BÀI TẬP LẦN 2 NGUYỄN THANH PHONG

Trần Tiến Anh Ngày nộp: 06/03/2014 BÀI TẬP 1-4 Introduction to Structural Dynamics and Aeroelasticity Problem 1: Show that the equation of motion for longitudinal vibration pf a uniform

Sinh viên: Nguyễn Thanh Phong

MSSV: G1002398

Lớp: GT10HK

KHÍ ĐÀN HỒI

Giảng viên: TS Trần Tiến Anh Ngày nộp: 06/03/2014

BÀI TẬP 1-4 Introduction to Structural Dynamics and Aeroelasticity

Problem 1:

Show that the equation of motion for longitudinal vibration pf a uniform beam is the same ass that for a string, viz

Solution:

We have:

With

Such as the force of the rod vertically then u turn stretches over time, and v is constant (the derivative with time 0)

Equation (2.19)

v

y

dx

Let us presuppose the existence of a static-equilibrium solution of the string deflection so that

whereT0 and are constants and δ=l−l0 is the change in the length of the string between its stretched and unstretched states

If the steady-state tension T0 is sufficiently high, the perturbation deflections about the static-equilibrium solution are very small Thus, we can assume:

we find that the equations of motion can be reduced to two linear partial differential

equations:

Problem 2:

Show that the equation of motion for longitudinal vibration pf a uniform beam is the same ass that for a string, viz

Solution:

To solve problems involving the forced response of strings using Lagrange’s equation, we need an expression for the strain energy, which is caused by extension of

the string, viz

where, as before

and the original length is0 To pick up all of the linear terms in Lagrange’s equations, we must include all terms in the energy up through the second power of the unknowns Taking the pertinent unknowns to be perturbations relative to the stretched but undeflected string, we can again write

ForEAequal to a constant, the strain energy is:

Strain energy simplifies to

the strain energy becomes

We assuming that v is const and so we have:

Problem 3:

Show that the equation of motion for longitudinal vibration pf a uniform beam is the same ass that for a string, viz

Solution:

To solve problems involving the forced response of strings using Lagrange’s equation, we also need the kinetic energy The kinetic energy for a differential length of

string is

With:

so we have: