Engineering Vibrations

Introduction

Vibration

• is a branch of science of engineering that deals with repetitive motion of mechanical system from machine parts to large structures

• An interplay between potential energy and kinetic energy

Simple Harmonic Motion: Springs

One of the simplest types of vibrational motion is that of an object attached to a spring. Provided that damping forces are absent, the classic spring system becomes a system subjected to simple harmonic motion

• Note that while within the elastic limit of the spring the mass held at the other and causes a proportional elongation on the spring (Hooke’s Law)

F α xF = k x

Wherek = spring constant (N/m)F = force exerted by spring (N) x = spring elongation/compression

General Assumption on the loading of the spring

• When at rest, the system held in equilibrium because the force of the spring is equal to the weight or load

• When in motion, the unbalanced force on the spring systems causes that motion

Situation of Spring-Mass System in SHM

1. At rest (ƩF = 0)

fs = w = mg

2. Moving in downward direction

ss

ss

ss

ss

xk

mgx

k

kxmgx

kxkxmg

ffw

'

'

''

3. Moving in an upward direction

ss

ss

xk

mgx

kxmgkx

fswfs

'

'

'

Elastic Potential Energy

Sample Problem

1. A block with mass of 5.0 kg is attached to a horizontal spring with spring constant k = 4.00 x 102 N/m. The surface the block rests upon is frictionless. If the block is pulled out to xi = 0.0500 m and released,a. Find the speed of the block at the equilibrium pointb. Find the speed when x = 0.0250 m, andc. Repeat part (a) if friction acts on the block, with coefficient µk = 0.150.

2. A block weighing 96.5 lbf is dropped from a height of 4ft upon a spring whose modulus is 100 lb / in. Calculate the total deflection of the spring.

Homework1. A 50.0 kg acrobat drops from a height 2 meters straight down

onto a springboard with a force of constant 8.00 X 103 N/m, as shown. By what maximum distance does she compress the spring?

2. A 0.500 kg block rests on a horizontal, frictionless surface as shown in the figure. The block is pressed back against a spring having a constant of k = 625 N/m, compressing the spring by 10.0 cm to point A. Then the block is released. a. Find the maximum distance d the block travels up the frictionless incline if θ = 30.0°b. How fast is the block going when halfway to its maximum height?

Spring Connections

1. Springs attached in parallel

2. Spring attached in series

Prelude to SHM: The Trigonometric Functions

• Sine Function

• Cosine Function

Simple Harmonic Motion

Sample Problems

An object oscillates with SHM along the x-axis. Its position varies with time according to the eq. x = 4m cos (πt + π/4). Where t is in second and the angles in the parenthesis are in radian

a. Determine the amplitude, frequency and period.b. Calc. the vel. And acc. Of the object at any time (t)c. Det. the position, velocity and acceleration of the object

at t=1sd. Det. the max speed and acceleration of the object.e. Find the displacement of the object between t=0 and

t=1s

A 50kg block moves between vertical guides as shown. The block is pulled 40mm down from its equilibrium position and released. For each spring arrangement, determine the period of vibration, the maximum velocity and maximum acceleration of the block.

Seatwork1. A 50kg block moves between vertical guides as shown. The

block is pulled 40mm down from its equilibrium position and released. For each spring arrangement, determine the period of vibration, the maximum velocity and maximum acceleration of the block.

2. A block of unknown mass is attached to a spring with k=6.5 N / m. It undergoes SHM with an amplitude of 10cm. When the block is halfway between its equilibrium position and the endpoint, its speed is measured to be 30 cm/sec. Calculate a.) mass of block b.) the period c.) the maximum acceleration