Vibrations

3- A precision grinding machine is supported on an isolator that has a stiffness of 1 MN/m and a viscous damping constant of 1 kN-s/m. The floor on which the machine is mounted is subjected to a harmonic disturbance due to the operation of an unbalanced engine in the vicinity of the grinding machine. Find the maximum acceptable displacement amplitude of the floor if the resulting amplitude of vibration of the grinding wheel is to be restricted to 106 m. Assume that the grinding machine and the wheel are a rigid body of weight 5000 N. (20 Marks)

Problem 3 Recreate the response history from HW #2, Problem 3 using the CDM tool you created as part of Problem 2. Select a time step, At that will produce an error that is smaller than 2% compared to the analytical solution. Check the three requirements per Module 3, Slide 10! Note: As part of the accuracy requirement, you may plot the analytical solution in the same figure as your numerical solution. Additionally, you may also plot the difference between the two, which can be interpreted as the error.

1. Consider two masses m, connected to each other and to two walls by three springs, as shown in the figure. Both masses feel a damping force-bv. The three springs have the same spring constant k. Find the general solution for the positions of the masses as functions of time. Assume underdamping.

2. Same as in problem 1 above, but let us now assume there are no damping forces anymore. However, the left mass is now driven by a driving force F.cos(2wt), and the right mass has a driving force 2F, cos(2wt), where w is the square root of k/m. Find the particular solution for the positions of the masses as functions of time. Explain why your answer makes sense.

3. Fourier Series: (a) Consider the function f(t) which is periodic with period T = 2 and looks as follows from 0 to 2x:

4. Two horizontal frictionless rails make an angle with each other, as shown in the figure. Each rail has a bead of mass m on it, and the beads are connected by a spring with spring constant k and relaxed length zero. Assume that one of the rails is positioned a tiny distance above the other, so that the beads can pass freely through the crossing. Find the general solution for the positions of each mass as functions of time.

For the system shown in the figure a) determine the equation of motion of the equivalent linear system (at the mass "m" location) b) find the natural frequencies (wn and wa). c) the frequency ratio r. d) obtain the term X/ost using the graph (state the used points on the graph). Take: k=(1000+Z) N/m, where Z is the last digit of your PMU ID no. m=10 kg, c=380 N s/m, r1=0.2 m, r2=0.4 m. Jpulley = 1.0 kg m², forced vibration w=12 rad/sec

A cylinder of known mass m and known mass moment of inertia Jo= mr²/2 is free to roll without slipping but is restrained by a known damper c and two springs of known stiffness k, and k; as shown. a) Determine the equation of motion in terms of the rotation angle 8. b) Determine the value of a that maximizes the natural frequency of the system.

A bar of known length L and known mass m is connected to a spring of known stiffness k. A known moment Mo is applied as shown. The spring is unstretched when the bar is vertical and it has a free length of Lo. The spring always remains horizontal as the bar rotates. a) Determine the full nonlinear equation of motion for the bar in terms of its angle of rotation measured from the bar's vertical position. b) Linearize around the bar's vertical position.

The system shown has a natural frequency, fn, of 5Hz, m-10kg, J.-5kg-m², r=0.10m, and r=-0.25m. If the mass is displaced 0.01m to the right and released from rest, the amplitude of the resulting free vibration reduces by 80% in exactly 10 cycles. Gravity may be taken as being into the page. a) Determine the equation of motion in terms of the motion of the mass x(t). b) Determine the value of the stiffness k. c) Determine the value of the damping coefficient c.