Page 53 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

50 Chapter 1 Fundamentals oF Vibration
of ordinary differential equations for a discrete system and partial differential equations for
a continuous system. The equations may be linear or nonlinear, depending on the behavior
of the components of the system. Several approaches are commonly used to derive the
governing equations. Among them are Newton’s second law of motion, D’Alembert’s prin-
ciple, and the principle of conservation of energy.

Step 3: Solution of the Governing Equations. The equations of motion must be solved
to find the response of the vibrating system. Depending on the nature of the problem, we
can use one of the following techniques for finding the solution: standard methods of solv-
1
ing differential equations, Laplace transform methods, matrix methods, and numerical
methods. If the governing equations are nonlinear, they can seldom be solved in closed
form. Furthermore, the solution of partial differential equations is far more involved than
that of ordinary differential equations. Numerical methods involving computers can be
used to solve the equations. However, it will be difficult to draw general conclusions about
the behavior of the system using computer results.

Step 4: Interpretation of the Results. The solution of the governing equations gives
the displacements, velocities, and accelerations of the various masses of the system. These
results must be interpreted with a clear view of the purpose of the analysis and the possible
design implications of the results.

mathematical model of a motorcycle
example 1.1
Figure 1.17(a) shows a motorcycle with a rider. Develop a sequence of three mathematical models
of the system for investigating vibration in the vertical direction. Consider the elasticity of the tires,
elasticity and damping of the struts (in the vertical direction), masses of the wheels, and elasticity,
damping, and mass of the rider.

Solution: We start with the simplest model and refine it gradually. When the equivalent values of the
mass, stiffness, and damping of the system are used, we obtain a single-degree-of-freedom model of
the motorcycle with a rider as indicated in Fig. 1.17(b). In this model, the equivalent stiffness 1k eq 2
includes the stiffnesses of the tires, struts, and rider. The equivalent damping constant 1c eq 2 includes
the damping of the struts and the rider. The equivalent mass includes the masses of the wheels,
vehicle body, and the rider. This model can be refined by representing the masses of wheels, elastic-
ity of the tires, and elasticity and damping of the struts separately, as shown in Fig. 1.17(c). In this
model, the mass of the vehicle body 1m v 2 and the mass of the rider 1m r 2 are shown as a single mass,
m v + m r . When the elasticity (as spring constant k r ) and damping (as damping constant c r ) of the
rider are considered, the refined model shown in Fig. 1.17(d) can be obtained.
Note that the models shown in Figs. 1.17(b)–(d) are not unique. For example, by combining
the spring constants of both tires, the masses of both wheels, and the spring and damping constants
of both struts as single quantities, the model shown in Fig. 1.17(e) can be obtained instead of
Fig. 1.17(c).
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1 The basic definitions and operations of matrix theory are given in Appendix A.