DETERMINATION OF DYNAMIC CHARACTERISTICS OF A 2DOF SYSTEM. by Zoran VARGA, Ms.C.E.

DETERMINATION OF DYNAMIC CHARACTERISTICS OF A 2DOF SYSTEM by Zoran VARGA, Ms.C.E. Euro-Apex B.V. 1990-2012 All Rights Reserved.

The 2 DOF System Symbols m 1 =3m [kg] m 2 =8m m=10 [kg] l=2 [m] E=210000 [N/mm 2 ] I=19 [cm 4 ] DOF Degree of Freedom m i mass at level i j Degree of Freedom [Δ] Flexibility Matrix [K] Stiffness Matrix [M] Mass Matrix [I] Identity Matrix A Area of the Moment Diagram y the ordinate along the center of gravity of the moment diagram ω the pulse of the system T the period of the system f the frequency of the system k vibration mode φ jk Eigenvector Φ Mode Shape Matrix L k Modal Participation Factor M k Modal Mass m eff,k Effective Modal Mass In order to determinate the dynamic characteristics 2 methods were applied: FLEXIBILITY Method STIFFNESS Method

The FLEXIBILITY Method The general equation of the Eigen vibrations of a vibrating system using the Flexibility Matrix [Δ]: Δ M λ I φ jk = 0 (1) In order to have non-zero solutions for the equation the determinant must then be zero: Δ M λ I = 0 (2) where λ = 1 ω 2 The Mass Matrix [Δ] Flexibility Matrix [M] Mass Matrix [I] Identity Matrix M = m 1 0 0 m 2 M = m 3 0 0 8 (3)

The FLEXIBILITY Method Determination of the FLEXIBILITY Matrix Δ = δ 11 δ 12 δ 21 δ 22 (4) δ ij the displacement of the mass i in the j direction Mohr Maxwell Formula: δ ij = M i (x) M j (x) EI dx = A i y j EI (5) Moment diagrams obtained from applying the unit force in the direction of the DOF

The FLEXIBILITY Method Determination of the FLEXIBILITY Matrix The displacement of mass m 1 in the 1 st DOF direction: δ 11 = 1 A y EI δ 11 = 1 EI 1 2 1 l l 2 3 1 l δ 11 = l3 3EI The displacement of mass m 2 in the 2 nd DOF direction: δ 22 = 1 A y EI δ 22 = 1 EI 1 2 1 l 2 l 2 2 3 1 l 2 δ 22 = Δ = l3 24EI l3 48EI 16 5 The displacement of mass m 1 in the 2 nd DOF direction: δ 12 = δ 21 (Theorem of reciprocal unit displacements) δ 12 = 1 EI A 1 y 1 + A 2 y 2 + A 3 y 3 δ 12 = 1 EI 1 2 1 l 2 l 2 0 + 1 l 2 l 2 1 2 1 l 2 + 1 2 1 l 2 l 2 2 3 1 l 2 δ 12 = δ 21 = 5l3 48EI 5 2

The STIFFNESS Method The general equation of the Eigen vibrations of a vibrating system using the Stiffness Matrix [K]: K ω 2 M φ jk = 0 (6) In order to have non-zero solutions for the equation the determinant must then be zero: K ω 2 M = 0 (7) [K] Stiffness Matrix [M] Mass Matrix The Mass Matrix M = m 1 0 0 m 2 M = m 3 0 0 8

The STIFFNESS Method Determination of the STIFFNESS Matrix k = R = M φ (8) Loading conditions M A = M B = 6EI L 2 R A = R B = 12EI L 3 M A = 4EI L M B = 2EI L R A = R B = 6EI L 2 K = k 11 k 12 k 21 k 22 k 13 k 14 k 31 k 32 k 23 k 43 k 24 k 44 k 41 k 42 k 33 k 34 (9)

The STIFFNESS Method Determination of the STIFFNESS Matrix k 11 = k 21 = 12EI L 3 = 12EI l/2 3 = 96EI l 3 k 31 = k 41 = 6EI L 2 = 6EI l/2 2 = 24EI l 2 k 13 = k 23 = 6EI L 2 = 6EI l/2 2 = 24EI l 2 k 33 = 4EI L = 4EI l/2 = 8EI l k 12 = 96EI l 3 k 22 = 96EI l 3 k 32 = 24EI l 2 k 42 = 24EI l 2 + 96EI l 3 = 192EI l 3 k 43 = 2EI L = 2EI l/2 = 4EI l 24EI l 2 = 0 k 14 = 24EI l 2 k 24 = 0 k 34 = 4EI l k 44 = 16EI l

