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CHAPTER 8: Vibration and DynamicsAn unconstrained rigid body moves under load but doesn’t

deform (subject of undergraduate rigid-body dynamics).

A constrained rigid body doesn’t move and doesn’t deform under load.

An unconstrained elastic body moves under load and deforms (e.g., an airplane).

A constrained elastic body doesn’t undergo rigid-body motion but deforms under load.

If the load varies in time not very slowly or is suddenly applied, a vibration/dynamic analysis is required.

If the load is cyclic, so is the deformation. Such deformation is called vibration.

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Examples of Vibration; Quantities of Interest

If the load is transient, random, or applied suddenly, the motion ofthe elastic body is called the transient response.

Some engineering problems where vibration is of concern:inside a car due to pavement roughness,aircraft wings and fuselage due to atmospheric disturbances,vibration of a shop floor due to rotating machinery, vibration of buildings due to an earthquake.

Quantities of interest in a vibration analysis:the largest acceleration in the structure,the largest stresses, the largest displacement, whether a structure will resonate under the loading.

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Matrices Involved in Dynamic Analysis Elastic forces are involved

⇒ a stiffness matrix is used (as in static analysis)

Inertia forces are involved ⇒ a mass matrix is used

Vibration may die out rapidly or slowly (i.e., damping is involved) ⇒ a damping matrix is used

The simplest vibrating system is a mass hanging at the end of or attached to a spring.

It is very instructive to study such a system.

Knowledge of a single position variable is adequate to know the position of the whole system at any instant of time

→ single degree of freedom(sdof) system

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Undamped Single dof Systems

u(t) is the dof to describe the motion. r(t) is the external load.

Applying Newton’s law to the above free-body diagram,

This is the equation of motion.

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Damped Single dof Systems

Damping is represented by a viscous dashpot whose resistance to motion is proportional to velocity. (Think of a bicycle pump!!)

The equation of motion is

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Support Excitation

Occurs when a structure in which a system is mounted vibrates: an electronic package mounted on a fuselage structureearthquakea spring/mass type accelerometer

u, s: displacements of the mass relative to the support and to a fixed reference frame, respectively. Then,

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Multiple dof SystemsIn many practical systems, knowledge of multiple position

variables is required to describe the position of any point in the system at any time.

→ multiple degree of freedom(sdof) system

The equation of motion in terms of the displacement vector D is

where K: stiffness matrix

C: damping matrix

M: mass matrix

R: load vector (forces and moments applied at the nodes of the structure)

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Free Vibration of sdof Systems

Vibration when there is no forcing function r(t).

When a system is disturbed while at rest (i.e., when an initial condition is applied to it), it vibrates freely.

The system doesn’t just vibrate with an arbitrary frequency. It vibrates with a frequency which is an inherent property of the system.

This frequency is called the natural frequency of the system.

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Free Vibration of Undamped sdof Systems

The free vibration is a harmonic motion:

where are the amplitudeof motion and the natural frequency of vibration, respectively.

ω : radians/sec f = ω /2π = cyclic frequency in Hz (cycles/sec)T=1/f = period of vibration

To find the value of ω for the system, substitute u into the equation of motion:

ω and u

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Free Vibration of Damped sdof Systems

When damping is present, the vibration dies out so it is a transient motion.

If the viscous damping coefficient c is smaller than a critical value cc , the motion is oscillatory but the amplitude decays in time.

The damped natural frequency of vibration ωd is given by

where ξ = the fraction of critical damping = the damping ratio

ξ is usually small so that ωd ≅ω..

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Decay Rate

For small damping, the ratio of two consecutive displacement peaks is given by

If ξ =0.1, u2/u1 ≅ =0.5

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ExampleGiven a damped sdof system under base (support) excitation.

k = 1000 N/m, m = 10 kg, c = 20 kg/s, ug= 0.05sin30t

Derive the equation of motion and give the undamped and damped natural frequencies of the system.

Solution: Let u be the displacement of the mass relative to the system:

Newton’s law applies to absolute acceleration:⇒

Rearranging, we get the EOM:

maf =

)( guumuckumaf &&&&& +=−−⇒=

tuuu 30sin)30)(05.0(1010002010 2=++⇒ &&&

maf =

tuuu 30sin451002 =++ &&&

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Example (cont)

tuuu 30sin451002 =++ &&&

In this form, square of the undamped natural frequency is the coefficient of the u term in the EOM:

⇒ ω =(100)1/2 =10 rad/s , f = ω /2π = 1.59 Hz

cc=2mω =200 kg/s ⇒ ξ = c/cc = 20/200 = 0.1

rad/s 95.901.01101 2 =−=−= ξωωd

Hz58.12/ and == πωddf

CHAPTER 8: Vibration and DynamicsExamples of Vibration; Quantities of InterestMatrices Involved in Dynamic AnalysisUndamped Single dof SystemsDamped Single dof SystemsSupport ExcitationMultiple dof SystemsFree Vibration of sdof SystemsFree Vibration of Undamped sdof SystemsFree Vibration of Damped sdof SystemsDecay Rate