Given: 29/08/2014 (Wk 6)
Assignment 2
Due: 18/09/2014 (Wk 8)
ANALYTICAL MODEL OF THE ENGINE AND TRANSMISSION ASSEMBLY VIBRATING SYSTEM
RMIT SAMME AUTOMOTIVE SYSTEMS AND CONTROL (AUTO1029)
The engine and transmission assembly was modelled as a rigid body supported on resilient mounts as illustrated in Figure 3.2. Its motion was described in six degrees of freedom (6 DOF), with body translations and rotations about an axes system originating from the centre of mass.
ASSIGNMENT 2: GENERAL RESPONSE OF AUTOMOTIVE ENGINE SYSTENSION SYSTEM USING MATLAB/SIMULINK
The mounts were assumed to be linear elastic springs, with independent stiffness’s along their principal axes. Viscous damping in the mounts was assumed.
This assignment is due on Thursday, September 18. Submit via Blackboard. No late submission will be accepted unless an application for extension (in writing and on the approved form) has been lodged and approved before submission date. Refer to the details of the engine and transmission assembly for the four-cylinder motor vehicle (from J.K.Vethecan, MEng Thesis, RMIT, 1988) Consider the special case where e x = e y = 0 ; that is, the second order unbalance force acts through the c.g. of the engine and transmission assembly. Otherwise, the data is shown as in Table 1. The damping can be taken to be negligible.
1. 2. 3.
4. 5. 6. 7.
Treat the dynamic system as having six degrees of freedom, and solve for the natural frequencies (in both, rad/s and Hz) of the system using MATLAB Prove that 3rd and 6th modes are orthogonal with respect to both, mass and stiffness matrices. Calculate the receptance frequency response function z2/F0 (i.e displacement of engine Mount 2 divided by excitation force F0) by synthesising the normalized (with respect to [M]) mode shapes. Plot the receptance frequency response function for the range 5-25 Hz. Also calculate the force transmitted through Mount 2 at 5,000 rpm engine speed (magnitude only required). Calculate z2 response of the system, excited with the only external input force F z, applied to the centre of gravity of the engine. Discuss your results. Apply the FFT (Fast Fourier Transform) to the z2 response, using fft MATLAB command and comment on the findings.
©2014, RMIT, School of Aerospace, Mechanical & Manufacturing Engineering
The axes, X; Y and Z, were selected so as to be orthogonal with the principal elastic axes of the mounts. Consequently the shear stiffness components, or the coupling stiffness’s, cancel from the equations, reducing the number of non zero elements in the stiffness matrix. The foundation, namely the vehicle structure, was considered to be rigid such that base motion could be ignored. This assumption was justifiable as road -induced inputs are low frequency in comparison to the 100-200 Hz range being investigated here. THE EQUATION OF OF MOTION
Lagrange's equations were applied in the derivation of the equations of motion, and can be written in the matrix form as,
M q C q K q F ..
.
(1)
where [M]; [C] and[K] are the mass, damping and stiffness matrices respectively, all six by six symmetric matrices. Damping in the mounts was assumed to be viscous. The mass matrix is, 0 0 0 m 0 0 0 m 0 0 0 0 0 0 m 0 0 0 M 0 0 0 I xx I xy I xz 0 0 0 I xy I yy I yz 0 0 0 I xz I yz I zz where M is the mass of the engine and transmission assembly and I ij (i, j = x, y, z) are its mass moments of inertia.
(AUTO-1029)
Lecturer: Prof. Pavel M.Trivailo
Given: 29/08/2014 (Wk 6)
Assignment 2 2
The stiffness matrix is, k xi 0 0 K 0 k xi a zi k xi a yi
0
k
yi
0
0
k yi a zi
0
k
k yi a zi
k a
xi zi
k a
k zi a yi
2
yi zi
k zi a xi
k zi a xi a yi
0
k yi a xi a zi
yi xi
k zi a xi
zi yi
zi yi
k a
0
k a
zi
0
k a
0
2
k zi a xi a yi
k a
zi xi
2
k xi a zi 2
k xi a yi a zi
k xi a yi
k a yi xi 0 k yi a xi a zi k xi a yi a zi k xi a yi 2 k yi a xi 2
th
Here k xi, k yi, k zi are the stiffness’s of the i mount and a xi, ayi, azi are its coordinates from the centre of mass. The summation symbol applies from i = 1 to n, where n is the number of mounts. The damping matrix is identical in composition to the stiffness matrix, with the stiffness elements replaced with damping elements. The displacement vector {q} consists of the x, y, z translations and x; y; z rotations of the body about the X, Y, Z axes. {F} is the forcing vector, containing the unbalanced second order inertial forces and couples. The displacement vector, {q}, and the forcing vector, {F}, are,
q x
y z
F F x
F y
Due: 18/09/2014 (Wk 8)
x
y
F z
z
T
M y
Ixy = -0.386 Iyz = -1.686 Izx = -0.285
Mass (kg) Engine and transmission assembly mass: m = 167.0 Reciprocating mass: M = 0.753 Length (m) Connecting rod length: l = 0.136 Crank radius: r = 0.0398 Distance between cylinder centrelines: S = 0.100 Distance from centre of force to c.g.: * ex = 0.072 * ey = 0.050 *
In this assignment, assume these are zero.
Distance from c.g. to engine mounts: Mount No. x y 1 0.358 0.260 2 0.263 -0.210 3 -0.437 0.290 4 -0.352 -0.130
M x
Inertias (kg.m ) Ixx = 6.892 Iyy = 11.076 Izz = 10.349
M z
T
z 0.050 0.070 -0.070 0.005
THE F ORCING F UNCTION
Internally generated forces in an internal combustion engine result from unbalanced fluctuating inertia forces and couples generated by the reciprocating masses and the combustion process. These vibratory forces are transmitted through the mountings into the vehicle structure. In conventional in-line four cylinder engines, the even harmonics are unbalanced and are dominated by the second harmonic, which causes high vibration levels at twice the crankshaft rotational frequency.
Dynamic Spring Rates (N/m) Mount No. x y 1 60 000 270 000 2 309 000 90 000 3 20 000 100 000 4 180 000 530 000
When only the second order reciprocating forces are included, the vector {F} becomes
Table 1. Inertia, mass, geometric and stiffness data required for the model
F 0
0
F 0
e y F 0
e x F 0
z 135 000 180 000 30 000 240 000
0
T
where F0 = 4Mr 2/, = l/r. These quantities are given in Table 1. ©2014, RMIT, School of Aerospace, Mechanical & Manufacturing Engineering
(AUTO-1029)
Lecturer: Prof. Pavel M.Trivailo