4569846498
Share Embed Flag
Download ePaper

4569846498

lollypop413
01.05.2017 Views

Kinematics and dynamics of rigid bodies 59 ∫ 2 2 ) Izz ( x y d m In addition we can introduce the products of inertia I xy , I yz and I xz : (2.172) Ixy Iyx ∫ xy dm Iyz Izy ∫ yz d m Ixz Izx ∫ xz d m (2.173) (2.174) (2.175) This allows (2.167) to (2.169) to be arranged in matrix form as follows: ⎡Hx ⎤ ⎡Ixx Ixy Ixz ⎤ ⎡ x ⎤ ⎢ H ⎥ ⎢ ⎥ (2.176) ⎢ y ⎥ I I I ⎢ ⎢ xy yy yz ⎥ ⎢ ⎥ y ⎥ ⎣⎢ Hz ⎦⎥ ⎢I I I ⎥ ⎣ xz yz zz ⎦ ⎣⎢ z ⎦⎥ If we return to our earlier consideration of Body 2 shown in Figure 2.27 the matrix equation in (2.176) would lead to {H 2 } 1/1 [I 2 ] 2/1 { 2 } 1/1 (2.177) In writing the vectors {H 2 } 1/1 { 2 } 1/1 we revert to the full definition of a vector used here where the upper suffix indicates that the vector is measured relative to the axes of reference frame O 1 and the lower suffix indicates that the components of the vector are resolved parallel to the axes of frame O 1 . The matrix [I 2 ] 2/1 is the moment of inertia matrix for Body 2 about its mass centre G 2 located at frame O 2 . The use of the upper and lower suffix here indicates that the moments of inertia have been measured relative to frame O 2 but transformed to frame O 1 . This is necessary so that the vector operation in (2.177) is consistent. This is only possible if the vectors and matrix are referred to the same frame, which in this case is O 1 . Note that in this form (2.177) is not practical since the orientation of frame O 2 relative to frame O 1 will change as the body rotates requiring the recomputation of [I 2 ] 2/1 at each time step. The matrix [I 2 ] 2/2, or in simpler form [I 2 ] 2 , is constant since it is measured relative to and referred to a frame that is fixed in Body 2 and hence only needs to be determined once for an undeformable body. When considering the equation for the angular momentum of a body it is preferable therefore to consider all quantities to be referred to a frame fixed in the body, in this case frame O 2 : {H 2 } 1/2 [I 2 ] 2/2 { 2 } 1/2 (2.178) Before progressing to develop the equations used to describe the dynamics of rigid bodies translating and rotating in three-dimensional space the definition of the moments of inertia introduced here requires further consideration. 2.10 Moments of inertia From our previous consideration of the angular momentum of a rigid body we see that there are three moments of inertia and three products of inertia

60 Multibody Systems Approach to Vehicle Dynamics δm Y 2 {R} 2 Y 1 Body 2 O 2 G 2 X 2 GRF O 1 X 1 Fig. 2.28 Moment of inertia for plane motion the values of which must be specified to analyse the rotational motion of the body. Before considering the three-dimensional situation it is useful to start with the two-dimensional inertial properties associated with plane motion. For the Body 2 shown in Figure 2.28 we assume that a constraint has been applied that allows the body to move only in the X 1 Y 1 plane of frame O 1 , the ground reference frame. As such the body has three degrees of freedom, these being translation in the X 1 direction, translation in the Y 1 direction and rotation about the Z 1 -axis. From the expressions given earlier for the moments of inertia it can be seen that these are in fact the second moments of the mass distribution about the chosen frame fixed in the body. For the body shown in Figure 2.28 we are only interested in rotation about the Z 1 -axis and as such only require the I zz moment of inertia. To indicate that this is for Body 2 we will refer to this as I 2zz . Considering the particle of mass m located by a general position vector {R} 2 at point P we see that we are not only measuring the vector with respect to frame O 2 but we are also referring the vector to frame O 2 . Ignoring the z co-ordinate as this is for plane motion in X 1 Y 1 we can take the general case and say that the position of the element is given by {R} T 2 [x y0]. The moment of inertia I 2zz is found by summing the second moments of the elements of mass over the volume of the body: I 2zz Σ |R| 2 2 m Σ (x 2 y 2 ) m (2.179) In the limit as the volume of the element becomes infinitesimal we can obtain an expression for the moment of inertia I 2zz as follows: I ( x y ) dm (2.180) The moment of inertia I 2zz in (2.180) is therefore the integral of the mass elements each multiplied by the square of the radial distance from the z-axis. This distance is referred to as the radius of gyration, in this case k 2zz where this is given by k 2zz 2zz ∫ 2 I2 m zz 2 2 (2.181)

60 Multibody Systems Approach to Vehicle Dynamics<br />

δm<br />

Y 2<br />

{R} 2<br />

Y 1<br />

Body 2<br />

O 2 G 2<br />

X 2<br />

GRF<br />

O 1<br />

X 1<br />

Fig. 2.28<br />

Moment of inertia for plane motion<br />

the values of which must be specified to analyse the rotational motion of the<br />

body. Before considering the three-dimensional situation it is useful to start<br />

with the two-dimensional inertial properties associated with plane motion.<br />

For the Body 2 shown in Figure 2.28 we assume that a constraint has been<br />

applied that allows the body to move only in the X 1 Y 1 plane of frame O 1 ,<br />

the ground reference frame. As such the body has three degrees of freedom,<br />

these being translation in the X 1 direction, translation in the Y 1 direction<br />

and rotation about the Z 1 -axis. From the expressions given earlier for<br />

the moments of inertia it can be seen that these are in fact the second<br />

moments of the mass distribution about the chosen frame fixed in the body.<br />

For the body shown in Figure 2.28 we are only interested in rotation about<br />

the Z 1 -axis and as such only require the I zz moment of inertia. To indicate<br />

that this is for Body 2 we will refer to this as I 2zz . Considering the particle<br />

of mass m located by a general position vector {R} 2 at point P we see that<br />

we are not only measuring the vector with respect to frame O 2 but we are<br />

also referring the vector to frame O 2 .<br />

Ignoring the z co-ordinate as this is for plane motion in X 1 Y 1 we can take<br />

the general case and say that the position of the element is given by<br />

{R} T 2 [x y0]. The moment of inertia I 2zz is found by summing the second<br />

moments of the elements of mass over the volume of the body:<br />

I 2zz Σ |R| 2 2 m<br />

Σ (x 2 y 2 ) m (2.179)<br />

In the limit as the volume of the element becomes infinitesimal we can<br />

obtain an expression for the moment of inertia I 2zz as follows:<br />

I ( x y ) dm<br />

(2.180)<br />

The moment of inertia I 2zz in (2.180) is therefore the integral of the mass<br />

elements each multiplied by the square of the radial distance from the z-axis.<br />

This distance is referred to as the radius of gyration, in this case k 2zz where<br />

this is given by<br />

k<br />

2zz<br />

2zz<br />

<br />

∫<br />

2<br />

I2<br />

m<br />

zz<br />

2<br />

2<br />

(2.181)

logo

products

Resources

Company

Terms of service
Privacy policy
Cookie policy
Cookie settings
Imprint
Change language
Made with love in Switzerland
© 2024 Yumpu.com all rights reserved

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!