Page 236 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 7.5 Assumed Mode Summation 223

By interchanging the order of summation and letting
_ d r

be defined as the generalized force, the virtual work for the system, expressed in
terms of the generalized coordinates, becomes

(7.4-3)

7.5 ASSUMED MODE SUMMATION
When the displacement is expressed as the sum of shape functions (f)fx) multi­
plied by the generalized coordinates qft), the kinetic energy, the potential energy,
and the work equation lead to convenient expressions for the generalized mass, the
generalized stiffness, and the generalized force.
In Chapter 2, a few distributed elastic systems were solved for the fundamen­
tal frequency using an assumed deflection shape and the energy method. For
example, the deflection of a helical spring fixed at one end was assumed to be
(y/l)x, and for the simply supported beam, the deflection curve y = ymax^Xx/l)
—4(x/iy], (x/I) < was chosen. These assumptions when solved for the kinetic

energy led to the effective mass and the natural frequency of a 1-DOF system.
These assumed deflections can be expressed by the equation
u(x, t) = (/>(x)<7, ( i )

where is the single coordinate of the 1-DOF system.
For the multi-DOF system, this procedure can be expanded to
u( x , t ) =
i
where is the generalized coordinate, and (ffx) is the assumed mode function.
There are very few restrictions on these shape functions, which need only satisfy
the geometric boundary conditions.

Generalized M ass
We assume the displacement at position x to be represented by the equation
r ( x , t ) = (t>^(x)q^{t) + 4>2{x)q2{t) + ■■■ +4>,^{x)q,^{t)

(7.5-1)
i = 1
where </>,(-^) are shape functions of only x.