CHAPTER 8: Vibration and Dynamics
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1 CHAPTER 8: Vibration and Dynamics An 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.
Transcript of CHAPTER 8: Vibration and Dynamics
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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
