Grinding machine length size compensation processing method

Abstract

The application discloses a grinding machine length size compensation processing method, the actual length coordinates of each journal of a crankshaft are obtained through a measuring head, the remaining amount of the width of the file is calculated, a dynamic intercepting mathematical model is established in combination with the maximum processing amount and the minimum processing amount of a grinding wheel, the maximum interceptable length is calculated by comparing the relationship between the remaining amount of the width of the file and a processing threshold value, and the final compensation length is intelligently determined according to the difference between the actual length of a product to be processed and the length required by a process; the problem that the crankshaft to be processed cannot be automatically processed due to the length size of the crankshaft to be processed being out of tolerance under the traditional centering processing mode of a grinding machine is solved; meanwhile, for a semi-finished crankshaft whose length size is within the tolerance range, the processing reference, the drawing reference and the measurement reference can be unified by the method, the reference conversion error is close to zero, the size consistency of the finished crankshaft is improved, the length of the out-of-tolerance crankshaft is corrected to be within the tolerance range under the premise that the service life of the grinding wheel and the processing quality are guaranteed, and the qualified crankshaft is optimized to an ideal size.

Description

Technical Field

[0001] This invention relates to the field of mechanical manufacturing and processing, specifically to an intelligent compensation machining algorithm and control system for the length direction of a crankshaft grinding machine. Background Technology

[0002] In crankshaft grinding, especially for multi-cylinder crankshafts (such as eight-cylinder crankshafts), it is necessary to ensure the post-grinding length and width dimensions of multiple main journals and connecting rod journals. Existing grinding machines typically use a probe to measure reference points, establish a coordinate system, and center the grinding wheel according to the theoretical length of the process. However, due to errors in previous processes (such as quenching and semi-finish turning), the length dimensions of the semi-finished crankshaft often cannot be guaranteed to be within the tolerance range of ±0.15mm.

[0003] The traditional grinding machine inspection logic is as follows: ① After the probe detects the length of the semi-finished crankshaft, it is directly centered and machined without considering length tolerances (i.e., if the semi-finished crankshaft length is acceptable, it is acceptable after machining; if the semi-finished crankshaft length exceeds the tolerance, it is out of tolerance after machining); ② If the probe detects that the length of the semi-finished crankshaft is out of tolerance (exceeding ±0.15mm), an alarm will be triggered and machining will stop. The crankshaft must be manually removed for rework until the length is acceptable before it can be re-machined. This approach not only increases process flow time and labor costs, but if operators ignore the alarm and force machining, it is very easy to produce defective products. In addition, the fixed centering machining method cannot be fine-tuned according to the actual condition of the semi-finished crankshaft, making it difficult to stably machine the finished product to the center value required by the process.

[0004] For grinding machines using CBN high-speed grinding wheels, under the premise of the same efficiency, if the single grinding amount is too large (exceeding the maximum processing capacity of the grinding wheel), it will seriously affect the grinding wheel life and workpiece roundness, runout, and other quality indicators. If the grinding amount is too small, it may lead to "yin-yang" defects due to the perpendicularity of the quenched end face. Therefore, how to automatically process out-of-tolerance semi-finished products to meet the requirements through algorithm compensation while ensuring the safety of the grinding wheel and the processing quality has become an urgent problem to be solved. Summary of the Invention

[0005] The purpose of this invention is to provide a method and system for compensating for the length dimension of a grinding machine, aiming to achieve the following two objectives:

[0006] First, it solves the problem of out-of-tolerance machining caused by process fluctuations in the length of crankshafts to be processed, which prevents automatic machining. This allows out-of-tolerance products within a certain range to still meet process tolerance requirements after compensation machining, reducing rework, repairs, and scrap losses.

[0007] Second, for qualified semi-finished products whose length dimensions are within the tolerance range, an appropriate amount of allowance is automatically extracted through an algorithm to process the finished product dimensions as close as possible to the center value of the process length, thereby improving the consistency and accuracy of batch products.

