Question

Vibrations

Two masses are connected via three springs to each other and to the wall at left and right. The equations of motion are

-k_{1} x_{1}-k\left(x_{1}-x_{2}\right)=m_{1} \ddot{x}_{1}

-k_{2} x_{2}-k\left(x_{2}-x_{1}\right)=m_{2} \ddot{x}_{2}

Suppose the force constants of the left and right springs are k1= k2 =2 and that of the center spring is k=1 let the masses be m1=m2=1 the two masses are given initial displacements at t =0 of x1 ==1 and x2=0 the initial velocity of each mass at t=0 is zero

Your job is to calculate expressions giving the position of each mass as a function of time using the diagonalization method discussed in class. This classic problem can be solved in a number of different ways, but I will not give you credit unless you solve using the diagonalization method from the course.

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### Question 44800

Vibrations

A mass is attached at one end of a uniform bar of mass whose other end is pivote dat point O as shown in the following figure. Determine the natural frequency of vibration of the resulting pendulum for small angular displacements. (10 marks)2.
I_{1 \text { aroundo }}=m_{1} l^{2}
I_{2 \text { coundmiddle point }}=\frac{1}{12} m_{2} I^{2}

### Question 44799

Vibrations

1. Derive the constants of X, and & for an underdamped system, where 3 is the damping ratio, by using initial conditionsof x(0)=x, and *(0)=x, (5 marks)

### Question 44696

Vibrations

7. To familiarize yourself with the nature of the forced response, plot the solution of a forced response of equation mit kx =Fcos at with @=2 rad /s, for a variety of values of the initial conditions and @,, as given in the following chart:

### Question 44695

Vibrations

Consider the system in Fig.3, write the equation of motion,and calculate the response assuming that the system is initially at rest for the values k = 100N / m k, = 500N / m and9.m = 89 kg.

### Question 44694

Vibrations

8. Consider the system in Fig. 2, write the equation of motion,and calculate the response assuming (a) that the system is initially at rest, and (b) that the system has an initial displacement of 0.05 m.

### Question 44692

Vibrations

6. The free response of a 1000-kg automobile with stiffness ofk= 400,000N/m is observed to be of the form give in Figure 6. Modeling the automobile as a single-degree-of-freedom oscillation in the vertical direction, determine the damping coefficient if the displacement at is measured to be 2cm and0.22 cm at

### Question 44691

Vibrations

Repeated problem for the system in Figure 5.5.

### Question 44690

Vibrations

4. Calculate the natural frequency and damping ratio for the100system in Figure 4. It is given the values m =10 kg, c =kg/s, = 4000 N/m, = 200 N/m, = 1000 N/m. Ignore the friction between the cart and the ground. Is the system overdamped, critically damped, or underdamped?

### Question 44689

Vibrations

3. Set up the differential equation of motion for the system shown in Fig.2, Determine the expression for (a) the critical damping coefficient, and (b) the natural frequency of the damped oscillation

### Question 44688

Vibrations

2. The pendulum illustrated in Figure.1 consist of a rigid bar of mass ": and length /, Attached to the end is a lumped mass m. Find the pendulum's natural frequency of oscillation.