Appendix A Frequency Response Curves

There are four types of one DOF harmonically excited systems as shown in Fig. 3.1: 1. base excitation, 2. eccentric excitation, 3. eccentric base excitation, 4. forced excitation. The frequency responses of the four systems can be summarized, labeled and shown as follows: X S = F (A.1) 0 F/k 1 =  (A.2) (1 − r2)2 + (2ξr)2

˙ XF S1 = √ (A.3) F/ km r =  (A.4) (1 − r2)2 + (2ξr)2

¨ XF ZB XE ZR S2 = = = = (A.5) F/m Y eεE eεR r2 =  (A.6) (1 − r2)2 + (2ξr)2

˙ ˙ ˙ ZB XE ZR S3 = = = (A.7) ωnY eεEωn eεRωn r3 =  (A.8) (1 − r2)2 + (2ξr)2

R.N. Jazar, Advanced Vibrations, 667 DOI 10.1007/978-1-4614-4160-1, © Springer Science+Business Media New York 2013 668 A Frequency Response Curves

Fig. A.1 Frequency response for S0

Z¨ X¨ Z¨ S = B = E = R 4 2 2 2 (A.9) ωnY eεEωn eεRωn r4 =  (A.10) (1 − r2)2 + (2ξr)2

F X G = TF = B (A.11) 0 F Y  1 + (2ξr)2 =  (A.12) (1 − r2)2 + (2ξr)2

X˙ G = B (A.13) 1 ω Y n  r 1 + (2ξr)2 =  (A.14) (1 − r2)2 + (2ξr)2

  X¨ F F F m G = B = TB = TE = TR + b 2 2 2 2 1 (A.15) ω Y kY eω me eω me m n  n n r2 1 + (2ξr)2 =  (A.16) (1 − r2)2 + (2ξr)2 A Frequency Response Curves 669

Fig. A.2 Frequency response for S1

Fig. A.3 Frequency response for S2 670 A Frequency Response Curves

Fig. A.4 Frequency response for S3

Fig. A.5 Frequency response for S4 A Frequency Response Curves 671

Fig. A.6 Frequency response for G0

Fig. A.7 Frequency response for G1 672 A Frequency Response Curves

Fig. A.8 Frequency response for G2 Appendix B Trigonometric Formulas

Definitions in Terms of Exponentials eiz + e−iz cos z = (B.1) 2

eiz − e−iz sin z = (B.2) 2i

eiz − e−iz tan z = (B.3) i(eiz + e−iz)

eiz = cos z + isinz (B.4)

− e iz = cos z − isinz (B.5)

Angle Sum and Difference sin(α ± β) = sin α cos β ± cos α sin β (B.6) cos(α ± β) = cos α cos β ∓ sin α sin β (B.7)

tan α ± tan β tan(α ± β) = (B.8) 1 ∓ tan α tan β

cot α cot β ∓ 1 cot(α ± β) = (B.9) cot β ± cot α Symmetry sin(−α) =−sin α (B.10) cos(−α) = cos α (B.11) tan(−α) =−tan α (B.12)

R.N. Jazar, Advanced Vibrations, 673 DOI 10.1007/978-1-4614-4160-1, © Springer Science+Business Media New York 2013 674 B Trigonometric Formulas

Multiple Angles 2tanα sin(2α) = 2sinα cos α = (B.13) 1 + tan2 α

cos(2α) = 2 cos2 α − 1 = 1 − 2sin2 α = cos2 α − sin2 α (B.14)

2tanα tan(2α) = (B.15) 1 − tan2 α

cot2 α − 1 cot(2α) = (B.16) 2 cot α

sin(3α) =−4sin3 α + 3sinα (B.17)

cos(3α) = 4 cos3 α − 3 cos α (B.18)

− tan3 α + 3tanα tan(3α) = (B.19) −3tan2 α + 1

sin(4α) =−8sin3 α cos α + 4sinα cos α (B.20)

cos(4α) = 8 cos4 α − 8 cos2 α + 1 (B.21)

−4tan3 α + 4tanα tan(4α) = (B.22) tan4 α − 6tan2 α + 1

sin(5α) = 16 sin5 α − 20 sin3 α + 5sinα (B.23)

cos(5α) = 16 cos5 α − 20 cos3 α + 5 cos α (B.24)     sin(nα) = 2sin (n − 1)α cos α − sin (n − 2)α (B.25)     cos(nα) = 2 cos (n − 1)α cos α − cos (n − 2)α (B.26)

tan((n − 1)α) + tan α tan(nα) = (B.27) 1 − tan((n − 1)α) tan α

Half Angle    α 1 + cos α cos =± (B.28) 2 2    α 1 − cos α sin =± (B.29) 2 2 B Trigonometric Formulas 675    α 1 − cos α sin α 1 − cos α tan = = =± (B.30) 2 sin α 1 + cos α 1 + cos α

