Contents
xiii
8.3.1
Free Longitudinal Vibration of a Bar Clamped
at X = 0 and Free at X = L . . . . . . . . . . . . . . . . . . . . . . . 317
8.4
Free Torsional Vibration of the Shaft . . . . . . . . . . . . . . . . . . . . . . . 318
8.5
Free Flexural Vibration of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 322
8.6
Free Flexural Vibration of the Simply Supported Beam . . . . . . . 324
8.7
Free Flexural Vibration of Beams with Other End
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
8.7.1
Uniform Beam Having Both Ends Free . . . . . . . . . . . . 326
8.8
Free Flexural Vibration of Beams with General End
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
8.9
Orthogonality Properties of Normal Modes . . . . . . . . . . . . . . . . . 336
8.10 Effect of Rotary Inertia on the Free Flexural Vibration
of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
8.11 Free Vibration of the Shear Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 343
8.12 Effect of Axial Force on the Free Flexural Vibration
of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
8.13 Free Vibration of Beams Including Shear Deformation
and Rotary Inertia Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
8.14 Collocation Method for Obtaining Normal Modes
of Vibration of a Continuous System . . . . . . . . . . . . . . . . . . . . . . . 350
8.15 Rayleigh’s Quotient for Fundamental Frequency . . . . . . . . . . . . . 354
8.16 Rayleigh–Ritz Method for Determining Natural
Frequencies of Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . 356
8.17 Vibration of Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
8.18 Transverse Vibration of Rectangular Thin Plates . . . . . . . . . . . . . 363
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
9 Forced Vibration of Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . 371
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
9.2
Forced Axial Vibration of Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
9.3
Forced Vibration of the Shear Beam Under Ground
Motion Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
9.4
Forced Vibration of Flexural Member . . . . . . . . . . . . . . . . . . . . . . 378
9.5
Forced Transverse Vibration of Uniform Damped Beam . . . . . . 382
9.6
Forced Vibration of Flexural Member Subjected
to Ground Motion Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
9.7
Response of Beams Due to Moving Loads . . . . . . . . . . . . . . . . . . 386
9.7.1
Response of the Beam When the Mass
of the Vehicle is Large . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.7.2
Response of the Beam when the Mass
of the Vehicle is Small . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393