The STIFFNESS Method Determination of the STIFFNESS Matrix K = k 11 k 12 k 13 k 14 k 21 k 22 k 23 k 24 k 31 k 32 k 33 k 34 k 41 k 42 k 43 k 44 K = 8EI l 3 12 12 3l 3l 12 24 3l 0 3l 3l 3l 0 l 2 l 2 = /2 l 2 /2 2l 2 k tt k t0 k 0t k 00 (10) Condensed STIFFNESS Matrix: K = k tt k 0t T k 00 1 k 0t (11) K = 48EI 7l 3 2 5 5 16

The Dynamic Characteristics The FLEXIBILITY Method The STIFFNESS Method M λ I = 0 K ω 2 M = 0 l 3 48EI 16 5 5 2 m 3 0 0 8 λ 1 0 0 1 = 0 48EI 7l 3 2 5 5 16 ω2 m 3 0 0 8 = 0 48 40 15 16 48 40 15 16 α 1 0 0 1 λ 48EI l 3 m 1 0 0 1 = 0 = 0, where α = λ 48EI l 3 m 2 5 5 16 7l3 ω 2 m 48EI 3 0 0 8 = 0 2 5 5 16 α 3 0 0 8 = 0, where α = 7l3 ω 2 m 48EI 48 α 40 15 16 α = 0, 2 3α 5 5 16 8α = 0, α 2 64α + 168 = 0 α 1 = 61.2575 α 2 = 2.7425 24α 2 64α + 7 = 0 α 1 = 0.1143 α 2 = 2.5524 α = 1 ω 2 48EI l 3 m

The Dynamic Characteristics The FLEXIBILITY Method The STIFFNESS Method The pulsation of the system ω = 1 α 48EI l 3 m = 6.928 α EI l 3 m rad s ω = α 48EI 7l 3 m = 2.6186 α 48EI l 3 m rad s The pulsation of the system in the 1 st mode of vibration For α 1 = 61.26, For α 1 = 0.1143, ω 1 = 0.885 EI l 3 m = 0.885 21 1010 19 10 8 2 3 10 ω 1 = 0.885 EI l 3 m = 0.885 21 1010 19 10 8 2 3 10 ω 1 = 19. 76 [rad/s] ω 1 = 19. 76 [rad/s] The pulsation of the system in the 2 nd mode of vibration For α 2 = 2.74, ω 2 = 4.184 EI l 3 m For α 2 = 2.5524 ω 2 = 4.184 EI l 3 m ω 2 = 93. 44 [rad/s] ω 2 = 93. 44 [rad/s]

The Dynamic Characteristics The FLEXIBILITY Method The STIFFNESS Method The period and the frequency of the system (12) (13) T k = 2π [s] f ω k = 1 [Hz] k T k The period and the frequency of the system in the 1 st mode of vibration T 1 = 2π ω 1 f 1 = 1 T 1 T 1 = 0. 318 [s] f 1 = 3. 144 [Hz] The period and the frequency of the system in the 2 nd mode of vibration T 2 = 2π ω 2 f 2 = 1 T 2 T 2 = 0. 067 [s] f 2 = 14. 87 [Hz]

The Dynamic Characteristics The FLEXIBILITY Method Δ M λ I φ jk = 0 48 α 40 15 16 α φ 1k φ 2k = 0 48 α φ 1k + 40 φ 2k = 0 15 φ 1k + 16 α φ 2k = 0 The Eigenvectors The STIFFNESS Method K ω 2 M ϕ jk = 0 2 3α 5 5 16 8α φ 1k φ 2k = 0 2 3α φ 1k 5 φ 2k = 0 5 φ 1k + 16 8α φ 2k = 0 In the 1 st mode of vibration 48 α 1 φ 11 + 40 φ 21 = 0 48 61.2675 φ 11 + 40 φ 21 = 0 φ 11 = 1 φ 21 = 0. 3315 2 3α 1 φ 11 5 φ 21 = 0 2 3 0.1143 φ 11 5 φ 21 = 0 φ 11 = 1 φ 21 = 0. 3315 In the 2 nd mode of vibration 15 φ 12 + 16 α 2 φ 22 = 0 15 φ 12 + 16 2.7425 φ 22 = 0 φ 22 = 1 φ 12 = 0. 8838 5 φ 1k + 16 8α 2 φ 2k = 0 5 φ 12 + 16 8 2.5524 φ 22 = 0 φ 22 = 1 φ 12 = 0. 8838

