More Examples Of Generalized Coordinates

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1 Slides of ecture 8 Today s Class: Review Of Homework From ecture 7 Hamilton s Principle More Examples Of Generalized Coordinates Calculating Generalized Forces Via Virtual Work /3/98 /home/djsegal/unm/vibcourse/slides/ecture8.frm

2 Homework from ecture 7 In the most recent class, we derived the governing equations for the compound pendulum. The homework assignment was to verify the derivation and to linearize the resulting equations. θ R We found the governing equations to be m R θ m ( m + m )R θ + m R R θ cos( θ θ ) m R R θ sin ( θ θ ) + g ( m + m )R sin θ and m R θ + m R R θ cos( θ θ ) + m R R θ sin ( θ θ ) + gm R sin θ /3/98 /home/djsegal/unm/vibcourse/slides/ecture8.frm

3 Homework from ecture 7. Establish the location of stable equilibrium about which to θ θ linearize. By observation, that location is. s θ θ + θ s θ θ + θ. Substitute and into the governing equations and expand with Taylor series. ( m + m )R θ + m R R θ cos( θ θ ) m R R θ θ -θ sin ( θ θ ) + g ( m + m )R sin θ θ and m R θ + m R R θ cos( θ θ ) + m R R θ θ -θ θ sin ( θ θ ) + gm R sin θ /3/98 3 /home/djsegal/unm/vibcourse/slides/ecture8.frm

4 Homework from ecture 7 3. Delete all terms involving powers and products of,,, and θ θ θ θ ( m + m )R θ + m R R θ m R R θ ( θ θ ) + g ( m + m )R θ m R θ + m R R θ + m R R θ ( θ θ ) + gm R θ and /3/98 4 /home/djsegal/unm/vibcourse/slides/ecture8.frm

5 Homework from ecture 7 4. Group terms to form the mass and stiffness matrix. ( m + m )R m R R m R R m R θ θ + g ( m + m )R θ gm R θ Note that both the mass and stiffness matrices are symmetric and positive-definite. The mass equation will be positive so long as we avoid massless degrees of freedom, that is we always want to choose our degrees of T freedom so that. q r The stiffness matrix will always be non-negative definite. If there are no rigid-body modes, it will be positive definite. We shall discuss this more later. /3/98 5 /home/djsegal/unm/vibcourse/slides/ecture8.frm

6 Hamilton s Principle We consider a mechanical system whose configuration at any time is characterized by the N generalized coordinates. The system is subject to potential energy and additional forces and evolves over the interval according to the agrange equations d dt T q r T q r (, t ) { q } A V { F r } V A + F q r r r for each. We can imagine the evolution of the system configuration over that interval by picturing the motion of a point whose coordinates are { q ( t ) } in an N-dimensional Cartesian system. q 3 q( ) q(t) q q(t ) q /3/98 6 /home/djsegal/unm/vibcourse/slides/ecture8.frm

7 Hamilton s Principle We contract these scalar equations with test functions η r which are as yet undetermined except for the conditions for each r, and then sum them. η r ( ) η r ( t ) N t d T T V A + F dt q r q r q r η r ( t ) d t r r /3/98 7 /home/djsegal/unm/vibcourse/slides/ecture8.frm

8 Hamilton s Principle Integration by parts yields N r t T q r N r V A + F q r η r ( t ) d t r t d dt T ηr q r T η r d t q r + η r ( ) η r ( t ) Recalling that, the above integral simplifies N t T V A T + F q r q r η r ( t ) η r r q r d t r /3/98 8 /home/djsegal/unm/vibcourse/slides/ecture8.frm

9 Hamilton s Principle The above is true for all test functions η r. et η r ( t ) q r ( t ) q r ( t ) { q ( t ) } δq r ( t ) where is another path { q ( ) } q t { ( ) } from to near { q ( t ) } the path taken by. q 3 {q(t)} q( ) ~ {q(t)} q q(t ) Our integrals can now be written: q N T δqr + T V A q r δq r + F q r q r δq r r d t r t Observe that the terms involving potential energy are a complete differential. /3/98 9 /home/djsegal/unm/vibcourse/slides/ecture8.frm

10 Hamilton s Principle Rearranging the above: t N T δqr + T V A q r δq r + + F q r q r δq r r d t r t ( δt δv + δw ) d t A δw ( t ) F r ( t ) ( q r ( t ) q r ( t ) ) Where. This form of Hamilton s principle asserts that the actual path is one about which t ( δt δv + δw ) d t /3/98 /home/djsegal/unm/vibcourse/slides/ecture8.frm

