In this example, a model of a trailer being towed behind a massive vehicle will be developed. In this case, it is logical to use an inertial coordinate system rather than a body-centered coordinate system because the towing vehicle is assumed to move at constant velocity along a straight path in inertial space no matter what the trailer does. Thus a coordinate system moving with the towing vehicle or on the ground aligned with the path of the towing vehicle can be considered an inertial coordinate system.
As shown in Fig. 5.1, the angle h(t) between the centerline of the trailer and the path of the towing vehicle completely describes the motion of the trailer, which is considered to be a rigid body moving in plane motion. When a single position variable such as h and its derivatives suffice
to describe the dynamics of a system, the system is said to have a single degree-of-freedom.
The trailer is assumed to be a rigid body pivoted without friction at the hitch point. The mass of the wheels is included in the total mass of the trailer. The trailer is assumed to act as though it had no suspension and thus to move in plane motion. Although the height of the center of mass of the trailer above the ground does affect real trailers, particularly those with suspension systems, this simple model essentially assumes that all forces and accelerations lie in the ground plane. The new aspect in this example is the consideration of the interaction of pneumatic tires and the road surface. In Chapter 4, the tire–road interaction was discussed in more detail than is necessary for the present purposes. Here only the elementary concept of a cornering coefficient will be used for the analysis.
Fig. 5.1is a sketch of the simplest model of a trailer to be studied in this chapter. The towing vehicle is assumed to move with constant forward velocity U and with no lateral velocity even if there is a lateral force from the trailer on the body at the hitch. The trailer is considered to be pivoted frictionlessly at the hitch. This means that there is no moment exerted on the trailer at the hitch.
Parameters include a and b, the distances from the center of mass to the hitch point and the axle, respectively; m, the trailer mass; Ic, the centroidal moment of inertia; l, the track of the wheels; and U, the constant towing speed. The single geometric degree-of-freedom variable describing the motion of the trailer is the angle h(t), which implies that all velocities and accelerations necessary to write dynamic equations for the trailer can be expressed in terms of h and its time derivatives.
To find a dynamic equation for h, one must consider another variable, the force, from the ground acting on the two tires.Fig. 5.2shows a single tire rolling with velocity U, and being acted upon by a lateral force F. Note that, in Fig. 5.2, F represents a force on the axle of the wheel tending to push the wheel and tire to the right. If one neglects the acceleration of the wheel mass, there is an equal and opposite force on the bottom of the tire from the road that is not shown.
For a given tire at a given normal load, it is found experimentally that the relation between the lateral force, F, and the slip angle, a, has the gen- eral form shown in the right side of Fig. 5.2 (seeFig. 4.2in the previous chapter). For small values of a, the force is nearly proportional to the slip angle and the slope of the relationship is defined as the cornering coefficient Ca. The linear range typically applies up to a slip angle of about 5j or 10j. In the linear range, the side velocity y is much smaller than the rolling
velocity U, and thus a in radians is nearly _y=U rather than the tangent of y/U.
ai y=U ð5:1Þ
The force relationship can be written approximately as
F¼ Caa ð5:2Þ
We now consider the basic motion to be one in which h and _h are zero, and the trailer simply follows the towing vehicle. For a stability analysis, we assume a perturbed motion in which both h and _h are small. This means that small angle approximations can be used for trigonometric functions of h and that the lateral velocity of the tires and thus the slip angles will also turn out to be small.
The major task in developing the equation of motion is to derive an expression for the slip angle of the tires in terms of h and _h. The first step is to make a generic sketch, Fig. 5.3, showing the velocity components at the wheels.
FIG. 5.3 Velocity components at a wheel. FIG. 5.2 Definition of cornering coefficient.
The procedure is to express the velocity at the wheel as a known velocity, U at the hitch point plus~x~rABwhere~rABis the vector from the hitch point to the wheel. This type of calculation as was previously performed for the shopping cart inChapter 1,Eqs. (1.15) and (1.16). In this plane motion case, it is easy to write down components ofx~~rAB using elementary considerations instead of establishing unit vectors and evaluating the expression formally.
