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    Help with equations of motion

    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...
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    Quadratic form of strain energy

    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...
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    Assumed modes and their units

    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...
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    Mathematica Symbolic vector differentiation with Mathematica

    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...
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    Computing virtual work

    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...
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    Multi-DOF vibration systems

    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?
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    Multi-DOF vibration systems

    ^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?
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    Multi-DOF vibration systems

    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...
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    Equivalent stiffness of a beam

    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?
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    Equivalent stiffness of a beam

    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...
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    Virtual displacements and Virtual Work

    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...
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    Virtual displacements and Virtual Work

    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...