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Common transmission failures result from the glazing of clutch surfaces by deposition of oil oxidation and decomposition products. Both the oxidation and decomposition processes depend on temperature histories of the surfaces. Because it is difficult to measure these surface temperatures during operation, it is useful to develop models to predict clutch-interface thermal behavior. The relative velocity between mating clutch plates, from the initial engagement to the zero-sliding (lock-up) condition, generates heat that is transferred to the plates. The relative velocity decreases at a constant rate during this period, producing a heat flux that is initially very large and decreases linearly with time, until lock-up occurs. Accordingly, \(q_{f}^{\prime \prime}=q_{o}^{\prime \prime}=\left[1-\left(t / t_{\mathrm{lu}}\right)\right]\), where \(q_{o}^{\prime \prime}=1.6 \times 10^{7} \mathrm{~W} / \mathrm{m}^{2}\) and \(t_{1 \mathrm{u}}=100 \mathrm{~ms}\) is the lock-up time. The plates have an initial uniform temperature of \(T_{i}=40^{\circ} \mathrm{C}\), when the prescribed frictional heat flux is suddenly applied to the surfaces. The reaction plate is fabricated from steel, while the composite plate has a thinner steel center section bonded to low- conductivity friction material layers. The thermophysical properties are \(\rho_{s}=\) \(7800 \mathrm{~kg} / \mathrm{m}^{3}, c_{\mathrm{s}}=500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k_{s}=40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) for the steel and \(\rho_{\mathrm{im}}=1150 \mathrm{~kg} / \mathrm{m}^{3}, c_{\mathrm{fm}}=1650 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k_{\mathrm{fm}}=4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) for the friction material. (a) On \(T-t\) coordinates, sketch the temperature history at the midplane of the reaction plate, at the interface between the clutch pair, and at the midplane of the composite plate. Identify key features. (b) Perform an energy balance on the clutch pair over the time interval \(\Delta t=t_{\mathrm{lu}}\) to determine the steadystate temperature resulting from clutch engagement. Assume negligible heat transfer from the plates to the surroundings. (c) Compute and plot the three temperature histories of interest using the finite-element method of FEHT or the finite-difference method of IHT (with \(\Delta x=0.1 \mathrm{~mm}\) and \(\Delta t=1 \mathrm{~ms}\) ). Calculate and plot the frictional heat fluxes to the reaction and composite plates, \(q_{\mathrm{rp}}^{\prime \prime}\) and \(q_{\mathrm{cp}}^{\prime \prime}\), respectively, as a function of time. Comment on features of the temperature and heat flux histories. Validate your model by comparing predictions with the results from part (b). Note: Use of both \(F E H T\) and \(I H T\) requires creation of a look-up data table for prescribing the heat flux as a function of time.

Step by step solution

01

(a) Sketch Temperature History

To sketch the temperature history at the midplane of the reaction plate, the interface between the clutch pair, and the midplane of the composite plate, we need to consider the applied heat flux, initial conditions and materials properties. At the beginning of the sliding process, the heat flux is high, causing temperatures to increase rapidly. Then, as the sliding process approaches lock-up, the heat flux decreases and the temperature increase slows down.

02

(b) Energy Balance to Find Steady-State Temperature

Using the supplied heat flux and thermophysical properties, we will perform an energy balance over the clutch pair over the specified time interval. This will involve determining the heat input, heat storage within the materials, and heat loss within the clutch based on the given properties and conditions. We will then use the energy balance to determine the steady-state temperature of the clutch after lock-up.

03

(c) Numerical Calculation of Temperature Histories and Heat Fluxes

Employing finite-element or finite-difference methods, we will compute and plot the temperature histories of the midplane of the reaction plate, the interface between the clutch pair, and the midplane of the composite plate. We will use the given \(\Delta x\) and \(\Delta t\) to discretize the problem. Additionally, we will calculate and plot the frictional heat fluxes to the reaction and composite plates as a function of time, to better understand the thermal behavior of the clutch. Finally, we will validate our model by comparing our computed results with the steady-state temperature from part (b).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Finite-Element Method

The Finite-Element Method (FEM) is a numerical approach used to solve complex problems in engineering and physics, including the heat transfer in clutch systems. It allows the calculation of temperature distributions by breaking down a model into smaller, simpler parts called elements. Each element represents a specific component of the clutch system, such as a small section of the reaction plate or a portion of the friction material.

FEM uses the material properties and boundary conditions provided, like thermal conductivity and specific heat, to compute the temperature at each node. Nodes are the key points at the corners or interfaces of elements. This approach is crucial for simulating how temperature changes within different sections of the clutch over time.

Employing FEM involves compiling detailed geometric data of the clutch materials and defining initial conditions and constraints, such as the frictional heat flux. This heat flux is responsible for increasing the temperature transiently, which is then spread through surrounding areas. By using time step discretization, the method can simulate the progression of temperature history across the entire system, aiding in understanding the thermal behavior under operating conditions.

Temperature History

Temperature history in clutch systems is essential for predicting the thermal response during operation. As the clutch engages and slips, a high initial heat flux is generated, raising the temperature of the materials involved.

Initially, the temperature rises rapidly due to the large initial heat flux. As the relative velocity between clutch plates decreases, this heat flux diminishes linearly until reaching lock-up. During this period, the temperature increases more slowly until it stabilizes.

Key points to illustrate in the temperature history:

• Midplane of the reaction plate, where the conduction predominately occurs.
• Interface between clutch pair, facing the maximum direct heat impact.
• Midplane of the composite plate, showing indirect heating due to lower conductivity.

Understanding these temperature profiles helps in assessing possible temperature-induced failures like glazing, which affects clutch performance.

Thermal Model Validation

Thermal model validation is a critical step to ensure that the simulated results accurately reflect the real-world behavior of the clutch system. It involves comparing the model's predictions with known or calculated benchmarks, such as the steady-state temperature derived from energy balance methods.

To validate the clutch's thermal model, we compare the calculated temperatures and heat flux distributions using FEM or FDM with the steady-state results. Differences between these results can suggest inaccuracies in the model assumptions or parameters, such as material properties or boundary conditions.

A properly validated model will:

• Accurately follow the trends of initial rapid heating followed by stabilization.
• Predict the right temperature levels at various sections of the clutch system.
• Provide insights to adapt and improve the design to minimize overheating and potential clutch failures.

Thus, validation ensures confidence in the model's capability to predict real phenomena, crucial for optimizing clutch designs.