2tan α sin α = 2 (B.31) + 2 α 1 tan 2

1 − tan2 α cos α = 2 (B.32) + 2 α 1 tan 2

Powers of Functions 1  sin2 α = 1 − cos(2α) (B.33) 2

1 sin α cos α = sin(2α) (B.34) 2

1  cos2 α = 1 + cos(2α) (B.35) 2

1  sin3 α = 3sin(α) − sin(3α) (B.36) 4

1  sin2 α cos α = cos α − 3 cos(3α) (B.37) 4

1  sin α cos2 α = sin α + sin(3α) (B.38) 4

1  cos3 α = cos(3α) + 3 cos α (B.39) 4

1  sin4 α = 3 − 4 cos(2α) + cos(4α) (B.40) 8

1  sin3 α cos α = 2sin(2α) − sin(4α) (B.41) 8

1  sin2 α cos2 α = 1 − cos(4α) (B.42) 8

1  sin α cos3 α = 2sin(2α) + sin(4α) (B.43) 8

1  cos4 α = 3 + 4 cos(2α) + cos(4α) (B.44) 8 676 B Trigonometric Formulas

1   sin5 α = 10 sin α − 5sin(3α) + sin(5α) (B.45) 16

1   sin4 α cos α = 2 cos α − 3 cos(3α) + cos(5α) (B.46) 16

1   sin3 α cos2 α = 2sinα + sin(3α) − sin(5α) (B.47) 16

1   sin2 α cos3 α = 2 cos α − 3 cos(3α) − 5 cos(5α) (B.48) 16

1   sin α cos4 α = 2sinα + 3sin(3α) + sin(5α) (B.49) 16

1   cos5 α = 10 cos α + 5 cos(3α) + cos(5α) (B.50) 16

1 − cos(2α) tan2 α = (B.51) 1 + cos(2α)

Products of sin and cos

1 1 cos α cos β = cos(α − β)+ cos(α + β) (B.52) 2 2

1 1 sin α sin β = cos(α − β)− cos(α + β) (B.53) 2 2

1 1 sin α cos β = sin(α − β)+ sin(α + β) (B.54) 2 2

1 1 cos α sin β = sin(α + β)− sin(α − β) (B.55) 2 2

sin(α + β)sin(α − β) = cos2 β − cos2 α = sin2 α − sin2 β (B.56)

cos(α + β)cos(α − β) = cos2 β + sin2 α (B.57)

Sum of Functions α ± β α ± β sin α ± sin β = 2sin cos (B.58) 2 2

α + β α − β cos α + cos β = 2 cos cos (B.59) 2 2 B Trigonometric Formulas 677

α + β α − β cos α − cos β =−2sin sin (B.60) 2 2 sin(α ± β) tan α ± tan β = (B.61) cos α cos β

sin(β ± α) cot α ± cot β = (B.62) sin α sin β

+ sin α + sin β tan α β = 2 (B.63) sin α − sin β α−+β tan 2

sin α + sin β −α + β = cot (B.64) cos α − cos β 2

sin α + sin β α + β = tan (B.65) cos α + cos β 2

sin α − sin β α − β = tan (B.66) cos α + cos β 2

Trigonometric Relations

sin2 α − sin2 β = sin(α + β)sin(α − β) (B.67)

cos2 α − cos2 β =−sin(α + β)sin(α − β) (B.68) Appendix C Unit Conversions

General Conversion Formulas

Nambsc ≈ 4.448a × 0.3048b × lbaftbsc ≈ 4.448a × 0.0254b × lbainbsc lbaftbsc ≈ 0.2248a × 3.2808b × Nambsc lbainbsc ≈ 0.2248a × 39.37b × Nambsc

Conversion Factors

Acceleration

1ft/s2 ≈ 0.3048 m/s2 1m/s2 ≈ 3.2808 ft/s2

Angle 1deg≈ 0.01745 rad 1 rad ≈ 57.307 deg

Area 1in2 ≈ 6.4516 cm2 1cm2 ≈ 0.155 in2 1ft2 ≈ 0.09290304 m2 1m2 ≈ 10.764 ft2 1 acre ≈ 4046.86 m2 1m2 ≈ 2.471 × 10−4 acre 1 acre ≈ 0.4047 hectare 1 hectare ≈ 2.471 acre

Damping

1Ns/m ≈ 6.85218 × 10−2 lbs/ft 1 lbs/ft ≈ 14.594 Ns/m 1Ns/m ≈ 5.71015 × 10−3 lbs/in 1 lbs/in ≈ 175.13 Ns/m

