Chapter 3 Student

7/24/2019 Chapter 3 Student

1/25

Chapter 3Forced Vibrations of SDOF

Dr.-Ing. Azmi Mohamed Yusof

Faculty of Mechanical Engineering

At the end of this chapter students will be able to state, derive and apply

the fundamental principle of vibration involving:-

Forced vibration of undamped system with harmonic excitation

Forced vibration of damped system with harmonic excitation

Various applications, specifically the system with rotating unbalanced

and method for vibration isolation

MEC 521 VIBRATIONS : Dr. Azmi M.Yusof 2

Course outcome


2/25

Undamped system with harmonic oscillation

Consider the simplest vibrating system as shown infig. (a)

The time varying external force is given by

cos

Applying Newtons second law for the FBD shown in

fig. (b),

cos

or

cos (1)

Where is the natural frequency of the mass, while is the frequency of the external force


Forced Vibrations of SDOF

=0

=0

The solution to this second order, linear, nonhomogeneous ODE containcomplementary solution , and particular solution

---- (2)

The complementary solution :

0

cos sin ---- (3)

Let assume the particular solution as :

cos ---- (4)

Obtaining the velocity and acceleration term for eqn. 4 gives

cos

sin

cos




3/25

Substituting the above equations into eqn. (1)

cos

cos

cos

;

From eqn. (2), , the final solution gives

cos sin

cos



The complementary solution defines the freevibration (see figure (a)) and it will typically dampen

out.

Therefore xc is referred to as transient

The particular solution describes the force vibration

caused by the applied load (fig. (b))

There resultant vibration is shown in fig. c

As the free vibration will in time dampen out, the

remaining vibration will be the force vibration

Thus, is called steady state (see fig. (d)).




4/25

We need to determine the constant D & E by using

initial conditions.

Suppose 0 and 0 , then wehave

and

By substituting into equation 1

cos

sin

cos

If the static deflection of the mass due to force Fo is

given by

, the maximum amplitude can be

written as,

1

1



The ratio, is known as the magnification factor

(M)

M The ratio of the amplitude of vibration to the

amplitude of zero frequency deflection

The plot

or Magnification factorversus the

frequency ratior

is shown in fig. (a)

The asymptote occurs at r =1, thus the system

response can be of three types

When 0 1

When

= 1




5/25

Case 1, when 0 natural frequency

of forced vibration

The amplitude of forced response > the static deflection

The harmonic response of the system xp(t) is in phase with the external

force. See the plots as shown in the figure



External force excitation System response

Case 2, when > 1

The natural frequency of the free vibration

response < natural frequency of forced

vibration

cos , where

and has opposite sign, 180o out ofphase

When

, then X 0 : the response at

very high frequency is close to zero.



External force excitation

System response


6/25



External force excitation

System response

Case 3, when

= 1

The condition when = n is called resonance

Amplitude of oscillation increases linearly with

time or infinite

Undesirable condition which may harm overall

system

The equation of motion is

The response of the system at resonance is:

Example

A weight of 50N is suspended from a spring of stiffness 5000 N/m and is

subjected to a harmonic force of amplitude 40N and frequency of 4 Hz.

Determine

a) the extension of the spring due to suspended spring

b) the static displacement of the spring due to the maximum applied

force

c) the amplitude of the forced motion of the weight




7/25

Solution



Example

A 50 kg mass is hanging from a spring of stiffness 5 x 104 N/m. A

harmonic force of magnitude 100 N and frequency 100 rad/s is applied to

the system. Determine

a) the amplitude of the forced response

b) the natural frequency of the system




8/25

Solution



Example

A reciprocating pump, having a mass of 68 kg, is mounted at the middle of a

steel plate of thickness 1 cm, width 50 cm, and length 250 cm, clamped along

two edges as shown in the figure below. During operation of the pump, the plate

is subjected to a harmonic force, 220 cos62.832 N. Determine theamplitude of vibration of the plate. (Take the plate equivalent stiffness of the

beam as

, and Youngs modulus of the beam as 200 Gpa).