The Dynamic Characteristics The verification of the Eigenvectors m i φ i,1 φ i,2 = 0 (14) Modal Participation Factor L k = m i φ ki [kg] (15) m 1 φ 11 φ 12 + m 2 φ 21 φ 22 = 0 3m 1 0.8838 + 8m 0.3315 1 = 0 2.652m + 2.652m = 0 0 = 0 The Mode Shape Matrix Φ = 1 0.8838 0.3315 1 1 st vibration mode: L 1 = m 1 φ 11 + m 2 φ 12 L 1 = 3m 1 + 8m (0.3315) L 1 = 56.52 [kg] 2 nd vibration mode: L 2 = m 1 φ 21 + m 2 φ 22 L 2 = 3m 0.8838 + 8m 1 L 2 = 53.48 [kg] Modal Mass M k = m i φ ki 2 [kg] (16) 1 st vibration mode: M 1 = m 1 φ 2 2 11 + m 2 φ 12 M 1 = 3m 1 2 + 8m 0.3315 2 M 1 = 38.79 [kg] 2 nd vibration mode: M 2 = m 1 φ 2 2 21 + m 2 φ 22 M 2 = 3m 0.8838 2 + 8m 1 2 M 2 = 103.44 [kg]

The Dynamic Characteristics Effective Modal Mass RESULTS 1 st vibration mode: 2 nd vibration mode: m eff,k = L k 2 M k m eff,1 = L 1 2 m eff,1 = (56.52)2 38.79 M 1 m eff,2 = L 2 2 m eff,2 = (53.48)2 103.44 [kg] = 82.35 [kg] M 2 = 27.65 [kg] (17) Eigen mode Pulsation [rad/s] Period [s] Frequency [Hz] Modal Mass [kg] Effective Modal Mass [kg] 1 19.76 0.318 3.144 38.79 82.35 2 93.44 0.067 14.87 103.44 27.65 m eff,1 + m eff,2 = 82.35 + 27.65 = 110 [kg] Sum of the effective masses equals the total system mass.

Modeling a 2DOF System with FEM Graitec Advance Design Dlubal - RFEM Modal analysis Eigenvalues Mode Pulsation Period Frequency Energy Modal masses Damping N (Rad/s) (s) (Hz) (J) X (%) kg (%) 1 19.71 0.32 3.14 194.24 82.54 ( 75.04) 5.00 2 91.46 0.07 14.56 4182.75 27.46 ( 24.96) 5.00 residual 0.00 ( 0.00) Total 4377.00 110.00 (100.00)

DLUBAL 100 GRAITEC 100 ANALYTIC 100 COMPARING THE RESULTS Eigen mode 1 Eigen mode 2 1 Eigen mode 2 1 2 Node Eigenvectors Pulsation Period Frequency 1 1 2 0.3315 Node Eigenvectors Pulsation Period Frequency 1-0.8838 2 1 1 0.1604 2 0.05342 Node Eigenvectors Pulsation Period Frequency 1 0.08724 2-0.09821 1 1 2 0.33145 1-0.88384 2 1 Modal Part. Factor Modal Part. Factor Modal Part. Factor Modal Mass Modal Mass Modal Mass Effective Modal Mass [rad/s] [s] [Hz] [kg] [kg] [kg] [%] [%] 19.76 0.318 3.144 56.52 38.79 82.35 74.9 Effective Modal Mass 93.44 0.067 14.87 53.48 103.44 27.65 25.1 [rad/s] [s] [Hz] [kg] [kg [kg] [%] [%] 19.71 0.32 3.14 - - 82.54 75.04 Effective Modal Mass 91.46 0.07 14.56 - - 27.46 24.96 [rad/s] [s] [Hz] [kg] [kg [kg] [%] [%] 19.769 0.318 3.146 56.52 38.79 82.34 74.9 93.425 0.067 14.869 53.48 103.44 27.66 25.1 The agreement between the analytical (theoretical) method and the results of the industrial FEM packages is as expected very good.