R.N. Jazar, Advanced Vibrations, 679 DOI 10.1007/978-1-4614-4160-1, © Springer Science+Business Media New York 2013 680 C Unit Conversions

Energy and Heat

1Btu≈ 1055.056 J 1 J ≈ 9.4782 × 10−4 Btu 1 cal ≈ 4.1868 J 1 J ≈ 0.23885 cal 1kWh≈ 3600 kJ 1 MJ ≈ 0.27778 kWh

Force 1lb≈ 4.448222 N 1 N ≈ 0.22481 lb

Length 1in≈ 25.4mm 1cm≈ 0.3937 in 1ft≈ 30.48 cm 1 m ≈ 3.28084 ft 1mi≈ 1.609347 km 1 km ≈ 0.62137 mi

Mass 1lb≈ 0.45359 kg 1 kg ≈ 2.204623 lb 1slug≈ 14.5939 kg 1 kg ≈ 0.068522 slug 1slug≈ 32.174 lb 1 lb ≈ 0.03.1081 slug

Moment and Torque

1lbft≈ 1.35582 Nm 1 Nm ≈ 0.73746 lbft 1lbin≈ 8.85075 Nm 1 Nm ≈ 0.11298 lbin

Moment of Inertia

1lbft2 ≈ 0.04214 kgm2 1 kgm2 ≈ 23.73 lbft2

Power 1Btu/h ≈ 0.2930711 W 1 W ≈ 3.4121 Btu/h 1hp≈ 745.6999 W 1 kW ≈ 1.341 hp 1hp≈ 550 lbft/s1lbft/s ≈ 1.8182 × 10−3 hp 1lbft/h ≈ 3.76616 × 10−4 W1W≈ 2655.2lbft/h 1lbft/min ≈ 2.2597 × 10−2 W1W≈ 44.254 lbft/min

Pressure and Stress 1lb/in2 ≈ 6894.757 Pa 1 MPa ≈ 145.04 lb/in2 1lb/ft2 ≈ 47.88 Pa 1 Pa ≈ 2.0886 × 10−2 lb/ft2 C Unit Conversions 681

Stiffness

1N/ m ≈ 6.85218 × 10−2 lb/ft 1 lb/ft ≈ 14.594 N/m 1N/m ≈ 5.71015 × 10−3 lb/in 1 lb/in ≈ 175.13 N/m

Temperature   ◦C = ◦F − 32 /1.8 ◦F = 1.8 ◦C + 32

Velocity

1mi/h ≈ 1.60934 km/h1km/h ≈ 0.62137 mi/h 1mi/h ≈ 0.44704 m/s1m/s ≈ 2.2369 mi/h 1ft/s ≈ 0.3048 m/s1m/s ≈ 3.2808 ft/s 1ft/min ≈ 5.08 × 10−3 m/s1m/s ≈ 196.85 ft/min

Volume 1in3 ≈ 16.39 cm3 1cm3 ≈ 0.0061013 in3 1ft3 ≈ 0.02831685 m3 1m3 ≈ 35.315 ft3 1gal≈ 3.785 l 1 l ≈ 0.2642 gal 1gal≈ 3785.41 cm3 1l≈ 1000 cm3 References