One can consider the vector distance from the hitch point to the upper wheel,~rAB, inFig. 5.3as the sum of two vectors. One vector lies along the centerline of the trailer and reaches from the hitch to the center of the axle. The length of this vector is (a+b). A second vector lies between the center of the axle and the upper wheel. It has a length of l/2. Because _h plays the role of x, the result of the velocity components contributed byx~ ~rAB is the two components proportional to _h shown in Fig. 5.3. The distance (a+b) multiplied by _h gives a lateral velocity to the wheel and the distance (l/2) multiplied by _h gives a (small) forward velocity to the wheel. When the vector velocity with magnitude U is combined with the two components proportional to _h, the total velocity of the wheel has been represented in the sketch.
From Fig. 5.3, one can now see how to express the velocity of the wheel as the sum of two other components. The first component is called the rolling velocity and is the component of the velocity in the direction the wheel is pointing. The other component is the lateral velocity and is the component of the velocity perpendicular to the direction the wheel is pointing. These are the two velocity components shown inFig. 5.2as U and
y, respectively.
The rolling velocity in Fig 5.3 is
Ucos hþ ðl=2ÞhiU ð5:3Þ
in which the fact that both h and _h are small have been used (cos hi1, (l/2) _hbU).
The lateral velocity is
Usin hþ ða þ bÞ _hiUh þ ða þ bÞ _h ð5:4Þ
where the approximation sin hih has been used.
A combination of Eqs. (5.3) and (5.4) yields the approximate expression for the slip angle as the ratio of the lateral velocity to the rolling velocity as was indicated in Eq. (5.1).
Note that since the contribution to the rolling velocity,ðl=2Þ _h, was neglected, the expression for slip angle applies equally well to both wheels. (The velocity component associated with the half-track would point in the backward direction on the lower wheel inFig 5.3,but it would be neglected for the rolling velocity anyway. The lateral velocity components would be identical for both wheels.)
We are now in a position to derive an equation of motion for the trailer. The force F inFig. 5.1is the force on the wheels from the ground. When the slip angle is positive as shown in Fig. 5.3, the force on the tires from the ground actually points in the direction shown in Fig. 5.1. Equation (5.5) can now be combined with Eq. (5.2) to yield an expression for the force exerted on the trailer tires
F¼ Caa¼ Caðh þ ða þ bÞ _h=UÞ ð5:6Þ
Although inFig. 5.2,the cornering coefficient was for a single tire, in Eq. (5.6) the coefficient is meant to include the net force being generated by both tires. This is possible because the approximate value for the slip angle in Eq. (5.5) applies equally to both tires.
It will be left as an exercise, to use Newton’s laws to find the equation of motion. The procedure is straightforward. The sum of the lateral forces must equal the mass times the acceleration of the center of mass, mah¨. The sum of all moments about the center of mass must equal the moment of inertia times the angular acceleration, Izh¨. While the procedure is easily described, it does require the introduction of a side force at the hitch in addition to the total tire force. This hitch force is eventually eliminated to produce the final equation of motion. Here an alternative method for deriving the equation of motion will be introduced.
A. Use of Lagrange’s Equations
We will use this opportunity to show how Lagrange’s equations can be used to derive the equation of motion (Crandall et al., 1968). In many cases, it is easier to use this method than to use Newton’s laws that typically require manipulation to eliminate internal variables such as the hitch force in the case at hand. The algebra required for the elimination of internal variables when using Newton’s laws is often subject to human error and the Lagrange equation method eliminates this problem. In any case, it is useful to have available two quite different methods to derive equivalent equa- tions of motion so that one can compare them to check the final result for
correctness. It is assumed that the reader has some familiarity with Lagrange’s equations or will skip this section and derive the equation using Newton’s laws.
Because we have assumed that the hitch has no friction, the connec- tion at the hitch is a so-called workless constraint. Because there is assumed to be no friction moment at the hitch, no energy is gained or lost as the trailer rotates. Also, because the hitch has no lateral velocity, the lateral force at the hitch does no work. Generally, forces and moments associated with workless constraints do not have to be considered when using Lagrange’s equations. For the present trailer model, the hitch force need not be considered when using Lagrange’s equations although it must be considered when using Newton’s equations.