[0008] To achieve the above objectives, the technical solution of the present invention is as follows:

[0009] A method for compensating for the length dimension of a grinding machine, characterized by comprising the following steps:

[0010] Step S1: Measure the position of the reference point, establish the workpiece coordinate system, and obtain the actual length coordinates and actual width coordinates of each journal of the crankshaft;

[0011] Step S2: Calculate the remaining journal width B;

[0012] Step S3: Preset the maximum machining amount W and minimum machining amount V of the grinding wheel;

[0013] Step S4: Determine the relationship between the remaining width B and V and W, and calculate the maximum cut length value L;

[0014] Step S5: Calculate the difference Y between the actual length Z of the semi-finished product and the length required by the post-grinding process, compare the calculated maximum cut length L with Y, and determine the final cut value Ls;

[0015] Step S6: Calculate the compensated target machining length Zs according to the formula, and control the grinding wheel to machine according to Zs;

[0016] Step S7: Compile and run a macro program in the machine tool CNC system to achieve fully automatic measurement, calculation, compensation and alarm functions.

[0017] In step S4, when V < B < W, the calculation of the maximum cut length value L follows the following rules:

[0018] If B < (W + V) / 2, then L = (B - V) / 2;

[0019] If B = (W + V) / 2, then L = (B - V) / 2 = (W - B) / 2;

[0020] If B > (W + V) / 2, then L = (W - B) / 2.

[0021] In step S5, the rule for determining the final truncation value Ls is as follows:

[0022] When L ≥ Y, Ls = Y;

[0023] When L < Y, Ls = L;

[0024] When B ≤ V or B ≥ W, Ls = 0, and if Zs exceeds the process length requirement, an alarm is triggered.

[0025] The formula for calculating the compensated length Zs is as follows: when the actual length coordinate is out of tolerance in the positive direction, Zs = Z - Ls; when the actual length coordinate is out of tolerance in the negative direction, Zs = Z + Ls.

[0026] The parameters in the formula are defined as follows:

[0027] Z: Actual length of the semi-finished product (measured by the probe);

[0028] B: Remaining width allowance;

[0029] W: Maximum machining capacity of the grinding wheel (preset value);

[0030] V: Minimum machining allowance of the grinding wheel (preset value);

[0031] Y: The difference between the actual length Z of the semi-finished product and the length required by the process;

[0032] L: The calculated maximum cut-out length;

[0033] Ls: Final truncation value;

[0034] Zs: Target machining length after compensation.

[0035] This invention calculates the dynamic relationship between the actual width and a set threshold, correcting out-of-tolerance dimensions towards the tolerance center as much as possible without damaging the grinding wheel. This achieves the effect of "even if the semi-finished product is out of tolerance, it can be processed into a qualified product in a single clamping operation." It solves the problem of non-automatic processing due to out-of-tolerance length dimensions of semi-finished crankshafts in traditional grinding machine centering methods. Furthermore, for semi-finished crankshafts with length dimensions within the tolerance range, this method can further optimize their processing to near the center value of the process length, improving the dimensional consistency of the finished crankshaft. This invention can correct the length of out-of-tolerance crankshafts to the tolerance range and optimize qualified crankshafts to ideal dimensions while ensuring grinding wheel life and processing quality, thus improving material utilization and automation levels. Attached Figure Description

[0036] Figure 1 : Diagram showing the remaining width B. Figure 2 Compensation diagrams for Embodiment 1 and Embodiment 2. Detailed Implementation

[0037] The present invention will be further described in detail below with reference to specific implementation methods and program logic. In this embodiment, during crankshaft grinding, the actual coordinates of a certain journal are measured by a probe, and the remaining width B of each journal is calculated accordingly. The calculation method of this value of B is as follows:

[0038] Taking a certain journal as an example, let the theoretical coordinate position of the left side width be X1, and the theoretical coordinate position of the right side width be X2. Using a probe to obtain the actual coordinate positions X1' of the left side width and X2' of the right side width, the formula for calculating the remaining width B is:

[0039] B = |X1 - X1'| + |X2 - X2'|

[0040] like Figure 1 As shown, in a specific measurement example, if the theoretical coordinate position of the left side width is 100.50mm and the theoretical coordinate position of the right side width is 200.50mm; while the actual measured coordinate position of the left side width is 100mm and the actual coordinate position of the right side width is 200mm, then the left side should be machined to 100mm and the right side to 200mm. Substituting these values ​​into the above formula...

[0041] B = |X1 - X1'| + |X2 - X2'|

[0042] =|100.5-100|+|200-200.5|=0.5+0.5=1

[0043] The actual width allowance B is calculated to be 1mm. This value of B will be used as a variable in the subsequent compensation algorithm.