Agyris J, Mlejnek HP (1991) Computational mechanics. Elsevier, New York Alkhatib R, Jazar RN, Golnaraghi MF (2004) Optimal design of passive linear mounts with genetic algorithm method. J Sound Vib 275(3–5):665–691 Balachandran B, Magrab EB (2003) Vibrations. Brooks/Cole, Pacific Grove Benaroya H (2004) Mechanical vibration: analysis, uncertainties, and control. Marcel Dekker, New York Benson D (2006) Music: a mathematical offering. Cambridge University Press, London Christopherson J, Jazar RN (2006a) Dynamic behavior comparison of passive hydraulic engine mounts, Part 1: mathematical analysis. J Sound Vib 290:1040–1070 Christopherson J, Jazar RN (2006b) Dynamic behavior comparison of passive hydraulic engine mounts, Part 2: finite element analysis. J Sound Vib 290:1071–1090 Coddington EA (1961) Ordinary differential equations. Prentice Hall, New York Del Pedro M, Pahud P (1991) Vibration mechanics. Kluwer Academic, Dordrecht Den Hartog JP (1934) Mechanical vibrations. McGraw-Hill, New York Deshpande S, Mehta S, Jazar RN (2006) Optimization of secondary suspension of piecewise linear vibration isolation systems. Int J Mech Sci 48(4):341–377 Dimarogonas A (1996) Vibration for engineers. Prentice Hall, New York Esmailzadeh E (1978) Design synthesis of a vehicle suspension system using multi-parameter optimization. Veh Syst Dyn 7:83–96 Golnaraghi MF, Jazar RN (2001) Development and analysis of a simplified nonlinear model of a hydraulic engine mount. J Vib Control 7(4):495–526 Harris CM, Piersol AG (2002) Harris’ shock and vibration handbook. McGraw-Hill, New York Hirsch MW, Smale S (1974) Differential equations, dynamic systems, and linear algebra. Aca- demic Press, New York Inman D (2007) Engineering vibrations. Prentice Hall, New York Jazar RN (2009) Vehicle dynamics: theory and application. Springer, New York Jazar RN (2010) Applied robotics: kinematics, dynamics, and control, 2nd edn. Springer, New York Jazar RN (2011) Advanced dynamics: rigid body, multibody, and aerospace applications. Wiley, New York Jazar RN, Golnaraghi MF (2002a) Engine mounts for automotive applications: a survey. Shock Vib Dig 34(5):363–379 Jazar RN, Golnaraghi MF (2002b) Nonlinear modeling, experimental verification, and theoretical analysis of a hydraulic engine mount. J Vib Control 8(1):87–116 Jazar RN, Kazemi M, Borhani S (1992) Mechanical vibrations. Ettehad Publications, Tehran (in Persian)

R.N. Jazar, Advanced Vibrations, 683 DOI 10.1007/978-1-4614-4160-1, © Springer Science+Business Media New York 2013 684 References

Jazar RN, Narimani A, Golnaraghi MF, Swanson DA (2003) Practical frequency and time optimal design of passive linear vibration isolation mounts. J Veh Syst Dyn 39(6):437–466 Jazar RN, Alkhatib R, Golnaraghi MF (2006a) Root mean square optimization criterion for vibra- tion behavior of linear quarter car using analytical methods. J Veh Syst Dyn 44(6):477–512 Jazar RN, Houim R, Narimani A, Golnaraghi F (2006b) Nonlinear passive engine mount, fre- quency response and jump avoidance. J Vib Control 12(11):1205–1237 Kneubuhl FK (1997) Oscillations and waves. Springer, Berlin Korenev BG, Reznikov LM (1993) Dynamic vibration absorbers: theory and technical applica- tions. Wiley, Chichester Meirovitch L (1967) Analytical methods in vibrations. Macmillan, New York Meirovitch L (1997) Principles and techniques of vibrations. Prentice Hall, New York Meirovitch L (2002) Fundamentals of vibrations. McGraw-Hill, New York Narimani A, Golnaraghi MF, Jazar RN (2004a) Frequency response of a piecewise linear system. J Vib Control 10(12):1775–1894 Narimani A, Jazar RN, Golnaraghi MF (2004b) Sensitivity analysis of frequency response of a piecewise linear system in frequency island. J Vib Control 10(2):175–198 Ogata K (2004) System dynamics. Prentice Hall, New York Palm WJ (2006) Mechanical vibration. Wiley, New York Rao SS (2003) Mechanical vibrations. Prentice Hall, New York Rayleigh JWS (1945) The theory of sound. Dover Publication, New York Roseau M (1987) Vibrations in mechanical systems. Springer, Berlin Shabana AA (1997) Vibration of discrete and continuous systems. Springer, New York Snowdon JC (1968) Vibration and shock in damped mechanical systems. Wiley, New York Thomson WT, Dahleh MD (1997) Theory of vibration with applications. Prentice Hall, New York Tongue BH (2001) Principles of vibration. Oxford University Press, New York Tse FS, Morse IE, Hinkle RT (1978) Mechanical vibrations theory and applications. Allyn and Bacon Inc, Boston Index

Symbols B 1/8 car model, 15, 32, 53 Bandwidth, 183 excitation frequency, 32 Base excitation, 360 periodic, 208 A Base excited, 222 Acceleration Based excitation, 197 angular, 88 acceleration, 204, 206 centripetal, 92 frequency response, 197 transmitted force, 206 kinematics, 87 velocity, 204, 206 matrix, 87 Basic lemma, 153 tangential, 92 Beating, 22, 23, 25 transformation, 91 Bernoulli, Johann and Jacob, 157 Admissible path, 151 Bernoulli equation, 424 Amplitude, 17, 19, 176 Bicycle car dynamic, 179 mode shape, 594–596, 598, 601, 604, 606, maximum, 182 608 ratio, 347, 364, 377, 391, 401 natural frequency, 594–596, 598, 601, 604, static, 179 606, 608 Angular acceleration, 87, 88 vibration, 592–594, 602, 606, 608 B-expression, 93, 94 Bounce mode, 597 matrix, 87 Bounded integral, 38, 39 relative, 91 Brachistochrone, 156 rotational transformation, 88, 90 Buckingham theorem, 191 vector, 87 Bushehr, 100 Angular lag, 199 Angular momentum, 103–106 C Angular speed, 80 Centrifugal moments, 109 Angular velocity, 80, 82, 84, 86, 88 Characteristic combination, 84 equation, 475 instantaneous, 83 parameters, 475 instantaneous axis, 80, 84 values, 477 matrix, 81 Characteristic equation, 236, 258 relative, 82 coefficients, 117 speed, 80 Characteristic matrix, 236 vector, 80 Constant of motion, 68