11 Hamilton s Principle: Special Case A For the special case where the generalized forces F r are prescribed loads, we can define the Potential Energy of oading N A A F r qr r V T V + A and the Total Potential Energy is. In this case, Hamilton s Principle becomes: The true path in configuration space of the system makes the quantity t J ( T V T ) d t stationary. /3/98 /home/djsegal/unm/vibcourse/slides/ecture8.frm

12 Hamilton s Principle: Example Beam Bending Equation The strain energy in an Euler- Bernoulli beam is V -- EI. The x potential energy of loading is y d x M Q p(x) M Q A M y x M y Q y ( ) + Q y ( ) p ( x )y ( x ) d x x T And the Kinetic Energy is where is the mass per unit length of the beam. -- m y d x m t /3/98 /home/djsegal/unm/vibcourse/slides/ecture8.frm

13 Hamilton s Principle: Example Beam Bending Equation ets evaluate the virtual quantities, beginning with Kinetic Energy: δt -- m ( y + δy ) t o m y t δy d x t d x -- m y d x t and t t y δt d t m δy d x t t d t t m t y δy ẏ δy d t t t d x /3/98 3 /home/djsegal/unm/vibcourse/slides/ecture8.frm

14 Hamilton s Principle: Example Beam Bending Equation t t δt d t m [ ẏδy ] t d x m t y δy d x d t t Strain Energy: δv -- EI EI y'' x ( y + δy ) d x δy x d x EI y'' δy x -- EI y x d x EI ( y'''δy ) + EI y IV δy d x /3/98 4 /home/djsegal/unm/vibcourse/slides/ecture8.frm

15 Hamilton s Principle: Example Beam Bending Equation The strain energy term becomes: t δv d t t EI y'' δy x EI ( y'''δy ) + EI y IV δy d x d t /3/98 5 /home/djsegal/unm/vibcourse/slides/ecture8.frm

16 Hamilton s Principle: Example Beam Bending Equation In a similar manner, we find the contribution from Potential Energy of oading t δa d t t M δy M x δy Q x δy ( ) + Q δy ( ) d t t p ( x )δy d x d t /3/98 6 /home/djsegal/unm/vibcourse/slides/ecture8.frm

17 Hamilton s Principle: Example Beam Bending Equation We can group terms. δy ( ) δy ( ) We start with the terms involving,, δy, and x x δy t M δy x M δy x Q δy ( ) + Q δy ( ) d t t EI y'' δy + EI ( y'''δy ) x d t /3/98 7 /home/djsegal/unm/vibcourse/slides/ecture8.frm

18 Hamilton s Principle: Example Beam Bending Equation [ Q E I y''' ( ) ]δy ( ) From which we deduce. y ( ) A geometric boundary condition specifying implies that δy ( ) Q EI y''' ( ). If displacement is not specified there, then. This is a natural boundary condition. [ Q E I y''' ( ) ]δy ( ) Similar interpretations are made of, [ M E I y'' ( ) ] δy x [ M E I y'' ( ) ] δy x and, /3/98 8 /home/djsegal/unm/vibcourse/slides/ecture8.frm

19 Hamilton s Principle: Example Beam Bending Equation Matching terms in the spacial integral we have t [ mẏ + EI y IV p ]δy ( x, t ) d x d t from which we conclude that mẏ + EI y IV p ( x, t ) /3/98 9 /home/djsegal/unm/vibcourse/slides/ecture8.frm

20 Hamilton s Principle Hamilton s principle is general and always works, though sometimes it is hard to evaluate. In particular, note how Hamilton s Principle is used to derive the partial differential governing equations. Also, we saw how to define the potential energy of loading and to use that with Hamilton s principle. We will see that we can also use it in with agrange s equations. /3/98 /home/djsegal/unm/vibcourse/slides/ecture8.frm

21 Change of Subject. The following is an introduction to the method of ASSUMED MODES. /3/98 /home/djsegal/unm/vibcourse/slides/ecture8.frm