[0044] The compensation logic of this invention is as follows:

[0045] 1. When B ≤ V, the width allowance is insufficient to guarantee the end face grinding, Ls=0, Zs=Z. If Zs exceeds the tolerance, an alarm will be triggered.

[0046] 2. When B ≥ W, the width allowance is too large and exceeds the single-pass capacity of the grinding wheel, Ls=0, Zs=Z. If Zs exceeds the tolerance, an alarm will be triggered.

[0047] 3. When V < B < W, execute the dynamic compensation algorithm:

[0048] Calculate the threshold comparison value (W + V) / 2;

[0049] Formula 1: When B < (W + V) / 2, L = (B - V) / 2;

[0050] Formula 2: When B = (W + V) / 2, L = (B - V) / 2 = (W - B) / 2;

[0051] Formula 3: When B > (W + V) / 2, L = (W - B) / 2;

[0052] Compare L and Y: If L ≥ Y, it means that the current margin is sufficient to correct back to the center value, then Ls = Y; if L < Y, it means that it cannot be completely corrected but a portion of the margin can be taken, then Ls = L.

[0053] Example 1 (Optimization within the normal acceptable range, such as...) Figure 2 (As shown) The crankshaft length tolerance requirement is ±0.15mm. Set the machine tool parameters: V=0.4mm, W=1.2mm, then (W+V) / 2 = 0.8mm. Measure the length of a journal: Z=100.1mm (process requirement 100mm, difference Y=100.1-100=0.1mm).

[0054] Case A: The calculated width allowance B = 0.7mm. 0.7 < 0.8, so Formula 1 applies. L = (0.7 - 0.4) / 2 = 0.15mm. Since L(0.15) > Y(0.1), Ls = Y = 0.1mm. The compensated length Zs = 100.1 - 0.1 = 100mm. The dimensional accuracy after machining is at the center value.

[0055] Case B: The measured width allowance B = 0.8mm. 0.8 = 0.8, so formula two applies. L = (1.2 - 0.8) / 2 = (0.8 - 0.4) / 2 = 0.2mm. Since L(0.2) > Y(0.1), Ls = Y = 0.1mm. The compensated length Zs = 100.1 - 0.1 = 100mm. The dimension after machining is at the center value.

[0056] Case C: The measured width allowance B = 0.9mm. 0.9 > 0.8, so Formula 3 applies. L = (1.2 - 0.9) / 2 = 0.15mm. Since L(0.15) > Y(0.1), Ls = Y = 0.1mm. The compensated length Zs = 100.1 - 0.1 = 100mm. The dimension after machining is at the center value.

[0057] Example 2 (Remedial processing of out-of-tolerance semi-finished products, such as...) Figure 2 (As shown) The crankshaft length tolerance requirement is ±0.15mm. Machine parameters are set: V=0.4mm, W=1.2mm, then (W+V) / 2 = 0.8mm. The journal length is measured as Z=100.3mm (process requirement 100mm, difference Y=100.3-100=0.3mm, exceeding the upper limit +0.15).

[0058] Case A: The measured width allowance B = 0.7mm. 0.7 < 0.8, so Formula 1 applies. L = (0.7 - 0.4) / 2 = 0.15mm. Since L(0.15) < Y(0.3), Ls = L = 0.15mm. The compensated length Zs = 100.3 - 0.15 = 100.15mm.

[0059] Case B: The measured width allowance B = 0.8mm. 0.8 = 0.8, so formula two applies. L = (1.2 - 0.8) / 2 = (0.8 - 0.4) / 2 = 0.2mm. Since L(0.2) < Y(0.3), Ls = L = 0.2mm. The compensated length Zs = 100.3 - 0.2 = 100.1mm.

[0060] Case C: The measured width allowance B = 0.9mm. 0.9 > 0.8, so Formula 3 applies. L = (1.2 - 0.9) / 2 = 0.15mm. Since L (0.15) < Y (0.3), Ls = L = 0.15mm. The compensated length Zs = 100.3 - 0.15 = 100.15mm. Although it cannot be corrected to the center value of 100 at this time, 100.15mm and 100.1mm are within the machine's intelligent automatic tolerance zone of ±0.15mm. The crankshaft that originally needed to be reworked is processed into a qualified product after logical calculation.

