Download Elastic Multibody Dynamics: A Direct Ritz Approach by Hartmut Bremer PDF

By Hartmut Bremer

This textbook is an creation to and exploration of a couple of center themes within the box of utilized mechanics. at the foundation of Lagrange's precept, a principal Equation of Dynamics is gifted which yields a unified view on latest equipment. From those, the Projection Equation is chosen for the derivation of the movement equations of holonomic and of non-holonomic structures. the strategy is utilized to inflexible multibody platforms the place the inflexible physique is outlined such that, via rest of the pressure constraints, you can actually at once continue to elastic our bodies. A decomposition into subsystems results in a minimum illustration and to a recursive illustration, respectively, of the equations of movement. utilized to elastic multibody platforms one obtains, besides using spatial operators, a straight-on process for the interconnected partial and traditional differential equations and the corresponding boundary stipulations. The spatial operators are ultimately utilized to a RITZ sequence for approximation. The ensuing equations then look within the similar constitution as in inflexible multibody structures. the most emphasis is laid on methodical in addition to on (graduate point) academic points. The textual content is observed through a great number of examples and functions, e.g., from rotor dynamics and robotics. The mathematical necessities are subsumed in a quick expedition into balance and keep an eye on.

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Extra resources for Elastic Multibody Dynamics: A Direct Ritz Approach (Intelligent Systems, Control and Automation: Science and Engineering)

Sample text

16) This kind of specification is suitable in dynamics, simply because revealing dynamical behavior always and only from the “inertial viewpoint” is too restrictive. 3) which are suitable in multibody kinematics. For instance, instead of assigning one scalar rotation anz gle to a rotation axis, one may use αq three different angles α, β, γ assiqβ gned to three corresponding unit vectors ei ∈ IR3 , i = 1, 2, 3. 4. Cardan Angles AIB = R = eα e1 eβ e2 eγ e3 . 17) The single exponential matrices shall be called (transposed) “elementary transformations”.

4), dr = ur dϕ ⇒ rT On Rotation dr = [rr]T u = 0. 7) yields the eigenvalues √ (i = −1). ). It follows that ro = r(ϕ = 0) = Xc, leading to r(ϕ) = Xdiag{exp(λi ϕ)}X−1 ro := Φro . 6)] has been used. One has thus found the mapping to be R = exp(uϕ). 11) This seemingly simple result has the drawback that the matrix exponential is only defined by its series expansion, R = E + uϕ + u2 ϕ2 ϕ3 ϕ4 + u3 + u4 + ··· . 2! 3! 4! 12) However, because u3 = −u ⇒ u4 = −uu etc. (verified by direct calculation), one may collect the powers of ϕ, R= E+u ϕ− ϕ3 ϕ5 ϕ2 ϕ4 + + · · · + uu − + ··· 3!

21) is: transformation from an inertial representation to a body-fixed one needs three subsequent rotations, first about the e1 -axis with α, followed by a rotation about the (new) e2 -axis with β, followed by a rotation about the (new) e3 -axis with γ. As a consequence, the ˙ γ) axes that the elementary rotations (α, ˙ β, ˙ refer to do not form an orthogonal frame (see Fig. 7). Notice that, once a sequence of rotations is chosen, the sequence of matrix multiplications is fixed since matrix multiplications are not commutative.

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