{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

DifferentialEquations_01_Eqns_of_Motion

# DifferentialEquations_01_Eqns_of_Motion - Section 1.1 1.1...

This preview shows pages 1–3. Sign up to view the full content.

Section 1.1 Solid Mechanics Part II Kelly 3 1.1 The Equations of Motion In Part I, balance of forces and moments acting on any component was enforced in order to ensure that the component was in equilibrium. Here, allowance is made for stresses which vary continuously throughout a material, and force equilibrium of any portion of material is enforced. One-Dimensional Equation Consider a one-dimensional differential element of length x Δ and cross sectional area A , Fig. 1.1.1. Let the average body force per unit volume acting on the element be b and the average acceleration and density of the element be a and ρ . Stresses σ act on the element. Figure 1.1.1: a differential element under the action of surface and body forces The net surface force acting is A x A x x ) ( ) ( σ σ Δ + . If the element is small, then the body force and velocity can be assumed to vary linearly over the element and the average will act at the centre of the element. Then the body force acting on the element is x Ab Δ and the inertial force is xa A Δ ρ . Applying Newton’s second law leads to a b x x x x xA a xA b A x A x x ρ σ σ ρ σ σ = + Δ Δ + Δ = Δ + Δ + ) ( ) ( ) ( ) ( (1.1.1) so that, by the definition of the derivative, in the limit as 0 Δ x , a b dx d ρ σ = + 1-d Equation of Motion (1.1.2) which is the one-dimensional equation of motion . Note that this equation was derived on the basis of a physical law and must therefore be satisfied for all materials, whatever they be composed of. The derivative dx d / σ is the stress gradient – physically, it is a measure of how rapidly the stresses are changing. Example Consider a bar of length l which hangs from a ceiling, as shown in Fig. 1.1.2. A x Δ ) ( x σ ) ( x x Δ + σ x x x Δ + a b ,

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document