R.N. Jazar, Advanced Vibrations, 685 DOI 10.1007/978-1-4614-4160-1, © Springer Science+Business Media New York 2013 686 Index

Coordinate Dido problem, 158 Cartesian, 121 Differentiating change, 278, 281 transformation formula, 87 generalized, 52, 274, 286, 287, 300 Directional improper, 289 cosine, 74, 76 natural, 274, 278, 296, 297, 327 Directions polar, 307 principal, 112 principal, 273, 276–278, 280, 281, Dissipation function, 135 283–286, 303, 460 Dunkerley formula, 271, 272 proper, 287 Dynamics, 51 relative, 278 Lagrange, 51 transformation, 290 Newton–Euler, 51 translational and rotational, 263 Newtonian, 59 Coordinates rigid body, 73 change of, 256 rotational, 103 Couple, 60 translational, 100 Coupling, 273, 286 variational, 151 coefficient, 291 dynamic, 291, 300 E dynamically, 273, 278 Eccentric base excitation, 137, 386 elastically, 273 frequency response, 220 inertially, 273 mass ratio, 218 Eccentric base excited, 222 mass, 273 Eccentric excitation, 137, 210, 373 static, 291 acceleration, 213 statically, 273, 278, 301 eccentric mass, 210 stiffness, 273 eccentricity, 210 Critically-damped frequency response, 210 vibration, 487, 489 mass ratio, 211 Cycloid, 157 transmitted force, 214 velocity, 213 D Eccentric excited, 222 Damper, 3 Eccentricity, 210 characteristic curve, 4 Eigenvalue damping, 4 complex, 461 linear, 4 Eigenvalue problem, 236 parallel, 7, 8 characteristic equation, 236 serial, 5, 6 Eigenvector viscous, 4 complex, 461 Damping first-unit, 237 energy loss, 16 high-unit, 237 proportional, 303 last-unit, 237 Damping ratio, 177, 475 normal form, 237 determination, 527 normalization, 237 Decibel, 26 Eigenvector problem, 236 Decoupling, 273, 279, 296, 303, 304, 329 Energy damping, 304 conservation, 66, 67 general, 304 damping loss, 16 Degree of freedom, 51–53 kinetic, 3, 61, 103, 107, 121, 123, 135 Derivative maximum, 64 transformation mechanical, 3, 62, 134 simple, 87 mode shape, 248 transformation formula, 87 multi DOF, 63, 64 Deviation moments, 109 potential, 3, 62, 126, 135 Index 687

Envelope, 484 Forced Equation transmitted, 187 fundamental vibration, 17 Forced base excited, 222 homogeneous, 457 Forced excitation, 176, 344 linear ODE, 427 acceleration, 186 Euler frequency response, 176 equation of motion, 103, 106, 108 transmitted force, 187, 189 Euler equation velocity, 186 Formula body frame, 106 derivative transformation, 87 Euler–Lagrange derivative transport, 93 equation of motion, 151, 153 Dunkerley, 271 Excitation relative acceleration, 90, 91 base, 175, 176, 197, 343, 344, 667 relative angular velocity, 85 eccentric, 175, 176, 343, 344, 667 Foucault pendulum, 98 eccentric base, 175, 176, 343, 344, 667 Fourier forced, 175, 176, 343, 344, 667 coefficients, 33, 36, 39 harmonic, 308 complex coefficients, 40 harmonically, 175, 343, 667 complex series, 39 position of, 398 series, 36, 37, 39 Fourier, Joseph, 36 F Fourier series Falling wheel, 65 complex, 195 First integral, 69 complex coefficients, 195 Frame First variation, 152, 160 central, 101 Forbidden umbrella, 485 principal, 103, 104, 106, 110, 112, 113 Force, 60 Free system, 233 body, 60 Frequency, 4, 188 complex, 195 angular, 5, 17 conservative, 62, 126 cyclic, 5 contact, 60 damped natural, 477 dissipative, 139 high, 188 elastic, 139 low, 188 external, 60 natural, 481 function, 58 nodal, 550 generalized, 121, 123, 126, 136, 139 ratio, 177 gravitational, 54 response, 175, 179 harmonic, 189, 196 Frequency ratio, 177 internal, 60 Frequency response, 175 moment of, 60 attractive, 190 periodic, 193 base excited, 224 phased-harmonic, 447 classification, 221 comparison, 203, 398 potential, 62, 126 first-order system, 440 proportionality, 55 two DOF, 343 resultant, 60 Front wheel mode, 597 rotating, 107 Full car system, 60 mode shape, 627 total, 60 natural frequency, 627 transmitted, 189, 206, 359, 372, 385, 396 vibration, 622–624, 627 two harmonic, 196 Function Force system, 60 dissipation, 135 equivalent, 60 even, 37, 39 688 Index