[0061] Program implementation logic (based on CNC macro program)

[0062] To implement the above compensation algorithm, the following macro program was developed and run in the machine tool CNC system to achieve fully automatic measurement, calculation, compensation, and alarm functions. The calculation part of the program and its comments are as follows:

[0063] /

[0064] #22=0.4 Minimum processing quantity

[0065] #23=1.2 Maximum processing capacity

[0066] #26=0.15 Length tolerance after machining

[0067] /

[0068] #1=1 8-link journal counter

[0069] #10 = 0.0 (Intermediate variable #10)

[0070] #11 = 0.0 (Intermediate variable #11)

[0071] WHILE [#1 LE 8] DO1 is a command that executes the command cyclically for 8 connecting rod shaft diameters.

[0072] /

[0073] N2010 IF [#[100+#1-1] GT #22]THEN GOTO 2011

[0074] GOTO 2012

[0075] If the width is greater than the minimum processing size, proceed to step 2011; otherwise, proceed to step 2012.

[0076] N2011 #10 = #[100+#1-1] - #22 width minus minimum processing quantity, assigned to intermediate variable #10

[0077] /

[0078] N2012 IF [#23 GT #[100+#1-1]] THEN GOTO 2013

[0079] GOTO 2014

[0080] If the maximum processing volume is greater than the width, proceed to 2013; otherwise, proceed to 2014.

[0081] /

[0082] N2013 #11 = #23 - [#[100+#1-1]] Maximum processing quantity minus the width, assigned to intermediate variable #11

[0083] N2014 IF [#10 GT #11] THEN #10 = #11 If the intermediate variable #10 is greater than the intermediate variable #11, then the intermediate variable #11 is assigned to the intermediate variable #10.

[0084] (If intermediate variable #10 is less than or equal to intermediate variable #11, intermediate variable #10 remains unchanged.)

[0085] /

[0086] /

[0087] N2020 #10 = #10 * 0.5 One-sided cutoff

[0088] IF [ABS[#[200+#1-1]] GT #10] THEN GOTO 2030 If the absolute value of the difference between the actual length of the semi-finished product and the required length is greater than the intermediate variable #10, then go to 2030.

[0089] #10 = ABS[#[200+#1-1]] (The absolute value of the difference between the actual length of the semi-finished product and the required length is less than or equal to the intermediate variable #10), and the absolute value of the difference between the actual length of the semi-finished product and the required length is assigned to the intermediate variable #10.

[0090] /

[0091] /

[0092] N2030 IF [#[200+#1-1] GT 0] THEN GOTO 2033 If the difference between the actual length of the semi-finished product and the required length is greater than 0 (length is a positive number), go to 2033.

[0093] IF [#[200+#1-1] LT 0] THEN GOTO 2034 If the difference between the actual length of the semi-finished product and the required length is less than 0 (the length is negative), go to 2034.

[0094] /

[0095] N2032 #12 = #[200+#1-1]

[0096] GOTO 2080

[0097] If the difference between the actual length of the semi-finished product and the required length is equal to 0, assign the value to intermediate variable #12, and go to 2080.

[0098] /

[0099] N2033 #12 = #[200+#1-1]- #10

[0100] GOTO 2080

[0101] The difference between the actual length of the semi-finished product and the required length is positive. Subtract intermediate variable #10, assign the value to intermediate variable #12, and return 2080.

[0102] /

[0103] N2034 #12 = #[200+#1-1] + #10

[0104] GOTO 2080

[0105] The difference between the actual length of the semi-finished product and the required length is negative. Adding intermediate variable #10, the value is assigned to intermediate variable #12, and then returned to 2080.

[0106] /

[0107] /

[0108] N2080 #[200+#1] = #12 Intermediate variable #12 is assigned to the difference between the actual length of the semi-finished product and the length required by the process.

[0109] #1 = #1 + 1 8-link shaft diameter counter

[0110] END1 - End of connecting rod shaft diameter cycle command

[0111] /

[0112] #1=9 9 main spindle counters

[0113] #10 = 0.0 (Intermediate variable #10)

[0114] #11 = 0.0 (Intermediate variable #11)

[0115] WHILE [#1 LE 17] DO1 Executes the instruction cyclically for 9 spindle diameters.

