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11 are defined as X 2 Xo tan- 1 (.

The configuration of a rigid body in space, however, can be determined using six coordinates, three coordinates define the translation of the body center of mass and three coordinates define the orientation ofthe body in a fixed coordinate system.

The particle momentum is a vector quantity defined as p mv (1.14) where p is the momentum of the particle, m is the mass, and v is the velocity vector.Therefore, these coefficients have a significant effect on the response ofthe mechanical and structural systems.19 can also be written as x wX sin ( wt /1 ) (3.21 ) which indicates that the velocity x has a phase lead angle of n/2 compared to the displacement x given.2 'c I cos3.156t c2 sin.1 56t x(t) XeO.10 cos 0 (1.38) Substituting these two equations into.I -2.0 V -3.0 I J -4.0 -5.0 -6.0.0.0.0.0.0.0.0.0.0.0 __ Time (s) FIG.The desired damping characteristic can, therefore, be obtained by changing these parameters.Example.4 Find the complete solution of the following second-order ordinary differential equation 5x 2x 50x 0 Solution.I 3: 2x IOt(1 sin 2t Xo 1.__

5(b one can show that F 3EI V 13 where F is the applied force, v is the transverse deflection of the end point and I, I, and E are the length, second area moment of inertia, and modulus of elasticity of the beam.

22(b) which has six degrees of freedom, three degrees of freedom describe the translation of the center of mass of the chassis and three degrees of freedom define the chassis orientation.Based on Galileo's work, Sir Isaac Newton (1642-1727) formulated the laws of motion in which the relationship between force, mass, and momentum is established.1.6 idealization OF mechanical AND structuralsvstems Modern mechanical and structural systems may consist of several components connected together by different types of joints.The planar motion of a particle is described by only two coordinates x and y, as shown in Fig.The motion is primarily the result of initial conditions, such as an initial displacement of the mass element of the system from an equilibrium position and/or an initial velocity.Complex Roots In this case, the roots conjugates which can be written as Pl P2 IX iP IX - iP Pl and P2 appear as complex.4.In this book an attempt has been made to provide the rational development of the methods of vibration analysis from their foundations and to develop the techniques in clearly understandable stages.In fact, as was shown in the examples presented in this chapter, the stability of the system depends on the roots of the characteristic equations.

Daniel Bernoulli also examined the vibration of a system of n point masses and showed that such a system has n independent modes of vibration.

1.2 elements OF THE vibration models Vibrations are the result ofthe combined effects ofthe inertia and elastic forces.

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