an introduction to mechanical vibration

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An Introduction to Mechanical

Vibration

Single Degree of Freedom System

A single degree of freedom system consists of a mass, a spring, and a

damper if the system is modeled as a damped system. The spring

is modeled as a linear spring, which provides a restoring force.

The damper is modeled as a viscous damper, which provides a

damping force proportional to a relative displacement and acting

in the direction against a velocity vector. If there is a driving

force acting on the mass, the system vibrates under the driving

force, which is called forced vibration. Otherwise, the system may

vibrate under initial displacement and/or initial velocity, which is

called free vibration.

Undamped Free Response

Damped Free Response

Forced Response

Undamped Free Response
http://www.mfg.mtu.edu/cyberman/machtool/machtool/vibration/free.htmlhttp://www.mfg.mtu.edu/cyberman/machtool/machtool/vibration/damped.htmlhttp://www.mfg.mtu.edu/cyberman/machtool/machtool/vibration/forced.htmlhttp://www.mfg.mtu.edu/cyberman/machtool/machtool/vibration/free.htmlhttp://www.mfg.mtu.edu/cyberman/machtool/machtool/vibration/damped.htmlhttp://www.mfg.mtu.edu/cyberman/machtool/machtool/vibration/forced.html


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Free vibration occurs because there is an initial disturbance to the

system. The initial disturbance is referred to as an initial condition,

which can be either a displacement of the mass, an initial velocity of the

mass, or both. Frequency of the system is determined by the mass and

spring constant of the system, initial conditions do not affect thevibration frequency of an undamped system, therefore it is called the

natural frequency. Vibration amplitude and the phase angle are also

determined by initial conditions. Theoretically, an undamped free

vibration system will vibrate forever once it is started.

DANIEL J. INMAN, Engineering Vibration, Prentice Hall, Englewood Cliffs, New Jersey,1994

Damped Free Response

Physically, there is no vibrating system that vibrates forever, that

means there is always some kind of damping in the system that

dissipates energy. For mathematical simplicity, the damping is modeled

as viscous damping. Depending on the magnitude of damping, a

damped system can be underdamped, critically damped or

overdamped. The critical damping coefficient is determined by the

system's mass and spring constant. Under critical damping, the

damping ratio is unity. Critical damping separates nonoscillatory

motion from oscillatory motion. When the damping ratio is greater than


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1, which is called overdamping, the system does not oscillate. For a

damping ratio less than 1, which is called underdamping, the system

oscillates with decaying magnitude, as shown in the figure below. For

most physical system, damping ratios are less than 1. Actually, most

physical systems have damping ratio less than 0.1. With damping in thefree vibration system, the mass always restores its equilibrium position

even it is disturbed. The greater the damping, the less time it takes to

restore its equilibrium position. So in most cases, adequate damping is

desireable.

DANIEL J. INMAN, Engineering Vibration, Prentice Hall, Englewood Cliffs, New Jersey,1994

Forced Response of SDOF


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When there is a driving force acting on SDOF system, the system's

vibrating frequency is determined by the driving force frequency and

damping ratio. We are mostly interested in the magnitude and phase

shift of the response. These characteristics are indicated in the system's

Frequency Response Function. When driving frequency is close to thesystem's natural frequency, forced vibration magnitude is large. When

driving frequency is equal to natural frequency, forced vibration

reaches its maximum magnitude, this is called resonance. Under

resonance, the response's phase shift is 90 degree. The damping ratio

also has an effect on FRF. The greater the damping ratio, the less the

forced vibration magnitude.


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Isolating Dynamic Forces from Transmitting to Ground


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Vibration Absorber

Vibration absorber is used to protect the primary system from steady-

state harmonic disturbance. By attaching the absorber to the

primary system which is modeled as a SDOF system, the new

system becomes two DOF system as shown in the model below.

Depending on the driving frequency on the original system, the

absorber needs to be carefully tuned, that is, to cho0se adequate

value of the absorber mass and stiffness, so that the motion of the

original mass is a minimum.

Vibration Absorber Tuning Simulation SoftwareVibration absorber model:

Vibration Damping

What is damping? Damping is the energy dissipation properties of a

material or system under cyclic stress.

