two degree of freedom systems...two degree of freedom –dynamic absorbernotes: 1. it can be seen,...

Two Degree of Freedom Systems

Dr./ Ahmed NagibDecember 16, 2019

Two Degree of Freedom Systems

Real systems can not be modelled as one degree

of freedom system, and are modelled by using

multiple degree of freedom systems.

We will extend the previous chapters for two

degree of freedom system.

Examples Two Degree of Freedom

Systems

Systems that require two independent coordinates

to describe their motion are called two degree of

freedom systems. Some examples of systems

having two degrees of freedom are shown.

Examples Two Degree of Freedom

Systems

Two Degree of Freedom – Undamped System

Example 1

Two Degree of Freedom – Undamped System

Example 2

Two Degree of Freedom – Forced Vibration


Example

Find The Steady state response of the following System


Example

Two Degree of Freedom – Dynamic Absorber


A machine or system may experience excessive vibration if

it is acted upon by a force whose excitation frequency

nearly coincides with a natural frequency of the machine or

system. In such cases, the vibration can be reduced by

using a vibration neutralizer or dynamic vibration absorber,

which is simply another spring-mass system. The dynamic

vibration absorber is designed such that the natural

frequencies of the resulting system are away from the

excitation frequency. The analysis of the dynamic vibration

absorber will be considered by idealizing the machine as a

single degree of freedom system.


We are primarily interested in reducing the amplitude of the machine (X1)

In order to make the amplitude of zero, the numerator of Eq. (X1) should

be set equal to zero.

This gives

If the machine, before the addition of the dynamic vibration absorber,

operates near its resonance, .Thus if the absorber is

designed such that

where


The equations of X1 and X2 can be rewritten as


As seen before, X1 = 0 at ω =ω2 =ωa. At this frequency :

𝑋2 = −𝑘1𝑘2

𝛿𝑠𝑡 = −𝑘1𝑘2

×𝐹𝑜𝑘1

= −𝐹𝑜𝑘2

This shows that the force exerted by the auxiliary spring is opposite to the impressed

force (k2 X2 = - Fo) and neutralizes it, thus reducing X1 to zero. The size of the dynamic

vibration absorber can be found from

k2 X2 = m2 ω2 X2 = - Fo

Thus the values of k2 and m2 depend on the allowable value of X2


The two peaks correspond to the two natural frequencies of the composite system. The

values of the two natural frequencies can be found by equating the denominator of the

following equation to zero

, which leads to:

1 +𝑘2𝑘1

−𝜔

𝜔𝑠

2

1 −𝜔

𝜔𝑎

2

−𝑘2𝑘1

= 0 𝑜𝑟,

𝜔4 − 𝜔𝑠2 + 𝜔𝑎

2 +𝑘2𝑚1

𝜔2 + 𝜔𝑠2. 𝜔𝑎

2 = 0


The roots of the previous equation is given by

which can be seen to be functions of and


Notes:

1. It can be seen, that ωn1 is less than and

ωn2 is greater than the operating speed

(ω) of the machine. Thus the machine

must pass through ωn1 during start-up

and stopping. This results in large

amplitudes.

2. Since the dynamic absorber is tuned to

one excitation frequency (ω), the steady-

state amplitude of the machine is zero

only at that frequency. If the machine

operates at other frequencies or if the

force acting on the machine has several

frequencies, then the amplitude of

vibration of the machine may become

large.


Notes:

3. It can be seen from equations (31) and

(32) that the amplitude of the absorber’s

mass (X2) is always much greater than that

of the main mass (X1). Thus the design

should be able to accommodate the large

amplitudes of the absorber mass.

4. Since the amplitudes of m2 are expected to

be large, the absorber spring (k2) needs to be

designed from a fatigue point of view.


Notes:

5. The variation of ωn1/ωa and ωn2/ωa as

functions of the mass ratio m2/m1 are shown

for three different values of the frequency

ratio ωa/ωs. It can be seen that the difference

between ωn1 and ωn2 increases with

increasing the ratio of m2/m1.


• If ω hits ωn1 or ωn1 resonance occurs

• Using X1

Xst<1, defines useful operating

range of absorber

• In this range some absorption still occurs

Absorber Zone


Absorber Zone

This illustrate that the useful

operating range of absorber

design is

(0.908 𝜔a < 𝜔 < 1.118 𝜔a) .

Hence if the driving

frequency drifts within this

range, the absorber design

still offers some protection to

the primary system by

reducing its steady state

vibration amplitude.


Example

two degree of freedom systems...two degree of freedom –dynamic absorbernotes: 1. it can be seen,...

Documents