- #1
dekarman
- 7
- 0
Hi,
I am solving the following nonlinear dynamical system using Energy Balance Method (EBM*). My intention is to arrive at an approximate analytical expression for the frequency of oscillation and the excitation force.
u''+u=A(1+2*u) with u(0)=u'(0)=0, where A is a constant (Physically it is like a Heaviside step loading).
I first write the Hamiltonian,
0.5*(u')^2+0.5*u^2=Au(1+2*u)
which implies the residue function R= 0.5*(u')^2+0.5*u^2-Au(1+2*u).
Then I assume the initial guess, which satisfies the initial conditions u=A*(1-cos(w*t)).
Then calculate R using the initial guess and collocate it at w*t=(pi/2) to obtain the relation between A and w.
However, the relation which I obtain is w=sqrt(1+4*A), which means that with increasing value of A, the frequency increases and hence indicates a hardening nature of the system stiffness.
However, by looking at the differential equation, this is clearly a softening system since the coefficient of u is (1-2*A).
I am not able to figure out this discrepancy.
Can somebody please point out where I am going wrong exactly.
---------------------------------------------------------------------------------------------------------------
*Reference of EBM, APPLICATION OF THE ENERGY BALANCE METHOD
FOR STRONGLY NONLINEAR OSCILLATORS, H. Pashaei et al., Progress In Electromagnetics Research M, Vol. 2, 47–56, 2008, which is available on internet.
Thank you very much in advance.
Manish
I am solving the following nonlinear dynamical system using Energy Balance Method (EBM*). My intention is to arrive at an approximate analytical expression for the frequency of oscillation and the excitation force.
u''+u=A(1+2*u) with u(0)=u'(0)=0, where A is a constant (Physically it is like a Heaviside step loading).
I first write the Hamiltonian,
0.5*(u')^2+0.5*u^2=Au(1+2*u)
which implies the residue function R= 0.5*(u')^2+0.5*u^2-Au(1+2*u).
Then I assume the initial guess, which satisfies the initial conditions u=A*(1-cos(w*t)).
Then calculate R using the initial guess and collocate it at w*t=(pi/2) to obtain the relation between A and w.
However, the relation which I obtain is w=sqrt(1+4*A), which means that with increasing value of A, the frequency increases and hence indicates a hardening nature of the system stiffness.
However, by looking at the differential equation, this is clearly a softening system since the coefficient of u is (1-2*A).
I am not able to figure out this discrepancy.
Can somebody please point out where I am going wrong exactly.
---------------------------------------------------------------------------------------------------------------
*Reference of EBM, APPLICATION OF THE ENERGY BALANCE METHOD
FOR STRONGLY NONLINEAR OSCILLATORS, H. Pashaei et al., Progress In Electromagnetics Research M, Vol. 2, 47–56, 2008, which is available on internet.
Thank you very much in advance.
Manish