For this problem, there is only a single generalized coordinate, h. The appropriate Lagrange equation is then as follows:
d dt BT B _h B T Ahþ B V Bh ¼ Nh ð5:7Þ
In the Eq. (5.7), T¼ Tðh; _hÞ is the kinetic energy, V=V(h)=0 is the potential energy (which happens to be zero in this case), and Nhis the generalized force for the h generalized coordinate. The generalized force can be evaluated by evaluating the work performed by all forces not asso- ciated with the kinetic or potential energy during a small virtual displace- ment of h,
yW ¼ Nhyh ð5:8Þ
The virtual displacement yh is an imaginary, infinitesimal change or variation in h. The work termyW represents the work carried out on the system by all forces not represented by changes in kinetic or potential energy terms included in Eq. (5.7). In this case, the generalized force will be used to account for the tire force of Eq. (5.6).
The general expression for kinetic energy for a rigid body in plane motion is
T¼ ð1=2Þmv2cþ ð1=2ÞIcx2 ð5:9Þ
in which the only square of the velocity of the center of mass and the square of the angular velocity are required. (For Newton’s laws, one needs to know the vector linear and angular accelerations, which are often compli- cated to evaluate.)
To derive linearized perturbation equations for stability analysis using Lagrange equations, the energy expressions should be correct approximations up to second-order terms in small quantities. It is often useful to find the expressions exactly and then later to make the second- order approximation. Fig. 5.4 shows the velocity components for the center of mass when h is not necessarily small.
Note that although a vector diagram is required to find the vector velocity of the center of mass~vc, only the square of the magnitude of the velocity is required to compute the kinetic energy. Again, the required velocity is the velocity of the hitch point plus the effect of rotation, which is a vector perpendicular to the line from the hitch point to the center of mass and with a magnitude a _h. Using either the law of cosines for one of the velocity triangles in Fig. 5.4, or by evaluating the sum of the squares of two perpendicular components of~vc, the expression for v2c is found to be the following:
vc2¼ U2þ ða _hÞ2þ 2Uða _hÞsin h ð5:10Þ
Now modifying the expression to retain terms in h and _h up to second order, the results required to evaluate the kinetic energy in Eq. (5.9) are
vc2¼ U2þ a2_h2þ 2Ua _hh; x2¼ _h2 ð5:11Þ
Note that second-order terms in small quantities include the squares of the quantities as well as product of the quantities. The constant term U2could actually be left out of Eq. (5.11) because the Lagrange equation, Eq. (5.7), involves only derivatives of the energy functions.
The generalized force Nh, is evaluated by computing the work performed by the force, F, exerted by the ground on the two tires when h is increased by the virtual displacementyh. In this case, the work performed by the tire force on the body is negative when yh is positive. One can imagine increasing h by the amountyh against the force F. In this process, F
would do negative work on the trailer. This negative work would be given by the force times the distance the force moves,
yW ¼ Fða þ bÞyh ð5:12Þ
Now substituting Eq. (5.6) into Eq. (5.12) and considering Eq. (5.8), we see that the generalized force is the coefficient multiplyingyh.
Nh ¼ Caða þ bÞh Caða þ bÞ2_h=U ð5:13Þ
In this particular case, the generalized force is actually a moment because the generalized coordinate h is an angle.
Substituting the quantities in Eq. (5.11) into Eq. (5.9), the kinetic energy is
T¼ mðU2þ a2_h2þ 2Ua _hhÞ=2 þ Ic_h2=2 ð5:14Þ
Finally, the Lagrange equation, Eq. (5.7), yields the single second-order equation of motion for the variable h.
ðIcþ ma2Þ ¨h þ
Caða þ bÞ2 U
.
hþ Caða þ bÞh ¼ 0 ð5:15Þ
This equation is of the form of a single nth order equation as discussed in
Chapter 3,Eq. (3.3) with n=2. Alternatively, it is of the form of sets of second-order equations, Eq. (3.7), but in this case of only a single generalized coordinate there, is only one equation and the matrices involved are actually only scalars. For multiple degrees-of-freedom, Lagrange equation formulations will produce equations of the form of Eq. (3.7).
