LECTURE NOTES FOR COURSE EML 4220 - Anil V. Rao

1 MECHANICAL VIBRATIONS: LECTURE NOTES FOR COURSE EML4220 ANIL V. RAOU niversity of FloridaSpring 2009iiAnil V. Raoearned his in mechanical engineering and in mathematics fromCornell University, his in aerospace engineering from theUniversity of Michi-gan, and his and in mechanical and aerospace engineering from PrincetonUniversity. After earning his , Dr. Rao joined the Flight Mechanics Departmentat The Aerospace Corporation in Los Angeles, where he was involved in mission sup-port for Air Force launch vehicle programs and trajectory optimization softwaredevelopment. Subsequently, Dr. Rao joined The Charles Stark Draper Laboratory, Inc.,in Cambridge, Massachusetts. As a Draper employee, Dr.

2 Rao led numerousprojectsrelated to trajectory optimization, guidance, and navigation of both space flight andatmospheric flight vehicles. Concurrently, from 2001 to 2006 Dr. Rao was an Ad-junct Professor of Aerospace and Mechanical Engineering at Boston University wherehe taught the core undergraduate engineering dynamics COURSE . While at Boston Uni-versity, Dr. Rao was voted the 2002 and 2006 Mechanical and Aerospace EngineeringFaculty Member of the Year and was voted 2004 College of EngineeringProfessor ofthe Year for outstanding Anil Vithala Rao 2006 Vakratunda Mahaakaaya Soorya Koti SamaprabhaNirvighnam Kuru Mein Deva Sarva Kaaryashu SarvadaaviContents1 Response of Single Degree-of-Freedom Systems to Initial Mass-Spring-Damper System.

3 General Solution of a Second-Order LTI Differential Equation .. General Solution to Second-Order Homogeneous LTI System .. 42 Forced Response of Single Degree-of-Freedom Response of Single Degree-of-Freedom Systems to Nonperiodic Inputs .. Physics of Impulsive Motion .. Impulse Response of Second-Order Linear System .. Step Response of Second-Order Linear System .. Response of Single Degree-of-Freedom Systems to Periodic Inputs .. Base Motion Isolation .. Fourier Series Representation of an Arbitrary Periodic Function .. Response of a Single Degree-of-Freedom System to an Arbitrary Periodic Input .. 533 Response of Multiple Degree-of-Freedom Systems to Initial Unforced Undamped Multiple Degree-of-Freedom Systems.

4 Unforced Damped Multiple Degree-of-Freedom Systems .. Non-Symmetric Mass and Stiffness Matrices .. 844 Forced Response of Multiple Degree-of-Freedom Generic Model for Forced Multiple Degree-of-Freedom System .. Response of Modally Damped Systems to Nonperiodic Inputs .. Response of Modally Damped Systems to Periodic Inputs .. Response of Systems with General Damping to Periodic Inputs .. Undamped Vibration Absorbers .. 103A Review of Linear Row Vectors, Column Vectors, and Matrices .. Types of Matrices .. Simple Algebra Associated with Matrices .. Null Space and Range Space of a Real Matrix .. Eigenvalues and Eigenvectors of a Real Square Matrix .. Eigenvalues and Eigenvectors of a Real Symmetric Matrix.

5 Symmetric Weighted Eigenvalue Problem .. Definiteness of Matrices .. 121 Bibliography121viiiContentsChapter 1 Response of Single Degree-of-FreedomSystems to Initial ConditionsIn this chapter we begin the study of vibrations of mechanical systems. Generally speaking avibration is a periodic or oscillatory motion of an object or a setof objects. Vibrating systemsare ubiquitous in engineering and thus the study of vibrationsis extremely most basic problem of interest is the study of the vibration of a one degree-of-freedom( , a system whose motion can be described using a single scalar second-order ordinary dif-ferential equation). The generic model for a one degree-of-freedom system is a mass connectedto a linear spring and a linear viscous damper ( , a mass-spring-damper system).

