Hi,
I am having a little bit of conceptual trouble with this problem and would appreciate your help. The problem setup is given in the figure. Let's say we have a slender uniform rigid arm(mass m, length l) in space, with a coordinate system B attached to the left end of the arm as shown. C is...
Hi,
I have some confusion about strain energy. When using Lagrange's equations to derive the EOM of vibrating structures, the strain energy is written as :
U = q^{T}Kq ; (q is the vector of generalized coordinates, and K is the stiffness matrix). Writing it in this form makes it easy to...
Hi,
I am trying to understand the assumed-modes approach to solving vibration problems. For example, the transverse deformation of a cantilever beam is using two assumed modes is given as
ζ(x,t) = ψ_{1}(x)q_{1}(t) + ψ_{2}(x)q_{2}(t)
ψ_{1}(x) = (x/L)^{2}; ψ_{2}(x) = (x/L)^{3}
In this...
Hi,
I am a new user of Mathematica (although I am reasonably familiar with MATLAB) and I am trying to differentiate a scalar wrt a vector Mathematica. ie, I want to check if
\begin{equation}
\phi = \textit{x}^{T} \textbf{A} \textit{y} \quad \mbox{where $\textit{x}$, $\textit{y}$ are vectors...
Hi all,
I've been trying to understand the vibration of a cantilever beam subjected to a forcing function using Lagrange's equation, but have got stuck at the virtual work part. I would appreciate your inputs here.
Using the assumed modes method, the transverse deformation is written as...
Agree with what you said. Let's say we didn't know about modes and we want to solve the EOM for a 2-DOF system. If we want to assume a solution to the equations, how would you proceed? What would prompt us to use the same frequency for both the masses?
^AlephZero,
Thanks for the reply. Does that mean if I give random initial displacements (and/or velocities) to both the masses and let go, and trace out q_{1} and q_{2}, we will see periodic variations in the response?
Hi all,
I have a question about Multi DOF vibrating systems. For free vibration of undamped MDOF systems, we have the equations of motion as :
M \ddot{q} + K {q} = {0} (1)
Where,
M - n x n mass matrix
K - n x n stiffness matrix
{q} - n x 1 vector of generalized coordinates
Most vibrations...
Sorry for the late response.
Thanks for clarifying that. Also, does that mean that if we know the moment acting at a location x, we can calculate the slope of the curved beam at that point due to the moment?
Hi all,
I am trying to understand the concept of equivalent stiffness of a beam. As I see it, the equivalent stiffness is the stiffness of a linear spring that would deflect the same amount under the same load. For a cantilever beam with a load P and a deflection \delta at the free end, if we...
Thanks Studiot and Bill for your answers. I think I am sort of getting it now. One thing that I don't get is this - At each instant, the position vector is perpendicular to the reaction force. If we consider an infinitesimal real displacement, then the real work done by the forces on the mass...
Hi all,
I am having trouble understanding Virtual displacements and related ideas and would appreciate your help!
Let's say we have a simple pendulum and we take a snapshot of this system at some instant of time :
A 'virtual displacement' is said to be :
- consistent with the constraints on...