Active damping and passive damping:

Active damping:

Active damping refers to energy dissipation from the system by

external means, such as controlled actuator, etc.

Passive damping:

Passive damping refers to energy dissipation within the structure

by add-on damping devices such as isolator, by structural joints

and supports, or by structural member's internal damping.

Material damping and system damping:

Material damping:

Energy dissipation in a volume of macro-continuous media.

System damping:

Energy dissipation in the total structure. in addition to damping

due to materials, it also includes energy dissipation effects of

joints, fasteners, and interfaces.
http://www.mfg.mtu.edu/marc/primers/machtool/vibration/%20tuneabsorb.html


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Damping Measurement:

There are many methods for measuring the damping of a vibtation

system. Logarithmic decrement method and bandwidth meathod are

introduced here.

Logarithmic decrement method is used to measure damping in time

domain. In this method, the free vibration displacement amplitude

history of a system to an impulse is measured and recorded. A typical

free decay curve is shown as below.Logarithmic decrement is the

natural logarithmic value of the ratio of two adjacent peak values of

displacement in free decay vibration.

To estimate damping ratio from frequency domain, we may use half-

power bandwidth method. In this method, FRF amplitude of the system

is obtained first.Corresponding to each natural frequency, there is a

peak in FRF amplitude. 3 dB down from the peak there are two point

corresponding to half power point, as shown in the figure below. The

more the damping, the more the frequency range between this two

point. Half-power bandwidth BD is defined as the ratio of the frequencyrange between the two half power points to the natural frequency at

this mode.


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changes the stiffness and damping between the source of

vibration and the device that is to be protected from vibration.

Isolating Vibration Transmitted from Ground

Another way to isolate vibration is to keep the vibration source from transmitting to

the base. This is done by placing an isolator between the vibration source and the

base. The model for this kind of vibration isolation is shown as below. We are most

interested in isolating the force from transmitting to the base. Force transmissibility is

a measure to characterize how good the isolator works.

Isolating Dynamic Forces from Transmitting to Ground

Vibration Absorber


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A vibration absorber is used to protect the primary system from steady-

state harmonic disturbance. By attaching the absorber to the

primary system which is modeled as a SDOF system, the new

system becomes a two DOF system as shown in the model below.

Depending on the driving frequency of the original system, theabsorber needs to be carefully tuned, that is, to choose adequate

values of absorber mass and stiffness, so that the motion of the

original mass is a minimum.

Vibration Absorber Tuning Simulation Software

Vibration absorber model:

Vibration Damping

What is damping? Damping is the energy dissipation properties of a

material or system under cyclic stress.

Active damping and passive damping:

Active damping:

Active damping refers to energy dissipation from the system by externalmeans, such as controlled actuator, etc.


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Passive damping:

Passive damping refers to energy dissipation within the structure by add-on

damping devices such as isolator, by structural joints and supports, or by

structural member's internal damping.

Material damping and system damping:

Material damping:Energy dissipation in a volume of macro-continuous media.

System damping:

Energy dissipation in the total structure. in addition to damping due to

materials, it also includes energy dissipation effects of joints, fasteners, and

interfaces.

Damping Measurement:

There are many methods for measuring the damping of a vibtation system.

Logarithmic decrement method and bandwidth meathod are introduced here.

Logarithmic decrement method is used to measure damping in time domain. In this

method, the free vibration displacement amplitude history of a system to an impulse is

measured and recorded. A typical free decay curve is shown as below.Logarithmic

decrement is the natural logarithmic value of the ratio of two adjacent peak values of

displacement in free decay vibration.

To estimate damping ratio from frequency domain, we may use half-power bandwidth

method. In this method, FRF amplitude of the system is obtained first.Corresponding

to each natural frequency, there is a peak in FRF amplitude. 3 dB down from the peak

there are two point corresponding to half power point, as shown in the figure below.

The more the damping, the more the frequency range between this two point. Half-

power bandwidth BD is defined as the ratio of the frequency range between the two

half power points to the natural frequency at this mode.


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an introduction to mechanical vibration

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