Motion of deformable bodies

Transcription

1 Karlstad university Faculty for health, science and technology FYGB08 Motion of deformable bodies Author: Axel Hedengren. Supervisor Professor Jürgen Fuchs. - January 23, 2017

2 Abstract Abstract This work analyses the motion of deformable bodies. The main point of departure is Lagrange s form of d Alembert s principle, which says that The total virtual work of the impressed forces plus the internal forces vanishes for reversible displacements. The first part of the theory states which coordinate systems involved and their relations. In the second part the different forces that could possibly act on or in the body and its effect on the different coordinate systems are stated, first in a general case for deformable bodies, after that in cases with constraints and at last reduced to a special case for the motion of a rigid body. 1

3 Contents Contents Abstract 1 Introduction 3 Background 4 4 Geometric representation Dynamical equations of motion General case: Elastic body Motion of deformable body with constraints Special case: Rigid body Conclusion 13 2

4 Introduction Introduction Deformation is defined as change of size or form of an object due to applied forces. These forces could be external forces or internal forces, for example internal forces due to temperature changes. For the calculations in this work two main assumptions have been adopted. The first, and probably the roughest, is that any atomic structure has been neglected and that the structure of the body is said to be continuous, this assumption is called the continuum principle. The second main assumption is called solidification principle and means that the body consists of mathematical points at infinitesimal distances, each point is said to act under rigid body conditions. This leads to Lagrange s form of d Alembert s principle, which states that The total virtual work of the impressed forces plus the internal forces vanishes for reversible displacements, (F i m i a i ) δr i = 0. (1) i This principle is the dynamical analogue to the principle of virtual work for applied forces in static systems, and is in fact more general than Hamilton s principle. The goal with this paper is to obtain the equations of motion for deformable bodies, not to solve them. 3

5 Background Consider a cat that is dropped upside down, if the cat is assumed to be a rigid body (for example a wood crafted cat) it will, as long as there are no external forces acting on it, land on its back. However if the cat is alive it will with the greatest probability land on its feet. To physically explain this phenomenon and other motions of deformable bodies we are forced to go deeper into the theory than for motions of rigid bodies. Geometric representation It is convenient to identify a reference configuration of the body which all subsequent configurations are referenced from. This reference configuration is often the initial configuration at t = 0, and is sometimes called undeformed configuration. One assumes a Cartesian coordinate system x 1, x 2, x 3 be assigned to a body, and lets the relative position of an arbitrary point P 0 of the body in reference configuration, in connexion to Origo of the body system, be represented by a vector The base vectors of this coordinate system are given by r 0 = r 0 (x 1, x 2, x 3 ). (2) J µ = r 0 x µ = r 0,µ µ = 1, 2, 3. (3) Further on, if one assumes that after some time t a deformation has occurred, the vector r has changed direction, at time t we are in a strained configuration, sometimes called deformed- or current-configuration. The position vector to the point P relative to the body frame is defined as r B = r B (x 1, x 2, x 3 ). (4) The base vectors of the strained configuration are given by L µ = r B x µ = r B,µ µ = 1, 2, 3. (5) The vector d from P 0 to P is called the elastic displacement vector. It is possible to express the position of the point P relative to the body frame as r B = r 0 + d. (6) Then the base vectors in the strained configuration can be expressed as The relations above are visualized in figure 1. L µ = (δ χ µ + d χ,µ)j χ. (7) 4

6 Figure 1: The point P 0 and P relative to a body system. If one assumes that the body system is connected to a coordinate system (X 1, X 2 X 3 ), that is fixed in the inertial space with unit vectors i 1, i 2, i 3, with a vector r I.B, then the vector to the point P from the Origo of the inertial frame can be expressed as r = r I.B + r 0 + d. (8) This relation is visualized in figure 2. From equation (8) one obtains that the change of r over time can be expressed as dr dt = dr ( ) I.B dd + + ω (r 0 + d). (9) dt dt [1][2][3]. r 5

7 Figure 2: The point P 0 and P relative to a inertial system. Dynamical equations of motion The relation between the inertial frame and the body frame can be expressed with an orthogonal matrix A: And from equation (7) one can obtain that General case: Elastic body J α = Ai β. (10) L α = ZJ γ = ZAi β. (11) By the theory of virtual work one can obtain that the internal virtual work must equal the external virtual work, such that In an elastic body, the possible virtual forces are External: Body surface forces (B(x i, t)). Body forces (F(x i, t)). δw I = δw E. (12) 6