[0116] /

[0117] N3010 IF [#[300+#1-1] GT #22] THEN GOTO 3011

[0118] GOTO 3012

[0119] If the width is greater than the minimum processing size, proceed to 3011; otherwise, proceed to 3012.

[0120] /

[0121] N3011 #10 = #[300+#1-1] - #22 width minus minimum processing quantity, assigned to intermediate variable #10

[0122] /

[0123] N3012 IF [#23 GT #[300+#1-1]]THEN GOTO 3013

[0124] GOTO 3014

[0125] If the maximum processing volume is greater than the width, proceed to 3013; otherwise, proceed to 3014.

[0126] /

[0127] N3013 #11 = #23 - [#[300+#1-1]] Maximum processing quantity minus the width, assigned to intermediate variable #11

[0128] N3014 IF [#10 GT #11] THEN #10 = #11 If the intermediate variable #10 is greater than the intermediate variable #11, then the intermediate variable #11 is assigned to the intermediate variable #10.

[0129] (If intermediate variable #10 is less than or equal to intermediate variable #11, intermediate variable #10 remains unchanged.)

[0130] /

[0131] /

[0132] N3020 #10 = #10 * 0.5 One-sided cutoff value

[0133] IF [ABS[#[400+#1-1]] GT #10] THEN GOTO 3030

[0134] If the absolute value of the difference between the actual length of the semi-finished product and the length required by the process is greater than intermediate variable #10, then go to 3030.

[0135] #10 = ABS[#[400+#1-1]] (The absolute value of the difference between the actual length of the semi-finished product and the length required by the process is less than or equal to the intermediate variable #10), and the absolute value of the actual length of the semi-finished crankshaft is assigned to the intermediate variable #10.

[0136] /

[0137] /

[0138] N3030 IF [#[400+#1-1] GT 0] THEN GOTO 3033 If the difference between the actual length of the semi-finished product and the required length is greater than 0 (length is a positive number), go to 3033.

[0139] IF [#400+#1-1] LT 0] THEN GOTO 3034 If the difference between the actual length of the semi-finished product and the required length is less than 0 (the length is negative), go to 3034

[0140] /

[0141] N3032 #12 = #[400+#1-1]

[0142] GOTO 3080

[0143] If the difference between the actual length of the semi-finished product and the required length is equal to 0, assign the value to intermediate variable #12, and go to 3080.

[0144] /

[0145] N3033 #12 = #[400+#1-1]- #10

[0146] GOTO 3080

[0147] The difference between the actual length of the semi-finished product and the required length is positive. Subtract intermediate variable #10, assign the value to intermediate variable #12, and return 3080.

[0148] /

[0149] N3034 #12 = #[400+#1-1] + #10

[0150] GOTO 3080

[0151] The difference between the actual length of the semi-finished product and the required length is negative. Adding intermediate variable #10, the value is assigned to intermediate variable #12, and then returned to 3080.

[0152] /

[0153] /

[0154] N3080 #[400+#1] = #12 Intermediate variable #12 is assigned to the difference between the actual length of the semi-finished product and the length required by the process.

[0155] #1 = #1 + 1 (9 main spindle counters)

[0156] END1 End of spindle diameter cycle command

[0157] /

[0158] N5000 #1=1 8-link journal counter

[0159] N5100 IF [#1 GT 8] THEN GOTO 6000 If the connecting rod journal counter is greater than 8, go to 6000.

[0160] IF [#[100+#1-1] LE #22] THEN GOTO 8001 If the width is less than or equal to the minimum machining allowance, go to 8001

[0161] IF [#[100+#1-1] GT #23] THEN GOTO 8002 If the width is greater than the maximum machining allowance, go to 8002

[0162] IF [ABS[#[200+#1]] GT #26] THEN GOTO 8101 If the absolute value of the compensated length is greater than the length tolerance after machining, proceed to 8101.

[0163] #1 = #1 + 1 8-link journal counter

[0164] GOTO 5100 8-link shaft diameter cycle determination complete

[0165] N6000 #1=9 9 main spindle counters

[0166] N6100 IF [#1 GT 17 ] THEN GOTO 9000 If the spindle counter is greater than 17, go to 9000.