B. Analysis of the Equation of Motion
Equation (5.15) resembles the equation for a torsional oscillator that might be written J ¨hþ Bh.þ Kh ¼ 0 , where J is a moment of inertia, B is a torsional damping constant, and K is a torsional spring constant. By comparing the two equations, one can give a physical interpretation to stability considerations for the trailer.
First, if the three coefficients of h, _h, and ¨h are positive, the results of Chapter 3 indicate that the trailer is stable and that the trailer will oscillate as a damped torsional oscillator would. The three coefficients in Eq. (5.15) are inherently positive with the possible exception of theh coefficient if a negative value for U were used. It appears that only if U were negative, that
is, if the towing vehicle were backing up, would the trailer act like an oscil- lator with negative damping coefficient and would be therefore be unstable. In fact, this is a case in which merely using a negative value for U does not correctly yield the equation for a reversed velocity because the slip angle expression of Eq. (5.5) does not correctly represent the slip angle for a trailer backing up when U is considered to be negative. (The use of a negative value for the velocity in the analysis of the shopping cart in Chapter 1 to represent backward motion was legitimate in that case.)
IfFig. 5.3is redrawn with U shown in the opposite direction, it will be seen that in the expressions for lateral velocity, Eq. (5.4), and the slip angle, Eq. (5.5), a negative sign appears in the term multiplying h, not in the term multiplying _h. What this means is that it is actually the last term in Eq. (5.15) that acquires a negative sign when the trailer is backing up, not the middle term. The conclusion that a backing up trailer is unstable at all speeds is correct but it is the ‘‘spring-like’’ term that is to blame rather than the ‘‘damping-like’’ term.
Secondly, the division by U in the _h term means that what appears to be a torsional damping term becomes very small at very high speed, and the trailer will tend to oscillate a long time if disturbed when U is large. This simplest trailer model never is truly unstable, but the trailer approaches an undamped, neutrally stable system as the speed of travel increases. This gives a hint that a more complete model might well show the possibility of instability at high speed. This is indeed the case as will be demonstrated below using a more complex trailer model.
Finally, one can consider the case at low speeds where one might expect the inertial effects in Eq. (5.15) to be unimportant and that the ¨h term might be neglected. Alternatively, one might imagine the case of very rigid tires with high values of Ca, so that any term missing a Cafactor in Eq. (5.15) could be neglected. Either way, if the ¨h term in Eq. (5.15) is suppressed the resulting equation is reduced to first order.
.
hþ U
ða þ bÞh¼ 0 ð5:16Þ
This equation, which might be called a ‘‘kinematic equation’’ rather than a dynamic equation, also implies that the slip angle a in Eq. (5.5) vanishes. Equation (5.16) has the same first-order form as the equation for the shopping cart studied previously in Chapter 1, Eq. (1.19). For a positive value of U, the single coefficient in Eq. (5.16) is positive, indicating that the trailer is stable. This certainly comes as no surprise because the equation only holds for slow motion.
In this case, simply considering a negative value for U does yield the correct equation for reverse motion. The coefficient is then negative and the trailer is unstable just as the backward moving shopping cart was in Chapter 1. The conclusion is that the trailer is stable for slow forward velocities and unstable backing up no matter how slowly. Anyone who has tried to maneuver a boat trailer will agree that this very simple model has some validity at least at low speeds.
III. TWO DEGREE-OF-FREEDOM MODEL
The previous analysis of the stability of a trailer assumed that it was a rigid body attached at the hitch to a towing vehicle that moved in a straight line at a constant velocity. The movement of the trailer could thus be described by a single geometric variable, the angle between the trailer centerline and the direction of motion of the towing vehicle. The result of the analysis was that the trailer never was unstable although it could oscillate with less and less damping as the speed increased. This leads to the suspicion that a more complex model of the trailer might indeed have the possibility of being unstable at high speeds. Real trailers do in fact sometimes exhibit insta- bility at high towing speeds. Therefore another degree-of-freedom will be