8 ( ) d 2 r Inertia forces ( ρ ),(This is actually not a force in strict meaning dt 2 but it has a force-like behaviour and are therefore treated as a force. The term comes from the right hand side of newtons second law F = ma). Internal: Elastic energy (σ µ (δr),µ ). This gives the dynamical virtual work expression ( ( )) d 2 r σ µ (δr),µ d = F(x i, t) ρ d + dt 2 By using that S B(x i, t) δrds. (13) r = r α i α F = F β J β (14) B = B γ J γ σ µ = σ µζ L ζ, (15) one can rewrite equation (13), to see each term more explicitly, as F β J β δr α i α d i + B γ J γ δr α i α ds ρ d2 r α S dt i 2 α δr α i α d σ µζ L ζ (δr α i α ),µ d = 0. (16) The virtual displacement is given by δr = δr I.B + δd + δθ (r 0 + d). (17) It can be rewritten in the same manner as for the virtual work, so that δr α i α = δr α I.Bi α + δd α j α + δθ α j α (r β 0 j β + d β j β ). (18) Now we have an expression for the virtual work and for the virtual displacement, but they are expressed in a mix of coordinate systems. By rewriting the equations in matrix form and using equations (10) and (11) one could express the equations for the virtual work and displacement in the inertial frame. One then obtains the equation for the virtual work as (A t {F }) t {δr}d + (A t {B}) t {δr}ds ρ{ d2 r S dt 2 }t {δr}d (A t Z t {σ µ }) t {δr,µ }d = 0, (19) and the equation for the virtual displacement as {δr} = {δr I.B } + A t {δd} + A t δθ({r0 } + {d}). (20) We can from equation (20) derive {δr,µ }, here is {r I.B } not a function of body axis variables and are therefore treated as a constant, {δr,µ } = A t {δd,µ } + A t δθ({r0,µ } + {d,µ }). (21) Now we want to analyse each integral in equation (19). 7

9 First we take the first integral in equation (19), rewrite it in the body frame system (J) and substitute equation 20, such that (A t {F }) t {δr}d = {F } t (A{δr I.B } + {δd} + { δθ}({r 0 } + {d}))d. (22) Recall that this part represents the virtual work due to the resultant applied forces, therefore we further on denote the force-term in this integral with an index B. To proceed one needs to determine the absolute velocity of the body frame in comparison with the inertial system. Suppose that we have n generalized coordinates. The velocity is {v I.B } = {ṙ I.B } = q j {r I.B } q j + t {r I.B}. (23) Defining a velocity coefficient in the inertial frame the virtual displacement of {r I.B } can be expressed as {γ I.B j } = q j {r I.B }, (24) {δr I.B } = {γ I.B j }δq j. (25) The velocity of {d} is a relative velocity. Expressed in body space we have { d} = q j {d} q j. (26) Define {γj d } = { d} q = {d}. (27) j q j The virtual displacement of {d} is This leads to the derivative {δd} = {δd,µ } = {γj d }δq j. (28) {γj,µ}δq d j. (29) 8

10 Now we almost have all tools to continue with the virtual work equation, we just have to obtain the virtual work due to inertial moments. Defining the angular velocity coefficient in body space as Then we can express a small virtual rotation as {β j } = {ω}. (30) q j {δθ} = {β j }δq j. (31) Now we can continue with the first integral. Inserting the results from equation (25), (28) and (31) into equation (22) gives (A t {F }) t {δr}d = {F B } t {γ j }d δq j, (32) where {γ j } = A{γ I.B j } + {γ d j } + { β j }({r o } + {d}). (33) The surface integral in equation (19) can be obtained by a similar procedure, where the external surface force is denoted by an index S, (A t {B}) t {δr}ds = {F S } t {γ j }dsδq j. (34) S S In the third integral we have a second order time derivative of r. equation (8) we have From {r} = {r I.B } + A t {r 0 } + A t {d}. (35) Where all terms varying with time except for r 0. We also have that Ȧ = ωa = ω t A. (36) The second order time derivative of r therefore becomes { r} = { r I.B } + A t ω{r 0 } + A t ω 2 {r 0 } + A t ω{d} + A t ω 2 {d} + 2A t ω{ d} + A t { d}. (37) By substituting this expression into the third integral of equation (19) and define the inertial force as F = ρ(a{ r I.B } + ω{r 0 } + ω 2 {r 0 } + ω{d} + ω 2 {d} + 2 ω{ d} + { d}), (38) one obtains ρ r t {δr}d = {F } t {γ j }d δq j. (39) 9

