Question

For a spring-mass system with m=1 kg, k =100 N/m, f1 (t ) =10 sin(2t ) (N) and f2(t) = 5sin(2t ) , the governing equations of the system have been derived as mx1 + 4 kx1 − kx2 = f1 (t) mx2 − kx1 + 4 kx2 = f 2 (t) (1) Put the equations of motion in matrix form, (2) Find the natural frequencies of the system, and (3) Determine the associated vibration mode shapes.

Name: for a spring mass system with m1 kg k 100 nm f1 t 10 sin2t n and f2t 5sin2t the governing equations of the system have been derived as mx1 4 kx1 kx2 f1 t mx2 kx1 4 kx2 f 2 t 1 put the equati 89346
Uploaded: 2023-06-18T08:02:05-08:00
Duration: 2 min 17 s
Channel: Adi S
Description: for a spring mass system with m1 kg k 100 nm f1 t 10 sin2t n and f2t 5sin2t the governing equations of the system have been derived as mx1 4 kx1 kx2 f1 t mx2 kx1 4 kx2 f 2 t 1 put the equati 89346

          For a spring-mass system with m=1 kg, k =100 N/m, f1 (t ) =10 sin(2t ) (N) and f2(t) = 5sin(2t ) , the governing equations of the system have been derived as
mx1 + 4 kx1 − kx2  = f1 (t)
mx2 − kx1 + 4 kx2  = f 2 (t)
(1) Put the equations of motion in matrix form,
(2) Find the natural frequencies of the system, and
(3) Determine the associated vibration mode shapes.

Added by Eva D.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Adi S

06/18/2023

Step 1

Let \( y_1 = x_1 \), \( y_2 = \dot{x}_1 \), \( y_3 = x_2 \), and \( y_4 = \dot{x}_2 \). Then, we have: \[ \begin{aligned} \dot{y}_1 &= y_2 \\ \dot{y}_2 &= \frac{1}{m}(f_1(t) - 4ky_1 + ky_3) = \frac{1}{1}(10\sin(2t) - 400y_1 + 100y_3) \\ \dot{y}_3 &= y_4 \\ Show more…

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For a spring-mass system with m=1 kg, k =100 N/m, f1 (t ) =10 sin(2t ) (N) and f2(t) = 5sin(2t ) , the governing equations of the system have been derived as mx1 + 4 kx1 − kx2 = f1 (t) mx2 − kx1 + 4 kx2 = f 2 (t) (1) Put the equations of motion in matrix form, (2) Find the natural frequencies of the system, and (3) Determine the associated vibration mode shapes.