Function (cont.) Kinetic energy, 61, 123 harmonic, 33 rigid body, 107 odd, 39 rotational body, 103 orthogonality, 32 Kronecker delta, 76, 110, 119 periodic, 33, 37, 39 Rayleigh, 135 L Functional, 151 Lagrange equation, 135, 136, 151, 153 equation of motion, 120–126, 151 G mechanics, 126 Galilei, Galileo, 157 method, 120, 135, 301 Generalized multiplier, 157 coordinate, 121, 123, 125, 127, 128 Lagrangean, 126, 127, 135, 136 force, 121, 123, 125, 126, 129, 139 equation, 126 Generalized coordinate, 52 function, 126 Generalized mass, 238 mechanics, 126 Generalized stiffness, 238 Law Geodesics problem, 154 second of motion, 67, 101 Gravitational force, 54 third of motion, 67 Libration, 26, 27 H Line of maxima, 182, 207, 215 Half car Linearization, 148 antiroll bar, 617, 621 Lissajous mode shape, 619, 620 curves, 30 natural frequency, 619, 620 motion, 31 vibration, 616–618 period, 31 Harmonic Lissajous curves, 30 motion, 28 Load, 60 oscillator, 27 Loudness, 19 Harmonic excitation, 308 Harmonic oscillator, 26, 27 M Hermitian form, 141 Manipulator one-link, 129 I Mass center, 60, 101 India, 100 Matrix Integrability, 424 damping, 304 Integral of motion, 68, 69 diagonal, 303, 454, 455 diagonal elements, 455 Integrating factor, 425, 427 dynamic stiffness, 325 Intensity, 19 exponential, 453, 456, 457 Inverse engineering, 186 flexibility, 297–299 Iran, 100 function, 459 global rotation, 79 K inverse, 311, 312, 457 Keyboard, 20 inverse modal, 301 black keys, 20 inversion, 311 flat keys, 20 local rotation, 79 keys, 20–22 modal, 274, 276, 277, 285, 301, 302 sharp keys, 20 non-diagonal, 456 stave, 20 non-singular, 307 white keys, 20 orthogonal, 73, 75 Kinematics orthogonality condition, 75 acceleration, 87, 88 receptance, 325 velocity, 80 similar, 307 Index 689

Matrix (cont.) Momentum, 60 similarity, 307 angular, 60, 103–106 singular, 307 linear, 60 stiffness, 294, 302, 303 translational, 60 symmetric, 148 Motion transformation, 73, 278 general, 428 McPherson suspension natural, 417, 418 equivalent vibrating model, 15 Music Mechanical amplitude, 19 energy, 134 frequency, 19 Mechanical energy, 62, 134 intensity, 19 Mechanics keyboard, 20–22 Newtonian, 59 length, 19 Mode loudness, 19 bounce, 597 notes, 19–22 front wheel, 597 pitch, 19 pitch, 597 rhythm, 19 rear wheel, 597 sound, 19 Mode shape, 234, 280, 285, 302, 406 spectrum, 19 complex, 271 timber, 19 design, 267 wave, 19, 20 length, 285 normalized, 302, 303 N orthogonality, 269 Natural frequency, 177, 181, 233, 481 rigid, 260, 261 determination, 529 Moment, 60 rigid mode, 260 external, 106 smallest, 271 resultant, 60, 106 zero, 260 total, 60 New Delhi, 100 Moment of inertia, 109 Newton about a line, 119 equation in body frame, 102 about a plane, 119 equation of motion, 67, 101, 102, 108, 120 about a point, 119 equations of motion, 123 about the origin, 120 Lagrange form, 123 characteristic equation, 117 Node, 188, 204, 403 diagonal elements, 109, 117 amplitude, 406 eigenvalues, 112, 116 frequency, 405 eigenvectors, 116 Notes elements, 109 flat, 20 frame-dependent, 110 frequency, 20–22 Huygens–Steiner theorem, 112 keyboard, 20 matrix, 109 sharp, 20 off-diagonal elements, 109 parallel-axes theorem, 110–112 O polar, 109 Objective function, 151 principal, 110–113, 118 One-eighth car model, 557 principal axes, 103 absolute acceleration, 559 principal invariants, 118 absolute displacement, 559, 561, 562 product, 109 damping ratio, 557 rigid body, 103, 105, 110 design curve, 585 rotated-axes theorem, 110–112 equation of motion, 557 Moment of momentum, 60 frequency response, 559, 562 Moments of inertia hard suspension, 570, 571 determination, 529 natural frequency, 557 690 Index