9/25

Solution



Forced vibration with viscous damping

Consider a viscously damped SDOF spring-mass

system as shown in fig. (a).

Suppose the external force applied is in the form of

sin

Using Newtons second law of motion for FBD shown in

fig. (b),

cos or

cos

The particular solution can be assumed in the form

cos sin The velocity and acceleration terms become,

sin

cos




10/25

Substitute these results into equation 1,

cos sin cos

Extracting the algebraic equation involving term A and B,

and 0

Dividing both equations with k

1 2 and 2 1 0

Where

,

and

Solve the above equation for A and B yield,

,

The result for a steady state equation

1 21 cos 2 sin



It can also be written in a generalized form

Introducing the magnification factor M,

Where the phase angle is:

The plot for M versus r, and versus r are shown in the figure




11/25


Finally, The complete solution is given by

cos cos

and can be determined from the initial conditions 0 0

See textbook page 275 for complete solution (long equation).




12/25

Example

A SDOF spring-mass-damper system is subjected to a harmonic force.The amplitude is found to be 25 mm at resonance and 10 mm at a

frequency 0.75 times the resonant frequency. Determine the damping

ratio for the system.

Solution



Example

A SDOF damped system is composed of a mass of 10 kg, a spring

having a constant of 2000 N/m, and a dashpot with damping constant of

50 Ns/m. The mass of the system is acted on by a harmonic force F = Fo

sin t having a maximum value of 250 N and a frequency of 5 Hz.

Determine the complete solution for the motion of the mass.

Solution




13/25



Example

A 25 kg mass is mounted on an oscillator pad whose stiffness is 5 x 105

N/m. When the system is subjected to a harmonic excitation of

magnitude 300 N and frequency 100 rad/s, the phase different between

the excitation and the steady state response is 25o. Determine:

a) the damping ratio of the isolator pad

b) the isolator pads maximum deflection due to this excitation




14/25



Response due to harmonic motion ofthe base/support motion

Sometime the base or support of a

spring-mass-damper-system undergoes

harmonic motion.

Example of this situation is the car

moving on bumpy road as shown in the

figure.



Let y(t) denote the displacement of the base and x(t) the displacement of

the mass

The system is simplified in the figure shown. The subsequent analysis is

based on the FBD shown. The net elongation of the spring is

The relative velocity between two ends of the damper is

The equation of motion according to the FBD is obtained as:

k 0


15/25

Rearrange the equation

If sin, then cos sin

If sin and - and substitute into eqn. 1 then sin

where , and

Recall our steady state response for a spring-mass-damper system

sin

sin

Therefore,

sin

sin

Where



The amplitude of oscillation is obtained as

or

Rearrange the equation,

If Z is the relative displacement between the mass and the base, then




16/25

The maximum force transmitted to the base is given by



Example

The simple model of motor vehicle

is shown in the figure, that can

vibrate in the vertical direction

while traveling over a rough road.

The vehicle has a mass of 1200

kg. The suspension system has a

spring constant of 400 kN/m and a

damping ratio of 0.5. If the vehicle

speed is 20 km/h, determine the

displacement amplitude of the

vehicle. The road surface varies

sinusoidally with an amplitude of Y= 0.05m and a wavelength of 6m.




17/25

Solution



Class exercise

A 50 kg mass is attached to the base through a spring in parallel with a

damper as shown in fig. below. The base undergoes a harmonic

excitation of y(t) = 0.20 sin 30t. The stiffness of the spring is 30000 N/m

and the damping constant is 200 Ns/m. Determine a) the amplitude of the

masss absolute displacement, b) the amplitude of its displacement

relative to its base [ans: a) 0.38m; b) 0.56m]



m

k c

y(t) = 0.20 sin 30t

x(t)


18/25

Solution



Class exercise

A racing car is modeled as a SDOF damped system vibrating in the

vertical direction. The elevation of the road is assumed to vary

sinusoidally. The distance from peak to through is 0.2 m and the distance

between peaks is 70m. The natural frequency of the system is 2 Hz and

the damping ratio of the damper is 0.15. Determine the amplitude of

vibration of the racing car at a speed of 120 km/h. [ans: 1.06 m]




19/25

Solution



Rotating unbalanced

Unbalanced forces in many rotating

mechanism is the common sources of

vibration excitation.

