Chapter 19 MECHANICAL VIBRATIONS Consider the free vibration of a particle, i.e., the motion of a particle P subjected to a restoring force proportional to the displacement of the particle - such as the force exerted by a spring. If the displacement x of the particle P is measured from its equilibrium position O, the resultant F of the forces acting on P (including its weight) has a magnitude kx and is directed toward O. Applying Newton’s second law (F = ma) with a = x, the differential equation of motion is -xm O x P Equilibrium .. +xm + .. mx + kx = 0 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

mx + kx = 0 x + wn2x = 0 x = xm sin (wnt + f) + .. .. -xm setting wn2 = k/m .. x + wn2x = 0 O The motion defined by this expression is called simple harmonic motion. x P Equilibrium The solution of this equation, which represents the displacement of the particle P is expressed as x = xm sin (wnt + f) +xm + where xm = amplitude of the vibration wn = k/m = natural circular frequency f = phase angle

x + wn2x = 0 x = xm sin (wnt + f) 2p wn Period = tn = wn 1 2p .. x + wn2x = 0 -xm x = xm sin (wnt + f) The period of the vibration (i.e., the time required for a full cycle) and its frequency (i.e., the number of cycles per second) are expressed as O x P Equilibrium 2p wn Period = tn = +xm 1 tn wn 2p Frequency = fn = = + The velocity and acceleration of the particle are obtained by differentiating x, and their maximum values are vm = xmwn am = xmwn2

The oscillatory motion of the particle P may be represented by the projection on the x axis of the motion of a point Q describing an auxiliary circle of radius xm with the constant angular velocity wn. The instantaneous values of the velocity and acceleration of P may then be obtained by projecting on the x axis the vectors vm and am representing, respectively, the velocity and acceleration of Q. xm f O x QO a wnt am= xmwn2 Q P wnt + f v vm= xmwn x

While the motion of a simple pendulum is not truely a simple harmonic motion, the formulas given above may be used with wn 2 = g/l to calculate the period and frequency of the small oscillations of a simple pendulum. The free vibrations of a rigid body may be analyzed by choosing an appropriate variable, such as a distance x or an angle q , to define the position of the body, drawing a diagram expressing the equivalence of the external and effective forces, and writing an equation relating the selected variable and its second derivative. If the equation obtained is of the form .. .. x + wn2x = 0 or q + wn2q = 0 the vibration considered is a simple harmonic motion and its period and frequency may be obtained.

The principle of conservation of energy may be used as an alternative method for the determination of the period and frequency of the simple harmonic motion of a particle or rigid body. Choosing an appropriate variable, such as q, to define the position of the system, we express that the total energy of the system is conserved, T1 + V1 = T2 + V2 , between the position of maximum displacement (q1 = qm) and the position of maximum velocity (q 2 = qm). If the motion considered is simple harmonic, the two members of the equation obtained consist of homogeneous quadratic expressions in qm and qm , respectively. Substituting qm = qm wn in this equation, we may factor out qm and solve for the circular frequency wn. . . . . 2

mx + kx = Pm sin wf t mx + kx = kdm sin wf t .. .. The forced vibration of a mechanical system occurs when the system is subjected to a periodic force or when it is elastically connected to a support which has an alternating motion. The differential equation describing each system is presented below. dm sin wf t dm x Equilibrium wf t = 0 wf t P = Pm sin wf t .. mx + kx = Pm sin wf t .. x mx + kx = kdm sin wf t Equilibrium

mx + kx = Pm sin wf t mx + kx = kdm sin wf t xpart = xm sin wf t .. .. The general solution of these equations is obtained by adding a particular solution of the form x Equilibrium xpart = xm sin wf t x P = Pm sin wf t Equilibrium to the general solution of the corresponding homogeneous equation. The particular solution represents the steady-state vibration of the system, while the solution of the homogeneous equation represents a transient free vibration which may generally be neglected.

xm xm dm Pm /k 1 1 - (wf /wn )2 mx + kx = Pm sin wf t .. mx + kx = kdm sin wf t .. xpart = xm sin wf t Dividing the amplitude xm of the steady-state vibration by Pm/k in the case of a periodic force, or by dm in the case of an oscillating support, the magnification factor of the vibration is defined by x Equilibrium P = Pm sin wf t xm Pm /k xm dm dm sin wf t Magnification factor = = dm wf t = 0 wf t 1 1 - (wf /wn )2 = x Equilibrium

Magnification factor = = mx + kx = Pm sin wf t .. mx + kx = kdm sin wf t .. xpart = xm sin wf t xm Pm /k xm dm Magnification factor = = x Equilibrium 1 1 - (wf / wn)2 = P = Pm sin wf t The amplitude xm of the forced vibration becomes infinite when wf = wn , i.e., when the forced frequency is equal to the natural frequency of the system. The impressed force or impressed support movement is then said to be in resonance with the system. Actually the amplitude of the vibration remains finite, due to damping forces. dm sin wf t dm wf t = 0 wf t x Equilibrium

The equation of motion describing the damped free vibrations of a system with viscous damping is .. . mx + cx + kx = 0 where c is a constant called the coeficient of viscous damping. Defining the critical damping coefficient cc as k m cc = 2m = 2mwn where wn is the natural frequency of the system in the absence of damping, we distinguish three different cases of damping, namely, (1) heavy damping, when c > cc, (2) critical damping, when c = cc, (3) light damping, when c < cc. In the first two cases, the system when disturbed tends to regain its equilibrium position without oscillation. In the third case, the motion is vibratory with diminishing amplitude.

mx + cx + kx = Pm sin wf t mx + cx + kx = kdm sin wf t .. . .. . The damped forced vibrations of a mechanical system occurs when a system with viscous damping is subjected to a periodic force P of magnitude P = Pm sin wf t or when it is elastically connected to a support with an alternating motion d = d m sin wf t. In the first case the motion is defined by the differential equation .. . mx + cx + kx = Pm sin wf t In the second case the motion is defined by the differential equation .. . mx + cx + kx = kdm sin wf t

[1 - (wf / wn)2]2 + [2(c/cc)(wf / wn)]2 The steady-state vibration of the system is represented by a particular solution of mx + cx + kx = Pm sin wf t of the form .. . xpart = xm sin (wf t - f) Dividing the amplitude xm of the steady-state vibration by Pm/k in the case of a periodic force, or by dm in the case of an oscillating support, the expression for the magnification factor is xm Pm/k xm dm 1 [1 - (wf / wn)2]2 + [2(c/cc)(wf / wn)]2 = = where wn = k/m = natural circular frequency of undamped system cc = 2m wn = critical damping coefficient c/cc = damping factor

[1 - (wf / wn)2]2 + [2(c/cc)(wf / wn)]2 xpart = xm sin (wf t - f) xm Pm/k xm dm 1 [1 - (wf / wn)2]2 + [2(c/cc)(wf / wn)]2 = = In addition, the phase difference j between the impressed force or support movement and the resulting steady-state vibration of the damped system is defined by the relationship 2(c/cc) (wf / wn) 1 - (wf / wn)2 tan j = The vibrations of mechanical systems and the oscillations of electrical circuits are defined by the same differential equations. Electrical analogues of mechanical systems may therefore be used to study or predict the behavior of these systems.