Vibrations Homework Help

For the following system, a) obtain the equivalent system b) derive the equation of motion using Energy Method, where, KE=½ m x², and PE=½ k x², c) find the natural frequency.

In the following Figure a system consists of a block with mass m, spring K and a damper C; is attached to the floor where the block has a vertical oscillation x(t), and floor is vibrating vertically y(t) at 30 Hz. 1) For the case which system damping ratio is zero, what should be the natural frequency of the block vibration in order to reduce the relative motion of the block to floor by 12 dB. [2 marks] 2) If we consider a value for damper; C, which makes it equal to 10% of critical damping then what would beX/Y) ratio in this case. [1 marks] 3) Explain and justify whether adding more damping can reduce the vibration of the block in response to floor vibrating at 30 Hz [2 marks]

THREE SPRINGS 2 A MASS ARE ATTACHED TO A RIGID WEIGHTLESS BAR AB FIND THE NATURAL FREQ OF THE SYSTEM

(Forced Vibration)Determine the amplitude of the steady state displacement of the mass for forced vibration.

A fragile instrument and its housing, which together weigh 10 lbs, are mounted on four symmetrically located identical springs as a means of isolating the instrument from the vibration of the table supporting it. Measurements indicate that the table reaches a maximum amplitude of 0.004 inch somewhere in the frequency range of 20 to 30 Hz.Determine the spring constant k of each spring so that the amplitude of vibration of the instrument will be limited to 0.0002 inch as a maximum.

During an earthquake, the one-story frame structure shown below is subjected to a ground vibration amplitude of 0.1 m at 10 Hz. a) Determine the amplitude of motion of the girder.Assume the girder is rigid and the structure has a damping ratio of 0.03. b) What is the amplitude of acceleration of the structure?

A pump system is to be installed as shown in the diagram below. The pump and the basehave a combined mass of 150 kg. The pump operates at 1750 rpm with a maximum unbalance of 0.002 kg-m. The isolation system has an equivalent stiffness of 7.3 x 106 N/m,and a damping ratio of 0.1. Determine: a) The steady state displacement amplitude of the pump b) The force transmissibility ratio c) The force transmitted to the ground d) Should the physical damping of the isolation be decreased or increased to reduce the force transmitted to the base at the operating speed?

A SYSTEM HAS THE FOLLOWING CHARACTERISTICS: m =10 kg k=4000N/m 3=10% Xozc, *.=0 A forcing function F(t) = 200 t sin (10t) is applied DETERMINE THE xp (t) for the system

m=10kg k=4000n/m c=400n-s/m r= 0.1 m F(t) = 300 sin (20t) EVALUATE THE STEADY STATE RESPONSE Oss(t)

For the system shown below, determine the amplitude of the steady state displacement of point A for forced vibration.

A steel shaft of diameter d and length I is fixed at one end and carries a propeller of mass m and aradius of gyration of r, at the other end (see Figure 1). Data: d = 5 cm,l=1m, m = 100 kg, r, = 10 cm 1- Determine the fundamental natural frequeney of vibration of the shaft in axial vibration Determine the fundamental natural frequency of vibration of the shaft in torsional Task 1.3- Is it possible to change the system parameters (d, l, m, r,) such that the fundamentalfrequencies for both axial and torsional vibration become identical?[30%] If yes, what are the new parameter values? Explain the method and report your detailedanalysis leading to the new values. If no, explain in detail why it is impossible. I.Note 1: In Tasks 1.1 and 1.2, derive boundary conditions, start from the relevant equation of motionand apply the boundary conditions, find characteristic equation, solve it for natural frequencies, andfind the fundamental frequency.

m l^{2} \ddot{\theta}+c l^{2} \dot{\theta}+m g l \sin \theta=T \sin \Omega t

Compute the static deflection (mm) of the beam in Figure 01 neglecting thebeam mass. By constructing an equivalent single degree of freedom lumpedparameter spring-mass model, estimate the natural frequency (Hz) of freevibration of the beam-mass system.

Problem 3 (5 points, bonus) Using MATLAB's App Designer, build an app to study the forced vibration of a damped SDOF system with arbitrary initial conditions. The app should allow you to adjust the values of mass, stiffness, damping, driving force amplitude, driving force frequency.Plot the displacement as a function of time for the following conditions: 1- m = 1 kg, k = 100², c = 0,f(t) = 5N (zero driving frequency).

Test your car's suspension system. Is it under damped, over damped, or else? Is it time to replace them yet? Why?

Create an analytical 2DOF model of the pitching and bouncing of the car. Try to show the motion law (x(t)and theta(t) as the solution to the Equation of motion]. [Hint: you can find derivations online to help you.]

Develop the analytical model, EOM and x(t) and theta(t) for rolling of the car. What can trigger rolling?

Determine the pitching natural frequency of your car. [Hint: keep braking periodically and have an observer video record the pitching of the car. Wherever it's maximum...]