22 More On Generalized Degrees of Freedom Distributed Displacement ets consider an Euler Bernoulli beam simply supported at each end. Initially, we assume that all forces are conservative. We postulate a displacement distribution of the sort y ( x, t ) A ( t ) x ( x ) A t ( ) x ( x ) A ( t ) f x ( ) + A t ( ) f ( x ) We shall derive agrange equations for the evolution of and A ( t ). These are our generalized degrees of freedom. A ( t ) /3/98 /home/djsegal/unm/vibcourse/slides/ecture8.frm

23 More On Generalized Degrees of Freedom Distributed Displacement Kinetic Energy: T -- mẏ d x -- m Ȧ ( t ) f [ ( x ) + Ȧ ( t ) f ( x ) ] d x -- [ ( Ȧ ) I + ȦȦI + ( Ȧ ) I 3 ] I m f ( x ) d x I m f x where, m ( ) f x d x m and I 3 m f ( x ) d x m /3/98 3 /home/djsegal/unm/vibcourse/slides/ecture8.frm

24 More On Generalized Degrees of Freedom Distributed Displacement Strain Energy: V where -- EI ( y'' ) d x -- EI A [ ( t ) f '' ( x ) + A ( t ) f '' ( x ) ] d x -- A [ ( ) I 4 + A A I 5 + ( A ) I 6 ] I 4 EI f '' ( x ) d x I 6 EI f '' x EI I 5 EI f '' ( x ) f '' ( x ) d x ( ) d x, and EI EI /3/98 4 /home/djsegal/unm/vibcourse/slides/ecture8.frm

25 More On Generalized Degrees of Freedom Distributed Displacement agrange Equations: d dt T Ȧ T A V + Ȧ I A + Ȧ I + A I 4 + A I 5 and d dt T Ȧ T A V + Ȧ I A + Ȧ I 3 + A I 5 + A I m Ȧ EI In matrix form: A A Ȧ Note that both matrices are symmetric, positive definite. /3/98 5 /home/djsegal/unm/vibcourse/slides/ecture8.frm

26 Generalized Forces, Calculated by Method of Virtual Work Recall that the generalized force associated with the generalized coordinate is q r F r Fn n xn q r We examine the incremental work associated with increments of q r : xn δw F r δq r Fn δqr q r Fn δxn n n The generalized force associated with the generalized coordinate q r is δw F r δq r /3/98 6 /home/djsegal/unm/vibcourse/slides/ecture8.frm

27 Generalized Forces, Calculated by Method of Virtual Work ets consider an Euler Bernoulli beam simply supported at each end. p(x) M M We consider moments M and M applied at the ends and a distributed traction applied along the length of the beam. We postulate a displacement distribution of the sort y ( x, t ) A ( t ) x ( x ) A ( t ) x ( x ) A ( t ) f x ( ) + A t ( ) f ( x ) ets calculate the generalized forces associated with the generalized A A coordinates and. /3/98 7 /home/djsegal/unm/vibcourse/slides/ecture8.frm

28 Generalized Forces, Calculated by Method of Virtual Work The work done to the structure by the forces acting through a virtual displacement δy ( δa ) f ( x ), is δw δa M f ' ( ) + M f ' ( ) + p ( x ) f ( x ) d x so F A M f ' ( ) + M f ' ( ) + p ( x ) f ( x ) d x F A can be calculated similarly. /3/98 8 /home/djsegal/unm/vibcourse/slides/ecture8.frm

29 Homework for ecture 8 A numerical experiment with linearization Many times we have derived the equations for a spring κ g θ θ mr + -- sin θ R reinforced pendulum:. κ The linearized form is where κ ω mr g + -- R. θ + ω θ θ m R We use the linearized frequency to non-dimensionalize the time parameter. Define, define, and define α g R ω τ ω t φ ( τ ) θ ( τ ω ) /3/98 9 /home/djsegal/unm/vibcourse/slides/ecture8.frm

30 Homework for ecture 8 continued d φ Then and d τ d t d θ ω d φ d τ + [ ( α )φ + α sin φ ]. Solve numerically the dimensionless governing equation for the φ ( ) π initial conditions: and over the period (, 6π ) α α α dφ dτ for the three cases:,, and.. Do the same as above but for the initial conditions and dφ dτ φ ( ) π Compare and discuss the your results for parts and. /3/98 3 /home/djsegal/unm/vibcourse/slides/ecture8.frm

31 Next Time Example problems students choice and discussion of past material. Discussion of mid-term exam. /3/98 3 /home/djsegal/unm/vibcourse/slides/ecture8.frm