1. HOMEWORK 01C Eigenvalues Problem 1: Problem 2: Problem 3: Problem 4: Lecture 1 Problem 5: Problem 6:

2. The solution of Problem 1 in Homework 01B gives the equation motion below, Problem 1: Click for answers. Find the eigenvalues of the system for the values of m=2 kg, k=3240 N/m ve c=380 Ns/m. Is the system stable? Write the form of free vibration response. Determine the values of Δt ve t ∞ for the system’s response to be analyzed. Here, f(t) is the input, x(t) is the generalized coordinate. Eigenvalue equation: 2.5ms 2 +2cs+2k=0 System is stable, form of the response: Δt=0.0022, t ∞ =0.7

3. The solution of Problem 2 in Homework 01B gives the equation motion below, Problem 2: f 0 =63.6943 Hz, ξ=0.3375, Δt=0.7854 x 10 -3 s, t ∞ =0.0465 s Find the eigenvalues of the system for the values of m=1.8 kg, L=0.42m, k=32000 N/m, c=486 Ns/m. Is the system stable? Find undamped natural frequency and damping ratio. Determine the values of Δt ve t ∞ for the system’s response to be analyzed. Here, T(t) is the input, θ(t) is the generalized coordinate. Click for answers.

4. Problem 3: f 0 =4.33 Hz, ξ=0.648, Δt=0.0041 s, t ∞ =0.36 The solution of Problem 3 in Homework 01B gives the equation motion below. Here, f(t) ve x 1 (t) are the inputs, x A (t) ve θ(t) are the generalized coordinates. Find the eigenvalues of the system for the values of m=20 kg, L=0.6 m, k=42000 N/m, c=2000 Ns/m. Is the system stable? Write the form of free vibration response. Find undamped natural frequency in terms of Hz and damping ratio. Determine the values of Δt ve t ∞ for the system’s response to be analyzed. Click for answers.

5. G y2y2 y1y1 kck c L1L1 L2L2 m,I yAyA yByB Problem 4: m=1050 kg, I=670 kg-m 2 k=35300 N/m, c=2000 Ns/m L 1 =1.7 m, L 2 =1.4 m For the system shown in the figure, y A are y B the inputs, y 1 and y 2 are the generalized coordinates. Find undamped natural frequencies in terms of Hz and damping ratio of the system. Determine the values of Δt ve t ∞ for the system’s response to be analyzed. f 1 =1.2968 Hz, f 2 =2.5483 Hz s 1,2 =1.8809±7.9283i (ξ=0.2308), s 3,4 =-7.2627±14.2698i (ξ=0.4536) Δt=0.0196 s, t ∞ =3.34 s Solution: Click for answers.

6. G y2y2 y1y1 kck c L1L1 L2L2 m,I yAyA yByB m=1050 kg, I=670 kg-m 2 k=35300 N/m, c=2000 Ns/m L 1 =1.7 m, L 2 =1.4 m Solution:

7. Problem 5: m 1 =5.8 kg, m 2 =3.2 kg, k 1 =4325 N/m, k 2 =3850 N/m, k 3 =3500 N/m, c 1 =37.2 Ns/m, c 2 =33.5 Ns/m, c 3 =32 Ns/m. m2m2 f2f2 x2x2 k3k3 c3c3 m1m1 f1f1 x1x1 k2k2 c2c2 k1k1 c1c1 f 1 =4.6549 Hz, f 2 =8.4979 Hz s 1,2 =-3.76±29.01i (ξ=0.129), s 3,4 =-12.57±51.89i (ξ=0.235) Δt=0.0059 s, t ∞ =1.67 s In the system shown in the figure, f 1 and f 2 are the inputs, x 1 and x 2 are the generalized coordinates. Find undamped natural frequencies in terms of Hz and damping ratio of the sytem. Determine the values of Δt ve t ∞ for the system’s response to be analyzed. Click for answers. Solution:

8. Problem 6: m 1 =250 kg, m 2 =350 kg, k=37000 N/m, c=1500 Ns/m, L 1 =1.2 m m 1, L 1 θ yAyA m2m2 k c f=1.4692 Hz, s 1,2 =-1.7308 ± 9.0768i (ξ=0.1872). Solution: In the system shown in the figure, y A is the input, θ(t) is the generalized coordinate. Find undamped natural frequencies in terms of Hz and damping ratio of the system. Click for answers.