One-eighth car model (cont.) Foucault, 98 optimal characteristics, 574 inverted, 56 optimal damping, 574 oscillating, 124 optimal design chart, 575 simple, 94, 124 optimal design curve, 565, 574, 577 spherical, 95, 97, 128 optimal stiffness, 573 velocity, 94 optimal suspension, 572 Period, 4 optimization, 565 Persian Gulf, 100 optimization strategy, 566 Phase, 17, 176, 199 relative displacement, 559, 561, 562 Pitch mode, 597 soft suspensions, 570, 571 Potential step input, 583 energy, 61, 62, 126 suspension clearance, 570 field, 61 suspension room, 570 force, 62, 126 suspension travel, 570 function, 61 time response, 583, 585 kinetic, 127 trade-off, 577 Power wheel travel, 569, 570 half point, 183 steady state vibrations, 180 working frequency range, 567 Principal Optimal control axes, 103 Lagrange equation, 151 coordinate frame, 103 Optimization invariants, 118 alternative method, 580 mass moments, 118 cost function, 581 rotation matrix, 116 design curve, 647 Principle one-eighth car, 556, 565 conservation of energy, 61, 62 quarter car, 647 superposition, 59 RMS, 565, 647 work and energy, 61, 62 time response, 583, 585 work-energy, 61 transient response, 583, 585 Problem trivial, 577 brachistochrone, 156 vehicle suspension, 573 Dido, 158 vibration, 345, 545–553 eigenvalue, 236 wheel travel, 656 eigenvector, 236 Orlando, 100 geodesic, 154 Orthogonality condition, 74, 75 minimization, 153 Over-damped shortest path, 154 vibration, 487, 489 Projectile, 484 umbrella, 485 P Path Q admissible, 151 Quadratic form, 147 minimizing, 151, 152 Quadrature optimal, 153, 154, 160 asymmetric, 141 shortest, 154 Quadratures, 140, 141 variable, 151 Quarter car, 144 Pendulum model, 53 acceleration, 94 natural frequency, 241 chain, 132 sprung mass, 241 compound, 129 unsprung mass, 241 constraint, 70 Quarter-car model, 629 double, 130, 300 3-D frequency response, 635 first integral, 70 body bounce frequency, 643 Index 691

Quarter-car model (cont.) vibration, 260 coefficient matrix, 634 Road dimensionless characteristics, 632 wave, 32 equations of motion, 630 Rolling disc, 130 frequency response, 632–634, 641, 644 Rotating vectors, 184 history, 632 Rotation invariant amplitude, 639 acceleration transformation, 88, 90 invariant frequency, 637, 639, 643 general, 73 main suspension, 629 local axes, 79 mathematical model, 629 multiple, 77–79 natural frequency, 637, 639, 642 successive, 78, 79 nodal amplitude, 640 successive global axes, 78 nodal frequency, 639, 640 successive local axes, 79 optimal characteristics, 655 Rotational dynamics, 103 optimal design curve, 647, 651 Rule optimization, 647 relative angular acceleration, 90 optimization strategy, 647 relative angular velocity, 90 principal natural frequency, 643 resonant frequency, 639 S sprung mass, 629 Second variation, 152, 160 street cars, 634 Series solution, 428 tire damping, 631 Solution unsprung mass, 629 exponential, 185 wheel hop frequency, 643 harmonic, 17, 19 wheel travel, 656 natural, 458 working frequency range, 648 nontrivial, 236 particular, 429 R series, 428 Rayleigh steady state, 309 dissipation function, 135 trivial, 236 Rear wheel mode, 597 wave, 17 Reciprocity, 325 weighted harmonic, 17 Resonance, 241 Sound Resonance zone, 179 decibel, 26 Response density, 26 steady state, 308 power density, 26 time, 417 Spatial integral, 61 transient, 476 Spherical Rest position, 235 pendulum, 95, 97 Ride, 591 Spring, 3 Ride comfort, 591 characteristic curve, 4 Rigid body different length, 10 angular momentum, 104, 105 displaced, 15 Euler equation, 106 kinetic energy, 13 kinetic energy, 107 length of, 11 moment of inertia, 103, 105, 110 linear, 4 principal rotation matrix, 116 massive, 12 rotational kinetics, 103 parallel, 7, 8, 10 steady rotation, 108 serial, 5, 6 translational, 100 stiffness, 4 Rigid body dynamics, 73 tilted, 13–15 Rigid mode Step input, 495 elimination, 261 Step response, 495 number of, 261 overshoot, 497 692 Index