6 Because ofits mathematical form, the mass-spring-damper system will be used as the baseline for analysisof a one degree-of-freedom system. In particular, the differential equation of motion will bederived for the mass-spring-damper system. It will then be shown that thetime response ofthis system is the sum of thezero input responseand thezero initial condition response. In thischapter we will focus attention on the zero input response, , the response of the system to agiven set of initial conditions. Several examples of single degree-of-freedom systems will thenbe given. In each of these examples the differential equation will be derived and will be shownto have the same mathematical form as the generic mass-spring-damper Mass-Spring-Damper SystemThe most basic system that is used as a model for vibrational analysis isa block of massmconnected to a linear spring (with spring constantKand unstretched length 0) and a viscousdamper (with damping coefficientc).

7 In addition, an external forceP(t)is applied to the blockand the displacement of the block is measured from the inertiallyfixed pointO, whereOis thepoint where the spring is unstretched. Finally, the spring and damper are both attached at theinertially fixed pointQ. This system is shown in Fig. 1 1 Denoting unit vector in the directionfromOtoQasExand the inertial reference frame of the ground byF, the inertial accelerationof the block is given asFa= xEx(1 1)Next, the forces exerted by the spring and damper are given, respectively, asFs= K( 0)us(1 2)Ff= cvrel(1 3)First, because the spring is attached at pointQ, we have =kr rQk(1 4)2 Chapter 1. Response of Single Degree-of-Freedom Systems to Initial Conditions 0cPgmxKOQF igure 1 1 Block of massmsliding without friction along a horizontal surface con-nected to a linear spring and a linear viscous the positions of the block and the attachment points of the spring, respec-tively.

8 Using a coordinate system with its origin at pointOatExas the first principal direction,we haver=xEx(1 5)rQ= 0Ex(1 6)Therefore, =kxEx 0 Exk=k(x 0)Exk=|x 0|(1 7)Then, becausex < 0we have|x 0|= 0 x(1 8)Finally, the unit vector in the direction from the attachment point of the spring to the positionof the block isus=r rQkr rQk=(x 0)Ex 0 x= Ex(1 9)The force in the linear spring is then given asFs= K( 0 x 0)( Ex)= KxEx(1 10)Next, because the ground is already assumed to be inertial, the relative velocity between theblock and the ground is simply the velocity of the block, ,vrel=Fv= xEx(1 11)Therefore, the force exerted by the viscous damper is obtained asFf= c xEx(1 12)The resultant external force acting on the particle is then obtained asF=P+Fs+Ff=PEx KxEx c xEx=(P Kx c x)Ex(1 13)Applying Newton s second law to the particle, we obtain(P Kx c x)Ex=m xEx(1 14)DroppingExfrom Eq.

9 (1 14) and rearranging, we obtain the differential equation of motion asm x+c x+Kx=P(1 15) General Solution of a Second-Order LTI Differential Equation3 Now historically it has been the case that the differential equation has been written in a formthat is normalized by the mass, , we divide Eq. (1 15) bymto obtain x+cm x+Kmx=Pm=p(t)(1 16)wherep(t)=P(t)/m. Furthermore, it is common practice to define the quantitiesK/mandc/mas follows: 2n=Km2 n=cmThe quantities nand are called thenatural frequencyanddamping ratioof the system,respectively. In terms of the natural frequency and damping ratio, the differential equation ofmotion for the mass-spring-damper system can be written in the so calledstandard formas x+2 n x+ 2nx=p(t)(1 17)It is seen that Eq.

10 (1 17) is a second-order linear constant coefficient ordinary differential equa-tion. Often, the term constant coefficient is replaced with thetermtime-invariant, , wesay that Eq. (1 17) is a called a second-orderlinear time-invariant(LTI) ordinary differentialequation. The terminology time invariant stems from the fact that, for a given inputp(t)anda given set of initial conditions(x(t0), x(t0)=(x0, x0)at the initial timet=t0is the same asthe solution to the inputp(t+ )for the initial conditions(x(t0+ ), x(t0+ )=(x0, x0)atthe (shifted) initial timet=t0+ . Because of this fact associated with an LTI system, withoutloss of generality we can assume that the initial time is zero, ,t0=0.))