Consider the rotating component is

mounted in bearing and rotates

counterclockwise.

Variables:

M = total mass of the system

m = eccentric mass located at e from the

center of rotation.

x = the displacement of the machine in

vertical direction

= angular speed

t = time


Forced Vibrations of SDOF - Applications


20/25

The component of the displacement of the eccentric mass in vertical

direction is given by:

sin

the acceleration is obtained as

sin

The total inertia forces of the machine

Applying Newtons second law of motion

and using equation 3 yield,

sin



Comparing this equation with fundamental equation

We obtain that

Recall the steady state solution for the system

sin , where

,

and

Finally the steady state response is obtained as:

/

1 2



Basic damped forced equation Equation 4

sin

sin


21/25

This equation can also be written as

1 2sin

Thus the amplitude of vibration is obtained as

1 2

Peak deflection of the mass M at resonance is given from

; thus

The force transmitted to the base

1 2

1 2sin



Example

An electric generator weighing 981 N and operating at 600 rpm is

mounted on four parallel springs of stiffness 5000 N/m each. Determine

the maximum permissible unbalance in order to limit the steady state

deflection to 6 mm peak-to-peak.

Solution




22/25



Example

An electric motor of mass M, mounted on an elastic

foundation, is found to vibrate with a deflection of

0.15 m at resonance. It is known that the unbalanced

mass of the motor is 8% of the mass of the rotor due

to manufacturing tolerances used, and the damping

ratio of the foundation is 0.025, Determine:

a) the eccentricity or radial location of the

unbalanced mass (e).

b) the peak deflection of the motor when the

frequency ratio varies from resonance

c) the additional mass to be added uniformly to the

motor if the deflection of the motor is to be reduced

to 0.1 m.




23/25

Solution



Vibration isolation and force transmissibility

To minimize excessive vibration excitation that may

contribute to system failure, an isolator is installed to

support the structure

The vibration isolator is typically designed on the

machine with flexible support

Good isolator design must consider proper selection of

the stiffness and damping coefficients.

Consider a spring-mass- damper system (fig.(a)) with

external force applied in the form of sin

The FBD for the system and the velocity triangle isshown in fig. (b) and (c).

The force transmitted to the support can be written as



sin

sin


24/25

Transmissibility, is defined as the ratio of the transmitted force to thatof the disturbing force

When the damping is negligible, then

1

1

To reduce the amplitude X without changing Td,

Isolated mass m can be mounted on larger mass M

The stiffness k must be increased to keep the ratio

constant



M

mk

The plot Td versus r is shown in the figure below

At region Td >1 for r < 1.41, the amplitude of transmitted force is greater

than the amplitude of applied force.

For r < 1.41, the transmitted force to the support can be reduced by

increasing the damping factor.

Vibration isolation is best accomplished by an isolator composed only of

spring elements for which r > 1.41 with no damping element used in the

system.



Transmiss

ibility,

Td


25/25

Example

A machine of mass 100 kg is mounted on springs anddamper as shown in the figure. The total spring stiffness is

50,000 N/m and the damping factor is 0.20. A harmonic

force, F = 200 sin 13.2t acts on the mass. Determine:-

a) the amplitude of the motion of the machine

b) its phase with respect to the existing force

c) the transmissibility

d) the maximum dynamic force transmitted to the foundation

e) the maximum velocity of the motion



Solution



Documents

Chapter 3 Student