Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 585

10

Dynamic Responses

In the previous chapters, the responses have been “stati c ” as the load does not vary with
time. In discrete form, the governing equations are KV P , in which K is the global stiffness

matrix of the structure, V is the displa ce ment vector, and P is the loading vector. If the force
changes with time, the relation KV

=

P does not adequately describe the movement of th e
structure. The inclusion of inertia forces changes the governing equations to M V

=

+ KV = P

in which M is the mass matrix. An analysis leads to a dynamic response.
Dynamic response problems fit into two broad classes. In one, the natural frequencies of
the vibration and the corresponding mode shapes are desired. In the other, the motion of a
structure subject to a prescribed load is sought.

10.1 Mass

A dynamic loading generates accelerations

2 u
∂

∂

t 2 =

u

in a structu re . This acceleration field can be considered to produce d'Alembert 1

inertia

forces, e.g., p V =− γ is the mass density, in the direction opposite to the accel-

eration. If inertia forces are included, the principle of virtual work of Chapter 6, Eq. (6.20)

u , where γ

S p δ

V δ T σ dV

u T p V dV u T p dS − δ W = −

V δ

− = 0

becomes

V δ

T σ dV u T u T p dS − δ W =

V δ

+ γ u dV −

S p δ

= 0 (10.1)

1 Jean Le Rond D'Alembert (1717-1783) was named after a church in Paris, France where he, as the illegitimate
son of a society hostess, was abandoned. His step parents were of sufficient means to ensure that he received a
formal education. He graduated from Mazarin College in 1735 and published in 1743 the topic Traite de dynamique ,
in which the well-known d'Alembert Principle was proposed. He is considered to be the founder of the theory of
partial differential equations.

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