[0167] IF [#[300+#1-1] LE #22] THEN GOTO 8003 If the width is less than or equal to the minimum machining allowance, go to 8003

[0168] IF [#[300+#1-1] GT #23] THEN GOTO 8004 If the width is greater than the maximum machining allowance, go to 8004

[0169] IF [ABS[#[400+#1]] GT #26] THEN GOTO 8102 If the absolute value of the compensated length is greater than the length tolerance after machining, go to 8102.

[0170] #1 = #1 + 1 (9 main spindle counters)

[0171] GOTO 6100 9-journal cycle determination completed

[0172] N8001 #3000=#1 Output connecting rod shaft diameter too small alarm

[0173] N8002 #3000=#1 Output connecting rod shaft diameter range too large alarm

[0174] N8003 #3000=#1 Output spindle diameter range too small alarm

[0175] N8004 #3000=#1 Output spindle diameter range too wide alarm

[0176] N8101 #3000=#1 Output connecting rod shaft diameter cut-off length out-of-tolerance alarm

[0177] N8102 #3000=#1 Output spindle diameter cut-off length out-of-tolerance alarm

[0178] N9000 #1=#1 Program End

[0179] Through the above program logic, the machine tool automatically completes the calculation after the measurement is completed, using the compensated Zs value as the new target position to guide the grinding wheel feed. The program includes comprehensive anomaly detection logic at the end; if the gauge width is insufficient, excessive, or still exceeds the tolerance after compensation, a corresponding alarm number will be output to prompt operator intervention.

[0180] Summarize

[0181] During normal production, the value of B for crankshafts in the same batch will not vary much. Based on the value of B, the values ​​of V and W can be adjusted accordingly using the formula mentioned above to achieve various changes in the intercepted values.

[0182] This method, without increasing hardware costs, only upgrades the software algorithm to endow grinding machines with "intelligent grinding" functionality. On one hand, it solves the problem of frequent downtime alarms caused by out-of-tolerance dimensions of semi-finished crankshafts, enabling crankshafts that would otherwise require rework to be corrected and processed online, significantly improving automation and material utilization. On the other hand, for semi-finished crankshafts with acceptable dimensions, this algorithm can dynamically calculate the optimal cutting amount, further converging the length of the finished crankshaft to the center area of ​​the process tolerance, improving the dimensional distribution and enhancing the overall precision level of the finished crankshaft. This method is particularly suitable for crankshaft production lines with multiple gears and high precision requirements.

Claims

1. A method for compensating for the length dimension of a grinding machine, characterized in that, Includes the following steps: Step S1: Measure the position of the reference point, establish the workpiece coordinate system, and obtain the actual length coordinates and actual width coordinates of each journal of the crankshaft; Step S2: Calculate the remaining journal width B; Step S3: Preset the maximum machining amount W and minimum machining amount V of the grinding wheel; Step S4: Determine the relationship between the remaining width B and V and W, and calculate the maximum cut length value L; Step S5: Calculate the difference Y between the actual length Z of the workpiece to be processed and the length required by the grinding process, compare the calculated maximum cut length L with Y, and determine the final cut value Ls; Step S6: Calculate the compensated target machining length Zs according to the formula, and control the grinding wheel to machine according to Zs; Step S7: Compile and run a macro program in the machine tool CNC system to achieve fully automatic measurement, calculation, compensation and alarm functions.

2. The grinding machine length compensation machining method according to claim 1, characterized in that, In step S4, when V < B < W, the calculation of the maximum cut length value L follows the following rules: If B < (W + V) / 2, then L = (B - V) / 2; If B = (W + V) / 2, then L = (B - V) / 2 = (W - B) / 2; If B > (W + V) / 2, then L = (W - B) / 2.

3. The grinding machine length compensation machining method according to claim 1, characterized in that, In step S5, the rule for determining the final truncation value Ls is as follows: When L ≥ Y, Ls = Y; When L < Y, Ls = L; When B ≤ V or B ≥ W, Ls = 0, and an alarm is triggered if Zs exceeds the process length requirement; the above materials do not include measurement standard compensation, grinding wheel dressing and shedding compensation, and linear compensation for crankshaft temperature effects.

4. The grinding machine length compensation machining method according to claim 1, characterized in that, The formula for calculating the compensated length Zs is as follows: when the actual length coordinate is out of tolerance in the positive direction, Zs = Z - Ls; when the actual length coordinate is out of tolerance in the negative direction, Zs = Z + Ls.

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