When a spring-mounted body is disturbed from its equilibrium position, its ensuing motion in the absence of any imposed external forces is termed free vibration. In every actual case of free vibration, there exists some retarding or damping force which tends to diminish the motion. Common damping forces are those due to mechanical and fluid friction. In this article we first consider the ideal case where the damping forces are small enough to be neglected. Then we treat the case where the damping is appreciable and must be accounted for. Equation of Motion for Undamped Free Vibration We begin by considering the horizontal vibration of the simple frictionless spring-mass system of Fig. 8/1a. Note that the variable x denotes the displacement of the mass from the equilibrium position, which, for this system, is also the position of zero spring deflection. Figure 8/1b shows a plot of the force necessary to deflect the spring versus the corresponding spring deflection for three types of springs. Although nonlinear hard and soft springs are useful in some applications, we will restrict our attention to the linear spring. Such a spring exerts a restoring force -kx on the mass—that is, when the mass is displaced to the right, the spring force is to the left, and vice versa. We must be careful to distinguish between the forces of magnitude which must be applied to both ends of the massless spring to cause tension or compression and the force of equal magnitude which the spring exerts on the mass. The constant of proportionality k is called the spring constant, modulus, or stiffness and has the units N/m or lb/ft. The equation of motion for the body of Fig. 8/1a is obtained by first drawing its free-body diagram. Applying Newton's second law in the form gives The oscillation of a mass subjected to a linear restoring force as described by this equation is called simple harmonic motion and is characterized by acceleration which is proportional to the displacement but of opposite sign. Equation 8/1 is normally written as where is a convenient substitution whose physical significance will be clarified shortly. Solution for Undamped Free Vibration Because we anticipate an oscillatory motion, we look for a solution which gives x as a periodic function of time. Thus, a logical choice is or, alternatively, Direct substitution of these expressions into Eq. 8/2 verifies that each expression is a valid solution to the equation of motion. We determine the constants A and B, or C and ψ, from knowledge of the initial displacement and initial velocity of the mass. For example, if we work with the solution form of Eq. 8/2 Free Vibration of Particles Figure8/1 (8/1) (8/2) (8/3) (8/4) (8/5)Page 1 of 19Free Vibration of Particles12/4/2008http://edugen.wiley.com/edugen/courses/crs1653/pc/meriam9316c08/meriam9316c08_3.xf...8/4 and evaluate x and at time , we obtain Substitution of these values of A and B into Eq. 8/4 yields The constants C and ψ of Eq. 8/5 can be determined in terms of given initial conditions in a similar manner. Evaluation of Eq. 8/5 and its first time derivative at gives Solving for C and ψ yields Substitution of these values into Eq. 8/5 gives Equations 8/6 and 8/7 represent two different mathematical expressions for the same time-dependent motion. We observe that and . Graphical Representation of Motion The motion may be represented graphically, Fig. 8/2, where x is seen to be the projection onto a vertical axis of the rotating vector of length C. The vector rotates at the constant angular velocity which is called the natural circular frequency and has the units radians per second. The number of complete cycles per unit time is the natural frequency and is expressed in hertz . The time required for one complete motion cycle (one rotation of the reference vector) is the period of the motion and is given by . We also see from the figure that x is the sum of the projections onto the vertical axis of two perpendicular vectors whose magnitudes are A and B and whose vector sum C is the amplitude. Vectors A, B, and C rotate together with the constant angular velocity . Thus, as we have already seen, and . Equilibrium Position as Reference As a further note on the free undamped vibration of particles, we see that, if the system of Fig. 8/1a is rotated 90° clockwise to obtain the system of Fig. 8/3 where the motion is vertical rather than horizontal, the equation of motion (and therefore all system properties) is unchanged if we continue to define x as the displacement from the equilibrium position. The equilibrium position now involves a nonzero spring deflection . From the free-body diagram of Fig. 8/3, Newton's second law gives At the equilibrium position , the force sum must be zero, so that Thus, we see that the pair of forces and mg on the left side of the motion equation cancel, giving which is identical to Eq. 8/1. (8/6) (8/7) Figure8/2 Page 2 of 19Free Vibration of Particles12/4/2008http://edugen.wiley.com/edugen/courses/crs1653/pc/meriam9316c08/meriam9316c08_3.xf...The lesson here is that by defining the displacement variable to be zero at equilibrium rather than at the position of zero spring deflection, we may ignore the equal and opposite forces associated with equilibrium.* Equation of Motion for Damped Free Vibration Every mechanical system possesses some inherent degree of friction, which dissipates mechanical energy. Precise mathematical models of the dissipative friction forces are usually complex. The dashpot or viscous damper is a device intentionally added to systems for the purpose of limiting or retarding vibration. It consists of a cylinder filled with a viscous fluid and a piston with holes or other passages by which the fluid can flow from one side of the piston to the other. Simple dashpots arranged as shown schematically in Fig. 8/4a exert a force whose magnitude is proportional to the velocity of the mass, as depicted in Fig. 8/4b. The constant of proportionality c is called the viscous damping coefficient and has units of or lb-sec/ft. The direction of the damping force as applied to the mass is opposite that of the velocity . Thus, the force on the mass is -c . Complex dashpots with internal flow-rate-dependent one-way valves can produce different damping coefficients in extension and in compression; nonlinear characteristics are also possible. We will restrict our