{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

SD-Lecture10-2DOF-Systems

SD-Lecture10-2DOF-Systems - CEG CEG-5065C Soil Dynamics...

This preview shows pages 1–6. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CEG CEG-5065C Soil Dynamics 5065C Soil Dynamics Lecture 10 Lecture 10 Systems with Two-Degrees of Freedom Luis A. Prieto-Portar 2009 The figure at right shows a simple non-damped mass and spring system with two-degrees of freedom. This simple system can be excited into vibration in two different ways: (1) A sinusoidal force is applied to the mass m 1 , thereby resulting in a forced vibration of the system, or (1) The system is set to vibrate by applying an impact force on the mass m 2 . Calculation of the system’s natural frequency. Consider the free-body diagram on the previous slide. The differential equations of motion are, 1 1 1 1 2 1 2 2 2 2 2 1 1 2 and Backsubstituting these solutions into the basic differential equations, + +- = +- = = = && && o o m z k z k ( z z ) and m z k ( z z ) Let z Asin t z B sin t ω ω ω ω ω ω ω ω 2 1 2 1 2 2 2 2 2 and Since A and B are not zero, the n +-- =- +- = o o A( k k m ) k B Ak ( k m )B ω ω 2 2 2 1 2 1 2 2 2 4 2 1 1 2 2 2 1 1 2 1 2 1 2 on-trivial solution is, +-- = & ¡ + + ∴- + = ¢ £ ¤ ¥ o o o o ( k k m )( k m ) k k m k m k m k k m m m m ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω 1 2 1 2 1 2 2 1 2 1 1 2 2 4 2 2 2 2 2 The equation for the natural frequency of the system can be simplified by setting and and which yields, 1 1 = = = +- + + + + = o ol ol o ol ol o ol ol w m k k m m m m ( )( ) ( )( )( ) η ω ω η ω ω η ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω Case 1: The Amplitude of vibration for a force on the mass m 1 . Consider the case when a vibration is induced on the system through a force Q o sin & t acting upon the mass m 1 . The differential equations of motion are now, 1 1 1 1 2 1 2 2 2 2 2 1 and + +- = +- = && && m z k z k ( z z ) Q sin t m z k ( z z ) ω 1 1 2 2 2 1 1 1 2 2 2 2 2 2 2 1 2 Let and Substituting backinto the differential equation, = =- + +- =-- = z A sin t z A sin t A ( m k k ) A k Q A ( k m ) A k ω ω ω ω ω ω ω ω ω ω ( )...
View Full Document

{[ snackBarMessage ]}

Page1 / 17

SD-Lecture10-2DOF-Systems - CEG CEG-5065C Soil Dynamics...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online