ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α Bθ EIz_ad p_ad R_ad m_ad κGA_ad α Aθ EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 (2^* - (:

Free-Free Boundary conditions - terms of Transfer Matrix

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR31 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1λ2 2^EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMplusAMλ1sq * 1 - (* EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMminAMλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad cminch * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α DenCol1 / :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR32a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQplusAQλ1sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMminAMλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * sin * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR32b EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMplusAMλ1sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQminAQλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * sinh * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR32 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR32a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR32b + (EIz_ad p_ad R_ad m_ad κGA_ad k_ad α DenCol2 / :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR41a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 (EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMplusAMλ1sq * 1 - (EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQminAQλ2sq * * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * sin * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR41b EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQplusAQλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 (* EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMminAMλ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * sinh * * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR41 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR41a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR41b + (* EIz_ad p_ad R_ad m_ad κGA_ad k_ad α DenCol1 / :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR42 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQplusAQλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQminAQλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad cminch * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α DenCol2 / - :

Free-free boundary conditions - characteristic equation

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad Z_free_free EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR31 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR42 * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR32 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR41 * - :

Other terms of the Transfer Matrix

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR33a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 2^EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMminAMλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * cos * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR33b EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 2^EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMplusAMλ1sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * cosh * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR33 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR33a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR33b + EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_min_λ2sq * + (* / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR34a 1 - (EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 (* EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθplusAθλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMminAMλ2sq * * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * sin * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR34b 1 - (EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 (* EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMplusAMλ1sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθminAθλ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * sinh * * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR34 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR34a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR34b + EIz_ad p_ad R_ad m_ad κGA_ad k_ad α DenCol2 / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR43a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 (EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQminAQλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * sin * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR43b EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQplusAQλ1sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * sinh * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR43 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR43a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR43b + EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_min_λ2sq * + (* / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR44a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθplusAθλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQminAQλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * cos * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR44b EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQplusAQλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθminAθλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * cosh * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR44 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR44a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR44b - EIz_ad p_ad R_ad m_ad κGA_ad k_ad α DenCol2 / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR11a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 2^EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMminAMλ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * cosh * (* :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR11b EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 (2^EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMplusAMλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * cos * (* :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR11 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR11a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR11b + EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_min_λ2sq * + (* / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR12a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQplusAQλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * sin * (* - :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR12b EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQminAQλ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * sinh * (* :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR12 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR12a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR12b + EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AθBQ_minus_AQBθ * / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR13 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad cminch EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_min_λ2sq * + (* / - :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR14a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθplusAθλ1sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * sin * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR14b EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθminAθλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * sinh * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR14 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR14a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR14b - EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AθBQ_minus_AQBθ * / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR21a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMplusAMλ1sq * 1 - (EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθminAθλ2sq * * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * sin * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR21b EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθplusAθλ1sq * 1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BMminAMλ2sq * * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * sinh * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR21 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR21a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR21b + (* EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_min_λ2sq * + (* / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR22a 1 - (EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQplusAQλ1sq * 1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθminAθλ2sq * * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * cos * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR22b 1 (EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθplusAθλ1sq * 1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BQminAQλ2sq * * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * cosh * :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR22 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR22a EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR22b + EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AθBQ_minus_AQBθ * / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR23 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθminAθλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ2 l_ad * sin * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθplusAθλ1sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1 l_ad * sinh * + EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BM EIz_ad p_ad R_ad m_ad κGA_ad AM EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_min_λ2sq * + (* / :

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EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR24 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθplusAθλ1sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α BθminAθλ2sq * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad cminch * EIz_ad p_ad R_ad m_ad κGA_ad k_ad α λ1sq_plus_λ2sq EIz_ad p_ad R_ad m_ad κGA_ad k_ad α AθBQ_minus_AQBθ * / :

EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad Trad EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR11 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR12 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR13 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR14 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR21 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR22 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR23 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR24 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR31 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR32 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR33 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR34 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR41 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR42 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR43 EIz_ad p_ad R_ad m_ad κGA_ad k_ad α l_ad TR44 44 mat :