Step response (cont.) Vector peak time, 497 angular acceleration, 87 peak value, 497 length invariant property, 77 rise time, 497 Vehicle model settling time, 497 1/8, 32 steady-state, 498 Vehicle vibration, 591 Stiffness matrix alternative optimization, 580 elements, 294, 296 antiroll bar, 617, 621 meaning, 294 base excited model, 556 Superposition, 27, 28, 317 bicycle car, 592, 594–596, 598, 601, 604, linear vibrations, 27 606, 608 wave functions, 28 body pitch, 592 Suspension body roll, 616–618 McPherson, 15 bounce, roll, and pitch, 622 optimization, 556 dissipation function, 135 vibration, 556 driver, 144 Switching point, 204 frequency response, 559 System full car, 622–624 coupled first-order, 449, 463 half car, 616–618 first-order, 417, 445 Lagrange equation, 135 free, 181 Lagrange method, 135 second-order, 463 mode shape, 592, 619, 620, 627 third-order, 442 natural frequency, 592, 619, 620, 627 undamped, 181 one-eighth model, 557 optimal design curve, 565 T optimization, 556 Theorem optimization strategy, 566 Buckingham, 191 quadrature, 140 flexibility, 300 quarter car, 144, 629 Huygens–Steiner, 112 sprung mass, 557 parallel-axes, 110, 112 time response, 583, 585 rotated-axes, 110 wheel travel, 569, 570 Time constant, 420, 429 working frequency range, 567 Time response Velocity homogeneous, 473 kinematics, 80 initial condition, 489 Vibration non-homogeneous, 473, 474 1/8 car model, 53 Torque, 60 absorber, 545 Total differential, 423 amplitude, 176 Transformation angular frequency, 5 general, 73 angular lag, 179 Translational dynamics, 100 application, 527 Turning point, 188 base excitation, 175, 197, 343, 667 Turning wheel, 66 beating, 22 characteristic equation, 236 U cyclic frequency, 5 Under-damped damping ratio, 177 vibration, 487, 489 discrete model, 51 displaced spring, 15 V dynamic amplitude, 179 Variation dynamics, 51 first, 152 eccentric base excitation, 175, 343, 667 second, 152 eccentric excitation, 175, 343, 667 Variational dynamics, 151 eigenvalue problem, 236 Index 693

Vibration (cont.) tilted spring, 15 eigenvector problem, 236 transient, 5 elements, 3 transmitted force, 189, 206 equilibrium position, 52 trivial solution, 235 equivalent systems, 54 two dimensional, 28, 30, 31 excitation, 5 two-DOF base excited, 56 forced, 5, 179 unstable, 52 forced classification, 221 vehicle, 591 forced excitation, 175, 343, 667 work of a harmonic force, 499 Frahm absorber, 345, 545–553 Vibrations Frahm damper, 345, 545–553 bandwidth, 182 free, 489 quality, 182 free system, 233 rotating vectors, 184 frequency ratio, 177 Viration frequency response, 175, 179, 186 critically-damped, 482 fundamental equation, 17 damped natural frequency, 477 fundamentals, 3 forced, 474 gravitational force, 54 free, 474 harmonic, 5 natural frequency, 481 initial condition, 489 over-damped, 482 isolator, 545 under-damped, 482 kinematics, 3, 17 Virtual lumped model, 51 displacement, 123 measurement, 527 work, 123 mechanical, 3 mechanical elements, 3 W natural frequency, 177 Wave Newton’s method, 51 amplitude, 19 nontrivial solution, 235 duration, 19 optimization theory, 345, 545–553 equation, 22 orthogonality functions, 32 frequency, 19 periodic, 5 height, 32 phase, 176, 194, 209 length, 19, 32 quarter car model, 53 periodic, 36 random, 5 road, 32 resonance zone, 179 square, 36 rest position, 235 Wheel travel, 569 ride comfort, 591 lower, 570 stable, 52 upper, 570 static amplitude, 179 Work, 61 steady-state solution, 175 virtual, 123 step input, 495 Work-energy principle, 61