1-CEE106 Ch1 Fundamentals Ms

7/29/2019 1-CEE106 Ch1 Fundamentals Ms

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Chapter 1

Fundamentals of Vibrations

- Professor Masoud Sanayei 1


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Harmonic Motion:

The simplest form of periodic oscillatory motion is harmonic motion.

Periodic Motion

where, = Period [sec]

x(t)

tA

m

= period k

Optic filmstrip recorder

x t x t x t n

2


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Consider the following notations as:Frequency, [Hz] or cps

Circular frequency, [rad/sec]

Amplitude of vibration

Then, the Simple harmonic equation can be written as:

2

x t A sin t A sin t

Cos tP

A15

2

3

4

O

Sin t

=t 1

2

3

45 t

x(t)A

O

t x(t)

1 0 0

2 /4 A

3 /2 0

4 3/4 -A

5 0

1f

22 f

A

3


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Time history relation between displacements, velocities and accelerations:Displacement

Velocity

Acceleration

Let

x t A sin t

x t Acos t A sin t2

2 2x t A sin t A sin t

4

1.57 Rad / Sec

4 Sec

f 0.25 Hz


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Vector phase relationship between displacement, velocity and acceleration:(Assume = 1.57 rad/sec)

Also note:

(1) D.E. for free vibration

2

x x

2x x 0

5


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Exponential form: Trigonometric functions and exponential functions are

related as:

Harmonic motion can be represented as:

Complex sinusoid

Z satisfies differential equation (1)

Conjugate of z is z*

z* is rotating in the negative direction with angular speed of. Then

ie cos i sin

i tz Ae A cos t i sin t z x iy =t

Re

Im

z=A eitO

2 i t 2z A e z

* i tz x iy Ae A cos t i sin t

* i t1

x z z Acos t Re Ae2

* i ti

y z z ASin t Im Ae2

y

ZxZ+Z*

Att

Z*6


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Mathematical advantages of exponential operations:

Let and

Then,

Exponential representation is for mathematical convenience.

1i

1 \1z A e 2i

2 2z A e

1 2i1 2 1 2z z A A e

1 2i1 1

2 2

z Ae

z A

n n inz A e 1 1

in n n

z A e

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1-CEE106 Ch1 Fundamentals Ms

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