Mechanical Vibration Analysis: Displacement, Acceleration, and Forces

This document presents a comprehensive solution to a mechanical vibration homework assignment. It addresses the problem of determining displacement and maximum acceleration for a mass subjected to a force, incorporating concepts like damping factor, natural frequency, and Newton's second law of motion. The solution details the step-by-step calculations, including the determination of the damping ratio, natural frequency of damped vibration, and the force generated by a bronze ball. The analysis also includes the calculation of displacement and acceleration using the provided formulas and data. The assignment explores the concept of impulse response and how to break up complicated forces into sums of simpler forces, compute the response and add to get the total solution. The final results for displacement and acceleration are presented, offering a clear understanding of the mechanical vibration problem and its solution.

MECHANICAL VIBRATION

Given data,
M = 2 Kg
mo = 0.15 kg
K = 95 N/ m
C = 1.95
h = 2.15
The above figure is flow diagram of problem, which is base one mechanical problem. With
the help two given data, the two thinks are required to determine; the first one is resulting
displacement and maximum acceleration for the mass m. In order to determine the respective
value, following steps are required followed are as follow:
Damping factor: it can be defined as the ratio of the load impedance to the amplifier output
input impedance. With the help of this, it is easy to determine the ability of amplifier to stop
the cone from moving. For the following problem,
Where,
Ccritical = damping coefficient of a system
£ = System damping ratio

£ = C/ Cc
= 1.95 ------------ (1)
On another hand, the maximum acceleration produce in the mass M can be determined with
the help of Newton 2nd law. According to newton second law of motion, it can be stated that
the acceleration of an object is rely on two factors; the mass of the object and the net force
acting upon it. Mathematically, it can be presented as
F= m* a
Where,
M = mass of an object
a = acceleration produce into the body
F= force asserted on an object
For the following problem,
a = F/m
a = F/2 --------------- (2)
The next steps is to determine natural frequency of damped vibration, which can be
elucidated as a set of motion in a resonant mechanical structure and left to its own motion
than it will continue oscillate. Mathematically, it can be presented as
Where,
Wd = undammed natural frequency of a system
Wn = Damped natural frequency of a system
For the following problem, it can be stated that,

Wn = root of k/m
= root of 95/2
= 6.9 rad/s ------------------- (3)
From the above data, it is required to determine force generate d by the bronze ball, which
can be calculated by applying newton second law:
F = m*g
= 0.15* 9.8
= 1.47 N ------------------- (4)
By evaluating equation (4) & (2), we get
a= f/m
= 1.47/2
= 0.735 m/s2
At last, in order to determine resulting displacement, it is necessary to determine frequency
produced into the spring, which can be determined through formula i.e.,
fn = 1/2*3.14 root of k/m
= 1/2*3.14 root of 95/2
= 1/6.28*6.9
= 1.09 --------- (5)
Through evaluation, equation (5), (1) and (3), it can easily obtained the displacement into the
system, which result
Displacement = 0.52 m

Due to exerted force caused by bronze ball, a perfectly plastic collision occurred that give a
acceleration to body mass M. It has been witnessed that the exerted forces produce a kinetic
enrgy in the spring, which can be determined through k*x2 where x is a displacement
occurred due to exerted force. Through critical analysis of mechanical vibration the result
that obtained are as follow:
Displacement = 0.52 m
Acceleration = 0.735 m/s2