11 In the last integral of equation (19) one can use the equation for the derivative of the virtual displacement, equation (21), and also the virtual displacement of d and of virtual rotation, equations (29) and (31). This gives, in body frame components, that (A t Z t {σ µ }) t {δr µ }d = {F µ E }t {γ d j,µ}d δq j, (40) where the elastic force is denoted by an index E, and is defined as If we define: {F µ E } = Zt {σ µ }. (41) The generalized external applied forces Q j = {F B } t {γ j }d + {F S } t {γ j }ds. (42) The generalized inertia forces Q j = The generalized elastic force Q E j = By use d Alembert s principle one obtains S {F } t {γ j }d. (43) {F µ E }t {γ d j,µ}d. (44) (Q j + Q J + Q E j )δq j = 0. (45) Further, if one assumes that the various q j are linearly independent, then each coefficient of δq j must vanish, and we obtain Q j + Q J + Q E j = 0, (j = 1, 2,.., n). (46) And in greater detail we have Q j = ρ(a{ r I.B } + ω{r 0 } + ω 2 {r 0 } + ω{d} + ω 2 {d} + 2 ω{ d} + { d}) t {γ j }d + (Z t {σ µ }) t {γj,µ}d, d (j = 1, 2,.., n). (47) which is the matrix form of the general dynamical equations of motion.[2][3][4]. 10

12 Motion of deformable body with constraints Equation (47) gives the equations of motion for a dynamical body without any holonomic or nonholonomic constraints on the generalized coordinates. It can somehow be an advantage in the computation to introduce constraints if there is possible to detect in the abstraction of the problem. Therefore it is preferable to determine the dynamical equations of motion with constraints. In general for a system with n degrees of freedom we need n generalized coordinates, say q 1, q 2,.., q n, to describe the system. However assume that there exist m independent constraints such that a ji (q, t) q i + a jt (q, t) = 0, j = 1, 2,.., m. (48) i=1 Introduce a set of n m independent parameters u j, called generalized speeds: u j = ψ ji (q, t) q i + ψ jt (q, t), j = 1, 2,.., n m. (49) i=1 By using the generalized speeds in the velocity- and angular velocity coefficients, such that {γ I.B j } = u j {ṙ I.B }, {γ d j } = u j { d} (50) {γ d j,µ} = u j { d,µ } {β j } = u j {ω}, (51) equation (47) does not change except for that j = 1, 2,.., n m, and we have Q j = ρ(a{ r I.B } + ω{r 0 } + ω 2 {r 0 } + ω{d} + ω 2 {d} + 2 ω{ d} + { d}) t {γ j }d + (Z t {σ µ }) t {γj,µ}d, d (j = 1, 2,.., n m). (52) [2][3][4]. Special case: Rigid body From the dynamical equations of motions we can now go to the equations of motion for a rigid body. First assume that the body frame system is fixed in the body. Further, the mass of the body is m and the distance from the origin of the body frame system is r c.m. We also have the correspondence between the total mass and the density such that ρd = m. (53) And we also have ρr o d = mr c.m. (54) 11

13 With these limitations one can show that equation (19), with some algebraic manipulations, in vector form, reduces to m( r I.B + r c.m ) γ I.B j + (I ω + ω I ω + mr c.m r I.B ) β j = Q j, j = 1, 2,.., n, (55) where n is the number of independent coordinates and the inertia term I is taken with respect to the origin of the body system, [5]. 12

14 Conclusion Conclusion In this work, a general formalism for the motion of deformable bodies has been explained. However, the method is based on the theory of infinitesimals, whereas real materials have atomic structure that has been neglected. So the calculations, that mathematically are exact, are invalid for real deformable bodies. However the method gives a good approximation, and it can have the same accuracy as for finite models. The equations of motion for deformable bodies that have been derived here are, except for very simple special cases, very difficult or even impossible to solve analytically and are often only numerically treatable. The formalism that has been explained in this paper could be expanded to cover the motion of multibody systems. 13

15 References References [1] Terzopoulos Demetri, Witkin Andrew, Deformable Models: Physically Based Models with Rigid and Deformable Components, IEEE computer graphics and applications, ol. 8 No. 6: 41-51, [2] Escalona J.L., alverde J., Mayo J., Dominiguez J., Reference motion in deformable bodies under rigid body motion and vibration. Part I: theory, Journal of sound and vibration, No. 264: , [3] Weng Shui-Lin, Greenwood Donald T, General Dynamical Equations of Motion for Elastic Body Systems, Journal of guidance control and dynamics, ol. 15 No. 6: , [4] Tong David, Classical dynamics, University of Cambridge, 2004, (accessed ). [5] Greenwood D.T.,Principles of dynamics, 2nd ed., Prentice-Hall, New Jersey,