WEEK 4 Dynamics of Machinery References Theory of Machines and Mechanisms, J.J.Uicker, G.R.Pennock ve J.E. Shigley, 2003 Prof.Dr.Hasan ÖZTÜRK Dr.H.ÖZTÜRK-2010

An ideal indicator diagram for a four-cycle engine. Prof.Dr.Hasan ÖZTÜRK An ideal indicator diagram for a four-cycle engine. Experimentally, an instrument called an engine indicator is used to measure the variation in pressure within a cylinder. The instrument constructs a graph, during operation of the engine, which is known as an indicator diagram. An indicator diagram for the ideal air-standard cycle is shown in the below Figure for a fourstroke- cycle engine. During compression the cylinder volume changes from 1 to 2 and the cylinder pressure changes from p1 to p2. The relationship, at any point of the stroke, is given by the polytropic gas law as: The polytropic exponent ,k, is often taken to be about 1.30 for both compression and expansion,

DYNAMICS OF RECIPROCATING ENGINES Prof.Dr.Hasan ÖZTÜRK we designate the crank angle as t, taken positive in the counterclockwise direction, and the connecting-rod angle as , taken positive when the crank pivot A is in the first quadrant as shown. A relation between these two angles is seen from the figure: where r and l designate the lengths of the crank and the connecting rod, respectively. Designating the piston position by the coordinate x from trigonometric identities

Note: For most engines the ratio r/l is about 1/4, and so the maximum value of the second term under the radical is about 1/16, or perhaps less If we expand the radical using the binomial theorem and neglect all but the first two terms, we obtain: Differentiating this equation successively to obtain the velocity and acceleration gives Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK GAS FORCES: we assume that the moving parts are massless so that gravity and inertia forces and torques are zero, and also that there is no friction. Now, using the binomial expansion and only the first two terms have been retained, we find that ‘ Similarly,

we can neglect those containing second or higher powers of r/l with only a very small error. The equation then becomes Prof.Dr.Hasan ÖZTÜRK

Equivalent masses: (c) Prof.Dr.Hasan ÖZTÜRK Equivalent masses: In analyzing the inertia forces due to the connecting rod of an engine, it is often convenient to picture a portion of the mass as concentrated at the crankpin A and the remaining portion at the wrist pin B. The reason for this is that the crankpin moves on a circle and the wrist pin on a straight line. Both of these motions are quite easy to analyze. However, the center of gravity G of the connecting rod is somewhere between the crankpin and the wrist pin, and its motion is more complicated and consequently more difficult to determine in algebraic form. The mass of the connecting rod m3 is assumed to be concentrated at the center of gravity G3. We divide this mass into two parts; one, m3p, is concentrated at the center of percussion P for oscillation of the rod about point B. This disposition of the mass of the rod is dynamically equivalent to the original rod if the total mass is the same, if the position of the center of gravity G3 is unchanged, and if the moment of inertia is the same. Writing these three conditions, respectively, in equation form produces Solving Eqs. (a) and (b) simultaneously gives the portion of mass to be concentrated at each point (c)

In the usual connecting rod, the center of percussion P is close to the crankpin A and it is assumed that they are coincident. Thus, if we let lA = lp, the above equations reduce to Prof.Dr.Hasan ÖZTÜRK

INERTIA FORCES Dividing the crank mass into two parts, at O2 and A, regarding the static equivalence conditions. Using the methods of the preceding section, we begin by locating equivalent masses at the crankpins and at the wrist pin. Thus, Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK the position vector of the crankpin relative to the origin O2 is Differentiating this equation twice with respect to time, the acceleration of point A is The inertia force of the rotating parts is then Because the analysis is usually made at constant angular velocity ( = 0), this equation reduces to Acceleration of the piston has been found as, Thus, the inertia force of the reciprocating parts is  = 0

secondary inertia force The total inertia force for all of the moving parts (for constant angular velocity). The components in the x and y directions are: primary inertia force secondary inertia force Prof.Dr.Hasan ÖZTÜRK

INERTIA TORQUE By taking moment about the crank center The inertia force caused by the mass at the crankpin A has no moment about O2 and, therefore, produces no inertia torque. By taking moment about the crank center Inertia torque is a periodic function, including the first three harmonics Prof.Dr.Hasan ÖZTÜRK

BEARING LOADS IN A SINGLE- CYLINDER ENGINE The resultant total bearing loads are made up of the following components: 1. Gas-force components, designated by a single prime; 2. Inertia force caused by the mass m4 of the piston assembly, designated by a double prime; 3. Inertia force of that part m3B of the connecting rod assigned to the piston-pin end (wrist-pin end), designated by a triple prime; 4. Connecting-rod inertia force of that part m3A at the crankpin end, designated by a quadruple prime. Prof.Dr.Hasan ÖZTÜRK

1- Gas force (examined at the beginning, page 7-8.) 2. Inertia force caused by the mass m4 of the piston assembly, designated by a double prime;   Prof.Dr.Hasan ÖZTÜRK

3. Inertia force of that part m3B of the connecting rod assigned to the piston-pin end (wrist-pin end), designated by a triple prime;   R Prof.Dr.Hasan ÖZTÜRK

Whereas a counterweight attached to the 4. Connecting-rod inertia force of that part m3A at the crankpin end, designated by a quadruple prime. Whereas a counterweight attached to the crank balances the reaction at O2, it cannot make F’’’’32 zero. Thus Prof.Dr.Hasan ÖZTÜRK

Superposition Prof.Dr.Hasan ÖZTÜRK

CRANKSHAFT TORQUE The torque delivered by the crankshaft to the load is called the crank shaft torque, and it is the negative of the moment of the couple formed by the forces F41 and Fy21,. Therefore, it is obtained from the equation The torque delivered by the crankshaft to the load. Prof.Dr.Hasan ÖZTÜRK

SHAKING FORCES OF ENGINES (due to only reciprocating masses) The inertia force caused by the reciprocating masses is illustrated acting in the positive direction in the below Figure (a). In Figure (b) the forces acting upon the engine block caused by these inertia forces are illustrated. Shaking Force: (Linear vibration in x direction) (Torsional vibration about crank center) Shaking Couple:

Circle diagram illustrating inertia forces. The total shaking force is: Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK