Ch1-1(1-2)c

# Ch1-1(1-2)c - MAE351 Mechanical Vibrations Lecture 1(Chap...

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1 MAE351 Mechanical Vibrations Lecture 1 (Chap. 1.1-1.2) 1 Basic Mechanical Elements of Vibrations x m c k 2 m = mass k = stiffness c = damping (Courtesy of Dr. D. Russell)

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2 Stiffness • From strength of materials recall: g x 0 x 1 x 2 x 3 f k nonlinear (or F) 3 10 3 N 0 20 mm x yield point quasi-linear linear (w/o spring offset) (or x) Free-body diagram and equations of motion x • Newton’s Law: y x k c m Friction-free surface 4 mg N f k f c
3 Stiffness and Mass Vibration is cause by the interaction of two different forces one related to position (stiffness) and one related to acceleration (mass). k x Displacement Stiffness ( k ) Mass ( m ) Proportional to displacement statics 5 m Mass Spring Proportional to acceleration dynamics Equation of Motion From Newton’s Law for this simple mass-spring system, the two forces must be equal i.e. f m = f k . m k x Displacement 6 Mass Spring This is a 2nd order differential equation (ODE) and all phenomena that have differential equations of this type for their equation of motion will exhibit oscillatory behavior .

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4 Examples of Single-Degree-of- Freedom Systems Pendulum Shaft and Disk m l =length Gravity g Torsional Stiffness k Moment of inertia J 7 Vibration = “Mechanical oscillation about a reference position, which is a result of dynamic forces of that system” Definition of Vibration X Total Deflection Vibratory Component, x(t) t Static Deflection, x st X total (t) = x st + x(t) Time
5 Solution to 2nd order DEs ) t A t x n sin( ) ( Lets assume a solution: x(t) Differentiating twice gives: ) ( sin( ) ( cos( ) ( t x t A t x t A t x n n n n n 2 2 - ) ) Substituting back into the equations of motion gives: t 9 Summary of simple harmonic motion x(t) Period 2 Amplitude A ) t A t x n sin( ) ( t 0 x Slope here is v 0 n T Maximum

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Ch1-1(1-2)c - MAE351 Mechanical Vibrations Lecture 1(Chap...

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